À Loïc

Pierre Antonio

À Mathieu

Gérard Marie Pierre

Emmanuelle, Nicolas, Lingsam Vincent Rocio

Merci À Laura

Etienne Dominika

Camille Claire

Roxanne Pistache

i Ce manuscrit marque la fin de trois années de recherche, cependant, il y a une grandeur quantifiable qui surpasse de loin le nombre de découvertes scientifiques effectuées : le nombre de rencontres et de moments inoubliables qui ont marqué ce beau voyage, voici donc quelques mots pour remercier les personnes avec qui je les ai partagés.

Je tiens dans un premier temps à exprimer toute ma gratitude à mon directeur de thèse, Benjamin Rotenberg, pour son expertise et son encadrement. Il m'a appris que pour faire de la bonne science, il ne suffisait pas de juste savoir résoudre les équations les plus compliquées, mais surtout de savoir pourquoi le faire, les interpréter et aller au fond de ses idées. De plus, je n'étais pas la personne la plus organisée, mais Benjamin a eu la patience de me montrer la bonne voie par son éthique de travail exemplaire. Il m'en a autant appris sur le savoir scientifique, que savoir en être un bon. J'ai aussi eu la chance d'être entouré par l'équipe SENSES dont les nombreuses discussions m'ont grandement aidé dans mon travail. Merci à Iurii, Giovanni, Jeongmin, Swetha, Nicolas, Matthieu et Sleeba pour leurs conseils et leur enthousiasme.

Certains projets n'auraient pu aboutir sans de précieuses collaborations, je souhaite donc remercier Gabriel et Sophie pour leurs apports ainsi que pour les agréables discussions, j'ai énormément appris grâce à eux.

Ces trois années se seraient certainement déroulées d'une manière très différente si je ne les avais pas effectuées au sein du laboratoire PHENIX. Ce cadre quotidien m'a permis de m'épanouir autant scientifiquement que personnellement, je souhaite donc remercier chaleureusement les personnes avec qui j'ai interagi, autant devant un tableau qu'autour d'un déjeuner ou une bière (ou encore un mètre de shooter).

Ionic transport at the nanoscale

Most electronic devices and energy storage cells in their modern forms, rely on the electrochemical properties of microscopic charge carriers, namely ions, to convey electrical power. Conventional devices such as batteries, relying on redox reactions to store energy, make use of the motion of charge carriers in the process of charging and discharging. During the last decades, technological development in material sciences, and in particular nanotechnologies, lead to rapid improvement of tailored designs with specific materials and geometries, increasing the performance of conventional batteries for example. But with the extreme miniaturization of those designs, going to the sub-micrometre scale has also led to the development of radically different technologies in the field of energy storage, such as high-power/density super-capacitors, relying solely on the properties of electrical double layers and the adsorption of ions in nanoporous electrodes [START_REF] Simon | Perspectives for electrochemical capacitors and related devices[END_REF]. While ubiquitous in biological systems, the use of ionic transport at the nanoscale could lead to major technological advances. Furthermore, with the versatility of nanotechnologies, new applications can be found in various fields. In particular, the transport of charged solutions through nanopores offers solutions to blue energy harvesting, water desalination, or single molecule sensing [START_REF] Siria | New avenues for the large-scale harvesting of blue energy[END_REF][START_REF] Gubbiotti | Electroosmosis in nanopores: computational methods and technological applications[END_REF]. Regarding the latter, it is now possible to accurately monitor the electrical current created by electrolyte flow across a single nanotube, such that the change of current upon inserting a DNA chain in the nanotube, serves as an efficient sequencing tool [START_REF] Liu | Ultrashort singlewalled carbon nanotubes in a lipid bilayer as a new nanopore sensor[END_REF]. But these promising applications based on intricate designs rely on the understanding of ionic transport at the nanoscale [START_REF] Kavokine | Fluids at the nanoscale: From continuum to subcontinuum transport[END_REF][START_REF] Faucher | Critical knowledge gaps in mass transport through singledigit nanopores: A review and perspective[END_REF]. Moreover, due to the nanometric size of these devices and the low number of charge carriers, electrical fluctuations play a key role in the electrical properties of these systems.

In fact, seemingly unrelated experiments such as ionic current measurements through single nanopores, nanoscale electrochemistry, NMR relaxometry, and Surface Force Balance experiments, all probe electrical fluctuations [START_REF] Hoang | Electrical noise in electrolytes: a theoretical perspective[END_REF]: of the electric current, the charge and polarization, the field gradient (for quadrupolar nuclei) and the coupled mass/charge densities. The fluctuations of such various observables arise from the same underlying microscopic dynamics of the ions and solvent molecules. In this work we will be focusing on the dynamics of the ions in electrolytes, trying to rationalize electrical current noise in nanopores.

Electrical current noise in nanopores

Electric current measurements in nanopores have received great attention for the last decade, due to the evolution of nanopore sensing technologies. With the recent advances in manufacturing processes, it is now possible to accurately measure electrical current across single nanotubes. Drilling [START_REF] Dekker | Solid-state nanopores[END_REF][START_REF] Ulrich | Direct force measurements on DNA in a solid-state nanopore[END_REF], electrochemical etching [START_REF] Feng | Single-layer MoS2 nanopores as nanopower generators[END_REF][START_REF] Feng | Observation of ionic coulomb blockade in nanopores[END_REF], or self-assembly techniques for the growth of nanotubes [START_REF] Kavokine | Fluids at the nanoscale: From continuum to subcontinuum transport[END_REF], allow building the nanotube with various specificities, such as material, length, shape, radius or coating. Then, the challenge of embedding the nanotube in a microfluidic device, can be met with also various strategies [START_REF] Siria | Giant osmotic energy conversion measured in a single transmembrane boron nitride nanotube[END_REF][START_REF] Secchi | Scaling behavior for ionic transport and its fluctuations in individual carbon nanotubes[END_REF][START_REF] Liu | Ultrashort singlewalled carbon nanotubes in a lipid bilayer as a new nanopore sensor[END_REF][START_REF] Pang | Origin of giant ionic currents in carbon nanotube channels[END_REF][START_REF] Kavokine | Fluids at the nanoscale: From continuum to subcontinuum transport[END_REF], such as nanomanipulation in Fig. 1.1a. Experimental set-ups such as the one illustrated in Fig. 1.1 allow to precisely measure electrical current time signals; I(t), with varying parameters such as the gradients of electrical potential, pressure, or concentration. At steady state, the current shows an average value Ī, but due to its nanometric size, the system is highly sensitive to microscopic fluctuations. Therefore the electrical signal also shows tiny time variations δI(t) = I(t)-Ī. For technological applications, fluctuations have long been regarded as parasitic noises, which are naturally amplified at the nanoscale, but an intriguing observation arises. One can analyze these fluctuations using spectral analysis, by taking the power spectral density (PSD) of the signal I(t):

S I (f ) = ˆ∞ -∞ δI(t) e -i2π f t dt 2 .
(1.1)

The electrical noise PSD is defined as the Fourier transform of its autocorrelation function, or conversely the square modulus of the Fourier transform of the noise signal (Eq. (1.1)). This function, well known in the field of signal processing, quantifies the correlation patterns, translated as frequency density, which are present in the signal. While parasitic noise would seemingly be random, corresponding to white noise with a flat spectrum meaning uncorrelated in time, most biological and artificial nanopores display a ubiquitous 1/f scaling law at low frequency, whose origins remain not fully explained [START_REF] Hofman | The validity of hooge's law for 1/f noise[END_REF][START_REF] Hoogerheide | Probing surface charge fluctuations with solid-state nanopores[END_REF][START_REF] Heerema | 1/f noise in graphene nanopores[END_REF][START_REF] Gravelle | Adsorption Kinetics in Open Nanopores as a Source of Low-Frequency Noise[END_REF][START_REF] Marbach | Intrinsic fractional noise in nanopores: The effect of reservoirs[END_REF][START_REF] Secchi | Scaling behavior for ionic transport and its fluctuations in individual carbon nanotubes[END_REF][START_REF] Matthew R Powell | Nonequilibrium 1/f noise in rectifying nanopores[END_REF]. This unexpected behaviour reflects correlations arising from microscopic interactions, highlighting missing pieces in the fundamental understanding of the latter. Low frequency S I (f ) ∝ 1/f noise, also named pink noise or fractal noise or flicker noise, has been observed in numerous seemingly unrelated fields, such as meteorology, gravitational waves, social sciences and semiconductors [START_REF] Handel | The nature of fundamental 1/f noise[END_REF]. In all these contexts, the PSD of some observables properties exhibit the low-frequency scaling

S I (f ) ∝ 1/f α , (1.2)
with an exponent α ∼ 1. While white noise (α = 0) is sometimes referred to as pure noise, due to its lack of information, and Brownian Noise (α ∼ 2) simply is the integral of the latter, several origins of pink noise have been found within various topics such as self-organized criticality, with Pareto principle and Zipf distribution the most famous representatives [START_REF] Bak | Self-organized criticality: An explanation of the 1/f noise[END_REF]. But for electrical noise in conductors, apart from Hooge's phenomenological law [START_REF] Hooge | 1/f noise in the conductance of ions in aqueous solutions[END_REF][START_REF] Hofman | The validity of hooge's law for 1/f noise[END_REF][START_REF] Sikula | Advanced Experimental Methods For Noise Research in Nanoscale Electronic Devices[END_REF], stating that

S I (f ) G 2 = K N 1 f (1.3)
where G is the conductance of the system, N the number of free charge carriers, and K a constant number, no general microscopic origin has been deciphered. Unlike anterior findings of Nyquist that will be discussed in Sec. 1.3.6 for the Johnson-Nyquist noise in conductors, Eq.(1.3) is unsatisfactory from the microscopic origin point of view [START_REF] Hofman | The validity of hooge's law for 1/f noise[END_REF], the prefactor K seems to depend experimentally on various properties of the system from experimental measurements, namely the properties of the nanopores, with its material, geometry and coating, or the concentration and the pH of the solution, or the surface charge and the applied voltage [START_REF] Dekker | Solid-state nanopores[END_REF][START_REF] Hoogerheide | Probing surface charge fluctuations with solid-state nanopores[END_REF][START_REF] Heerema | 1/f noise in graphene nanopores[END_REF][START_REF] Stuart F Knowles | Current fluctuations in nanopores reveal the polymer-wall adsorption potential[END_REF][START_REF] Gravelle | Adsorption kinetics in open nanopores as a source of low-frequency noise[END_REF] (see Fig. 1.2 and Fig. 1.3). Recently the electrical noise has been directly linked to adsorption on the pore walls [START_REF] Gravelle | Adsorption Kinetics in Open Nanopores as a Source of Low-Frequency Noise[END_REF][START_REF] Stuart F Knowles | Current fluctuations in nanopores reveal the polymer-wall adsorption potential[END_REF]. This observation for the electric current fluctuations in nanopores is but one of the many illustrations of electrical fluctuations in electrolyte measurements. Therefore it is of paramount importance to model and understand electrical fluctuations in electrolytes at the nanoscale in order to interpret those experiments. Uncovering the whole picture of these sensitive electrical fluctuations in electrolytes might be of great interest as a fundamental tool to probe, interpret and deduce properties of transport at the nanoscale [START_REF] Hoang | Electrical noise in electrolytes: a theoretical perspective[END_REF]. In addition, this understanding could lead to the improvement of many technological designs, with the increase of signal-to-noise ratio in single-molecule sensing for biophysics and biotechnologies [START_REF] Dekker | Solid-state nanopores[END_REF].

The present work aims to model the electrical fluctuations in bulk and confined electrolytes, from the microscopic dynamics of the ions, which play a key role in these phenomena, as implied by Hooge's law (1.3). At the intersection of many different disciplines, the modelling of ionic transport at the nanoscale combines various features, such as electrokinetics, chemistry, and surface effects, and, in particular, due to the small size of the system, fluctuations. Combined, modelling these contributions constitutes a great challenge, that can be approached through analytical and simulation techniques. In the remainder of this introductory chapter, we review preceding studies on the modelling of electrolytes, ionic transport, and some related topics.

Modelling electrical current noise in electrolytes

Apart from electrical fluctuations in nanotubes, because of the ubiquity of electrolytes both naturally and in technological applications, broad theoretical frameworks were developed to model them. Because these theories employ various analytical tools that were available at the time they were developed, we briefly provide a historical picture.

Historical approaches

Poisson-Boltzmann and Debye-Hückel, the mean-field approximation

This historical journey of the modelling of electrolytes begins in the early 20th century. At this point, statistical mechanics was emerging and developing thanks to the combined pioneering works of Ludwig Boltzmann, James Clerk Maxwell, and Josiah Willard Gibbs (and many more). One result of paramount importance is the derivation of the Boltzmann distribution

p B ( X) = 1 Z exp - E( X) k B T , Z = ˆΩ exp - E( X) k B T d X . (1.4)
This result stipulates that, for a system at thermal equilibrium in the canonical ensemble, the probability of finding that system in a given configuration X is related to the energy of that configuration E( X) by Eq. (1.4), where k B is the celebrated Boltzmann constant, T the temperature of the system, and Ω the set of all accessible configurations. Complementary to that, the laws of electrostatics in continuous media stipulate that the electrostatic energy of a given charge distribution in space ρ(r), is related to the Coulomb potential V (r) by the Poisson equation

∆V (r) = - ρ(r) 4πε , (1.5)
which depends on the permittivity ε of the medium. By combining Eq. (1.4) and Eq. (1.5), one can, for example, treat the restricted case of a point charge q self-interacting through its own electric field, therefore associate the energy of a configuration to the work required to move the charge across the distance r in that potential, then one gets to the selfconsistent Poisson-Boltzmann equation for the electrostatic field ∆V (r) = -q 4πε exp -q V (r) k B T .

(1.6)

This modelling using both statistical physics and electrostatics was proposed independently by Louis Georges Gouy in 1910 and David Leonard Chapman in 1913, with the objective of describing an ionic solution at thermal equilibrium, close to a solid charged interface. Extensively used in electrochemistry, many variations [START_REF] Herrero | Poisson-boltzmann formulary: Second edition[END_REF] around the Poisson-Boltzmann model Eq. (1.6) were developed to model specific situations, mainly different spatial regions of an electrical double layer, such as Helmholtz, Stern or Grahame models, and also conjointly in colloids chemistry with Derjaguin Landau Verwey Overbeek (DLVO) theory [START_REF] Dean | Electrostatics of Soft and Disordered Matter[END_REF]. By resolving these models one predicts the electrical potential in the electrolyte solution, which is the desired goal for most fundamental and technological applications around electrochemistry. Furthermore, modifications or refinement can be made to Eq. (1.6) in order to describe various situations, making it an extremely powerful tool for modelling electrolyte solutions. Poisson-Boltzmann theory is however limited to static properties. The understanding of transport has another long history, that we will quickly skim through.

Phenomenological approach to transport

Independently of ionic solutions or electrochemistry, transport phenomena are some of the most prolific sources of modelling in Physics and Mathematics. As for the previous part, we begin with the historical model that is most relevant to the subject of this thesis, allowing the introduction of important notions. First developed by Joseph Fourier in 1822 for the purpose of modelling how a quantity, such as heat, diffuses in a material, the Fourier heat equation reads ∂u(r, t) ∂t = D∆u(r, t) .

(1.7)

Of apparent simplicity, this equation elegantly states that the variation of heat, represented by the scalar field u(r, t), which depends on time and space, is proportional to its second spatial derivative, with the phenomenological constant D, the diffusion coefficient, describing the rate of transport. The most striking feature of that equation is that it applies to countless transport phenomena. Indeed, this equation can be derived with little specification of the studied system: one only has to assume that a quantity of interest U(t) (being heat, mass, charge, or a population of wild animals ...), is conserved in a given volume Ω, such that the evolution of U(t) is equal to the associated flux j(r, t) passing through the boundaries ∂Ω of that volume

U(t) + ‹ ∂Ω j(r, t) • d 2 Σ = 0 . (1.8)
Using Stokes' theorem, one rewrites that balance in the local form ∂u(r, t) ∂t + ∇ • j(r, t) = 0 , (1.9)

where u(r, t) is the spatial distribution of U(t) i.e its density, such that U(t) = ˆΩ u(r, t) d 3 r (1.10)

Eq. (1.9) is called a continuity/conservation equation, a common element in almost all transport theories. Then the next step is to use Fick's law, formalized by Adolf Fick in 1855, establishing a simple relation between the flux j(r, t) and the observed density u(r, t). In diffusive processes, one expects inhomogeneities to vanish in time, leading to flat homogeneous distributions due to the second law of thermodynamics. This property translates to Fick's law: j(r, t) = -D∇u(r, t) , (1.11) the regularizing flux must be equal to the opposite of the gradient of u(r, t), up to a strictly positive coefficient D, called the diffusion coefficient. Finally by combining Eq. (1.9) and Eq. (1.11), one arrives at the diffusion equation Eq.(1.7) as Fourier did for heat transfer in the early 19th century. Then, solving that mathematical equation allows to predict countless physical situations, with only the underlying principles of first and second thermodynamic laws. Anecdotally, while a mathematical formulation usually gives insights into concrete physical situations (at least for us), the roles can sometimes be reversed [START_REF] Satya | When random walkers help solving intriguing integrals[END_REF]. The entropic construction introduced by Boltzmann serves as a precious analytic tool for transport theory [START_REF] Perelman | The entropy formula for the ricci flow and its geometric applications[END_REF].

Derivations of Eq. (1.7), requiring apparently very few generic arguments, can be applied to numerous transport phenomena, ranging from material to social or economical sciences [START_REF] Bouchaud | De la physique statistique aux sciences sociales[END_REF]. Furthermore, this prototype of transport equation, while being only the tip of the iceberg, can be generalized and extended to some of the most prolific topics in Mathematics, Physics, or other disciplines. For example, one could ask the questions: why does the diffusion coefficient have to be constant, can it depend on the position in space ? Or even the local density ? What if there is an advective flux ? What about the stability if a small random perturbation flux is added ? Is Fick's law the only flux possible etc... Without going too far into that digression, we will actually focus on a model that can be seen as a direct legacy of Eq. (1.7), namely the Stochastic Dynamical Density Functional Theory (briefly introduced in Sec. 1.3.5.2 and presented in detail in Chap. [START_REF] Liu | Ultrashort singlewalled carbon nanotubes in a lipid bilayer as a new nanopore sensor[END_REF]), and apply it to electrolyte solutions. Because it is based on macroscopic/phenomenological considerations, Eq. (1.7), doesn't highlight the underlying microscopic mechanisms in the system, which should be the forces at play that drive transport. Therefore, to fully understand the properties of ionic transport arising at the macroscopic scale, one must also investigate their microscopic origins.

A challenge of size

An often forgotten, and yet striking observation in Physics textbooks, is that we know the properties of matter at the microscopic scales with phenomenal accuracy: what are its elementary components, and how it behaves. With the advent of modern quantum theories in the late 20th century, such as quantum field theory, the standard model etc the microscopic modelling of matter sounds like an almost closed debate, and yet many phenomena, such as the fluctuations of the ionic current in nanopores, remain with unanswered questions. Worst, many of the mechanisms explored at this scale, should be rationalizable in terms of classical theories. In our case, the main obstacle to our goal comes from the immense gap between the dynamics of ions at the nanoscale and the formulation of transport at the macroscopic scale. Nevertheless, we have already shown a way to relate both scales using statistical mechanics in Sec. 1.3.1, but only in the static case, and with several undisclosed approximations. In particular, in the study of ionic solutions, microscopic theories to model the liquid state [START_REF] Hansen | Theory of Simple Liquids[END_REF] are, without electrostatics, already complex and rich, because they consist in interacting N -body problems.

In the framework of nanofluidics, which has emerged as its own field in the last 20 years, an additional difficulty has arisen in the modelling of liquid matter. At the nanoscale, the long-established models such as the ones attributed to Bernoulli and Euler from the early 18th century to the one of Navier and Stokes in the 19th, relying on a continuous description of fluids, break down [START_REF] Kavokine | Fluids at the nanoscale: From continuum to subcontinuum transport[END_REF]. At this scale, the nowadays known discrete structure of matter cannot be ignored anymore, as it gives rise to significant changes to the transport properties, allowing coupling with quantum effects, atomistic correlations, surface chemistry, electrostatics, and thermal fluctuations (see Fig. 1.4 [START_REF] Kavokine | Fluids at the nanoscale: From continuum to subcontinuum transport[END_REF]). Then one has to infer all the relevant mechanisms at play, which potentially occur at different length and time scales, and combine them into a single model. After giving an overview of recent results concerning the microscopic modelling of electrolytes, starting with analytical descriptions and moving on to the numerical simulation methods, we will present a motivated strategy to investigate ionic transport at the nanoscale.

Electrostatic screening in electrolytes

Regarding the equilibrium thermodynamic descriptions of electrolytes, following the work of Gouy [START_REF] Gouy | Sur la constitution de la charge électrique à la surface d'un électrolyte[END_REF], Chapman [START_REF] Leonard | Li. a contribution to the theory of electrocapillarity[END_REF], Debye, Hückel [START_REF] Hückel | The theory of electrolytes: I. lowering of freezing point and related phenomena[END_REF], Onsager [START_REF] Per | The Collected Works of Lars Onsager[END_REF] and others, enormous progress has been made in both the description of ionic solutions and melts in the bulk or at interfaces. The specificity of Coulombic interactions leads to electrostatic screening, characterized, for sufficiently weak interactions (in particular in the case of dilute solutions), by the Debye length

λ D = k B T ε e 2 i z 2 i C i , (1.12)
where C i is the partial ionic concentration of species i, carrying the formal charge q i = z i e and ε is the dielectric constant of the pure solvent.

Figure 1.5: Electrostatic screening: On the left, schematics of electrostatic screening in a bulk electrolyte system, submitted to an external electric field E ext , which induces an inhomogeneity of charges, represented by an excess of anions in blue ρ -and cations in red ρ + . The separation of charges creates an induced electric field in the opposite direction E int , characterized by a typical size given by the Debye length λ D (Eq. (1.12)). On the right panel, schematics of screening at interfaces, with an electrical double layer at a charged wall (z being the distance from the wall, Σ its surface charge density, ε the solvent dielectric constant, and λ D the Debye length of the electrolyte Eq. (1.12)). a): Profile of the electrostatic potential ϕ Eq.(1.5) and ϕ s its value at the position of the interface. b) Profiles of the density of cations n + and anions n -, with a fixed value n 0 in the bulk. Ref. [START_REF] Herrero | Poisson-boltzmann formulary: Second edition[END_REF] Electrostatic screening is the manifestation of the competition between thermal fluctuations of the ions in the solvent, and the electrostatic forces that apply to them. If an electrostatic field is applied to an electrolyte solution that was locally neutral, the displacement of the ions induces a separation of anions and cations, leading to an induced dipole moment. This dipole then creates an additional induced electric field in the opposite direction, which will compensate the original one. This means that a charge in an electrolyte does not generate the usual long-range (∼ 1 r ) Coulombic interaction around itself, but an effective screened short-range (∼ e -r/λ D r ) interaction due to the mobile charges in the solution. Quantifying the entropic cost to reorganise the ions, statistical physics predicts that the screening occurs on length scales of the Debye length λ D , which can be computed from the linearized Poisson-Boltzmann equation Eq. (1.6). This result, first derived in the linearized mean-field approximation, is still the source of many ongoing studies, due to the fact that numerous experimental observations, such as underscreening and overscreening, remain partly unexplained [START_REF] Gebbie | Long-range electrostatic screening in ionic liquids[END_REF][START_REF] Lee | Underscreening in concentrated electrolytes[END_REF][START_REF] Smith | The electrostatic screening length in concentrated electrolytes increases with concentration[END_REF].

While several models still rely on variations of the Poisson-Boltzmann Eq. (1.6) [START_REF] Ben-Yaakov | Beyond standard poisson-boltzmann theory: ion-specific interactions in aqueous solutions[END_REF][START_REF] Adar | Bjerrum pairs in ionic solutions: A Poisson-Boltzmann approach[END_REF][START_REF] Adar | Electrostatics of patchy surfaces[END_REF][START_REF] Adar | Dielectric constant of ionic solutions: Combined effects of correlations and excluded volume[END_REF][START_REF] Adar | Osmotic pressure between arbitrarily charged planar surfaces: A revisited approach[END_REF], modern statistical mechanics offers new rigorous tools to tackle that problem, in particular addressing the effect of ionic correlations beyond the mean-field approximation. One can for example treat the short-range atomistic repulsion between ions, by adding hard-core repulsion similar to hard-spheres [START_REF] Wertheim | Exact solution of the percus-yevick integral equation for hard spheres[END_REF], resulting in the Primitive Model (PM). While this type of N -body problems used to be impossible to handle from the analytical point of view for decades, perturbative and nonperturbative treatments are now possible with the use of Landau Fluctuation theory [START_REF] Rotenberg | Underscreening in ionic liquids: a first principles analysis[END_REF] of renormalization-group techniques [START_REF] Kholodenko | Theory of symmetric electrolyte solutions: Field-theoretic approach[END_REF] for example. For general liquids, a significant leap was also made with the development of classical Density Functional Theory (DFT) [START_REF] Hansen | Theory of Simple Liquids[END_REF][START_REF] Chandler | Introduction to Modern Statistical Mechanics[END_REF][START_REF] Evans | Statistical mechanics of nonequilibrium liquids[END_REF], leading to systematic techniques to treat complex fluids, and especially charged ones for our case. In order to use DFT, one has to design a free energy functional corresponding to a microscopic model. For the electrolyte case, the Mean Spherical Approximation model was used as a closure relation to the Ornstein-Zernike integral equation [START_REF] Percus | Approximation methods in classical statistical mechanics[END_REF], leading to quantitative corrections to the electrostatic screening length obtained by Debye in the dilute regime (see Fig. 1.12) as a function of the particle radius σ also normalized by the Debye length for a 1:1 electrolyte. Flat solid line representing the Debye-Hückel prediction λ S = λ D , dashed line Landau fluctuation theory combined with MSA results for thermodynamic functions. Symbols are also computed from pair correlation function theory using the MSA approximations but as exact poles of the structure factors, with various parameters of the primitive model (Ref. [START_REF] Rotenberg | Underscreening in ionic liquids: a first principles analysis[END_REF]).

Transport in electrolytes

Regarding transport properties, an important result was formulated by Nernst and Einstein for an ideal system under an external drive. Combining Fick's diffusion Eq. (1.11), with an advective flux generated by a conservative force, and assuming the Boltzmann distribution Eq. (1.4) at stationary state, one concludes that a force F , applied on a thermal ideal particle, generates the velocity v with .13) This result is called the Einstein relation (we will give another derivation of that law in terms of fluctuation-dissipation relation from microscopic Langevin dynamics in Eq. (1.19)). Applying this relation to ideal mobile charges under an electric field readily yields a model for the conductivity of a strong electrolyte solution (with the same characteristics as in Eq. (1.12)):

v = D F k B T . ( 1
σ NE = e 2 k B T i z 2 i D i C i , (1.14)
where D i is the diffusion coefficient of ions of type (i). From Eq. (1.14), one finds that the electrical conductivity of an electrolyte is proportional to its concentration in salt i.e. σ = Λ m C s , where C s is the salt concentration and Λ m the equivalent molar conductivity, which is independent of the concentration in Nernst-Einstein formulation Eq. (1.14). But in the second half of the 19th century, by conducting accurate experiments on electrolytes, Friedrich Kohlrausch established a law carrying his name:

Λ m = Λ 0 m -K C s , (1.15)
where Λ 0 m is the molar conductivity at infinite dilution, and K is a constant called the Kohlrausch coefficient, predicting a deviation from the ideal regime with the scaling √ C s . Indeed, along the derivation of the Nernst-Einstein conductivity Eq. (1.14), we have assumed that each mobile charge behaves as independent, and summed their individual contribution. This assumption, while valid under an infinite dilution regime, neglects the many interactions occurring between ions (like for ideal gas and real gas). An early fruitful approach was done by Lars Onsager in 1927, who used Debye-Hückel theory (see Sec. 1.3.1.1) to incorporate electrostatic relaxation in the motion of the ions, and theoretically derived the Kohlrausch's law Eq.(1.15) from microscopic considerations, leading to Debye-Hückel-Onsager theory [START_REF] Onsager | Report on a revision of the conductivity theory[END_REF]. Several other thermodynamic approaches were extensively used to compute other transport properties such as ionic self-diffusion, osmotic pressure, or compressibility using fluctuation theories and incorporating ionic correlations at different levels of description [START_REF] Dufrêche | Ionic self-diffusion in concentrated aqueous electrolyte solutions[END_REF][START_REF] Bernard | Conductance in electrolyte solutions using the mean spherical approximation[END_REF][START_REF] Dufrêche | Transport equations for concentrated electrolyte solutions: Reference frame, mutual diffusion[END_REF][START_REF] Dufrêche | Analytical Theories of Transport in Concentrated Electrolyte Solutions from the MSA[END_REF].

But with the development of nonequilibrium thermodynamics, a variety of transport theories has been adapted from different fields to study electrolytes. In particular, in the scope of soft matter and biological physics with dynamical density functional theory [START_REF] Michael Te Vrugt | Classical dynamical density functional theory: from fundamentals to applications[END_REF], or mode coupling theory [START_REF] Liesbeth | Mode-coupling theory of the glass transition: A primer[END_REF] developed for glass transitions dynamics and applied to close relatives of electrolyte solutions which are molten salts and plasmas. These systems, thoughtfully considered by the physics community, give rise to questions also closely related to electrolyte solutions.

While it is easy to get lost in the wide range of formalisms to describe transport at the microscopic scale, it is important to focus on the ones that were applied to electrolytes. In particular, the modelling of a frequency-dependent ionic conductivity has been explored to understand the Debye-Falkenhagen effect [START_REF] Janssen | Transient dynamics of electric double layer capacitors: Exact expressions within the Debye-Falkenhagen approximation[END_REF], using the mode coupling theory (MCT) [START_REF] Banerjee | Ions' motion in water[END_REF][START_REF] Dufrêche | Ionic self-diffusion in concentrated aqueous electrolyte solutions[END_REF][START_REF] Chandra | The frequency dependent conductivity of electrolyte solutions[END_REF][START_REF] Chandra | Frequency dependence of ionic conductivity of electrolyte solutions[END_REF][START_REF] Roy | Mode coupling theory analysis of electrolyte solutions: Time dependent diffusion, intermediate scattering function, and ion solvation dynamics[END_REF][START_REF] Dufrêche | Electrostatic relaxation and hydrodynamic interactions for self-diffusion of ions in electrolyte solutions[END_REF]. In particular, from MCT, a critical element in the origin of the frequency-dependence, is the frequency-dependent friction of the electrolyte arising from hydrodynamic interactions. Indeed in meso/macroscopic transport theories, such as Poisson-Nernst-Planck (PNP, see below), the dynamics are often modelled after a coarse-graining of the degrees of freedom of the water molecule: under the adiabatic approximation, they relax instantly, giving rise to viscous effects on the ions. MCT is a framework allowing the description of the coupling between the underlying solvent dynamics, regrouped under hydrodynamic interactions, and ions dynamics, guided by ionic interactions.

Models were also developed from the other end of the spectrum, starting from these hydrodynamic models, and then refined to take into account thermal fluctuations and the dynamics of the ions beyond the continuum description. This is in particular the case of fluctuating hydrodynamics (FHD), originally designed for fluid flow at the nanoscale, where thermal fluctuations cannot be neglected (see. Fig. 1.4) [START_REF] Kavokine | Fluids at the nanoscale: From continuum to subcontinuum transport[END_REF][START_REF] Çois | Thermal fluctuations in nanofluidic transport[END_REF][START_REF] Detcheverry | Thermal fluctuations of hydrodynamic flows in nanochannels[END_REF][START_REF] Marbach | Transport and dispersion across wiggling nanopores[END_REF]. Once applied to electrolyte solutions [START_REF] Péraud | Fluctuation-enhanced electric conductivity in electrolyte solutions[END_REF][START_REF] Donev | Fluctuating hydrodynamics and debye-hückel-onsager theory for electrolytes[END_REF][START_REF] Donev | Fluctuating hydrodynamics of electrolytes at electroneutral scales[END_REF], the coarse-grained approach of the FHD formalism allows to recover properties of electrolyte predicted by Debye-Hückel-Onsager theory for transport [START_REF] Péraud | Fluctuation-enhanced electric conductivity in electrolyte solutions[END_REF].

Closely related to the last cited approaches: MCT and FHD, Dynamical Density Functional theories (DDFT) were recently developed as a dynamical extension of DFT (both on the quantum and classical level, but only the second for us). Now celebrated as a cornerstone of modern statistical physics [START_REF] Michael Te Vrugt | Classical dynamical density functional theory: from fundamentals to applications[END_REF], its broad framework goes beyond the scope of this thesis. DDFT possesses the quality of being derived from first principles straightforwardly (see Chap. 3), and is relatively well understood in terms of phenomenological interpretation. One advantage of its broad formulation is that one can choose which interactions are to be included in the model, in the same fashion as for DFT. For electrolytes, DDFT at the mean-field approximation level has actually been introduced as one of the early models for dynamics of charged fluids (way before DDFT was formalized), describing electrokinetics as an advection-diffusion-migration process in the so-called Poisson-Nernst-Planck (PNP) model [START_REF] Kavokine | Fluids at the nanoscale: From continuum to subcontinuum transport[END_REF][START_REF] Michael Te Vrugt | Classical dynamical density functional theory: from fundamentals to applications[END_REF]. For an electrolyte with the same characteristics as for Eq. (1.12) and Eq. (1.14), the local ionic densities ρ i (r, t), carrying the charge density q i ρ i respectively, evolve according to

               ∂ρ i (r, t) ∂t + ∇ • j i (r, t) = 0 j i (r, t) = -D i ∇ρ i + ρ i u + D i q i k B T ρ i (E ext -∇V ) ∆V (r, t) = - i q i ϵ 0 ϵ r ρ i (r, t) (1.16)
where u(r, t) is the velocity field of the fluid, and V (r, t) the electrostatic potential satisfying the Poisson equation. While not analytically solvable in the general case, when completed with a model for the hydrodynamic velocity field u and boundary conditions, the PNP model yields a rich phenomenology, allowing the exploration of charge correlation, charge relaxation, and other transport processes (even in the nonlinear regime) [START_REF] Frusawa | Electric-field-induced oscillations in ionic fluids: a unified formulation of modified poisson-nernst-planck models and its relevance to correlation function analysis[END_REF][START_REF] Kavokine | Fluids at the nanoscale: From continuum to subcontinuum transport[END_REF][START_REF] Michael Te Vrugt | Classical dynamical density functional theory: from fundamentals to applications[END_REF][START_REF] Palaia | Charging dynamics of electric double layer nanocapacitors in mean-field[END_REF]. Once linearized, PNP has also been used to describe transient fluctuations in electrolytes [START_REF] Mahdisoltani | Transient fluctuation-induced forces in driven electrolytes after an electric field quench[END_REF], long-range fluctuations-induced forces [START_REF] Mahdisoltani | Long-range fluctuation-induced forces in driven electrolytes[END_REF], charging in nanocapacitors and electrical double layers (EDL) [START_REF] Lobaskin | Diffusive-convective transition in the non-equilibrium charging of an electric double layer[END_REF][START_REF] Bazant | Diffuse-charge dynamics in electrochemical systems[END_REF][START_REF] Kondrat | Charging dynamics and optimization of nanoporous supercapacitors[END_REF][START_REF] Palaia | Charging dynamics of electric double layer nanocapacitors in mean-field[END_REF], the electrical conductivity of strong electrolytes [START_REF] Démery | The conductivity of strong electrolytes from stochastic density functional theory[END_REF][START_REF] Avni | Conductance of concentrated electrolytes: Multivalency and the Wien effect[END_REF][START_REF] Avni | Conductivity of Concentrated Electrolytes[END_REF], the explicit temporal response [START_REF] Bonneau | Temporal response of the conductivity of electrolytes[END_REF] or recently with the ionic current fluctuation spectrum [START_REF] Zorkot | Current fluctuations across a nano-pore[END_REF]. The incorporation of electrostatics, confinement, external driving forces, hydrodynamics, finite ionic size, and thermal fluctuations in DDFT, allows for the efficiently recover of many of the previously obtained results on electrolytes such as the Wien effect or Kohlrausch's (Eq.(1.15)) correction for the ionic conductivity.

While recent (stochastic or modified PNP) DDFT versions have been extensively used to compute macroscopic transport properties, contrary to MCT approaches, the emphasis on couplings at intermediate length and time scales, such as frequency-dependent fluctuations and conductivity, has rarely been investigated except for the recent thesis of Mira Zorkot [START_REF] Zorkot | Current fluctuations across a nano-pore[END_REF], and the analysis of dielectric properties of colloidal suspensions in electrolytes, with the computation frequency-dependent conductivity and permittivity, in order to interpret dielectric spectroscopy experiments. In particular, Refs. [START_REF] Rotenberg | Frequency-dependent dielectric permittivity of salt-free charged lamellar systems[END_REF][START_REF] Chassagne | Polarization between concentric cylindrical electrodes[END_REF][START_REF] Chassagne | Compensating for electrode polarization in dielectric spectroscopy studies of colloidal suspensions: Theoretical assessment of existing methods[END_REF][START_REF] Chassagne | Dielectric response of colloidal spheres in non-symmetric electrolytes[END_REF][START_REF] Chassagne | The dielectric response of a colloidal spheroid[END_REF] report the construction of response functions, which are the microscopic counterparts of transport coefficients, closing the gap between microscopic and macroscopic transport descriptions. The link between microscopic dynamics and macroscopic transport is of great importance for this thesis, and will be detailed in Sec. 1.3.6, and formally addressed in this manuscript in Sec. 2.3.

Simulation methods

With the recent progress in computer sciences, fundamental and applied material sciences made a leap forward, with the numerical resolution of models that were impossible to handle otherwise. Macroscopic numerical simulations, such as computational fluid dynamics, mechanical stress analysis, meteorological simulations etc now have a counterpart that can tackle the topic of material sciences, through the use of the phenomenal knowledge that we possess on the microscopic description of matter at the atomistic level. Solutions to N -body problems can eventually be obtained with a substantial number of elements, by integrating the microscopic equations of motion for a desired duration, from which macroscopic properties can be deduced using statistical mechanics and thermodynamics. While solving models such as the Schrödinger equation for the electrons and nuclei in a given material, with the least amount of approximations, also called ab initio methods, yields very accurate results in principle, this approach is rather limited because of the enormous entailed computational cost. Therefore ab initio methods are generally used for limited numbers of particles and very short timescales. In order to model larger systems and longer times scales, one needs to simplify the description, then one can do the process of zooming out of the frame of the picture. A set of particles very close to each other viewed from very far away might be pictured as a single collection by uniting them together, such as electrons and nuclei to form atoms, then molecules. The properties of these collections are to be defined in a rigorous way, namely coarse-graining techniques [START_REF] Pagonabarraga | Recent advances in the modelling and simulation of electrokinetic effects: bridging the gap between atomistic and macroscopic descriptions[END_REF]. By successively integrating out microscopic degrees of freedom, one establishes new equations of motion for the new grains, that can be solved with relevant simulation techniques. Similarly to experimental techniques, each modelling and simulations ones are adapted to a certain range of length and time scales as illustrated in Fig. 1.7. Simulating high-energy nuclei structures might not be relevant to the description of ionic dynamics. Starting from the left side of this axis, we will explore the different existing methods and their contributions to the understanding of electrolytes.

Microscopic and macroscopic simulations

Molecular dynamics (MD) simulations, based on the resolution of Newton's equations of motion for atoms modelled by force fields, have been successful at describing equilibrium properties of fluids, leading to molecular insights in many fields such as reaction pathways in chemistry [START_REF] Zhou | Molecular dynamics simulations, reaction pathway and mechanism dissection, and kinetics modeling of the nitric acid oxidation of dicyanamide and dicyanoborohydride anions[END_REF], dynamic properties of macro-molecules [START_REF] Gartner | Modeling and simulations of polymers: A roadmap[END_REF] or protein folding [START_REF] Sabab Hasan Khan | Protein folding: Molecular dynamics simulations and in vitro studies for probing mechanism of urea-and guanidinium chloride-induced unfolding of horse cytochrome-c[END_REF]. For charged fluids close to a charged interfaces, ubiquitous in electrochemistry or energyrelated topics, MD simulations have been extensively studied for aqueous electrolytes solutions [START_REF] Limmer | Charge Fluctuations in Nanoscale Capacitors[END_REF][START_REF] Scalfi | Charge fluctuations from molecular simulations in the constant-potential ensemble[END_REF][START_REF] Scalfi | Molecular Simulation of Electrode-Solution Interfaces[END_REF][START_REF] Pireddu | Frequency-dependent impedance of nanocapacitors from electrode charge fluctuations as a probe of electrolyte dynamics[END_REF][START_REF] Pireddu | A molecular perspective on induced charges on a metallic surface[END_REF] and complex ionic liquids [START_REF] Bedrov | Molecular dynamics simulations of ionic liquids and electrolytes using polarizable force fields[END_REF][START_REF] Merlet | Computer simulations of ionic liquids at electrochemical interfaces[END_REF][START_REF] Salanne | Simulations of room temperature ionic liquids: from polarizable to coarse-grained force fields[END_REF][START_REF] Salanne | Ionic liquids for supercapacitor applications[END_REF][START_REF] Salanne | From molten salts to room temperature ionic liquids: Simulation studies on chloroaluminate systems[END_REF][START_REF] Berthin | Solvation of anthraquinone and TEMPO redoxactive species in acetonitrile using a polarizable force field[END_REF][START_REF] Berthin | Nanostructural organization in a biredox ionic liquid[END_REF][START_REF] Bacon | On the key role of electrolyte-electrode van der waals interactions in the simulation of ionic liquids-based supercapacitors[END_REF] for energy storage. Regarding the role of microscopic ionic interactions in electrolyte transport, MD simulations have also been used to investigate the nonlinear response of bulk electrolytes to large electric fields [START_REF] Lesnicki | On the molecular correlations that result in field-dependent conductivities in electrolyte solutions[END_REF][START_REF] Lesnicki | Field-Dependent Ionic Conductivities from Generalized Fluctuation-Dissipation Relations[END_REF], as well as the electrokinetic response of confined electrolytes [START_REF] Mangaud | Chemisorbed vs physisorbed surface charge and its impact on electrokinetic transport: Carbon vs boron nitride surface[END_REF][START_REF] Yoshida | Molecular dynamics simulation of electrokinetic flow of an aqueous electrolyte solution in nanochannels[END_REF][START_REF] Xie | Liquid-solid slip on charged walls: The dramatic impact of charge distribution[END_REF][START_REF] Lorenz | Molecular dynamics of ionic transport and electrokinetic effects in realistic silica channels[END_REF]. While MD gives good structural insights, in particular into the solvation of charged species [START_REF] Fyta | Ionic force field optimization based on single-ion and ion-pair solvation properties: Going beyond standard mixing rules[END_REF], slow collective dynamical properties are rather difficult to obtain, due to the limited number of simulated ions and trajectory lengths [START_REF] Frenkel | Understanding molecular simulation: from algorithms to applications[END_REF][START_REF] Rotenberg | Electrokinetics: insights from simulation on the microscopic scale[END_REF]. Using typical time-steps of ∼ 1 fs, low-frequency correlations such as the one experimentally observed in Fig. 1.2, are a priori unreachable using conventional molecular dynamics simulations.

On the other side of the spectrum, from a macroscopic perspective, transport properties of confined electrolytes can be modelled with efficient continuous equations, of the same class as Eq. (1.7), and solved using a variety of computational fluid dynamics tools such as finite difference, finite volume, finite element or spectral element methods [START_REF] Ern | Theory and practice of finite elements[END_REF]. Unlike these methods relying on macroscopic considerations of transport, the Lattice Boltzmann Method (LBM) is rooted in microscopic models and mesoscopic kinetic equations for fluids using Boltzmann Equation and the Chapman-Enskog method. Combined with a model for electrostatics, such as the Poisson equation Eq. (1.5), Lattice Boltzmann Electrokinetics (LBE) has been successfully applied to electrolyte systems, in bulk and at interfaces [START_REF] Capuani | Discrete solution of the electrokinetic equations[END_REF][START_REF] Sylvain Reboux | Lattice-boltzmann simulations of ionic current modulation by dna translocation[END_REF][START_REF] Adelchi | Lattice Boltzmann electrokinetics simulation of nanocapacitors[END_REF][START_REF] Adelchi | Transient hydrodynamic finite-size effects in simulations under periodic boundary conditions[END_REF].

Stochastic dynamics for ions in an implicit solvent

Between the microscopic and the macroscopic pictures, mesoscopic simulation methods, such as LBE, are set at time and length scales where numerous coupled phenomena occur [START_REF] Kavokine | Fluids at the nanoscale: From continuum to subcontinuum transport[END_REF]. The understanding of the fluctuations arising from thermal motion, electrostatics, hydrodynamics interactions, and confinement remains a great challenge for modelling. In this subsection, we present the main simulation method that we use in this thesis. Traditionally used for colloidal suspensions, stochastic dynamics was formulated by Paul Langevin in 1908 to describe the Brownian motion, first observed by the botanist Robert Brown in 1827 for pollen beads in water, then studied experimentally and theoretically by Louis Bachelier, Albert Einstein, and Jean Perrin in the early 20th century. Stochastic dynamics, also called Langevin dynamics, is historically a formulation of an equation of motion where there is a clear separation of time scales between the observed object, and the thermalized 'background' in which the first one is bathing, such as colloidal suspen-sions or polymer solutions. In this framework, while some continuous solvent approaches describe the system as athermal, Langevin dynamics explicitly take thermal fluctuations into account. In this case, the coarse-graining of the solvent leaves a residue on the equations of motion, which takes the form of a traditional Newtonian equation for the studied object of mass m, with a friction term -γ ṙ and random force F rand originating from the collisions with the solvent molecules:

mr = -γ ṙ + F rand , (1.17)
with F rand having the statistical properties of zero average and no correlations in time.

Due to the properties of systems at equilibrium, the energy injected by the random force must be balanced by the one dissipated by the friction term:

⟨F rand (t)⟩ = 0 F rand (t) • F rand (t ′ ) = 6 γ k B T δ(t -t ′ ) , (1.18)
where δ(τ ) is the Dirac delta function. By observing that the object has a diffusing behaviour at long time scales, one can characterize the diffusion coefficient D, and identify it to the microscopic constant in Eq. (1.18), yielding the fluctuation-dissipation relation

D = k B T γ . (1.19)
In addition, one can associate the friction coefficient with Stokes' law for viscous drag, leading to the Stokes-Einstein equation:

D = k B T 6πη a , (1.20)
where η is the dynamic viscosity of the fluid, and a the radius of the object assumed spherical. This equation combining many important physical parameters of the system is extensively used experimentally to deduce one by measuring the others. Eq. (1.17) can be further specified or generalized to describe other systems, many bodies, nonequilibrium, memory effect etc. While separation of scales between simple ions and water molecules might not appear evident, the use of stochastic simulations for ionic solutions has been introduced in Refs. [START_REF] Turq | Brownian dynamics: Its application to ionic solutions[END_REF][START_REF] Trullàs | Langevin dynamics study of NaCl electrolyte solutions at different concentrations[END_REF], followed by the work of many, including in the PHENIX laboratory [START_REF] Jardat | Transport coefficients of electrolyte solutions from smart brownian dynamics simulations[END_REF][START_REF] Jardat | Brownian dynamics simulations of electrolyte mixtures: computation of transport coefficients and comparison with an analytical transport theory[END_REF][START_REF] Jardat | Brownian simulations contribution to the study of ionic dynamics in aqueous solutions[END_REF][START_REF] Jardat | Self-diffusion of ions in charged nanoporous media[END_REF][START_REF] Dufrêche | Mutual diffusion coefficient of charged particles in the solvent-fixed frame of reference from Brownian dynamics simulation[END_REF][START_REF] Grün | Relaxation of the electrical double layer after an electron transfer approached by Brownian dynamics simulation[END_REF]. Indeed, while the dynamics of generic monoatomic solute particles might not be always deconvoluted from the water molecules, ionic dynamics are driven by electrostatic effects, occurring at longer time-scales. Electrostatic relaxation arising from the Coulomb interactions occurs around the Debye time τ Debye , equal to the time for one ion to diffuse over the Debye length:

τ Debye = λ 2 D /D . (1.21)
This relaxation time, of the order of a nanosecond for aqueous NaCl at C s = 0.1 M, is best suited to be investigated in the framework of mesoscopic simulations such as Langevin dynamics. This approach has proven efficient for computing transport coefficients such as the effective diffusion coefficient (see Fig. 1.9) of ions, or the electrical conductance (see Fig. 1.10), agreeing with experimental measurements. In particular, Refs. [START_REF] Jardat | Transport coefficients of electrolyte solutions from smart brownian dynamics simulations[END_REF][START_REF] Yamaguchi | Brownian dynamics simulation of a model simple electrolyte in solvents of low dielectric constant[END_REF] show the importance of taking hydrodynamic interactions into account for transport coefficients in electrolytes. More recently, the effectiveness of Langevin and Brownian dynamics to couple interacting systems with thermal fluctuations has been fruitfully exploited for other mechanisms and properties for charged fluids and colloidal systems such as the alternating dielectric response of colloids in electrolyte solutions [START_REF] Zhou | Dielectric response of nanoscopic spherical colloids in alternating electric fields: a dissipative particle dynamics simulation[END_REF][START_REF] Zhou | Computer simulations of charged colloids in alternating electric fields[END_REF][START_REF] Zhou | Dynamic and dielectric response of charged colloids in electrolyte solutions to external electric fields[END_REF], adsorption in nanopores [START_REF] Gravelle | Adsorption Kinetics in Open Nanopores as a Source of Low-Frequency Noise[END_REF][START_REF] Sf Knowles | Noise properties of rectifying and non-rectifying nanopores[END_REF][START_REF] Stuart F Knowles | Current fluctuations in nanopores reveal the polymer-wall adsorption potential[END_REF], or ionic current fluctuations in nanopores [START_REF] Zorkot | Current fluctuations in nanopores: The effects of electrostatic and hydrodynamic interactions[END_REF][START_REF] Zorkot | The Power Spectrum of Ionic Nanopore Currents: The Role of Ion Correlations[END_REF][START_REF] Zorkot | Current fluctuations across a nano-pore[END_REF]. Stochastic simulation techniques are also the source of development of new methodologies for describing and simulating electrolytes [START_REF] Delle Site | Multiscale Approaches and Perspectives to Modeling Aqueous Electrolytes and Polyelectrolytes[END_REF] such as adaptative time-step for Brownian dynamics simulations using Monte Carlo methods [START_REF] Rossky | Brownian dynamics as smart Monte Carlo simulation[END_REF][START_REF] Jardat | Transport coefficients of electrolyte solutions from smart brownian dynamics simulations[END_REF], fluctuating hydrodynamics [START_REF] Ladiges | Discrete ion stochastic continuum overdamped solvent algorithm for modeling electrolytes[END_REF][START_REF] Ladiges | Modeling electrokinetic flows with the discrete ion stochastic continuum overdamped solvent algorithm[END_REF] for electrokinetics or [START_REF] Tyagi | Electrostatic layer correction with image charges: A linear scaling method to treat slab 2d+h systems with dielectric interfaces[END_REF][START_REF] Arnold | Efficient Algorithms for Electrostatic Interactions Including Dielectric Contrasts[END_REF][START_REF] Barros | Efficient and accurate simulation of dynamic dielectric objects[END_REF] for dielectric interfaces for implicit solvent models beyond electrolyte solutions.

Hydrodynamic effects, as for analytical DDFT, have to be treated with great care in implicit solvent particle simulations. For stochastic methods, other techniques have been have been developed, such as the recent advances in generalized Langevin equations (GLE) [START_REF] Lesnicki | Field-Dependent Ionic Conductivities from Generalized Fluctuation-Dissipation Relations[END_REF][START_REF] Cui | Generalized langevin equation and fluctuationdissipation theorem for particle-bath systems in external oscillating fields[END_REF]. Instead of taking the friction force to be a constant drag times the instantaneous velocity of the object, such as in Eq.(1.17), one can consider a generalized friction that could be dependent on the history of the trajectory, factoring elastic/nonlinear/relaxing/active properties of the solvent into a memory kernel. In this case, the GLE for a free Brownian motion reads

mr(t) = - ˆt 0 K (t -s; r, ṙ) ṙ(s) ds + F rand (t) . (1.22)
Closely related to the frequency-dependent friction used in Ref. [START_REF] Chandra | The frequency dependent conductivity of electrolyte solutions[END_REF][START_REF] Chandra | Frequency dependence of ionic conductivity of electrolyte solutions[END_REF][START_REF] Banerjee | Ions' motion in water[END_REF] in MCT, the identification of the memory kernel K is of great importance to establish good and systematic stochastic descriptions of coarse-grained models [START_REF] Lisý | Generalized langevin equation and the fluctuationdissipation theorem for particle-bath systems in a harmonic field[END_REF][START_REF] Vroylandt | Position-dependent memory kernel in generalized Langevin equations: Theory and numerical estimation[END_REF][START_REF] Vroylandt | Likelihood-based non-markovian models from molecular dynamics[END_REF][START_REF] Ayaz | Generalized langevin equation with a nonlinear potential of mean force and nonlinear memory friction from a hybrid projection scheme[END_REF][START_REF] Carof | Accurate Quadrupolar NMR Relaxation Rates of Aqueous Cations from Classical Molecular Dynamics[END_REF][START_REF] Lesnicki | Molecular hydrodynamics from memory kernels[END_REF], and can be estimated with microscopic simulation techniques, such as explicit solvent MD.

In the field of complex fluids, many theoretical tools such as collective variables, GLE, metadynamics, or machine-learning force fields are breeding grounds for intensive research.

1.3.6 Fluctuations: a link between microscopic dynamics and macroscopic response Until now, we have only vaguely stated the relation between microscopic modelling and the possibility to extract the transport properties of electrolytes. Similarly to thermodynamic observables, the experimentally accessible transport properties of a system is a manifestation of its microscopic response, which is usually not directly measured. One of the first links that was developed around this idea, was formalized in its modern form by the pioneering works of Onsager, Kubo, Green, and Le Chatelier in the framework of Linear Response Theory (LRT) [START_REF] Risken | Fokker-Planck Equation[END_REF], and encoded in the fluctuation-dissipation theorem (FDT). As its name implies, the usual FDT relates the equilibrium fluctuations of an observable, encoded in its spectral density, with the response functions of that same observable to a small external perturbation. This means that the study of the thermodynamic fluctuations of physical variables at equilibrium (or close to equilibrium), allows to predict response functions and transport coefficients. This statement is the main motivation for modelling microscopic dynamics, by analytical or numerical means. With the advent of microscopic and mesoscopic simulation methods, the importance of that thermodynamic link has been strengthened with the development of Green-Kubo [START_REF] Hansen | Theory of Simple Liquids[END_REF][START_REF] Risken | Fokker-Planck Equation[END_REF] and Kirkwood-Buff [START_REF] John | The statistical mechanical theory of solutions. i[END_REF][START_REF] Dawass | Kirkwood-buff integrals from molecular simulation[END_REF][START_REF] Krüger | Kirkwood-buff integrals for finite volumes[END_REF] integrals, allowing the extraction of thermodynamic and transport properties of a system using only equilibrium simulation techniques. The Green-Kubo formulae express transport coefficients as a limit of a time integral of correlations of microscopic fluctuations, while Kirkwood-Buff integrals allow to compute response functions from fluctuations in finite volumes. Other techniques such as fitting transport coefficients in hydrodynamic models (MCT) have notably been used for neutral fluids such as Lennard-Jones fluids or charged ones such as the one component plasma (OCP) or molten salts [START_REF] Giaquinta | Collective dynamics of charge fluctuations in ionic conductors[END_REF][START_REF] Cheng | Computing the heat conductivity of fluids from density fluctuations[END_REF][START_REF] Cheng | Computing chemical potentials of solutions from structure factors[END_REF].

For electrolytes, these Green-Kubo formulae have been extensively used, combined with various equilibrium simulations, to compute viscosity, electrical conductivity, or diffusivity [START_REF] Frenkel | Understanding molecular simulation: from algorithms to applications[END_REF][START_REF] Jardat | Transport coefficients of electrolyte solutions from smart brownian dynamics simulations[END_REF][START_REF] Lesnicki | Field-Dependent Ionic Conductivities from Generalized Fluctuation-Dissipation Relations[END_REF][START_REF] Lesnicki | On the molecular correlations that result in field-dependent conductivities in electrolyte solutions[END_REF], while the Kirwood-Buff integrals for charged systems have been mainly studied from the theoretical point of view [START_REF] Ph | The charge fluctuations in classical coulomb systems[END_REF][START_REF] Young | Charge fluctuations and correlation lengths in finite electrolytes[END_REF][START_REF] Bekiranov | Fluctuations in electrolytes: The lebowitz and other correlation lengths[END_REF][START_REF] Young C Kim | Screening in ionic systems: simulations for the lebowitz length[END_REF][START_REF] Kalcher | Structure-thermodynamics relation of electrolyte solutions[END_REF][START_REF] Leblé | The two-dimensional one-component plasma is hyperuniform[END_REF][START_REF] Ghosh | Fluctuations, large deviations and rigidity in hyperuniform systems: a brief survey[END_REF] and less implemented for microscopic simulations [START_REF] Fyta | Ionic force field optimization based on single-ion and ion-pair solvation properties: Going beyond standard mixing rules[END_REF]. The Kirkwood-Buff integrals are directly related to the study of charge fluctuations in finite volumes (see Sec. 4.2).

Deducing microscopic properties from macroscopic fluctuations

Conversely, response theory also implies that the fluctuations carry a large amount of information about the microscopic properties of the system. One of the first manifestations of that relation was established in a field closely related to this work, namely the thermal agitation in conductors. Thermal origins of electrical noise were modelled with the Johnson-Nyquist formula in 1928 [START_REF] Nyquist | Thermal agitation of electric charge in conductors[END_REF][START_REF] Johnson | Thermal agitation of electricity in conductors[END_REF]:

S I (f ) = 4k B T η(f ) Re [Z(f )] , (1.23)
where Z(f ) is the impedance of the system, and η(f ) a factor close to 1 in a certain range. Surprisingly, this century-old equation is similar to the modern form of the FDT and precedes by 20 years the derivations made by Callen, Welton, and Kubo. Furthermore, having used the Boltzmann equipartition law for harmonic oscillators, Nyquist proposed in Ref. [START_REF] Nyquist | Thermal agitation of electric charge in conductors[END_REF] that at high frequencies or low temperature, the η(f ) factor should be taken as:

η(f ) = h f k B T 1 e hf /k B T -1 , (1.24)
where h is the Planck constant, matching the earlier law established by Planck in 1901 for the blackbody radiation problem. This formula provides evidence that electrical fluctuations can serve as an incredible tool to understand fundamental physics. But in 1928, Nyquist did not have access to the full panel of analytical and simulations modelling armoury available now, in particular for confined electrolytes.

Main objectives and plan

Due to the ubiquity of electrical noise in electrolyte systems, and the relevance of the latter for technological applications at the nanoscale, we will on the one hand investigate the properties of fluctuations in electrolytes to deduce transport properties, both in bulk and confinement. On the other hand, like the implicit link that was given by Nyquist, we reverse that first objective and aim to investigate fundamental mechanisms taking place in the dynamics of ions by modelling electrical noise and exploiting their intricate links.

This manuscript is organized as follows. In Sec. 2.1, we will re-formalize the method of Brownian dynamics simulations that were used in the literature, and describe how we apply it to electrolytes. We will neglect hydrodynamic interactions at this level of description, to focus on electrostatic effects dominating the dynamics of ions at long times and in the dilute regime. We define important tools from continuous modelling in Sec. 2.2, in particular correlation functions and dynamical structure factors. We then demonstrate Green-Kubo relations adapted to Brownian dynamics, specifying and extending the work of Felderhof et al. [START_REF] Felderhof | Linear response theory of sedimentation and diffusion in a suspension of spherical particles[END_REF] to space and time-dependent electrical conductivity in Sec. 2.3. We then detail in Sec. 2.4 the numerical tools that were developed and used to analyse fluctuations from the simulations, which will serve as comparisons with analytical models.

Chapter 3 presents recent developments of Stochastic Dynamical Density Functional Theory (sDDFT) for electrolytes, after recalling a derivation from [START_REF] David S Dean | Langevin equation for the density of a system of interacting langevin processes[END_REF], will reconstruct this theory from Brownian particles dynamics model for electrolytes, highlighting the approximations made for the description of implicit solvent, and its interpretation in Sec. 3.1. We then follow a resolution method that was often used in the literature, to obtain a generalized formula for the dynamical structure factors in Sec. 3.2.

Chapter 4 starts with a phenomenological description of electrostatic relaxation, arising from the sDDFT computations of dynamical structure factors, and highlights its role in electrical fluctuations in space and time. The results are compared to Brownian Dynamics simulations, yielding good agreement on the effect of the salt concentration. In Sec. 4.2, as both a first approach to confinement for electrolytes, an overview of properties of the Kirwood-Buff integrals and their dynamical extension, with an abstract excursion on hyperuniformity and exotic kinetic signatures for charged systems, we demonstrate an important result on the difference between the number of charge carriers, often used in the literature as a proxy for the conductivity and total charge fluctuations. Sec. 4.3.1 compares implicit solvent models to Molecular Dynamics simulations with explicit water, showing limits of overdamped Markovian dynamics and the mean-field approximation, and reasserting the important role of ionic correlations and hydrodynamic effects, that were neglected. The short Sec. 4.4 shows the computations of out-of-equilibrium structure factors, showing excellent agreement between sDDFT and Brownian dynamics simulations. This section also puts to use the generalized dynamical structure factor results in the presence of an applied electric field and unequal mobilities leading to complex phenomenology.

In chapter 5, while losing the analytical description of local fluctuations of sDDFT, we model confined electrolytes in a slit pore (Sec. 5.1) and study the effects of confinement, geometry, diffusion, migration, electrostatic relaxation, and adsorption on the frequencydependent conductivity in Sec. 5.2. Making use of the tailored Green-Kubo formula for frequency-dependent conductivity, we also harness the use of ionic current fluctuations in the framework of linear response theory, and compare the results to nonequilibrium simulations, analytical resolution of the Fokker-Planck equation, and nonlinear models. This chapter underlines the importance of associated timescales, which serve as decoding tools for the rich microscopic mechanisms encoded in the electrical conductivity. Advances in fundamental physics allow to us write exact governing equations for elementary components of matter at the microscopic scale. But the gap between elementary particles such as nuclei or electrons and macroscopic matter such as a drop of water spans several orders of magnitude in length and time scales. Statistical mechanics and thermodynamics allow us to bridge that gap. This chapter summarizes and factorizes the main modelling tools originating from statistical physics, particle simulations, continuous modelling, numerical methods, and linear response theory, which are used to model the electrical charge and current fluctuations across this manuscript. Most of the claims in this chapter can be found rigorously derived and discussed in reference textbooks such as [START_REF] Hansen | Theory of Simple Liquids[END_REF][START_REF] Frenkel | Understanding molecular simulation: from algorithms to applications[END_REF][START_REF] David | Classical electrodynamics[END_REF][START_REF] Risken | Fokker-Planck Equation[END_REF]]. An exception is found in Sec. 2.3, which presents an explicit spacetime frequency-dependent Green-Kubo formula for the electrical conductivity, adapted to Brownian dynamics.

Brownian dynamics simulations

As explained in the last chapter, the increase in computational resources in the last decades has enabled the development of powerful methods to investigate matter, in particular the ways to solve equations with numerical means. These methods consist of solving numerically the governing equations for a chosen microscopic model and deducing properties of matter using tools from statistical mechanics and thermodynamics. A generic dynamical system representing the microscopic dynamics with a set of variables X(t) (see below) usually read:

d dt X(t) = f ( X, t) , (2.1)
where f ( X, t) is a function describing the time evolution of these variables. A macroscopic property of the system A is expected to be measured experimentally at time t, with the value denoted ⟨A(t)⟩. From statistical mechanics, this value can be expressed as:

⟨A(t)⟩ = ˆA( X, t) dµ( Xt ) , (2.2)
where A( X, t) is the microscopic operator of A, relating its measured value with a microscopic state of the system, and dµ( Xt ) a probability measure which encodes the statistics of the states of the system. In principle, the knowledge of this measure requires fully solving Eq. (2.1) for all initial conditions, which is not usually feasible from numerical simulations. But for an observable at equilibrium, following the pioneering work of Boltzmann and assuming ergodicity, one claims that:

⟨A⟩ ∼ lim T →∞ 1 T ˆT 0 A(X(t)) dt ∼ 1 N T N T n A n , (2.3)
where A n<N T is a sufficiently long time sampling of A( X, t). This sampling can be obtained by solving numerically the dynamical system for long times / large numbers of particles, without the need of fully knowing dµ( Xt ). The equivalence between the statistical ensemble average and the time average is at the core of molecular dynamics simulations, allowing us to estimate macroscopic quantities from numerical experiments.

A key question is: what do X(t) and f ( X, t) represent ? What are the degrees of freedom that we want to describe, and in which generalized coordinates is the system best represented ? One could choose for example to solve the most "exact model available", such as the Schrödinger equation for electrons and the nuclei with the least amount of approximations. This approach is referred to as first principles or ab initio calculations. While being extremely accurate, this method is extremely costly in terms of computational resources as well, exponentially scaling with the number of degrees of freedom described. Then if someone wishes to describe the flow of air around a golf ball with the Schrödinger equation, assuming this person has infinite computational power, he/she will realize that quantum effects play no role in the trajectory of the ball, and won't gain any knowledge to improve his swings. A good model for a studied phenomenon, while being as valid and accurate as possible, must keep the cause-and-effect path as short as possible. Finding the minimal cause for a phenomenon is an efficient way of understanding it, and in particular to exploit its properties in technological applications. For molecular simulations, one can do the process of uniting groups of elementary components together, such as reuniting electrons and nuclei into atoms, successively erasing internal degrees of freedom in the system and reducing drastically the computational cost. Following that process, one can arrive at an implicit solvent model for the electrolyte, modelled with an overdamped Langevin equation, which is the limit of Eq.(1.17) for many body systems, where we have assumed inertial effect to be relaxed, as we will be studying long time dynamical process for the ions. This is the simulation method used in this thesis, giving access to longer trajectories for larger systems than conventional molecular dynamics.

Stochastic dynamics

Brownian dynamics simulations to model aqueous electrolytes have been extensively used in the literature [START_REF] Jardat | Transport coefficients of electrolyte solutions from smart brownian dynamics simulations[END_REF][START_REF] Jardat | Brownian dynamics simulations of electrolyte mixtures: computation of transport coefficients and comparison with an analytical transport theory[END_REF][START_REF] Jardat | Brownian simulations contribution to the study of ionic dynamics in aqueous solutions[END_REF][START_REF] Jardat | Self-diffusion of ions in charged nanoporous media[END_REF][START_REF] Jardat | Diffusion of a tracer in a dense mixture of soft particles connected to different thermostats[END_REF][START_REF] Dufrêche | Electrostatic relaxation and hydrodynamic interactions for self-diffusion of ions in electrolyte solutions[END_REF][START_REF] Chandra | The frequency dependent conductivity of electrolyte solutions[END_REF][START_REF] Chandra | Frequency dependence of ionic conductivity of electrolyte solutions[END_REF][START_REF] Chandra | The frequency dependent conductivity of electrolyte solutions[END_REF][START_REF] Chandra | Frequency dependence of ionic conductivity of electrolyte solutions[END_REF][START_REF] Banerjee | Ions' motion in water[END_REF][START_REF] Mondal | Anomalous dielectric response of nanoconfined water[END_REF][START_REF] Zorkot | Current fluctuations in nanopores: The effects of electrostatic and hydrodynamic interactions[END_REF][START_REF] Zorkot | The Power Spectrum of Ionic Nanopore Currents: The Role of Ion Correlations[END_REF][START_REF] Zorkot | Current fluctuations across a nano-pore[END_REF][START_REF] Hashemi | Computing hydrodynamic interactions in confined doubly-periodic geometries in linear time[END_REF][START_REF] Donev | Fluctuating hydrodynamics and debye-hückel-onsager theory for electrolytes[END_REF][START_REF] Gravelle | Adsorption Kinetics in Open Nanopores as a Source of Low-Frequency Noise[END_REF][START_REF] Ladiges | Discrete ion stochastic continuum overdamped solvent algorithm for modeling electrolytes[END_REF][START_REF] Ladiges | Modeling electrokinetic flows with the discrete ion stochastic continuum overdamped solvent algorithm[END_REF][START_REF] Lesnicki | Field-Dependent Ionic Conductivities from Generalized Fluctuation-Dissipation Relations[END_REF][START_REF] Lesnicki | On the molecular correlations that result in field-dependent conductivities in electrolyte solutions[END_REF][START_REF] Chassagne | Compensating for electrode polarization in dielectric spectroscopy studies of colloidal suspensions: Theoretical assessment of existing methods[END_REF], in order to compute different quantities, such as effective self-diffusion coefficients, collective diffusion coefficients, electrical conductivity, pair correlation functions etc... This method have been used with various levels of refinement and descriptions, sometimes including external drive, hydrodynamic interactions, frequency or concentration dependent mobilities, etc. Here we (re)formalize our approach, by succinctly introducing the principles and notations that will be used to study electrical current and charge fluctuations.

The electrolyte system is described by the trajectories of 2N cations and anions, carrying the formal charge ±e respectively, and bathing in a continuous medium at temperature T in a fixed volume V , which is endowed with periodic boundary conditions to approximate a macroscopically large sample. This model corresponds to an aqueous binary monovalent electrolyte at concentration C s = N/V . We consider that the collection of particles evolves according to the following overdamped Langevin equation:

d dt R = -βD • ∇ R V ( R, t) + (2D) 1/2 • η , (2.4)
where the instantaneous positions of all the particles are represented by the vector R = {r i } i<N ∈ R 2N . Its evolution depends on the potential energy of the system V : R 2N → R, describing all the conservative interactions acting on the particles, such that the force exercised on a particle (i) reads F i = -∇ r i V ( R). In the overdamped limit, the force generates a velocity according to the mobility of the particle, which in the case of neglected hydrodynamics, is encoded in a strictly positive and diagonal matrix containing the self-diffusion coefficients of each particle: µ = βD = β Diag(D i ), with β = 1/k B T . At this coarse-grained level of description, the effect of collisions of water molecules with the ions is introduced as a random force, modelled with a normalized standard isotropic Gaussian white noise process η, with zero average ⟨η⟩ = 0 and delta correlated in time ⟨η(t) • η⊺ (t ′ )⟩ = Diag(δ(tt ′ )). For balance reasons, the amount of energy dissipated by the friction must be related to the one injected by the random force. Then the fluctuation-dissipation theorem stipulates that the prefactor of that force random force is the matrix (2D) 1/2 .

In terms of formal stochastic differential equations, one recasts Eq. (2.4) as:

d Rt = -βD • ∇ R V ( Rt , t) dt + (2D) 1/2 • d Ŵ t , (2.5)
with Ŵ t the standard Wiener process. An observable θ of that system is a function of the microscopic configuration, and possibly of time θ( R, t). Its measurable value, equal to its ensemble average over the system, is assimilated to the expectation value for the stochastic process of the system's configuration. Similarly to Eq. (2.2), one rewrites it in terms of probability measure notation :

⟨θ(t)⟩ = E Rt θ( R, t) . (2.6)
The target is to numerically solve the stochastic differential system Eq. (2.5). Using finite difference in time, with a chosen time step of δt, one can compute the position of the system Rn+1 at time t + δt = (n + 1)δt, from the previous position Rn at time t = nδt as :

Rn+1 = Rn -βD • ∇ R V ( Rn , n) δt + (2δt D) 1/2 • ωn , (2.7)
with ωn ∈ R 2N a vector of identically independently distributed (i.i.d.) normal random numbers, with zero mean and unit variance, that can be generated numerically. This discretization is referred to as the forward Euler-Maruyama scheme, analogous to the Euler methods for deterministic differential equations. Similarly to deterministic equations, there are many ways to discretize a stochastic equation, especially in splitting the Wiener process. Instead of the intuitive Euler-Maruyama scheme, we use the overdamped BAOAB scheme [START_REF] Leimkuhler | Rational Construction of Stochastic Numerical Methods for Molecular Sampling[END_REF]:

Rn+1 = Rn -βD • ∇ R V ( Rn , n) δt + 1 2 δt D 1/2 • [ ωn + ωn+1 ] , (2.8)
which performs with better precision at equal time step than the previous scheme Eq. (2.7).

Computing Rn for a sufficiently large number of time steps allows to sample the probability distribution and compute the observable as a discrete time average:

⟨θ(t)⟩ = E Rt θ( R, t) ∼ lim N T →∞ 1 N T N T n θ( Rn , t) , (2.9)
and for a variable not dependent on time, such as an observable at thermodynamic equilibrium:

⟨θ⟩ = E Rt θ( R) ∼ lim N T →∞ 1 N T N T n θ( Rn ) .
(2.10)

Force fields and diffusion coefficients

We have described the general procedure of computing the microscopic dynamics using numerical simulations. We now detail the parameters of the model, which contain the specificity of the studied system. In order to study electrical fluctuations in confined electrolytes, we start with the simple model of aqueous binary monovalent electrolytes. We model the potential energy through the force field:

V ( R, t) = V int ( R) + V ext ( R, t) , (2.11)
where the total potential energy can be decomposed in an internal term V int ( R) describing the interactions between the ions, which depends only on the configuration of the system, and an external term V ext ( R, t), which describes a possible external force acting on the electrolyte, such as an external electric field or confining walls. To model inter-ionic interactions, we introduce pairwise potentials, which only depend on the type of particles and the distance r ij separating them:

V int ( R) = i<j V pair ij (r ij ) .
(2.12)

For ionic species, a pairwise interaction is separated into two contributions, one arising from atomic repulsion of the ions: V sr (r) which is typically of range of the size of the ions, and the other V Coul. (r) arising from electrostatic forces due to their charge:

V pair ij (r ij ) = V sr (r ij ) + V Coul. (r ij ) . (2.13)
For atomic force fields, we use the Lennard-Jones potential:

V LJ (r ij ) = 4 ϵ ij σ ij r ij 12 - σ ij r ij 6 , (2.14)
where the range and strength of the interaction are tuned by the pair of coefficients (σ ij , ϵ ij ). The Lennard-Jones potential is the most classical model for atomistic interactions. It consists of a short-range repulsive part originating from the Pauli exclusion principle, and an attractive part modelling Van der Waals interactions arising from polarization. This potential is illustrated in Fig. 2.1 where the equilibrium position given by the minimum of the potential is located at:

arg min r V LJ (r) = 2 1/6 σ . (2.15)
The Lennard-Jones potential Eq. (2.14) can be simplified into a Weeks-Chandler-Andersen potential by truncating the attractive part and shifting the remaining repulsive part at its minimum:

V WCA (r ij ) = V LJ ij (r ij ) -V LJ ij (2 1/6 σ ij ) , r ij ≤ 2 1/6 σ ij 0 , r ij > 2 1/6 σ ij . (2.16)
In most sections, we study the electrolyte NaCl, with WCA parameters and diffusion coefficients taken from molecular dynamics simulations with SPC/E water from Ref. [START_REF] Koneshan | Solvent structure, dynamics, and ion mobility in aqueous solutions at 25 °c[END_REF],

where we have applied Lorentz-Berthelot mixing rules for unlike particles: Pair coefficients : For section Sec.4.2, we have chosen to study a binary symmetric electrolyte close to NaCl, with WCA potentials with, σ ++ = σ --= σ +-= 3.00 Å and ϵ ++ = ϵ --= ϵ +-= 0.100 kcal/mol, with symmetric diffusion coefficients: D + = D -= 1.5 × 10 -9 m 2 /s. This corresponds to approximating the different ions with the average pair properties, in order to focus on other physical properties. In Sec.4.3, we used the full Lennard-Jones potentials truncated at 9 Å, with force fields parameters described in Ref. [START_REF] Mester | Mean ionic activity coefficients in aqueous nacl solutions from molecular dynamics simulations[END_REF], and diffusion coefficients extracted from molecular dynamics simulations Ref. [START_REF] Hoang | Electrical noise in electrolytes: a theoretical perspective[END_REF], to explicitly compare with all-atoms molecular and Langevin dynamics, parameters in Tab. Classical electrostatic forces are prescribed by the Coulomb law, which models the potential energy between two point charges as inversely proportional to their distance, and proportional to the product of their charge:

   σ ij = σ ii + σ jj 2 ϵ ij = √ ϵ ii ϵ jj , (2.17
ϵ ij , σ ij , r cut ij , D ij Na + Cl - Na + 0.
V Coul. (r ij ) = q i q j 4πε 0 ε w 1 r ij . (2.18)
In continuous (linear) media, the strength of the electrostatic interaction is tuned by the relative permittivity or dielectric constant ε r > 1 which describes the ability of an insu-lating medium to shield an electric field. We set this value as the experimental [START_REF] Malmberg | Dielectric constant of water from 0 to 100 c[END_REF] static dielectric constant of water at T=298 K : ε w = 78.5. In principle, this value depends on microscopic thermodynamic and dynamical properties of the medium, therefore it should be modelled with the frequency-dependent permittivity ε w (ω) for example, to incorporate the dispersive, reactant and possible non-linear features arising from the molecular nature of water. Nevertheless, we make this approximation of a simple continuous medium to focus on the dynamics of the ions. It will be addressed in Sec. 4.3.

ϵ ++ = ϵ --= ϵ +-= ϵ -+ = 0.5 k B T , σ ++ = σ --= σ -+ = σ +-= 3 Å and q + = -q -= 1.6 × 10 -19 C. b)
Illustration of interionic potentials for a fully symmetric electrolyte, as combinations of the latter potentials. XXX -Coul. for cations/anions pairs and XXX + Coul. for equal sign charges pairs.

The sum of the two contributions to the interionic potentials from Eq. (2.16) or Eq. (2.14) for atomic potential and Eq. (2.18) for electrostatics are summed for all anions and cations as illustrated in Fig. 2.1. A possible third one for external contribution can be added to model an applied electric field or a confining potential such as solid walls (see Chap. 5). If bulk electrolytes are simulated, then the infinite boundaries of the system are approximated by 3D periodic boundary conditions (PBC), and the interactions are computed inside the simulated volume V and also with other infinitely replicated images, mimicking a large volume [START_REF] Frenkel | Understanding molecular simulation: from algorithms to applications[END_REF]. Finally, the full resulting potential is then used in Eq. (2.4) to evolve the microscopic dynamics of the ions in time.

Efficient force computation

At each time step, the computation of all forces is required to compute the configuration of the next one. This procedure is the most expensive part of the numerical simulation. Indeed, with pairwise potentials, one naively needs to compute the interactions for every pair of particles in the system, which amounts to a large number O N 2 . For shortrange interactions, this cost can be safely (exactly for the WCA potential) reduced by truncating the pair potential at a certain cutoff distance, because they decay fast enough (< 1/r 1+ϵ ) such that the tail contribution modelling the forces between two ions which are far away, can be neglected. Short-range interactions only need to consider pairs of particles within a small neighbourhood, reducing the complexity to O (N ). This is not the case for electrostatic potential, decaying as 1/r, the tail contribution cannot be neglected, so every pair of particles must be accounted for. To alleviate that numerical cost, we make use of the Particle-Particle Particle-Mesh (P3M) algorithm [START_REF] Hockney | Computer simulation using particles[END_REF][START_REF] Pollock | Comments on p3m, fmm, and the ewald method for large periodic coulombic systems[END_REF]. The P3M method is a variant of the Ewald summation method [START_REF] Frenkel | Chapter 12 -long-range interactions[END_REF], which relies on computing the longrange contribution of the PBC system in Fourier space. Making use of the Fast Fourier Transform (FFT) algorithm, the complexity is reduced to O (N ln N ). All simulations were implemented using the simulation code Large-scale Atomic/Molecular Massively Parallel Package Simulation (LAMMPS), where we chose r cut = 15 Å (see Fig. 2 The use of Brownian dynamics simulation with implicit solvent model, using Eq. (2.8), allows to explore substantially larger system sizes and longer trajectories compared to allatoms classical molecular simulations which are usually limited to a few nanoseconds a day [START_REF] Frenkel | Understanding molecular simulation: from algorithms to applications[END_REF] (see Fig. 2.2). Without water molecules, a much larger time step is possible thanks to the lower density of the system, ∼ 50 fs for BD instead of ∼ 1 fs for MD, the reduction of the number of particles also decreases the cost of force computations. LAMMPS is an efficient molecular dynamics simulation package that offers Message Passing Interface parallelization. The relatively long simulated trajectories, give access to inversely low-frequency phenomena ranging from ∼ 10 MHz up to high frequencies ∼ 1 THz. While this simulation method cannot reach the sub-KHz range where the pink noise is observed experimentally in nanopores (see Sec. 1.3), the GHz domain, where one observes electrostatic relaxation (see Sec. 1.3.4.2) in the dynamic of ions can be precisely studied.

Limitations of this model

While being more efficient than Molecular dynamics, and retaining more of the microscopic description of ionic interactions compared to continuous modelling under mean-field approximation; such as the Poisson-Boltzmann model (See Sec 1.3.1.1), the exposed model implicit solvent possesses several limitations regarding the description of electrolytes. First, the overdamped limit is to be taken under the assumption of relaxation of inertial processes, meaning that we have neglected the mass-acceleration term in the Langevin equation Eq. (1.17). The study of the dynamics of the ions is therefore restricted to time scales larger than the inertial relaxation time:

τ relax = m γ = Dm k B T , (2.19)
which is about 20 fs for the ions Na + or Cl -. Secondly, the implicit solvent approximation, with constant frictions and uncorrelated Langevin forces, which lead to Einstein relation µ i = β D i with no inter-diffusion coefficients, is only valid in the case of neglected hydrodynamic interactions. While these interactions have been shown to have important effects on diffusion and conductivity in bulk electrolytes (see Ref. [START_REF] Jardat | Transport coefficients of electrolyte solutions from smart brownian dynamics simulations[END_REF][START_REF] Dufrêche | Analytical Theories of Transport in Concentrated Electrolyte Solutions from the MSA[END_REF], Fig. 1.9 and Fig. 1.10), leading to the so-called electrophoretic effect. Thirdly, as water molecules exhibit electrostatic dipole moments, the orientational degrees of freedom of water should also lead to consequent frequency-dependence of the solvent permittivity [START_REF] Hoang | Electrical noise in electrolytes: a theoretical perspective[END_REF], and solvation effects. Experimentally, the dielectric constant of water decreases significantly for frequencies larger than 10 GHz. We will address these three points in Sec. 4.3, by comparing Brownian dynamics simulations to explicit solvent molecular dynamics simulations. More generally the coarse-graining of the degrees of freedom related to the water molecules should be included in electrolyte models in order to yield accurate results. This tedious task can be done in the bulk case for Brownian dynamics under the Stokesian flow assumption, by incorporating hydrodynamic interactions in the diffusion matrix using the Rotne-Prager tensor [START_REF] Jardat | Transport coefficients of electrolyte solutions from smart brownian dynamics simulations[END_REF], or the Oseen tensor [START_REF] Bonneau | Temporal response of the conductivity of electrolytes[END_REF], or also in the framework of fluctuating hydrodynamics [START_REF] Ladiges | Discrete ion stochastic continuum overdamped solvent algorithm for modeling electrolytes[END_REF][START_REF] Ladiges | Modeling electrokinetic flows with the discrete ion stochastic continuum overdamped solvent algorithm[END_REF], and lastly, a frequency-dependent permittivity should be included for the high-frequency electrostatics in water (we leave discussion of electrokinetic effects at interfaces for Chap. 5). Generalized Langevin Equations can also be used to reconstruct the dynamics (see Sec. 1.3.4.2). In this work, we will keep on proceeding without these mentioned electrokinetic effects, to restrict the study to the dynamic of ions at low frequencies [START_REF] Banerjee | Ions' motion in water[END_REF] and in the diluted regime, dominated by electrostatic interactions, and layering groundwork for further study as perspectives on electrolytes.

Density formalism

Solving discrete particle dynamics using computer simulations allows one to access numerous properties of the electrolyte system, but in order to interpret them, one can also establish analytical theories on an easier level of description and try to solve them by hand. Indeed one can continue the coarse-graining process, but this time not by uniting a number of ions into another grain, but all of them under a unique density field. The manipulation of continuous fields instead of dynamical systems of discrete coupled particles offers more analytical means. Furthermore, the interplay between analytical and numerical modelling permits to assess their respective relevance and validity. Here we explore the main tools used for continuous modelling of electrolyte dynamics.

Fokker-Planck equation

Generally, any Markovian stochastic process i.e. which evolution only depends on its instantaneous state, driven by a Wiener process, which is the case of Eq. (2.5), reads :

d Rt = µ( Rt , t) dt + σ( Rt , t) • d Ŵ t , (2.20)
where µ( Rt , t) is a drift vector and σ( Rt , t) is a matrix, can be reformulated into a deterministic equation for the probability density defined as :

p( R, t) d R = Probability that Rt = R at time t . (2.21)
This probability density allows us to rewrite the expectation value, of any function of the variables describing the state of the system, that can potentially depend on time explicitly, as an integral over the probability density measure:

⟨f (t)⟩ = E Rt f ( R, t) = ˆf ( R, t) p( R, t) d R . (2.22)
By applying the Chapman-Kolmogorov equation to Eq. (2.20) [START_REF] Risken | Fokker-Planck Equation[END_REF], and making a forward time expansion, one finds the forward Kolmogorov equation for the density p( R, t), also known as the celebrated Fokker-Planck equation:

∂ t p( R, t) = L FP ( R, t) p( R, t) , (2.23)
with the linear Fokker-Planck operator defined as:

L FP ( R, t) p = -∇ R • -µ( R, t) p + ∇ 2 R D( R, t) p , (2.24)
and D( Xt , t) = 1 2 σ σ ⊺ the diffusion matrix. A wide panel of methods exist in the literature to handle the Fokker-Planck equation [START_REF] Risken | Fokker-Planck Equation[END_REF]. We expose here the observables of interest that will be studied through the manuscript.

Applying this formalism to Eq. (2.5), which has trivial diffusion coefficients and additive noise, one writes:

∂ t p( R, t) = L FP ( R, t) p( R, t) , (2.25)
with the Fokker-Planck decomposed in the same fashion as the potential:

L FP ( R, t) p = L eq FP ( R) + δL( R, t) , (2.26)
which respectively act on p( R, t) as:

   L eq FP ( R) p = D • ∇ R • β ∇ RV int ( R) p + ∇ R p δL( R, t) p = D • ∇ R • β ∇ RV ext ( R, t) p .
(2.27)

A system under no external drive and at equilibrium is described by the distribution p B ( R) which satisfies at stationary state:

L eq FP ( R) p B ( R) = 0 . (2.28)
One then identifies the solution as the Boltzmann distribution:

     p B ( R) = Z -1 e -βV int ( R) Z = ˆe-βV int ( R) d R , (2.29)
where Z is called the partition function of the system. A configuration R is as likely as its energy is low. At T = 0, the energy must be at a global minimum R is at a ground state, then increasing the temperature allows higher energy states, whose distribution depends on the thermal energy k B T = β -1 .

Correlation functions

At steady state, one can study dynamics properties using correlation functions of two observable A and B, defined as:

⟨A(t) B⟩ = ¨A( X) B( X0 ) dµ( X, t ∩ X0 , 0) , (2.30)
where dµ( X, t∩ X0 , 0) is the joint probability measure that both Xt = X and Xt=0 = X0 , which can be expressed as a product of the stationary Boltzmann distribution times the transition probability:

dµ( X, t ∩ X0 , 0) = p( X, t| X0 , 0) p B ( X0 ) d X d X0 , (2.31)
with p( X, t| X0 , 0) the solution of the Fokker-Planck equation:

∂ t p( X, t) = L F P ( X) p( X, t)
with the initial condition p( X, t = 0) = δ( X -X0 ). Formally, this reads:

p( X, t| X0 , 0) = e L F P ( X) t δ( X -X0 )
so that performing the summation over all initial conditions X0 yields:

⟨A(t) B⟩ = ˆA( X) e L F P ( X) t B( X) p B ( X) d X. (2.32)
Using ergodicity, time correlation functions can be alternatively defined using time averages:

⟨A(t) B⟩ = lim T →∞ 1 T ˆT/2 -T /2 A(t -t ′ ) B(t ′ ) dt ′ , (2.33)
which will be used for numerical simulations. Time correlation functions for equilibrium stationary processes have a number of appreciable properties arising from time translation invariance and time reversibility:

• Time reversal symmetry:

⟨A(-t) B⟩ = ϵ A ϵ B ⟨A(t) B⟩ (2.34)
with ϵ A , ϵ B the time-reversal signature of the two variables

• Time translation invariance:

⟨A(t 2 ) B(t 1 )⟩ = ⟨A(t 2 -t 1 ) B(0)⟩ := ⟨A(t 2 -t 2 ) B⟩ (2.35)
• Time derivative of an observable:

Ȧ(t) B = -A(t) Ḃ , in particular Ȧ(t) A = 0 (2.36)
• Total time derivative:

d 2 dt 2 ⟨A(t) B⟩ = -Ȧ(t) Ḃ (2.37)
• Two variables infinitely separated in times are uncorrelated:

lim t→∞ ⟨A(t) B⟩ = ⟨A⟩ ⟨B⟩ (2.38)
Finally, regarding the last property, it is more convenient to define the correlation functions as:

C AB (t) = [A(t) -⟨A⟩] B -B (2.39)
such that C AB (t) decays to 0 at long times. The use of the complex conjugate B makes the operator definite positive for complex observables. The case of observables correlated with themselves, called auto-correlation functions (ACF), takes an important place in the next parts.

Density field operators

Another formalism can be adopted by studying the one-point density functions of the system such as the total number density:

ρ(r, t) = i δ 3 (r -r i (t)) , (2.40)
where the sum runs over all particles (i). If one is interested in local transport properties, for example of a quantity A that is microscopically distributed between the particles of the system, such that particle (i) carries the quantity a i , one defines the instantaneous microscopic density:

ρ A (r, t) = i a i δ 3 (r -r i (t)) . (2.41)
In particular, we will be focusing on the concentration and charge (adimentionalized by the elementary charge) densities of an electrolyte, where the ion (i) carries the charge

q i = z i e:        ρ n (r, t) = i δ 3 (r -r i (t)) ρ q (r, t) = i z i δ 3 (r -r i (t)) , (2.42)
ρ n (r, t) and ρ q (r, t) describe the local concentration of ion and charge density at time t respectively. We will also focus on the current densities associated with these fields, defined as:

       j n (r, t) = - i βD i ∇ r i V ( R, t) δ 3 (r -r i (t)) j q (r, t) = - i βz i D i ∇ r i V ( R, t) δ 3 (r -r i (t)) , (2.43)
where j n (r, t) and j q (r, t) are the particles and electric current densities respectively. The derivation of dynamical models for these operators in the frame of Density Functional Theory will be described in Chap. 3.

Dynamical Structure Factors

The dynamics and fluctuations of density functions can be studied through their correlation functions. For two observables A(r, t) and B(r, t), the time-and space-dependent correlation function reads:

C AB ( r -r ′ , t -t ′ ) = δA(r, t) δB * (r ′ , t ′ ) (2.44)
where we have generalized Eq. (2.32) to non-local correlations, with C AB (r, t) again only depending on the space-time separation between the two observables in a stationary bulk situation. Exploiting this symmetry, it is convenient to study the correlations in reciprocal Fourier space. For a field A(r) of zero average, we define its space Fourier transform and its inverse transform as:

       A(k) = ˆR3 A(r) e -ik•r d 3 r A(r) = 1 (2π) 3 ˆR3 A(k) e ik•r d 3 k . (2.45)
The space Fourier transforms of number and charge density fields simply read:

       ρ n (k, t) = i e -ik•r i (t) ρ q (k, t) = i z i e -ik•r i (t) . (2.46)
Then the correlation functions of these Fourier components are defined as:

C AB (k, k ′ , t) = δA(k, t) δB(-k ′ , 0) . (2.47)
One can check that this corresponds to the Fourier transform of Eq. (2.44). A special case of these space-time correlations will be studied, the ones associated with the density field operators defined in Eq. (2.46) (2.43). We will refer to them as intermediate scattering functions defined as:

F AB (k, t) = 1 N ⟨ρ A (k, t) ρ B (-k, 0)⟩ = 1 N i a i e -ik•r i (t)   j b j e +ik•r j (t)  
(2.48) with A and B being the either the concentration denoted with subscript n or the charge with q. F nn(/qq) (k, t) are the (charge-charge) intermediate scattering functions, where we have chosen the total number of particles as a normalizing factor, rendering the function dimensionless and a priori intensive:

             F bb (k, t) = 1 N ij e -ik•[r i (t)-r j (0)] F qq (k, t) = 1 N ij z i z j e -ik•([r i (t)-r j (0)]
(2.49)

For zero time delay, F AB (k, t = 0) the intermediate scattering function is also called the structure factor S AB (k):

S AB (k) := F AB (k, 0) .

(2.50)

The particle-particle and charge-charge structure factors can be computed at the meanfield level for an electrolyte, using the Random Phase Approximation (RPA) at the DFT level, or Debye-Hückel (DH) theory [START_REF] Hansen | Theory of Simple Liquids[END_REF], the result is

   S nn (k) = 1 S qq (k) = k 2 k 2 + κ 2 , (2.51)
where κ is an inverse correlation length. For the special case of DH theory (linearized-Poisson Boltzmann Eq. (1.6)), which only considers electrostatic interactions at the meanfield level, the correlation length is equal to

     λ D = κ -1 κ 2 = 4πl B i z 2 i ⟨ρ i ⟩ , (2.52)
defined as the inverse of the Debye parameter κ, where ⟨ρ i ⟩ is the average concentration of ionic species (i) of charge q i = z i e, and l B = βe 2 4πε 0 εw is the Bjerrum length, equal to the distance at which the energy between two elementary point charges in a medium is equal to the thermal energy. The Debye length is the length characterizing electrostatic screening. For an electrolyte at thermal equilibrium, due to the reorganization energy (see Eq. (2.29)), the electrostatic field created by a point charge is screened for distances larger than the Debye length (Sec. 1.3.2). Time correlations can be studied in frequency space as well, by defining time Fourier transforms as:

S AB (k, ω) = ˆR F AB (k, t) e -iωt dt .
(2.53)

S AB (k, ω), with a dimension of time, is the equivalent of power spectral density (Eq. (1.1)) for local density fluctuations. In particular, it encodes correlations between the spacetime fluctuations of observables A(r, t) and B(r ′ , t ′ ). For number and charge densities Eq. (2.46), these functions called number-number and charge-charge dynamical structure factors respectively read

             S nn (k, ω) = 1 N ˆR ij e -ik•[r i (t)-r j (0)] e -iωt dt S qq (k, ω) = 1 N ˆR ij z i z j e -ik•[r i (t)-r j (0)] e -iωt dt .
(2.54)

Linear response theory

As introduced in Sec. 1.3.5.1, there is a strong link between equilibrium fluctuations and response to a small external perturbation. The question to be answered is what is the observable-perturbation couple, and what is the explicit relation ? While Macroscopic relations are usually stated in the famous Maxwell's equation for continuous media, here we will relate them to microscopic quantities encoded in the charge-charge dynamical structure factor.

Charge dynamical structure factors and electrical linear response

Structure factors, such as Eq. 2.54, are extensively studied in the literature, both experimentally and theoretically, because they carry precious information on the microscopic correlations in the system, and consequently on the microscopic mechanisms. In particular, the charge-charge dynamical structure factor is directly connected to many electrical transport properties such as polarization or conductivity response. The knowledge of dynamical correlations in the source terms directly relates to their associated currents. The charge-charge dynamical structure factor will play a key role in describing the microscopic dynamics of electrolytes and will be studied through various methods in this manuscript.

The charge-charge dynamical structure factor relates to the response of the system to an external perturbation in a form of a monochromatic plane wave: E ext (k, ω). The response of the system is characterized by the density of electrical current j q and the density of polarization P q = i z i r i (t) δ(rr i (t)). In the linear response regime, they are proportional to the Maxwell field E(k, ω) which includes the induced electric field in the sample. Then the constitutive relations read in Fourier transforms:

j q (k, ω) = σ(k, ω) • E(k, ω) P q (k, ω) = ε 0 [ε(k, ω) -I] • E(k, ω) , (2.55)
with I the identity tensor. Current and polarization are related to E(k, ω) through the electrical conductivity and permittivity tensors respectively. Noticing that ∂ t P q (r, t) = j q (r, t), one related the tensors as:

σ(k, ω) = -iωϵ 0 [ε(k, ω) -I] (2.56)
forming the generalized conductivity tensor, or alternatively:

ε(k, ω) = I + i σ(k, ω) ωε 0 (2.57)
the generalized dielectric tensor. These are the main measured quantities for dielectric spectroscopy techniques for example, and are related to the charge-charge structure factors.

At linear regime, one also can demonstrate that the relevant dielectric response functions to the external field E ext , also known as susceptibilities, are [START_REF] Caillol | The Dielectric Constant and the Conductivity of an Electrolyte Solution at Finite Wave-Lengths and Frequencies[END_REF][START_REF] Branka | Computer simulation of wavevectordependent dielectric properties of polar and nondipolar liquids[END_REF]:

   χ l (k, ω) = 1 - 1 ε l (k, ω) χ t (k, ω) = ε t (k, ω) -1 (2.58)
with ε l (k, ω) and ε t (k, ω) the longitudinal/rotational-free and transverse/divergence-free parts of the 3D isotropic dielectric tensor:

ε(k, ω) = kk k 2 ε l (k, ω) + I - kk k 2 ε t (k, ω) . (2.59)
The same decomposition can be introduced for the conductivity σ(k, ω). For example, the fluctuation-dissipation theorem applied to the longitudinal function, for non-polarizable systems, stipulates that [START_REF] Giaquinta | Collective dynamics of charge fluctuations in ionic conductors[END_REF][START_REF] Madden | A Consistent Molecular Treatment of Dielectric Phenomena[END_REF][START_REF] Branka | Computer simulation of wavevectordependent dielectric properties of polar and nondipolar liquids[END_REF][START_REF] Hansen | Theory of Simple Liquids[END_REF]:

χ l (k, ω) = βe 2 N V ε 0 k 2 S qq (k) + iω 2 S qq (k, ω) , (2.60)
with S qq (k) and S qq (k, t) the static and dynamical charge structure factors given by Eq. (2.50) and Eq. (2.54) respectively. We will give a precise demonstration of that relation, in the form of generalized Green-Kubo relation for the electrical conductivity in next Sec. 2.3.3, for electrolytes modelled by Brownian dynamics. By exploiting relations of continuous media electrostatics, one can extend this type of relations for other observables such as the electric potential, field and field gradient, or induced charged on electrodes [START_REF] Hoang | Electrical noise in electrolytes: a theoretical perspective[END_REF]. This illustrates the key role of the charge-charge dynamical structure factor in modelling electrical fluctuations.

Linear response for overdamped Langevin dynamics

We now make use of the Fokker-Planck formalism to demonstrate Green-Kubo formulae for overdamped Langevin dynamics Eq.(2.4), that will be extensively used in the next chapters. This part of the work was done with the collaboration of Gabriel Stoltz. An observable for a system at equilibrium reads

⟨θ⟩ eq = ˆθ( R) p B ( R) d R , (2.61)
where the Boltzmann equilibrium distribution verifies L eq FP ( R) p B ( R) = 0. Out of equilibrium, the probability density of the system p neq ( R, t) follows a different Fokker-Planck equation:

∂ t p neq ( R, t) = L neq FP ( R, t) p neq ( R, t) , (2.62)
where the out of equilibrium FP operator L neq FP ( R, t), which explicitly depends time, can be decomposed as

L neq FP ( R, t) = L eq FP ( R) + δL( R, t) . (2.63)
One can decompose the solution as:

p neq ( R, t) = p B ( R) + δp( R, t) , (2.64)
where, assuming δL ≪ L eq FP (in a broad sense), we can expect that |δp ≪ p B |. Then expanding the Fokker-Planck equation to leading order in δp( R, t), one obtains

∂ t δp( R, t) = L eq FP ( R) δp( R, t) + δL( R, t) p B ( R) . (2.65)
The solution to that equation formally reads

δp( R, t) = ˆt 0 e L eq FP ( R) (t-s) δL( R, s) p B ( R) ds . (2.66)
This implies that the deviation of an observable from its equilibrium value: ∆θ(t) = ⟨θ(t) -⟨θ⟩ 0 ⟩, for a small perturbation, is equal to

∆θ(t) = ˆθ( R, t) ˆt 0 e L eq FP ( R) (t-s) δL( R, s) p B ( R) ds d R .
(2.67)

Frequency-dependent electrical conductivity

We now specify θ and δL, considering the electrical current arising from the application of a uniform time-dependent external electric field such that

V ext (R, t) = - i ez i r i • E(t) = -eZ R • E(t) (2.68)
where e is the elementary charge, Z is a diagonal matrix containing the valency of each ion (which commutes with D), and E(t) ∈ R 3 the uniform external electric field applied.

We want to study the macroscopic linear response of the total electrical current defined as

J el (t) = i eβz i D i F i (t) . (2.69)
The corresponding microscopic operator can be decomposed as

J el ( R, t) = i eβz i D i F i ( R, t) = -βeZD∇ RV ( R, t) = -βeZD∇ RV ext ( R, t) -βeZD∇ RV int ( R) = V σ NE E(t) + J eq el ( R) , (2.70)
where we have identified the Nernst-Einstein ideal conductivity (Eq.(1.14)) defined as

σ NE = βe 2 V 2N i z 2 i D i = βe 2 I∈{species} z 2 I D I C I , (2.71)
with C I the average concentration of ionic species I. Eq. (2.70) divides the current in the electrolyte into two contributions, the first one, given by the Nernst-Einstein conductivity, is the sum of the electric mobility of each ion in the overdamped regime: µ E i = ez i βD i . This ideal term is understood as the instantaneous response of one ion to an external field and then summed over all ions that are considered not interacting. The second term, arises from currents internally created only by the interactions, as captured by the internal potential V int , indirectly induced by the electric field on the configuration R of the system. Then from Eq. (2.67), the nonequilibrium electric current reads

∆J el (t) = ˆ V σ NE E(t) + J eq el ( R) p( R, t) d R = V σ NE E(t) + ˆJ eq el ( R) δp( R, t) d R = V σ NE E(t) + ˆJ eq el ( R) ˆt 0 e L eq FP ( R) (t-s) δL( R, s) p B ( R) ds d R .
(2.72)

In the present case, the modification in the Fokker-Planck drift operator Eq. (2.27) is:

δL( R, t)• = D • ∇ R • β ∇ RV ext ( R, t) • = -βeE(t) • ZD∇ R• , (2.73)
and explicitly acts on p B ( R) as the advective electric current itself (see. Eq. (2.29)):

δL( R, t) p B ( R) = -βeE(t) • ZD∇ Re -βV int ( R) = -βeE(t) • ZD -β∇ RV int ( R) p B ( R) = -βE(t) • J eq el ( R) p B ( R) .
(2.74)

Plugging this into Eq. (2.72) yields the final result

∆J el (t) = V σ NE E(t) -β ˆt 0 ⟨J el • J el (t -s)⟩ eq • E(s) ds , (2.75)
where we have identified the equilibrium autocorrelation function

⟨J el • J el (t -s)⟩ eq = ˆJ eq el ( R) e L eq PF (t-s) J eq el (R) p B ( R) d R . (2.76)
At stationary state,

∆J el (t) = V σ NE E(t) -β ˆ∞ 0 ⟨J el • J el (t -s)⟩ eq • E(s) ds , (2.77)
which can be conveniently rewritten in Fourier/Laplace transform:

J el (ω) = V σ(ω) • E(ω) (2.78)
where the frequency-dependent conductivity tensor is given by the Green-Kubo formula as:

σ αβ (ω) = σ NE δ αβ - β V ˆ∞ 0 J α el (0) J β el (t) eq e -iωt dt , (2.79)
where {α, β} ∈ {x, y, z} and δ αβ is the Kronecker delta function representing the components of the identity tensor. As for the electric current Eq. (2.72), the total electrical conductivity Eq. (2.79) is a sum of two contributions. The first term, known as the Nernst-Einstein conductivity, corresponds to an ideal electrolyte, which responds instantaneously to the external field in the overdamped regime. The second term is a delayed response arising from the interactions in the system. This delayed response is identified as the action of the current on itself (see Eq. (2.72) and Eq. (2.73)), and at linear regime, this self-correlation is captured by the equilibrium fluctuations. In other words, internal interactions of the system give rise to self-correlated electric currents, deviating from the ideal current. In particular, Eq. (2.79) states that the frequency-dependence of the electrical conductivity arises from the time relaxation of these correlations, making it also a tool to probe microscopic interactions using the frequency-dependent conductivity.

Wavenumber and frequency-dependent electrical conductivity

Furthermore, this frequency-dependent Green-Kubo formula that decomposes in an ideal and an interacting term can be extended to the case of a non-uniform electric field. This can be characterized by using the density operators defined in Sec.2.2.4, and the relation is more easily seen by studying them as plane-wave with wave vector k. The positiondependent external field reads as a vector:

Ê( R, t) = E 0 e i(k•r i +ωt) (2.80)
and the electrical current density operator at the corresponding wavevector reads:

j el ( R, k, t) = i βez i D i F i (t) e -ik•r i (t) , (2.81)
assuming small wavevectors/quasi homogeneity of the action of external electric field on the probability density, i.e.:

∇ R • Ê( R, t) p B ( R) ∼ Ê( R, t) • ∇ R p B ( R) , (2.82)
(which is verified for any divergence-free / transverse plane-wave form of the electric field), the analogous space-time frequency-dependent Green-Kubo formula holds with the conductivity tensor defined as:

j el (k, ω) = σ(k, ω) • E(k, ω) (2.83) σ αβ (k, ω) = σ NE δ αβ - β V ˆ∞ 0 j α el (k, 0) j β el (-k, t) e -iωt dt (2.84)
where we can identify the electrical current dynamical structure factor:

σ αβ (k, ω) = σ NE δ αβ -βe 2 C s S αβ j el (k, ω) . (2.85)
Formula Eq. (2.79) and (2.84) are similar to the celebrated Green-Kubo relation often used in molecular dynamics simulations which usually take the simpler form

σ MD = βe 2 3V ˆ∞ 0 J MD el (0) • J MD el (t) eq dt , with J MD el (t) = i q i v i (t) (2.86)
where v i is the velocity of ions i, which is ill-defined for overdamped Langevin dynamics simulations. They also reflect Onsager's regression hypothesis: a system doesn't see the difference between a small externally induced perturbation and thermally induced fluctuation at thermodynamic equilibrium. Out-of-equilibrium response to a small perturbation is related to the relaxation of equilibrium fluctuations, which is encoded in the autocorrelation functions. While these formulae have been (implicitly) derived [START_REF] Onsager | Report on a revision of the conductivity theory[END_REF][START_REF] Felderhof | Linear response theory of sedimentation and diffusion in a suspension of spherical particles[END_REF][START_REF] Felderhof | Linear response theory of the viscosity of suspensions of spherical brownian particles[END_REF] and used [START_REF] Jardat | Transport coefficients of electrolyte solutions from smart brownian dynamics simulations[END_REF][START_REF] Jardat | Brownian dynamics simulations of electrolyte mixtures: computation of transport coefficients and comparison with an analytical transport theory[END_REF][START_REF] Jardat | Brownian simulations contribution to the study of ionic dynamics in aqueous solutions[END_REF][START_REF] Jardat | Self-diffusion of ions in charged nanoporous media[END_REF] in the literature, even for the extended cases of nontrivial diffusion matrix, we will only use them in the simpler scope of trivial mobilities for the ions i.e. with no explicit hydrodynamic interactions. Furthermore, we introduce variants where we focus on the fact that space/time frequency-dependence, which directly follows from microscopic interactions, offers an innovative and resolved probe to the role of microscopic interactions in electrical fluctuations. Lastly, as we will see in the next section, the extraction and analysis of time and space correlations, appearing in formula Eq. (2.79) and Eq. (2.84), are smoothly accessible from Brownian dynamics simulation by making use of numerical methods.

Numerical methods

Choice of wavevectors

The computation of structure factors Eq. (2.54) from atomistic simulations reveals advantageous compared to working in real space with density distributions such as Eq. (2.46), especially in terms of data storage. The binning of the simulation box can be done in a manner justified by the properties studied. For example, if the target is to study electrostatic screening in a dilute electrolyte, the relevant length is the Debye length λ D ( Eq. (2.52)), with corresponding wavevector κ = λ -1 D , and the choice of considered wavevectors to study should be motivated by this value. But because the simulation is bounded by periodic boundary conditions which can only approximate bulk systems, the information we can obtain from numerical production is bounded by constraints: it is not possible to measure a property that spans larger distances than the simulation box L box . For the case of electrostatic screening, one needs L box to be much larger than λ D to access to the good range of wavevectors. This notion of finite size system translates in reciprocal space as a quantization of accessible wavevectors:

k ∈ (n x , n y , n z ) 2π L box ∈ R 3 (n x , n y , n z ) ∈ N * . (2.87)
The smallest accessible wavevector is 2π L box . This represents a fundamental barrier for numerical simulation, because an increase in the size of the simulated system, of a given factor, induces the same factor to the power three in terms of computation cost. The use of implicit/coarse-grained models such as BD is particularly relevant for this use. The simulation of sufficiently large systems allows us to extrapolate the hydrodynamic limit where length scales become macroscopic, which is equivalent to taking the vanishing wavevector limit. This method allows extrapolating macroscopic properties, by inferring the dependence of the observable as a function of the wavevector [START_REF] Giaquinta | Collective dynamics of charge fluctuations in ionic conductors[END_REF][START_REF] Cheng | Computing the heat conductivity of fluids from density fluctuations[END_REF][START_REF] Cheng | Computing chemical potentials of solutions from structure factors[END_REF].

On-the-fly computation

The usual procedure to exploit numerical particle dynamics simulations is to print out, during the iterative propagation of the dynamics, the instantaneous configuration of the system at a regular frequency, similar to taking snapshots. The stored sequence of snapshots called the trajectory of the system, is then post-processed by computing observables for each snapshot and performing statistical operations. This procedure requires the storage of large amounts of data. One way to circumvent this need is to directly store the value of the observable by computing it during the dynamic propagation. This method, called on-the-fly computation, can significantly reduce the amount of stored data, at the price of increasing the simulation time. But on-the-fly computations can also take advantage of the fact that many observable, such as forces, are already computed to generate the trajectory. The computation of structure factors greatly benefits from this procedure, as the Fourier transform of a density of interest ρ a (r, t) = i a i δ 3 (rr i (t)) simply reads:

ρ a (k, t) = N particles i a i e -ik•r i (t) , (2.88)
which can be straightforwardly computed for each configuration. with the collaboration of Jeongmin Kim, we implemented this method as subroutines for LAMMPS simulation scripts, and extended it to partial densities to evaluate cross-correlation between species:

ρ I a (k, t) = i∈I a i e -ik•r i (t) , (2.89)
with I a subset of the system e.g. only the cations. Brownian dynamics simulations allow to generate relatively long trajectories. We divided each simulation into sub-trajectories that are still much larger than the correlation times of the system T tot = N traj T such that sub-trajectories can be considered as statistically independent. For a quantity A (potentially complex) uniformly sampled with frequency f s = ∆t -1 , with ∆t = n s δt the sampling time (multiple of the time step of the simulation), for a duration T , one obtains the discrete sequenced composed of

N = f s T points (A n ) n<N ∈ C N associated with the discrete time series(t n ) n<N = n ∆t ∈ R N .
If the sampling frequency is sufficiently high, one can associate the discrete sequence with a continuous process A n ∼ A(t n ) and compute quantities such as time autocorrelation functions (Eq. (2.39)) or power spectral densities.

Spectral analysis

For this goal, various algorithms exist in the field of signal analysis, but due to the large amount of data, we chose methods based on the Fast Fourier Transform algorithm.

The main numerical methods used in this manuscript revolve around computing the time-delayed correlation properties of an observable sampled from BD simulations. The time series needs to be properly treated to compute the statistical properties of the system. Here we will explain the algorithms used in this work to compute direct time correlations and their Fourier transform, also called power spectral densities.

Assuming stationarity, the time autocorrelation Eq. (2.33):

C AA (t) = lim T →∞ 1 T ˆT/2 -T /2 A(t ′ ) A * (t -t ′ ) dt ′ (2.90)
can be approximated by the discrete convolution:

C AA (t n ) ∼ C AA,n = 1 N -n N -n-1 i=0 A i A * n-i . (2.91)
One essential issue of finite time sampling is the presence of finite boundary conditions, that can lead to significant changes in the outcome. One can extend the finite signal (A n ) 0≤n<N → (A m ) m∈Z using for example zero-padding:

(A m ) m∈Z : A m = A n , 0 ≤ m < N 0 , elsewhere . (2.92)
But a stationary signal must verify time translation symmetry (in the statistical senses). This cannot be satisfied with zero padding, but rather with periodic boundary conditions:

(A m ) m∈Z : A m = A n=m mod N . (2.93)
In this case, (A m ) m∈Z would appear stationary, but the properties spanning over larger time that T would still be incorrectly computed. Nevertheless, this allows us to exploit discrete Fourier transforms by writing the time signal as:

A n = 1 N N -1 p=0 Ãp e i 2π N pn , (2.94)
with coefficients the ( Ãp ) p<N computed as:

Ãp = N -1 n=0
A n e -i 2π N pn .

(2.95)

One can approximate, for a sufficiently long sampling duration and sufficiently high sampling frequency:

A(ω = p 2π N ) = ˆ+∞ -∞ A(t) e -iωt dt = lim T →∞ ˆT/2 -T /2 A(t) e -iωt dt = lim T →∞ ˆT 0 A(t) e -iωt dt = lim T →∞ lim N →∞ T N N -1 n=0 A(n T N ) e -iωn T N = lim T →∞ lim N →∞ ∆tA p , (2.96)
neglecting boundary terms. It follows that this is the discrete counterpart of the continuous Fourier transform. Apart from the mathematical properties, computationally, discrete Fourier transform can be computed in O(N ln N ) using the celebrated Fast Fourier Transforms (also exploited in Sec.2.1.3). This proves extremely useful, the naive computation of the autocorrelation function using the convolution product is of complexity O(N 2 ). By virtue of the convolution rule for the Fourier transform, the Wiener-Khintchin theorem states that:

S AA (ω) = lim T →∞ ˆ+T/2 -T /2 C AA (t) e -iωt dt = lim T →∞ 1 T |δA(ω)| 2 , (2.97)
the Power Spectral Density, i.e. S AA (ω), is the time Fourier transform of the autocorrelation function C AA (t) and the squared modulus of the signal spectrum |δA(ω)| 2 (up to a constant). The computation of C AA (t) and/or S AA (ω) can thus be reduced to complexity O(N ln N ), by computing the discrete Fourier transform of the signal, taking its modulus squared, and transforming it back to real-time space (see Fig. 2.3).

Figure 2.3:

Efficient computation for the Power Spectral Density/Autocorrelation function: using the Fast Fourier Transform. Relations between the signal A(t), its correlation function C AA (t), its Fourier transform A(ω) and its power spectral density S AA (ω).

With the help of Iurii Chubak, Jeongmin Kim and Giovanni Pireddu, we implemented efficient analysis codes to compute: self, partial, and cross structure factors and intermediate scattering functions from time series computed from LAMMPS of the form Eq.(2.89). This code was written in C++ using the efficient FFTW3 library.

Double time sampling

In Sec.2.3.1. we exposed an issue inherent to numerical molecular simulations, finite size effects. The same also holds for time signals analysis. Discrete and finite sampling of a time signal induce constraints on the quality of extracted properties, especially regarding dynamical properties such as spectral transforms. In the case of a signal sampled with discrete time, the accessible frequencies with relevant information are roughly given by the Shannon-Nyquist theorem [START_REF] Shannon | Communication in the presence of noise[END_REF]:

ω ∈ p 2π T ∈ R p ∈ N , 0 < p < N/2 . (2.98)
The accessible frequencies are linearly spaced, while for coupled systems such as electrolytes, the relevant ones are often separated by orders of magnitude. For example, a signal might carry information in the form of two separated frequencies (Fig. 2.4) coupled with noise. A quantitative study would thus require both a high sampling frequency and a long sampling duration. Then the computational cost of sampling becomes demanding, and the numerical processing cumulates precision errors, especially for integral transforms such as Fourier transforms. ). Due to the mixing of frequencies, the separation of time scales in the total signal is hard to achieve. On the right panel, a typical representation of a time correlation function with two characteristic relaxation processes, both the peaked short-time (blue) and slow decaying long-time (red) parts are relevant in this case. Double time sampling allows for a better convergence of a quadrature, which is used to compute the Power Density Spectrum, as displayed by the coloured rectangles.

0 + s 1 (t) τ 1 0 + s 2 (t) τ 2 0 + s r (t) 0 t 0 S(t) = 0 t 0 C SS (0) C SS (t)
To palliate this issue of ill-sampled integrand, we implemented another sampling method based on a double sampling of the ACF. The ACF is estimated at short times with a high-frequency sampling of the time signal, which is done only for a short period of time T s much smaller than T , and another parallel estimation is made by sampling the signal less frequently (but faster than T s ) during the full trajectory length T . Both overlapping ACF are concatenated to form a smoother function as displayed in Fig. 2.4, which is more favourable to compute Fourier transforms by performing the integral in Eq. (2.53). This last procedure is carried out using the integrate library of SciPy in Python language, while various methods such as the Filon-Lagrange were tried, the smoothness of the sampling allows the use of the straightforward trapezoidal quadrature method with accurate results.

Trajectory average, error estimate and convergence criteria

Finally, after processing all sub-trajectories, yielding independent estimates of the ACF/PSD, we average the results as

C AA,n = 1 N traj N traj -1 n traj =0 C n traj AA,n .
An error estimate is made for each time/frequency point individually, using the 95 % confidence interval of a normal distribution:

∆C AA,n = 2 N traj   1 N traj -1 N traj -1 n traj =0 C n traj AA,n -C AA,n 2   1/2
, where we have used an unbiased estimator for the standard deviation, often displayed as shaded regions in the figures across the manuscript. We implemented this procedure along with the efficient FFT spectral computation codes. Overall the implementation of these numerical methods generated reliable systematic tools, useful beyond the present PhD work.

Conclusion

In this first methodological chapter, we have introduced the main tools that will allow us to model and study electrical fluctuations from the dynamics of ions. First in Sec. 2.1, following the work of predecessors [START_REF] Jardat | Transport coefficients of electrolyte solutions from smart brownian dynamics simulations[END_REF][START_REF] Dufrêche | Transport equations for concentrated electrolyte solutions: Reference frame, mutual diffusion[END_REF], we have re-formalized the framework of Brownian dynamics for implicit solvent electrolytes, that offers a relevant description in term of length and time scales for the microscopic dynamics. In this part, we have introduced the relevant modelling parameters which are the inter-ionic force fields and the mobility/diffusion coefficients. After detailing the limitations of this model, in Sec 2.2 we defined the tools that will serve as observables for the rest of this manuscript, with the objective of relating microscopic dynamics, obtained by BD simulations, to thermodynamic properties of electrolytes. In Sec. 2.3 we bridge the gap between microscopic dynamics and macroscopic transport using linear response theory. While we show that the full picture of the electrical response of electrolytes can be obtained from the dynamics of the ions in Sec. 2.3.1, we focus on the electrical conductivity of the solution. Sec. 2.3.3 presents new results on that topic in Eq. (2.79) and Eq. (2.84). While Green-Kubo formulae are extensively used for Molecular dynamics, the overdamped Langevin dynamics version requires attention. We present a demonstration that on the one hand specifies the derivations of Ref. [START_REF] Risken | Fokker-Planck Equation[END_REF][START_REF] Felderhof | Linear response theory of sedimentation and diffusion in a suspension of spherical particles[END_REF][START_REF] Felderhof | Linear response theory of the viscosity of suspensions of spherical brownian particles[END_REF] for electrical conductivity, and on the other hand, extends its use to plane wave response for frequency-and wavevector-dependence. The separation between ideal and non-ideal response in the electrical conductivity spectrum is linked to the microscopic dynamics and fluctuations in the electrolyte. Lastly in Sec. 2.4, we give technical details on how the previously derived formulae are implemented numerically. In this chapter, we develop the analytical modelling of the fields introduced in Sec. 2.2.3, with the one-point particle and charge densities defined by Eq. (2.46). We will build a dynamical model for their evolution, by describing their associated fluxes Eq. (2.43) using Stochastic Dynamical Density Functional Theory. By solving this model, we will obtain insights into the electrical fluctuations, which will be compared to Brownian dynamics simulations in the next chapter. Classical Dynamical Density Functional theory (DDFT), is one of the cornerstones of modern statistical mechanics [START_REF] Michael Te Vrugt | Classical dynamical density functional theory: from fundamentals to applications[END_REF], used to describe dynamical properties. Originally developed for the description of fluids, DDFT has a wide panel of applicability, ranging from hydrodynamics and active colloids [START_REF] Rex | Dynamical density functional theory with hydrodynamic interactions and colloids in unstable traps[END_REF] to biology and [START_REF] Fang | Controlled release of proteins from polymer-modified surfaces[END_REF] social sciences [START_REF] Michael Te Vrugt | Effects of social distancing and isolation on epidemic spreading modeled via dynamical density functional theory[END_REF]. While DDFT can be seen as the dynamic generalization of the classical static Density Functional theory (DFT), it can also be thought of as a generalization of Fick's diffusion of interacting particles and can be rigorously derived following many distinct routes. For our purpose of describing aqueous electrolytes, we will use a stochastic version of the theory (sDDFT), that was established as an exact description for a system of Brownian particles, such as Eq. (2.5), by Dean [START_REF] David S Dean | Langevin equation for the density of a system of interacting langevin processes[END_REF]. After recalling its derivation from overdamped Langevin dynamics and limitations, we present a new solution to the sDDFT model for electrolytes in a linearized regime for arbitrary electrolytes at and close to equilibrium. We then make an emphasis on its predictions for fluctuation modes in binary electrolytes that will be analysed and tested in the next chapter.

Stochastic dynamical density functional theory

General dynamical density functional theory usually takes the form of an evolution of a density field ρ(r, t) that reads

∂ρ ∂t = ∇ • µ • ρ∇ δF[ρ] δρ . (3.1)
The equation can be interpreted as an extension of Fick's law for diffusion, where a density field ρ(r, t) of interest (heat, concentration, charge...) is conserved, with its associated flux arising from a local unbalance of chemical potential, modelled by the functional F[ρ], because the system is locally out of equilibrium: δF [ρ] δρ ̸ = 0. This part can also be interpreted as the dynamical generalization of classical density functional theory, where the equilibrium density ρ eq (r) such as

δF [ρeq] δρ
= 0 is sought, and for the time evolution around this state. Then the flux is assumed to be proportional to the chemical potential gradient, the local density and a mobility coefficient µ [ρ; r, t] which in its most general form is a matrix dependent of the density, location and time, but by neglecting hydrodynamics effects and assuming low concentration, we will restrict ourselves to constant diagonal mobility µ i = βD i . This very general formulation can be derived from first principles such as Hamiltonian mechanics using the projection operator formalism, or other coarse-graining strategies. To underline the relation with the Brownian dynamics simulation methods in Sec. 2.1, which we use for aqueous electrolytes, we present a derivation inspired from Ref. [START_REF] David S Dean | Langevin equation for the density of a system of interacting langevin processes[END_REF].

Derivation from the Langevin equation

The evolution equations for the densities can be obtained directly by combining the Eq. (2.5) and Eq. (2.46), as was done by Dean [START_REF] David S Dean | Langevin equation for the density of a system of interacting langevin processes[END_REF]. We recall here the main arguments. Consider an arbitrary smooth function f (r n , t) defined on one coordinate of the system r n , that is itself described with the n-th density operator ρ n (r, t) = δ 3 (rr n (t)). Then, using the definition of a density, the function f at time t reads

f (r n , t) = ˆρn (r, t)f (r, t) d 3 r. (3.2)
Using Îto chain rules for stochastic calculus for f (r n , t), one writes

df = ∂f ∂t dt + ∇ Rf • d Rt + 1 2 ∇ 2 Rf : D dt (3.3)
where d Rt is given by the Langevin equation Eq. (2.5), and A : B = Tr(A × B ⊺ ) is the double contraction of matrices A and B. Therefore one can write the time derivative of f (r n , t) as

df (r n , t) dt = ˆρn (r, t) ∇f (r, t) • -βD • ∇V (r, t) + (2D) 1/2 η n (t) + D : ∇ 2 f (r, t) + ∂f (r, t) ∂t d 3 r , (3.4)
and re-arranging gradient terms by integrating by parts

df (r n , t) dt = ˆf (r, t) ∇ • D • ∇ρ n (r, t) + βD • (ρ n (r, t)∇V (r, t)) -(2D) 1/2 ρ n (r, t)η n (t) d 3 r + ˆρn (r, t) ∂f (r, t) ∂t d 3 r . (3.5)
But from Eq. (3.2), one can also compute the total time derivative of f as

df (r n , t) dt = ˆ∂ρ n (r, t) ∂t f (r, t) d 3 r + ˆρn (r, t) ∂f (r, t) ∂t d 3 r . (3.6)
Then a comparison between the two equations yields

   ∂ρ n (r, t) ∂t + ∇ • j n (r, t) = 0 j n (r, t) = -D • [∇ρ n (r, t) + βρ n (r, t)∇V (r, t)] + (2D) 1/2 ρ n (r, t) η n (t) . (3.7)
At this point, it is important to recall that we have only an equation for the one body density operator of particle n, no ensemble average has been carried out. Then summing for a macroscopic subset of the system, such as one of the ionic species i (i.e. particles n are ions of type i), one obtains

       ∂ρ i (r, t) ∂t + ∇ • j i (r, t) = 0 j i (r, t) = -D • [∇ρ i (r, t) + βρ i (r, t)∇V (r, t)] + n∈{i} (2D) 1/2 ρ n (r, t) η n (t) . (3.8)
Finally, we define the global noise field for ionic species i:

n∈{i} ρ n (r, t) η n (t) ≡ ρ i (r, t) ξ i (r, t) , (3.9)
which is a multiplicative noise, with ξ i (r, t) assumed to be a normalized uncorrelated Gaussian process (see [START_REF] David S Dean | Langevin equation for the density of a system of interacting langevin processes[END_REF] for more details). The resulting equation of motion for the concentration field

ρ i (r, t) is    ∂ρ i (r, t) ∂t + ∇ • j i (r, t) = 0 j i (r, t) = -D • [∇ρ i (r, t) + βρ i (r, t)∇V (r, t)] + 2D i ρ i (r, t) ξ i (r, t) .
(3.10)

Eq. (3.10) is called the Dean-Kawasaki formalism and belongs to a class of models called Dynamical Density Functional Theories (DDFT). As explained above, there are different routes to arrive at this equation, but we have presented one explicitly arising from the overdamped microscopic Langevin equation Eq. (2.5). We can recognise the main features of Fickian diffusion for the dynamics encoded in the conservation equation driven by the divergence of a flux, and of Density Functional Theory from the form of the flux. The last term is a feature arising from the coarse-graining of the Langevin forces. As we want to apply this formalism to electrolytes, the next step is to explicit the interacting term of the model, encoded in the interaction potential Eq. (2.11).

Mean-field approximation

Until this point, Eq. (3.10) remains exact, in the sense that it is strictly equivalent to the microscopic equations of motion Eq. (2.5). We now make a further assumption in this model, the so-called mean field approximation. Instead of considering all the pairwise interactions between the finite size ions, we assume that the field ρ i (r, t) is submitted, not to all the interactions created by ions, but only to a single interacting field, created by the ionic densities. This averaging neglects the local ionic correlations and in particular the short-range interactions Eq. (2.13). The resulting potential is then only a function of the ionic densities

V MF [ρ i , r] = V int [ρ i , r] + V ext [ρ i , r],
where the two remaining terms are the internal electrostatic potential generated by the well known Poisson equation,

∆V int [ρ i ; r, t] = - e ε 0 ε w j z j ρ j (r, t) , (3.11)
which relates the Laplacian of the electrostatic potential to the total density of charges in the system, through the permittivity of the solvent. Finally, the external potential is such that:

E ext (r, t) = -∇V ext (r, t) . (3.12)
The main justification for this procedure is the assumption that the system is sufficiently dilute such that the finite size of the ions can be neglected. Then the remaining contributions to the interaction are the long-range ones. Combining Eq. (3.10), Eq. (3.11) and Eq. (3.12), sDDFT at the mean-field approximation results in the Poisson-Nernst-Planck model with an additional stochastic flux.

Phenomenological interpretation

This description applies to a bulk implicit solvent electrolyte, composed of fully dissociated ions at salt concentration C s , with different ionic species indicated by index i, and characterized by the following average concentrations, diffusion coefficients and valencies:

C i ≡ ⟨ρ i ⟩ , D i , z i .
In the framework of sDDFT, the coupled equations of motion for the partial ionic density ρ i (r, t) read

             ∂ρ i (r, t) ∂t + ∇ • j i (r, t) = 0 j i (r, t) = -D i ∇ρ i + βez i ρ i ∇V int -E ext + 2D i ρ i ξ i (r, t) ∆V int (r, t) = - e ε 0 ε w j z j ρ j , (3.13)
also called the stochastic Poisson-Nernst-Planck model (sPNP). The structure of these equations can be understood phenomenologically as a coupled diffusion-advection process with a white noise source term. For each ionic species, the mass conservation is described by the first equation. The second equation describes the fluxes as a sum of deterministic diffusion and advection terms, which can be obtained from the free energy functional derivative, and an additional noise term:

• The first term proportional to the gradient of the concentration density is the diffusion term: any inhomogeneity will be regularized by a flux due to the ideal part of the chemical potential δF id [ρ j ] δρ i (r, t) capturing entropy increase. The diffusion coefficient, following the fluctuation-dissipation theorem, is related to the mobility of an individual ion at infinite dilution in the solvent:

D i = k B T µ i .
• The second and third terms, proportional to the local concentration, are advective fluxes originating from the internal end external interactions captured by the excess part of the chemical potential δF exc [ρ j ] δρ i (r, t) . Here we have neglected the short-range contributions, such that the ionic interactions are modelled with the fields -∇V int and E ext at the mean-field level. The internal electrostatic field -∇V int created by the ions is mediated by the solvent and prescribed by the third equation, known as the Poisson equation. Combined with the external applied field E ext , they create an advective flux of species i, that is proportional to its electrical mobility

µ E i = ez i µ i = ez i βD i .
• The last term describes the fluctuations induced by the microscopic collisions between the ions and the solvent. The amplitude of that multiplicative noise is proportional to the square root of the local concentration and its diffusion coefficient, following the fluctuation-dissipation theorem. The statistics of that noise are carried by the stochastic field ξ i (r, t), which follows the standard isotropic Markovian assumptions:

-Zero average due to the isotropy of the system: the collisions cannot induce a global drift.

-Uncorrelated in space because the collisions are contact interactions, that are local in the framework of an implicit solvent.

-Uncorrelated in time due to the instantaneous relaxation of the solvent.

-Uncorrelated between species also because of independence of collisions.

Mathematically these properties translate to

   ⟨ξ i (r, t)⟩ = 0 ξ α i (r, t) ξ β j (r ′ , t ′ ) = δ ij δ αβ δ 3 (r -r ′ )δ(t -t ′ ) , (3.14)
where ξ β j denotes the complex conjugate of ξ β j .

Limitations

It is important to note that, being derived from the overdamped Langevin equation where we have assumed the equilibrium dissipation theorem by coarse-graining the degrees of freedom of the solvent, and assuming relaxation to be athermal, adiabatic and instantaneous, sDDFT suffers from the same limitations. This includes neglecting any kind of hydrodynamic effects, or dynamical dielectric properties of water. While we rapidly explore them in Sec .4.3, these effects have been taken into account within DDFT in the literature [START_REF] Bonneau | Temporal response of the conductivity of electrolytes[END_REF][START_REF] Donev | Dynamic density functional theory with hydrodynamic interactions and fluctuations[END_REF][START_REF] Avni | Conductivity of Concentrated Electrolytes[END_REF]. We restrict ourselves to the simpler model Eq. (3.13) since we mainly focus on long-time scales effects, assuming that the dynamics of ions, especially electrostatic relaxation that we will study in Sec. 4.1, dominate at long times and large length scales and that the system is diluted enough to be studied under the mean-field approximation, which cone previously in the literature [START_REF] Mahdisoltani | Long-range fluctuation-induced forces in driven electrolytes[END_REF][START_REF] Mahdisoltani | Long-range fluctuation-induced forces in driven electrolytes[END_REF][START_REF] Zorkot | Current fluctuations across a nano-pore[END_REF][START_REF] Démery | The conductivity of strong electrolytes from stochastic density functional theory[END_REF] 3.2 Resolution

Linearization

The sPNP model, Eq. (3.13) is a system of coupled nonlinear diffusion-advection equations with multiplicative noise, similar to any general sDDFT formulation. A general exact solution for ρ i (r, t) is out of reach. However, an approximate resolution is possible by linearizing around a state that is electroneutral [START_REF] Mahdisoltani | Long-range fluctuation-induced forces in driven electrolytes[END_REF]:

         ρ i (r, t) = ρ 0 i (r, t) + δρ i (r, t) |δρ i (r, t) | ≪ ρ 0 i (r, t) ρ 0 q = i z i ρ 0 i (r, t) = 0 . (3.15)
These assumptions correspond to considering that the local densities stay close to a value that is constrained by electroneutrality. The fields are allowed to fluctuate around this state but must stay globally electroneutral on average. One has to be careful when it comes to inhomogeneous/time-dependent external fields, in that case, one has to solve a standard uncoupled diffusion-advection model for the neutral solution:

     ∂ t ρ 0 i (r, t) = D i ∆ρ 0 i -z i βe∇ • (ρ 0 i E ext (r, t)) i z i ρ 0 i (r, t) = 0 . (3.16)
Furthermore, if we assume the system to be close to equilibrium, one can consider the background fields to be time-independent and homogeneous, under the condition that the action of the external field is sufficiently small, i.e. that E ext is divergence-free and/or evolving slowly compared to diffusion and electrostatic effects (adiabatic).

Under these assumptions of globally electroneutral, homogeneous and stationary state, ρ 0 i (r, t) = ρ 0 i . One can then linearize Eq. (3.13) for the perturbation, which now reads

             ∂ρ i (r, t) ∂t + ∇ • j i (r, t) = 0 j i (r, t) = -D i ∇ρ i + z i ρ 0 i ∇ϕ -z i ρ i βeE ext + 2D i ρ 0 i ξ i ∆ϕ (r, t) = -4πl B i z i ρ i , (3.17)
where we have introduced the adimentionalized electrostatic potential ϕ = βeV int , the reduced external field E = βeE ext and the Bjerrum length l B = βe 2 4πε 0 εw as in Sec. 2.2.4, and renamed δρ i (r, t) as ρ i (r, t) for simplicity of notation. The model now takes the form of a linear system with additive noise. We can now diagonalize the space-differential operator using the space Fourier transform defined with Eq. (2.45), which we recall as:

       f (k) = ˆR3 f (r) e -ik•r d 3 r f (r) = 1 (2π) 3 ˆR3 f (k) e ik•r d 3 k ,
where we have taken the abuse of notation by identifying f (k) = FT [f (r)](k) as the reciprocal space function. Applying the transform to Eq. (3.17), the dynamics take a simplified form:

∂ t ρ i (k, t) + L ij (k) ρ j (k, t) = 2D i ρ 0 i η i (k, t) , (3.18)
which is a linear inhomogeneous dynamical system (where we have adopted Einstein's summation convention over repeated indices). The two key features of this model are the dynamical matrix L ij (k) and the density noise source term η i (k, t). The first one is prescribed by:

L ij (k) = D i (k 2 -iz i βeE ext • k)δ ij + κ 2 i z j /z i . (3.19)
The diagonal elements are classical diffusion-advection terms due to entropy and external drive, and the off-diagonal elements are coupling effects due to the internal electrostatic fields created by the ionic densities. The strength of these couplings terms is set by the Debye parameters which are related to the Debye length:

κ 2 i = 4πl B ρ 0 i , i κ 2 i = κ 2 = λ -2 D . (3.20)
The right-hand-side in Eq. (3.18) is a source term proportional to an uncorrelated scalar density noise, resulting from the divergence of the stochastic flux:

η i (r, t) = ∇ • ξ i (r, t).
Making use of the properties of the Fourier transform on Eq.(3.14), one obtains:

η i (k, t) η j (k ′ , t ′ ) = (2π) 3 k 2 δ ij δ 3 (k + k ′ )δ(t -t ′ ) . (3.21)
Eq. (3.18) is sufficient to fully compute the time-dependent structure factors in the stationary and transient regimes by writing the formal solution

ρ i (k, t) = e -tL ij (k) • ρ j (k, 0) + ˆt 0 e +sL jl (k) • 2D l ρ 0 l η l (k, s) ds . (3.22)
The problem can be considered mathematically closed from there. Since we aim at computing the fluctuations frequency spectrum at stationary state only, it is convenient to also diagonalize the time differential operator using the time Fourier transform,

       f (t) = ˆR f (t) e -iωt dt f (ω) = 1 2π ˆR f (ω) e iωt dω , (3.23)
where we have used the same abuse of notation as for the space Fourier transform. Applying this transform to Eq. (3.18) one gets to the stationary linear system:

L ij (k, ω) • ρ j (k, ω) = 2D i ρ 0 i η i (k, ω) , (3.24)
where the dynamical matrix reads:

L ij (k, ω) = L ij (k) -iωδ ij , (3.25)
and the noise correlation spectrum satisfies:

η i (k, ω) η j (k ′ , ω ′ ) = (2π) 4 k 2 δ ij δ 3 (k + k ′ )δ(ω + ω ′ ) . (3.26)
In that case, the stationary solution is simply a product of the source term with the inverse of the dynamical matrix:

ρi (k, ω) = L -1 ij (k, ω) • 2D j ρ 0 j η j (k, ω) . (3.27)
Finally the dynamical structure factor matrix Eq. (2.53), correlating two density of species i and j, reads:

S ij (k, ω) = k 2 ρ tot L -1 ik (k, ω) • (2D k ρ 0 k ) • L -1 kj (k, ω) (3.28)
where we have used the relation between the product of Fourier transforms and correlation functions i.e. the Wiener-Khintchin theorem Eq. (2.97) for stationary translationalinvariant systems:

ρ j (k, ω)ρ j (k ′ , ω ′ ) = (2π) 4 ρ tot δ 3 (k + k ′ )δ(ω + ω ′ ) S ij (k, ω) , (3.29)
with ρ tot = l ρ 0 l , the average total density of the system.

This general mathematical expression for structure factors for an arbitrary multicomponent electrolyte allows us to explicitly compute them, but the formulae are quite lengthy and unsuitable for further physical interpretations at this point. To proceed further, we will restrict the development to binary monovalent electrolytes, consisting of only one anionic and one cationic species of valency |z ± | = 1, denoted as {+, -}.

Binary electrolytes and normal modes

The case of binary monovalent electrolytes takes a special place in the generality of the systems described by the mathematical model above, it is a good starting model to study complex mixture systems. Using the fact that electroneutrality imposes

ρ 0 + = ρ 0 -= C + = C -= C s , we rewrite κ 2 + = κ 2 -= κ 2 /2.
Then the two partial ionic densities ρ ± (k, ω) follow Eq. (3.24) with the 2D dynamical matrix

L {+,-} (k, ω) = D + k 2 -iβeE ext • k + κ 2 /2 -iω -D + κ 2 /2 -D -κ 2 /2 D -k 2 + iβeE ext • k + κ 2 /2 -iω .
(3.30) The next idea is to look for normal modes of the system where the two equations of motion will appear uncoupled. Apart from the conservation of partial ionic densities, the system also satisfies the conservation of total density and the conservation of total charge. The charge density being a central element in this thesis, we will proceed to switch from partial density for the cation and the anion basis {+, -} to the total density (referred as number or mass) with a subscript n and the charge with a subscript q basis, written {n, q}. In this case, as our main observable of interest are the number and the charge, the change of variable is

     ρ + = 1 2 (ρ n + ρ q ) ρ -= 1 2 (ρ n -ρ q ) -→ ρ n = ρ + + ρ - ρ q = ρ + -ρ - (3.31)
Under this linear transformation, it is convenient to define a change of variables for the diffusion coefficients, such that unequal mobilities will be written as a deviation with respect to the symmetric case with averaged diffusion coefficients:

D + = D(1 + γ) D -= D(1 -γ) -→      D = 1 2 (D + + D -) γ = D + -D - D + + D - (3.32)
For example, the case of Na + Cl -with D Na = 1.28×10 -9 m 2 /s and D Cl = 1.77×10 -9 m 2 /s, yields D = 1.53×10 -9 m 2 /s and the asymmetry factor γ = -0.161. This asymmetry factor varies with different binary electrolytes and can be generalized for arbitrary valencies and concentrations. For the binary case, under this change of variables, the problem now reads

L {n,q} (k, ω) • ρ n (k, ω) ρ q (k, ω) = 4DC s η n (k, ω) η q (k, ω) . (3.33)
In this formalism, the noises are not independent anymore, but correlated due to the difference of diffusivity between the anions and the cations, generating cross-correlations between number and charge:

η n (k, ω) η q (k, ω) • η n (k ′ , ω ′ ) η q (k ′ , ω ′ ) ⊺ = k 2 1 γ γ 1 (2π) 4 δ(ω + ω ′ )δ 3 (k + k ′ ) (3.34)
Moreover the dynamics exhibits the Debye length λ D and Debye relaxation time τ Debye defined as

λ D = 1/κ = (8πl B C s ) -1/2 τ Debye = 1/(Dκ 2 ) = λ 2 D /D (3.35)
as natural units. Hence the equations can be rescaled using the reduced units:

     K = k /κ Ω = ω /Dκ 2 E = βeE ext /κ . (3.36)
Therefore the dynamical matrix simplifies as:

L {n,q} (k, ω) = Dκ 2 ℓ {n,q} , (3.37)
where we have the adimensionalized dynamical matrix

ℓ {n,q} = K 2 -i(γE • K + Ω) γ(K 2 + 1) -iE • K γK 2 -iE • K K 2 + 1 -i(γE • K + Ω) . (3.38)
Using that the inverse of a 2x2 matrix can be explicitly written as:

a b c d -1 = 1 ad -cd d -b -c a ,
one can rewrite the dynamical structure matrix in the compact form:

Ŝ(k, ω) = K 2 Dκ 2 det l 2 ℓ qq -ℓ nq -ℓ qn ℓ nn 1 γ γ 1 ℓ qq -ℓ qn -ℓ nq ℓ nn (3.39)
The study and interpretation of these modes are left for the next chapter, along with the comparison with Brownian dynamics simulation results. We will focus on the influence of each parameter individually in a gradually complex phenomenology, starting from the electrostatic relaxation and influence of the salt concentration to the effect of external drive and asymmetric diffusion coefficients.

Conclusion

In this mathematically oriented chapter, we have first shown the microscopicmacroscopic link between the overdamped Langevin equation Eq. (2.5) and the sPNP model Eq. (3.13) in the framework of sDDFT, following the work of Dean [START_REF] David S Dean | Langevin equation for the density of a system of interacting langevin processes[END_REF] and applying it to electrolytes. After discussing the interpretation and limitations of that model, both from the physical and mathematical points of view, we solved it under linearization close to electroneutrality. While results similar to Eq. (3.22), Eq. (3.28) and Eq. (3.39) can be found in the literature [START_REF] Mahdisoltani | Transient fluctuation-induced forces in driven electrolytes after an electric field quench[END_REF][START_REF] Zorkot | Current fluctuations across a nano-pore[END_REF][START_REF] Démery | The conductivity of strong electrolytes from stochastic density functional theory[END_REF] for specific cases, we presented new results for the general case of electrolytes with arbitrary concentration, valencies, and diffusion coefficients. Lastly, we emphasized normal modes for binary electrolytes as they highlight the role of number and charge fluctuations. In this short chapter, we have established an analytical description for the implicit solvent electrolyte, that will be used to understand mechanisms of ionic charge and current fluctuations in electrolytes. Predictions of sDDFT will be interpreted and compared to Brownian dynamics simulations and molecular dynamics simulations in Chap. 4.

Chapter 4

Properties and fluctuations of bulk electrolytes In the two last chapters, we have introduced two methods to investigate aqueous electrolytes: Brownian dynamics simulations and stochastic dynamical density functional theory. The latter results from several approximations, and in particular, the mean-field approximation, picturing the ionic densities as continuous fields. In this chapter, following preceding studies in the literature [START_REF] Onsager | Deviations from ohm's law in weak electrolytes[END_REF][START_REF] Jardat | Transport coefficients of electrolyte solutions from smart brownian dynamics simulations[END_REF][START_REF] Young C Kim | Screening in ionic systems: simulations for the lebowitz length[END_REF][START_REF] Chandra | Frequency dependence of ionic conductivity of electrolyte solutions[END_REF][START_REF] Mahdisoltani | Transient fluctuation-induced forces in driven electrolytes after an electric field quench[END_REF][START_REF] Zorkot | Current fluctuations across a nano-pore[END_REF], we will investigate the properties of bulk electrolytes. The results shown here both aim to test sDDFT predictions, and establish new results regarding the time and space correlations in the fluctuations of charge. In particular, we will give a new view of electrostatic relaxation mechanisms through the charge-charge dynamical structure factor in Sec. 4.1 and ionic fluctuations in finite observation volumes in Sec. 4.2 Then to further test sDDFT and Brownian dynamics simulations, we will compare them to Langevin dynamics and molecular dynamics with explicit water in Sec. 4.3, from this we will discuss their respective limitations, and explore how to potentially improve analytical models. Finally, we will present a short part on preliminary results on dynamical structure factors in the nonequilibrium regime, and highlight the effect of unequal mobility for the anion and the cation in Sec. 4.4.

Diffusion and electrostatic relaxation

Predictions for a 1:1 symmetric electrolyte

The first study presents the main result of this thesis. We propose to start with a binary symmetric electrolyte with equal mobilities D + = D -at equilibrium (E ext = 0). The situation would apply to only a few real electrolytes such K + Cl -, but this model has the quality of briefly and qualitatively encapsulating the main effects of electrostatic interactions on electrolyte fluctuations. The roles of asymmetric diffusion coefficients and external drive will be examined later in this chapter, in Sec. 4.4. In the present case, Eq. (3.39) can be greatly simplified by setting E = 0 and γ = 0 and yields the total density and charge isotropic structure factors: In this simplified case, there is no coupling between the mass and the charge: S nq (k, ω) = S qn (k, ω) = 0, so that mass and charge are now referred to as normal modes. The formulae tell that the frequency-dependence of the dynamical structure factors is Lorentzian for both cases, but behave differently in terms of length scale. We note that the dynamical structure factor can be transformed back to intermediate scattering functions, where the behaviour corresponds to decaying exponential functions:

         S nn (k, ω) = 1 Dκ 2 K 2 Ω 2 + K 4 = Dk 2 ω 2 + (Dk 2 ) 2 S qq (k, ω) = 1 Dκ 2 K 2 Ω 2 + (K 2 + 1) 2 = Dk 2 ω 2 + (D (k 2 + κ 2 )) 2 , ( 4
     F nn (k, t) = e -Dk 2 t F qq (k, t) = k 2 k 2 + κ 2 e -D(k 2 +κ 2 )t . (4.2)
Mass diffuses according to regular isotropic Fickian diffusion without any characteristic length scale, but charge diffuses at small scales and relaxes for large ones, with the characteristic crossover mode at (|k| , ω) = (κ, τ -1 Debye ), being the inverse Debye length and Debye time respectively. This difference is represented in Fig. 4.1, which reports S nn and S qq in the (k, ω) plane for a binary monovalent symmetric electrolyte.

From Fig. 4.1a, one observes the characteristic features of fluctuations driven by free diffusion. The total mass/concentration fluctuations are unbounded and increase for decreasing time frequencies ω. The amplitude of the number fluctuations S nn (k, ω) is maximal when the wavelength and frequency of the fluctuation satisfy the dispersion relation ω = Dk 2 , characteristic of a diffusive plane wave. Fig. 4.1b shows a different mechanism: the charge fluctuations follow the same diffusive law for large frequencies, but are bounded by the global maximum located at

     arg max (k,ω) S qq (k, ω) = (κ, 0) max S qq (k, ω) = S qq (κ, 0) = 1 4Dκ 2 , (4.3)
and the local maxima are located along the dispersion relation ω = D(k 2 + κ 2 ). This is characteristic of relaxation, meaning that charge fluctuations that are larger than the Debye length λ D , have a vanishing amplitude and equivalently a time life that is of the order τ Debye . This is particularly important in the hydrodynamic limit:

       S nn (k, ω) ---→ ω→0 1 Dk 2 ---→ k→0 ∞ S qq (k, ω) ---→ ω→0 k 2 D(k 2 + κ 2 ) 2 ---→ k→0 0 (4.4)
where the diverging behaviour of S nn shows that mass fluctuation modes are of the propagating type and persist at macroscopic scales. While spatially extended mass inhomogeneities diffuse slower, it is not the case for charge fluctuations S qq that vanish due to electrostatic relaxation. Charge inhomogeneities larger than the Debye length λ D , relax with the corresponding transport process namely diffusion over the Debye length, characterized by the Debye time τ Debye = λ 2 D /D. Electrostatic relaxation is the dynamical counterpart of the static electric-field screening (see Sec. 1.3.3).

Simulation details

We have shown that under the approximations of mean-field and linearization around charge neutrality, justified for diluted solutions, the equilibrium dynamical charge-charge structure factor only depends on two relevant parameters: the diffusion coefficient D and the Debye length λ D = (8πl B C s ) -1/2 . To verify the influence of the latter, we proceed to a set of Brownian dynamics simulations for a range of concentrations, whose results are compared to sDDFT Eq. (4.1). The simulations are carried out using the methodology described in Sec. 2.1 with the force field parameters in Tab. 2.1. We vary the number of NaCl pairs (753, 3012, 7530, 30120) in a simulation box of size (500*500*500) Å 3 , resulting in the desired salt concentrations of Cs = (0.01, 0.04, 0.1, 0.4) M. Along with salt concentration setting the Debye time and length, one also changes the density of the system, characterized by the volume packing fraction, equal to the ratio between the volume occupied by the ions and the accessible volume of the simulation box (see Tab. 4.1). This last parameter plays no role in the theory presented in Chap. 3, as the ions are considered point-like in the mean-field approximation so short-range steric effects are not accounted for. This approximation, also present in the static Debye-Hückel, theory fails at large concentrations of about 0.01 M [START_REF] Hansen | Theory of Simple Liquids[END_REF], and is expected to serve as a tracking parameter of the validity of sDDFT. For these simulations the diffusion coefficients are taken as explained in Sec. 2.1.2, with the value D Na + = 1.28 × 10 -9 m 2 /s and D Cl + = 1.77×10 -9 m 2 /s, leading to average of the pair D = (D Na + +D Cl -)/2 = 1.53×10 -9 m 2 /s.

C s Mol/L λ D = (8πl B C s ) -1/2 Å τ Debye = λ 2 D /D ns ϕ = 8
3 πr ions C s is computed using the average radius used in the force fields: r ions ∼ 3 Å.

Comparison between sDDFT and BD simulations,

the effect of salt concentration Fig. 4.2 reports the comparisons for the charge-charge dynamical structure between sDDFT predictions and Brownian dynamics simulations. For each panel, the different curves correspond to S qq (k, ω) for a fixed wavevector, corresponding to a horizontal cross-section of Fig. 4.1b. sDDFT predictions are in excellent agreement with the numerical simulations. The characteristic crossover from diffusion at high frequencies S qq ∼ ω -2 to relaxation at low frequency S qq ∼ ω 0 is well described for a wide range of wavevectors and salt concentrations. We also observe a slight deviation at low frequency and wavevector for C s = 0.01 M, which may originate from finite size effects, because of the periodic boundary simulations, where 2π L box ≪ κ is not verified anymore.

A surprising agreement of linearized mean-field theory with numerical simulations, at rather high concentrations, is observed. While experimentally observed at 0.01 M [START_REF] Hansen | Theory of Simple Liquids[END_REF], in the framework of implicit solvent, the failure of the dilute approximation does not occur at 0.4 M. Although 0.4 M is a rather concentrated solution, the packing fraction is only of 5.4%, which may not be enough to observe the breakdown of the continuous mean-field approximation. One can expect, from the study of hard spheres models [START_REF] Frenkel | Understanding molecular simulation: from algorithms to applications[END_REF], that shortrange effects will only appear above packing fractions of 10%, which is much higher than experimentally observed. Consequently, the breakdown of Debye-Hückel theory and other models relying on the dilute approximation cannot be attributed to the finite size effects of the ions only, but rather association mechanisms or other phenomena related to the properties of the solvent (see Sec. 3.1.4) which are not accounted for in neither sDDFT nor BD simulations.

Ionic fluctuations in finite volumes

While the results presented in the last section arise from the well-known electrostatic relaxation, the consequences of the latter on a finite macroscopic domain are less obvious, as we showed that the amplitude of a fluctuation is strongly related to time and length scales. In order to slowly progress towards the modelling of confined electrolytes, one formally has to add boundary conditions and refine the model, which can be tedious due to the requirement for proper interfacial models, which are extremely varied and specific in real systems. To approximate these boundaries, while remaining as general as possible, one possible way of completing the sDDFT model is to add no-flux boundary conditions, as was done in Ref. [START_REF] Mahdisoltani | Transient fluctuation-induced forces in driven electrolytes after an electric field quench[END_REF]. One can also, more coarsely, restrict the fluctuation modes to finite wavevectors using cut-off values from geometrical considerations (see Ref. [START_REF] Zorkot | Current fluctuations across a nano-pore[END_REF]). In this work, done in collaboration with Sophie ch [START_REF] Hoang | Ionic fluctuations in finite volumes: fractional noise and hyperuniformity[END_REF], we will take a similar route. We propose to follow the ionic fluctuations in a finite region of a bulk electroneutral system, and follow their properties with varying observation time and length scales. We will first start with the total number of ions, and then study the total charge in an observed volume with the analytical tools derived in Sec. 3.2, then compare analytical predictions against Brownian particles simulations (see Fig. 4.3). Sec. 4.2.2 and Sec. 4.2.3 adopt a broad theoretical framework to describe particle correlations in charged system, allowing the generalization to binary Coulomb gas in arbitrary dimensions. We then go back to a real electrolyte, where we study the dynamical properties of ionic fluctuations, which exhibit intriguing time and frequency signatures.

Q(t)/q N(t)

Simulation details

The Brownian dynamics simulations are carried out with the methodology described in Sec. 2.1 for a symmetric electrolyte, with the parameters shown in Tab. 4.2 and fixed cubic simulation box of side L box = 160 Å. We consider a totally symmetric 1:1 electrolyte (see Sec. 2.1.2) with spherical radius r ions = 3 Å, diffusion coefficient of D = 1.5 × 10 -9 m 2 /s, constant relative permittivity ϵ w = 78.5, and temperature T = 300 K. Instead of studying the structure factors as in Sec. 4.1, we monitor the number of particles and charges counted inside cubic subvolumes (see Fig. 4.3), and compute their time correlation function C nn (t) and C qq (t) Eq. (2.39) .

N 0 C 0 (mM) λ D (nm)

Number and charge fluctuations in arbitrary dimensions

To model these observables, we start by using the definition of a density and write the instantaneous total number of counted elements (i) in a volume V, as an integral of its instantaneous density ρ i (r, t) over the observed space:

N i,V (t) = ˆV ρ i (r, t) d d r . (4.5)
At equilibrium, this quantity possesses a constant average value denoted ⟨N i,V ⟩, and fluctuates in time around that value as illustrated in Fig. 4.3. To track these fluctuations, we introduce its time correlation (see Eq. (2.39)) function:

C ii,V (t) = ⟨δN i,V (0) δN i,V (t)⟩ where δN i,V (t) = N i,V (t) -⟨N i,V ⟩ (4.6)
this quantity can be related to the dynamical structure factor using the definition of instantaneous density and properties of correlation functions and their Fourier transforms in a bulk system:

C ii,V (t) = ˆV δρ i,V (r, 0)δρ i,V (r ′ , t) d d r d d r ′ = ⟨N tot,V ⟩ ˆRd d d k (2π) d |G V (k)| 2 V F ii (k, t) (4.7) where G V (k) = ´Rd 1 V (x)e -ik•x d d
x is a geometrical factor equal to the Fourier transform of the observed volume, and F ii (k, t) is the associated intermediate scattering function (see Eq. (2.48)). This approach is noticeably similar to Ref. [START_REF] Zorkot | Current fluctuations across a nano-pore[END_REF], used to study electrical current fluctuations in a nanopore, by restricting accessible wavevectors by geometrical considerations.

Eq. (4.7) is a fundamental tool to relate the microscopic fluctuations and the macroscopic observable. Specifically, the class of integrals appearing in Eq. (4.7) are called Kirkwood-Buff integrals [START_REF] John | The statistical mechanical theory of solutions. i[END_REF] and are at the centre of recent studies [START_REF] Krüger | Kirkwood-buff integrals for finite volumes[END_REF][START_REF] Dawass | Kirkwood-buff integrals from molecular simulation[END_REF] pursuing new methodologies to extract thermodynamic properties of solutions from computer simulations. The advantage of this formulation is that, in contrast to the study in real space, where an observable A(r) is correlated with A(r ′ ), A(k) and A(k ′ ) are uncorrelated in a bulk system due to translational invariance. The nonlocality in real space can be deconvoluted as a direct sum which combines the geometrical property of the observed volume and the associated local properties of the microscopic fluctuations in Fourier space. To pursue the analytical development, the geometrical factor G V (k) can be computed for various shapes and forms, and scales as a volume-like quantity. We present here the computation for a hyper-cube of side L obs and a hyper-sphere of radius R obs in arbitrary dimension d. For a cube, the integrand factorises in cartesian coordinates:

|G V (k)| 2 V = L -d obs ˆRd 1 cube (x)e -ik•x d d x 2 = L d obs d i=1 ˆ+1/2 -1/2 e -iL obs k i u i du i 2 = L d obs d i=1 sinc 2 L obs k i 2 , (4.8)
whereas the hypersphere case can be integrated using hyperspherical coordinates using that the volume of a d-hypersphere of radius R is equal to

V(R, d) = π d/2 Γ(d/2+1) R d : |G V (k)| 2 V = π d/2 Γ (d/2 + 1) R d obs -1 ˆRd 1 sphere (x)e -ik•x d d x 2 = π d/2 Γ (d/2 + 1) -1 R d obs ˆ|u|<1 e -iR obs k•u d d u 2 = π d/2 Γ (d/2 + 1) k d -1 (2π) d J d/2 (R obs k) 2 , (4.9)
where J α (x) is the first kind Bessel Function of order α. Anticipating that the structure factor derived in Eq. ( 4.1) depends only on the norm of the wave-vector |k| because it describes an isotropic system, we focus on the spherical case possessing the same isotropic symmetry. Then, by inserting the form factor of the sphere Eq. (4.9) into the time correlation Eq. (4.7), we obtain

C ii,V (t) = ⟨N tot,V ⟩ ˆRd d d k π d/2 Γ (d/2 + 1) k d -1 J d/2 (R obs k) 2 F ii (k, t) . (4.10)
Once again using hyperspherical coordinates and isotropy, we can integrate over solid angle and rewrite:

´Rd f (k) d d k = ´∞ 0 S d-1 (k)f (k) dk, where the surface of a d-hypersphere of radius k is S d-1 (k) = 2π d/2 Γ(d/2) k d-1 : C ii,V (t) = ⟨N tot,V ⟩ ˆ∞ 0 dk 2π d/2 Γ (d/2) k d-1 π d/2 Γ (d/2 + 1) k d -1 J d/2 (R obs k) 2 ) F ii (k, t) = ⟨N tot,V ⟩ d ˆ∞ 0 J d/2 (R obs k) 2 k F ii (k, t) dk .
(4.11) This final result is reminiscent of a power spectral density written as a Hankel transform by virtue of the Parseval-Plancherel theorem. This result can be derived by exploiting the properties relating the Fourier transform of isotropic functions and Hankel transforms. Now we can insert the intermediate scattering functions given by sDDFT for a binary symmetric electrolyte for the particle density and the charge density Eq. (4.2). We note that at this point, the two results of Eq. (4.2) hold in an arbitrary dimension where the generalisation of the electrolyte would still interact through a Poisson type potential, that would yield a screened interaction (Yukawa potential in 3D) at linear order, and consequently the same structure factors. Then the full expressions for the correlation functions are:

         C nn,V (t) = ⟨N tot,V ⟩ d ˆ∞ 0 J 2 d/2 (R obs k) k e -Dk 2 t dk C qq,V (t) = ⟨N tot,V ⟩ d ˆ∞ 0 J 2 d/2 (R obs k) k k 2 + κ 2 e -D(k 2 +κ 2 )t dk (4.12)
General analytical developments beyond scaling laws become more difficult from this point. But the static situation t = 0 which corresponds to the variance of the number of particles/charges can be analytically computed from Eq. (4.12). We will study this result in the following.

Hyperuniformity and arbitrary spherical volume

Hyperuniformity [START_REF] Ghosh | Fluctuations, large deviations and rigidity in hyperuniform systems: a brief survey[END_REF][START_REF] Torquato | Hyperuniform states of matter[END_REF] refers to an abstract property of some systems which suppresses density fluctuations at large scales. Quantitatively, for a thermalized system, one would naively expect from the central limit theorem that the variance of an extensive observable carried by particles in a given volume would scale as the total number of elements contained, (consequently as its volume) i.e. δN 2 i,V ∝ ⟨N tot,V ⟩ ∝ V. But the central limit theorem, while being robust, is known to not hold anymore for correlated variables, which is the case in the presence of interactions as they can create long-range correlations such as electrostatics. Confirming and extending the existing work in the literature on charge hyperuniformity in electrolytes Ref. [START_REF] Ph | The charge fluctuations in classical coulomb systems[END_REF][START_REF] Jancovici | Charge fluctuations in finite coulomb systems[END_REF][START_REF] Young C Kim | Screening in ionic systems: simulations for the lebowitz length[END_REF][START_REF] Martin | Electrostatic fluctuations in cavities within polar liquids and thermodynamics of polar solvation[END_REF][START_REF] Joel | Charge fluctuations in coulomb systems[END_REF], we show that for of a globally neutral electrolyte, the variance of the total charge in a volume scales sublinearly to its volume in arbitrary dimension.

For the static t = 0 case where C nn/qq (t = 0) = C 0 nn/qq is the variance of the number of particles and charges respectively, the integrals in Eq. (4.12) can be carried out exactly, yielding the result:

         C 0 nn,V = ⟨N tot,V ⟩ d ˆ∞ 0 J 2 d/2 (R obs k) k dk = ⟨N tot,V ⟩ C 0 qq,V = ⟨N tot,V ⟩ d ˆ∞ 0 J 2 d/2 (R obs k) k k 2 + κ 2 dk = ⟨N tot,V ⟩ d I d/2 (R obs κ) K d/2 (R obs κ) (4.13)
where I α (x), K α (x) are the first and second kind modified Bessel functions of order α. Eq. (4.13) generalizes the result obtained by Kim and Fisher in 2008 [START_REF] Young | Charge fluctuations and correlation lengths in finite electrolytes[END_REF].

The first equation corresponds to the classical situation of a Gaussian stochastic process, where the variance exactly scales as the average value. The mass fluctuations are extensive, they scale as the average total number and consequently as the volume for a fixed average density. This result holding for a generic observation volume is a consequence of the scale invariance of a regular diffusion process. In contrast, the charge fluctuations show non-trivial behaviour because interactions introduce a finite correlation length that couples with the observation volume:l the scale invariance is broken. Considering macroscopic limits where the observation length becomes large, upon inserting asymptotic expansion of Bessel functions

I a (x)K a (x) ---→ x→∞ 1 2x + o 1 x (4.14)
in Eq. (4.14) into Eq. (4.13), one gets the exact law

   C 0 nn,V = ⟨N tot,V ⟩ C 0 qq,V ------→ R obs κ≫1 d 2R obs κ ⟨N tot,V ⟩ . (4.15)
This proves, from the generality of Eq. (4.11), that for any system showing a Debye-Hückel-like structure factor:

F (k, t = 0) = S(k) ∼ k 2 k 2 + ξ -2 , (4.16)
where ξ is the correlation length/screening length (ξ = λ D = κ -1 for an electrolyte), resulting from an effective screened Poisson potential, which is the case for an electrolyte interacting through a 3D Coulomb potential at the linearized mean-field level of description, hyperuniformity occurs when the observation scale is much larger than the correlation length i.e. R obs ≫ ξ. The fluctuations of charge vanish for large volumes. Moreover, Eq. (4.15) also shows that the charge fluctuations scale as the surface surrounding the observation volume, precisely, it scales as a shell with a thickness of d/2 times the correlation length developed around the volume, as illustrated in Fig. 4.4c for a square d = 2 case, meaning that fluctuations are located at the boundaries of the system. Our derivation coincides with the other approaches in Refs. [START_REF] Ph | The charge fluctuations in classical coulomb systems[END_REF][START_REF] Young | Charge fluctuations and correlation lengths in finite electrolytes[END_REF][START_REF] Bekiranov | Fluctuations in electrolytes: The lebowitz and other correlation lengths[END_REF][START_REF] Young C Kim | Screening in ionic systems: simulations for the lebowitz length[END_REF][START_REF] Kalcher | Structure-thermodynamics relation of electrolyte solutions[END_REF][START_REF] Ghosh | Fluctuations, large deviations and rigidity in hyperuniform systems: a brief survey[END_REF] for electrolyte systems, where the underlying property is the perfect electrostatic screening (see Stillinger-Lovett sum rules in Ref. [START_REF] Hansen | Theory of Simple Liquids[END_REF]), and extends to other types of interactions in arbitrary dimensions. We now close the mathematical excursion. Rescaled charge fluctuations

C Q (0)/L 2 obs ∏ D (2C 0 q 2 ) 10 1 10 2
Observation box size L obs (nm) Static charge fluctuations size L obs for increasing salt concentrations, going from purple to yellow. Dots: results from BD simulations; lines: Eq. (4.17). Error bars are one standard deviation from the mean and are smaller than dot sizes. (b) Rescaled (a) plot showing data collapse, highlighting "entropic" and "enthalpic" regimes. (c) Sketch of the origin of fluctuations, in 2D for simplicity, in both regimes (see text for details). Fluctuations growing as the area of the observation volume in the enthalpic regime are, by definition, hyperuniform [START_REF] Hoang | Ionic fluctuations in finite volumes: fractional noise and hyperuniformity[END_REF].

C Q (0) = hQ 2 i °hQi 2 (2C 0 q 2 ) C 0 =
For the case that is actually relevant to real-world electrolytes in a 3D world, we compare in Fig. 4.4a the results of Brownian dynamics simulations realized with various salt concentrations and cubic observation box sizes, against the theoretical model. By inserting Eq. (4.8) into Eq. (4.7) and taking d = 3 and t = 0, the explicit formula for the charge variance reads:

C 0 qq,V = ⟨N tot ⟩ L 3 obs ˆR3 i∈{x,y,z} dk i 2π k 2 sinc 2 L obs k i 2 k 2 + κ 2 (4.17)
and has no known closed form. The computation could be carried out numerically using quadrature methods in spherical coordinates, by writing:

C 0 qq,V = ⟨N tot ⟩ L 3 obs (2π) 3 ˆ∞ 0 k 2 k 2 + κ 2 A(k, L obs )k 2 dk where A(k, L obs ) = ˆπ 0 ˆ2π 0 sinc 2 L obs k sin θ cos ϕ 2 × sinc 2 L obs k sin θ sin ϕ 2 sinc 2 L obs k cos θ 2 dϕ sin θdθ . (4.18)
but A(k, L obs ) can also be estimated asymptotically for kL obs ≪ 1 and by noticing that:

A(kL obs ≪ 1) ∼ ˆπ 0 ˆ2π 0 dϕ sin θdθ = 4π ,
where we have used that lim x→0 sinc(x) = 1. Furthermore, because Eq. (4.18) must be convergent, the high-frequency limit must be to leading order in k:

A(kL obs ≫ 1) ∼ L 4 obs /k 4 .

Therefore we can approximate the factor by

A(k, L obs ) ∼ 4π 1 + αk 4 /L 4 obs
, where α is a numerical prefactor that could be computed by detailed expansion. Instead, we take α such that the unscreened limit, i.e κL obs ≪ 1 of Eq. (4.18), coincides with the unscreened result that was computed analytically for the mass variance: C qq,V κL obs ≪1 -----→ C 0 nn,V = ⟨N tot,V ⟩, because one can suppose that the number of charges behaves as the number of charge carriers with respect to thermal fluctuations at small scales. Hence α = (2/π 2 ) 2/3 /16. Then the charge variance is approximated by

C 0 qq,V ∼ ⟨N tot ⟩ L 3 obs 2π 2 ˆ∞ 0 k 2 k 2 + κ 2 k 2 1 + αk 4 /L 4 obs dk (4.19)
and finally integrated, with the final result:

C 0 qq,V ∼ ⟨N tot ⟩ 1 - 1 1 + 2(2π) 1/3 κL obs + 2(2π) 2/3 κ 2 L 2 obs .
(4.20)

The uniform and hyperuniform limits C 0 qq,V ∼ ⟨N tot ⟩ and C 0 qq,V ∼ ⟨N tot ⟩(16π) 1/3 λ D /L obs are recovered for L obs ≪ λ D and for L obs ≫ λ D respectively.

The comparison of this theoretical model with Brownian dynamics simulations shows excellent agreement in Fig. 4.4a, and the rescaling shown in Fig. 4.4b exhibits the transition from uniform to hyperuniform fluctuations as a function of the parameter L obs /λ D . As a direct consequence of Sec. 4.1 where we studied electrostatic screening and relaxation, at small scales relative to the interactions (L obs ≪ λ D ), the thermally induced fluctuations follow a classical Gaussian behaviour driven by extensive (∝ L 3 obs ) entropic diffusion laws (Fig. 4.4c). At scales larger than the Debye length (L obs ≫ λ D ), the thermally induced fluctuations of charges couple with the electrostatic interactions (enthalpy) which restrict them to "surface effects" (Fig. 4.4d). This result illustrates the fact that one cannot identify fluctuations of the number of charges carriers with the fluctuations of electrical current in an electrolyte, as is often done in the literature [START_REF] Gravelle | Adsorption Kinetics in Open Nanopores as a Source of Low-Frequency Noise[END_REF], such as in Hooge's phenomenological law [START_REF] Hofman | The validity of hooge's law for 1/f noise[END_REF], and should instead study the fluctuations of total number of charges. Regarding this last observation, in order to pursue the study of electrical current fluctuations, we focus on the kinetics of total charge fluctuations.

Time-correlations and dynamics in electrolytes

Now that we have a better understanding of the role of electrostatic interactions on the static correlations, C nn/qq,V (t = 0), confirming and extending existing results in the literature [START_REF] Ph | The charge fluctuations in classical coulomb systems[END_REF][START_REF] Jancovici | Charge fluctuations in finite coulomb systems[END_REF][START_REF] Young C Kim | Screening in ionic systems: simulations for the lebowitz length[END_REF][START_REF] Martin | Electrostatic fluctuations in cavities within polar liquids and thermodynamics of polar solvation[END_REF][START_REF] Joel | Charge fluctuations in coulomb systems[END_REF], we will extend this approach to the dynamical correlations C nn/qq,V (t). In contrast, resolving the fluctuating dynamics in finite volumes has received less attention. Indeed, as implied in Green-Kubo formulae, the study of time-resolved equilibrium fluctuations yields precious transport properties of the system (see Sec. 2.3). The work presented here can be perceived as a merge of Kirkwood-Buff and Green-Kubo integrals.

Particle number fluctuations

Inserting Eq. (4.8) and Eq. (4.2) into Eq. (4.7), taking d = 3 and performing the integral over the k-modes, the equation for the total number of ions in the observed volume takes the closed form:

C nn,V (t) = ⟨N tot,V ⟩ f t τ diff 3 , where τ diff = L 2 obs 4D and f (u) = u π e -1/u -1 + erf 1 u . (4.21)
We compare BD results with Eq. (4.21) in Fig. 4.5a for C s = 1.4 mM (symbols and lines, respectively). The excellent agreement shows that sDDFT is indeed well suited to predict particle number fluctuations. Eq. (4.21) shows that number fluctuations scale with the average number of particles in the observation box, ⟨N ⟩, and are determined by a single timescale τ diff = L 2 box /4D corresponding to particle diffusion across the observation box. This is confirmed in Fig. 4.5-b, which shows that all BD results collapse on a master curve, well described by Eq. (4.21) , when rescaled by ⟨N ⟩ and time by τ diff . Furthermore, we find numerically that the correlations decay algebraically as t -3/2 at long times. Expanding Eq. (4.21) we find C nn,V (t)/⟨N ⟩ = (τ diff /πt) 3/2 , which confirms the exponent of the algebraic decay. This slow relaxation of the correlations indicates that particle rearrangements are slow with time due to their diffusive or Brownian nature.

b. a.

c. (b) Rescaled (a) plot showing algebraic decay at long times as 1/t 3/2 . (c) Associated frequency spectrum with the 1/f 3/2 signature of fractional noise [START_REF] Benoit | Fractional brownian motions, fractional noises and applications[END_REF][START_REF] Marbach | Intrinsic fractional noise in nanopores: The effect of reservoirs[END_REF]. Here, C s = 104 mM; coloured legends are shared across (a-c) [START_REF] Hoang | Ionic fluctuations in finite volumes: fractional noise and hyperuniformity[END_REF].

1/f 3/2 1/t 3/2

Total charge fluctuations

We now turn to the relaxation of charge correlations by considering C qq,V (t) = ⟨δN z (t) δN z ((0)⟩. Fig. 4.6a displays the BD results for a fixed salt concentration and various observation volumes, rescaled by ⟨N tot ⟩. At early times, correlations collapse for small L obs (light orange) but not for large ones (dark red), a natural consequence of the above-discussed static (t = 0) hyperuniformity. Surprisingly, at long times, we observe the opposite behaviour: correlations collapse for large L box but not for small ones. Furthermore, the decay of charge correlations for large L box is not algebraic, in contrast with number fluctuations, but exponential (see Fig.

C qq,V (t) = ⟨N tot ⟩ e -Dκ 2 t L 3 obs ˆR3 i∈{x,y,z} dk i 2π k 2 sinc 2 L obs k i 2 k 2 + κ 2 e -Dk 2 t . (4.22)
Oppositely to Eq. (4.21), the integral of Eq. (4.22) over the k-modes is not expressible without further assumption. Box charge correlations Numerically integrated, Eq. (4.22) reproduces remarkably well the BD results (see Fig. 4.6, lines). For sufficiently large L box , the correlations decay exponentially with characteristic timescale τ Debye . Indeed, the relaxation of charge fluctuations is primarily driven by electrostatics: the transient local breakdown of electroneutrality induces an internal electric field driving the ions to restore electroneutrality. As displayed in Eq. (4.22), along with the Debye time τ Debye = 1/(Dκ 2 ) in the decaying exponential prefactor, there are other timescales involved expressed in the k-modes integral, combining electrostatics, diffusion and geometry (κ, D, L obs ). The interplay between τ Debye and τ diff can produce a variety of timescales that could all explain part of the behaviour. To understand the relaxation behaviour more systematically, we explore in Fig. 4.7 the relaxation of C qq,V (t)e t/τ Debye . Since BD results are well captured by sDDFT over a broad range of parameters, we use analytical expansions of Eq. (4.22) to quantify the dependence of the results on λ D and L obs . Fig. 4.7-a first reports the case of small observation volumes compared to the Debye length L obs ≪ λ D . Beyond the initial static regime where C qq,V ∼ L 3 obs , when t ≳ τ diff , we find, expanding Eq. (4.22), that the correlations decay as C qq,V ∼ L 3 obs (τ diff /t) 3/2 . This decay exactly follows that of the particle number decay in Fig. 4.5-b. At this observation length scale and timescale, electrostatics do not play any role, and the only relevant timescale appears to be τ diff . Eventually, at longer times, t ≳ τ Debye , correlations decay faster as C qq,V ∼ L obs λ 2 D e -t/τ Debye (τ diff /t) 5/2 , and the Debye timescale τ Debye appears to govern charge fluctuation relaxation. As explained above, this time scale emerges due to restoring electrostatic forces that damp fluctuations arising from diffusion. ) and t = 4τ Debye /π 2 , respectively. (c) Rescaled charge correlations at long times, with a scaling law as C qq,V ∼ L obs . In all panels: dots: results from BD simulations with shaded areas (or error bars) indicating one standard deviation around the mean; lines: Eq. (4.22) [START_REF] Hoang | Ionic fluctuations in finite volumes: fractional noise and hyperuniformity[END_REF].

C Q (t)/q 2 hN i = hQ(t)Q(0)i/q 2 hN i L obs =
C Q (t) (2C0q 2 ) L obs ≫ λ D L obs ≪ λ D t ∼ τ Diff t ∼ τ Diff t ∼ τ Debye t ∼ τ Debye L obs τ Diff = L 2 obs /4D τ Debye = λ 2 D /D × e -t/
L 3 o b s ( τ D iff t ) 3 /2 L obs λ 2 D ( τ Diff t ) 5/2 L o b s λ 2 D ( τ D i ff t ) 5 / 2 × λ 3 D (τ Diff /t)
How do these effects survive when the length scales, L obs ≫ λ D , and hence the timescales τ Debye ≪ τ diff are reversed? In Fig. 4.7-b, we show BD results with parameters L obs ≃ 2λ D , which is already hard to achieve with reasonable simulation times. Beyond the static hyperuniform regime where C qq,V ∼ L 2 obs λ D , for t ≳ τ Debye we find C qq,V ∼ L 2 obs λ D e -t/τ Debye (τ Debye /t) 1/2 . The decay of the correlations is apparently entirely due to electrostatic effects, with τ Debye the relevant timescale, and is faster than exponential. Finally, for t ≳ τ diff , when particles have had time to diffuse across the observation volume, τ diff appears in the dynamics, as C qq,V ∼ L obs λ 2 D e -t/τ Debye (τ diff /t) 5/2 .

Curiously, at long times, correlations decay as C qq,V ∼ L obs λ 2 D e -t/τ Debye (τ diff /t) 5/2 in both the L obs ≪ λ D and L obs ≫ λ D regimes. At such long timescales, particles have diffused over distances long enough that λ D and L obs appear comparably small. Remarkably, the amplitude of the fluctuations now scales with the perimeter L obs of the observation domain, which we verify numerically in Fig 4 .7-c. Note, that the collapse of the data onto the scaling law is not perfect, since we are limited in time with simulations and the time investigated is not always much bigger than τ Debye , τ diff for all parameters (C s ,L obs ). This extreme long-time scaling appears to be a case of hyperuniformity, where the dimensional degree of hyperuniformity is increased because fluctuations have relaxed. It is tempting to interpret this result in the following way: at long times, only boundary crossings in volume elements surrounding cube edges with area λ 2 D matter. This open interpretation could be formally addressed, for example, by investigating the spatial relaxation of fluctuations.

Implicit vs explicit solvent

Until this section, we have shown properties derived from stochastic density functional theory, and compared them with Brownian dynamics simulation. Both are overdamped dynamical models for ions in an implicit solvent and approximations of Langevin dynamics, which is itself an approximation of Hamiltonian/Newtonian dynamics. The investigation of this layered construction of approximations is essential to fully comprehend and assess the limitations of the implicit solvent model used in this thesis (see Sec. 2.1). This section, which work was done with the collaboration of Jeongmin Kim (who performed all the molecular dynamics simulations) [START_REF] Hoang | Electrical noise in electrolytes: a theoretical perspective[END_REF], is dedicated to addressing the elephant in the room. By comparing the different levels of description, we establish a quantitative domain of validity for BD and sDDFT, and take a look at physical processes that are beyond the scope of mean-field and implicit solvent models.

Molecular, Langevin and Brownian Dynamics

Molecular dynamics is a simulation method describing the microscopic motion of atoms following Newton's Laws. For our NaCl electrolyte case, it describes the motion of the ions bathing in water molecules described with a water model. Implicit solvent models forget about the molecular essence of water, consequently the related properties such as solvation mechanisms or hydrodynamics and electrokinetic effects (see Sec. 3.1.4). Here we compare predictions of three different simulation methods to assess the effect of solvent and short-time relaxation using BD, LD, and MD. The first method assumes implicit friction leading to instantaneous relaxation of the velocity of a particle, the second explicit friction with implicit water, and the last one explicit water. We start by analysing the static properties of the system at various length scales, then focus on dynamical properties, both at finite and zero frequencies for a concentrated solution of aqueous Na + Cl -at C s ∼ 1.23 M. We show that the large-scale/long-time behaviours of the ions are dominated by the electrostatic relaxation, and qualitatively well described by mean-field predictions. We also point out the double breakdown of sDDFT for intermediate wavevectors and frequencies, which are due to the finite size of the ions and water-related effects.

Simulation details

All MD, LD and BD simulations are performed with the LAMMPS simulation package [START_REF] Thompson | Lammps -a flexible simulation tool for particle-based materials modeling at the atomic, meso, and continuum scales[END_REF]. The Fourier components of the charge density, ρz ( ⃗ k, t) (Eq. 2.54), are sampled every 6 fs, 400 fs, and 1 ps for MD, LD and BD respectively, for selected wavevectors compatible with the periodic boundary conditions, satisfying |k| = nk min with k min = 2π/L box , where L box = 45.6 Å is the box size for MD simulations and n integers between 1 and 256. This covers length scales ranging between 0.2 and 45.6 Å, smaller than the particle size and larger than the typical correlation lengths in the electrolyte. For LD and BD simulations, with a box size 2L box , we also consider |k| = k min /2. The correlation function F qq (k, t) and its Fourier transform S qq (k, ω) are then computed from the time series of ρz (k, t) using fast Fourier transforms as explained in Sec. 2.4. In the absence of an external field, the three directions of space are equivalent: In order to improve the statistics we consider wavevectors in the x, y and z directions and average the results. The total simulation time for each production run is 5 µs for LD and 50 µs for BD. Each of them is then divided into 50 blocks considered independent for the analysis. In Fig. 4.8 we show the static structure factor for the three different simulation methods (BD, LD and MD) and Debye-Hückel and ideal-gas analytical predictions. All simulation methods coincide fairly well with Debye-Hückel theory at low wavevectors, which predicts that (Eq. (2.51))

S qq (k) = k 2 k 2 + κ 2 D .
This shows that the long-range structure is mainly a consequence of electrostatics. At high wavevectors all methods converge to S qq (k → ∞) = 1, i.e. for very small lengthscales the many-body system can be considered as a collection of independent particles, and follows the statistics of an ideal gas: there are no ionic correlations at these length scales. At wavevectors comparable with the particle sizes, the study of highly concentrated electrolytes reveals a mismatch between all predictions (except BD and LD which are identical at equilibrium). Debye-Hückel theory underestimates ionic correlations with respect to BD and LD: short-range attraction and repulsion, due to the Lennard-Jones potentials (Eq. (2.2)). This leads to small oscillations of the structure factor around k ∼ 2π/σ that are not predicted from mean-field approaches. These oscillations are all the more increased in the presence of explicit water, which forms solvation shells around the ions. Two ions then interact through this complex medium [START_REF] Hansen | Theory of Simple Liquids[END_REF], resulting in more complex structures. This mechanism is ignored in implicit solvent models such as LD, BD and Debye-Hückel theory. These characteristic oscillations in reciprocal space, here a synonym of strong ionic correlations, are equivalently studied in direct space through oscillations of radial distribution function [START_REF] Hansen | Theory of Simple Liquids[END_REF].

Dynamical properties

We now turn to the role of water description on dynamical properties. Fig. 4.9a shows the dynamical structure factors normalized by their zero frequency value in order to highlight dynamical processes, encoded in the frequency-dependency:

• A first observation is that the dominant process at low frequencies is the electrostatic relaxation described in Sec. 4.1, where the dynamical structure factor goes from the low-frequency plateau value to a Lorentzian decay following Eq. (4.1). The transition frequency is given by ω ∼ D(k 2 + κ 2 ). The three simulation methods confirm qualitatively the sDDFT predictions.

• Around the THz domain, a regime not accessible by BD simulation, we observe that the implicit solvent model described by LD still follows the Lorentzian decay, but MD simulations reveal additional features. The small bump appearing is a result of ion-water correlations: the ions dynamics at high frequencies are not a free diffusion process due to surrounding water. The complex dynamics of this solvation shell, which does not relax instantly, couple with the dynamics of the ions.

• The high-frequency domain (> 10 THz) is a range where MD simulations severely differ from other methods. The implicit solvent model predicts a 1/ω 2 decay, while MD shows a Gaussian-like decay. At this timescale, one cannot neglect the inertial effect of the ions anymore, which can be modelled as described below. The high-frequency statistics of a thermal ideal gas under implicit solvent approximation differs from the explicit solvent case. The first one assumes exponential relaxation of the velocities (instantaneous for BD), while the latter obeys kinetic the Maxwell-Boltzmann distribution due to the independence of velocities and the central limit theorem. Following conservation of momentum for Newtonian/Hamiltonian dynamics, the behaviour at high frequencies (ω → ∞) and small distances (k → ∞) can be modelled as a ballistic regime, provided that the probed distances are sufficiently short (typically a fraction of the distance between ions and molecules, which is also comparable to their size). In that case, one can also neglect correlations between ions, so that the correlation of the sum over ions reduces to a sum of self terms. The resulting dynamical structure factor can be derived from the Maxwell-Boltzmann distribution of velocities v = ω/k [START_REF] Hansen | Theory of Simple Liquids[END_REF] :

S Ball. qq (k, ω) = πβm + 2k 2 e -βm + ω 2 2k 2 + πβm - 2k 2 e -βm -ω 2 2k 2 (4.23)
where m + and m -are the masses of the cations and anions, respectively. This prediction is accurate for k = 256k min , as shown in Fig. 4.9a.

Fig. 4.9b shows the non-normalized zero frequency limit extrapolated from the simulation. This quantity is of great importance regarding the transport electrical properties of the system, as it corresponds to a usual low frequency/long time Green-Kubo integral (see Sec. 2.3). The prediction of this long-time finite size transport coefficient by sDDFT is fairly accurate for high and low wavevectors but again underestimated around the reciprocal correlation length κ. This is also the case of BD simulations compared to MD (the deviation at large wavevector for BD simulations comes from the undersampling of the time signal). We can make the same remarks as for the static correlations: mean-field theory accurately predicts the correlations due to electrostatics but neglects the effect of short-range exclusion due to the fact that ions have a finite size. BD and LD simulations allow us to explore that effect, but only MD takes into account the effect of water on correlations at these wavevectors.

By the observation that deviations in dynamical correlations (Fig. 4.9b) predicted from sDDFT reflect the same deviations as the ones in static correlations (Fig. 4.9a), we have shown an inherent weakness of DDFT formulations at the mean-field level of description [START_REF] Michael Te Vrugt | Classical dynamical density functional theory: from fundamentals to applications[END_REF]. But conversely, we have also gained insights on how to potentially refine it, as we will demonstrate in the next section.

Extensions of DDFT

We recall the general dynamical equation for DDFT (Eq. (3.1)):

∂ρ ∂t = ∇ • βD ρ ∇ δF [ρ] δρ , (4.24)
where we consider a generic density ρ(r, t) with constant mobility/diffusion, and unspecified free energy functional. By doing a linear expansion of that equation close an equilibrium state ρ(r, t) = ρ 0 + δρ(r, t), one arrives at the following equation for the excess density:

∂δρ ∂t = D∆δρ -Dρ 0 ∆ ˆc(2) ( r -r ′ ; ρ 0 ) δρ(r ′ , t) d 3 r ′ , (4.25)
where c (2) (r; ρ 0 ) is the direct pair correlation function for a homogeneous fluid of density ρ 0 , proportional to the second functional derivative of the free energy functional. Rewriting this equation in the reciprocal Fourier space, one obtains

∂δρ(k, t) ∂t = -Dk 2 1 -ρ 0 c (2) (k; ρ 0 ) δρ(k, t) . (4.26)
By invoking the Ornstein-Zernike equation, one identifies the static structure factor: The dashed lines in Fig. 4.9b and Fig. 4.9c show that the prediction for implicit solvent is quantitatively improved using Eq. (4.29). Once static correlations arising from short-range ionic interactions are taken into account, DDFT predictions are in excellent agreement with implicit solvent BD simulations. This is however not the case for explicit solvent simulations. We can explain this by the fact that water not only modifies the static interactions between the ions but also their dynamics: a particle moving in a fluid has an influence, mediated by water, on the motion of neighbouring particles. This is referred to as hydrodynamic effects. Fig. 4.9c shows the unnormalized dynamical charge-charge structure factor for the wavevector k = 4k min and finite frequencies, comparing the numerical simulations with the bare PNP and the modified DDFT approaches. The corrections to DDFT approach using static correlations extracted from BD simulations are again quantitatively well improved, which is not the case of MD, highlighting the presence of hydrodynamic effects.

1 -ρ 0 c (2) (k; ρ 0 ) = 1/S(
     F (k, t) = e -Dk 2 t/S(k) S(k) S(k, ω) = Dk 2 ω 2 + [Dk 2 /S(k)] 2 .
These multiple comparisons conclude that continuous modelling, which did not account for the finite size of the ions nor the hydrodynamic interactions, fails to describe the charge fluctuations at atomic scales, in particular when the wavevector is close to the inverse correlation length. The dynamic "particle" correction to mean-field theory can be extracted using the static structure factor from Brownian dynamics. This is not the case for explicit solvent descriptions, as the water molecules induce supplementary transport phenomena, such as hydrodynamics. In this case, the mobilities of the ions themselves are not trivial anymore. One possible protocol is to extract the relevant properties from MD simulations, namely a good description for the static ionic correlations with the free energy functional: F[ρ] and a proper description of the dynamics with a generalized mobility tensor: µ[ρ, r, t] [START_REF] Donev | Dynamic density functional theory with hydrodynamic interactions and fluctuations[END_REF][START_REF] Michael Te Vrugt | Classical dynamical density functional theory: from fundamentals to applications[END_REF]. The intimate link between structure and dynamics can be also found in the concept of De Gennes narrowing [START_REF] Kellouai | On de gennes narrowing of fluids confined at the molecular scale in nanoporous materials[END_REF][START_REF] Michael Te Vrugt | Classical dynamical density functional theory: from fundamentals to applications[END_REF][START_REF] De Gennes | Liquid dynamics and inelastic scattering of neutrons[END_REF][START_REF] Wu | Atomic dynamics in simple liquid: de gennes narrowing revisited[END_REF], which finds an illustration here in the case of electrolytes.

Driven regime and asymmetric diffusion coefficients

Finally, now that we have gone through the study of charge-charge correlations in the equilibrium case, we turn to the driven regime. While the results and interpretations shown in this section are preliminary, we identify new mechanisms appearing in the nonequilibrium regime for bulk electrolytes. Until now, the general result for the general sDDFT model that was detailed in Chap. 3 has not been fully exploited. We restricted ourselves to equilibrium (no applied field) and equal diffusion coefficients. We now relax some of the restrictions on Eq. (3.39), increasing the complexity and phenomenology in the studied systems. A more complete theoretical analysis can be found in Refs. [START_REF] Mahdisoltani | Transient fluctuation-induced forces in driven electrolytes after an electric field quench[END_REF][START_REF] Palaia | Charging dynamics of electric double layer nanocapacitors in mean-field[END_REF][START_REF] Bonneau | Temporal response of the conductivity of electrolytes[END_REF]. First, we model the electrolyte system as before, but with the addition of a constant external electric field E ext ̸ = 0. The dynamical matrix determining the dynamical structure factors from linearized sDDFT (Eq. (3.38)) reads in reduced units:

ℓ {c,z} = K 2 -iΩ -iK • E -iK • E K 2 + 1 -iΩ . (4.30)
This yields the concentration and charge dynamical structure factors:

         S nn (k, ω, E) = K 2 Dκ 2 Ω 2 + (K 2 + 1) 2 + (K • E) 2 |det(Ω, K, E)| 2 S qq (k, , ω, E) = K 2 Dκ 2 Ω 2 + K 4 + (K • E) 2 |det(Ω, K, E)| 2 , (4.31)
where the denominator is the modulus squared of the determinant of the dynamical matrix:

det(K, Ω, E) = (K 2 -iΩ)(K 2 + 1 -iΩ) + (K • E) 2 (4.32)
Compared with the equilibrium case Eq. (4.1), this equation shows anisotropic structure factors. In the direction orthogonal to the external field: k ⊥ E ext , due to the absence of hydrodynamic interactions or shear-related effects, the previous result remains unchanged, contrary to the direction along the external field. In particular in that direction, Eq. (4.31) shows that the dynamical structure factors are not monotonously decreasing functions of the frequency anymore, but possess a resonance peak located in the frequency

Ω * = EK 4K 2 (E 2 + K 2 ) -1 -K 2 (E 2 + K 2 ) . (4.33)
We compare these predictions to Brownian dynamics simulations in Fig. 4.10, which shows a remarkable agreement with sDDFT predictions Eq. (4.31). On the left panels displaying the dynamical structure factors in the directions perpendicular to the external field, the result remains unchanged compared to equilibrium results Fig. 4.2. On the right panels, the low-frequency plateau and high-frequency decay (∼ ω -2 ), features of diffusion and electrostatic relaxation, are still present, but the values of the plateau are decreased, and a peak close to the transition frequency Eq. ( 4.33) appears. This mechanism can be interpreted as decrease of the low-frequency fluctuations due to the external field because it induces long-range ordering in the system [START_REF] Zorkot | Current fluctuations across a nano-pore[END_REF][START_REF] Mahdisoltani | Transient fluctuation-induced forces in driven electrolytes after an electric field quench[END_REF], while the Debye mode (k = κ, ω = Dκ 2 ), characteristic of the transition between diffusion and electrostatic relaxation (see Sec. Lastly, we relax the assumption of equal diffusion coefficients. While this assumption is common in the literature [START_REF] Zorkot | Current fluctuations across a nano-pore[END_REF][START_REF] Mahdisoltani | Transient fluctuation-induced forces in driven electrolytes after an electric field quench[END_REF][START_REF] Bonneau | Temporal response of the conductivity of electrolytes[END_REF][START_REF] Démery | The conductivity of strong electrolytes from stochastic density functional theory[END_REF], and valid for a limited number of salts such as KCl, the case of unequal diffusion coefficients such as NaCl with D Na + = 1.28 × 10 -9 m 2 /s and D Cl -= 1.77×10 -9 m 2 /s has rarely been explicitly treated. In particular, the difference in mobilities can lead to drastic changes in the temporal response of electrolytes [START_REF] Hashemi | Computing hydrodynamic interactions in confined doubly-periodic geometries in linear time[END_REF]. We assess this effect using the general prediction for the linearized sDDFT model Eq. (3.39) with the explicit diffusion asymmetry factor γ ̸ = 0 that was introduced in Eq. (3.32). While at equilibrium the factor γ, quantifying the deviation from symmetric electrolytes, had only minor effects on the dynamical structure factors, it is not the case for nonequilibrium electrolytes. As shown in Fig. 4.11, the resonance peak previously predicted at the frequency Eq. (4.33), is quantitatively shifted. This result is confirmed by BD simulations of NaCl (γ = -0.161), which are in very good agreement with Eq. (3.39). The two last presented results raise interesting questions and perspectives beyond the present work, in particular in terms of the nonequilibrium response for bulk electrolytes [START_REF] Bonneau | Temporal response of the conductivity of electrolytes[END_REF][START_REF] Mahdisoltani | Long-range fluctuation-induced forces in driven electrolytes[END_REF][START_REF] Palaia | Charging dynamics of electric double layer nanocapacitors in mean-field[END_REF], both the stationary and the transient regime, which could be encoded in the dynamical structure factors. The validity of the sDDFT model should also be carefully studied in these regimes, as well as the nonequilibium thermodynamics of this system. We leave those open questions for future work.

Conclusion

Exploiting stochastic density functional theory, described in Chap. 3, and Brownian simulations of a model binary electrolyte in implicit solvent, detailed in Sec. 2.1.1, we have presented important features of correlations in the charge fluctuations in electrolytes, first at the global scale by observing density fluctuations modes encoded in the charge-charge dynamical structure factor in Sec. 4.1, then at the finite size scale, by studying the fluctuations of total number charges in a given finite volume without explicit boundaries in Sec. 4.2. In both cases, the main observation regarding charge fluctuations is that contrary to number fluctuations, thermal motion couples with electrostatic interactions, inducing electrostatic screening and relaxation for the dynamics of charges. Charge fluctuations are screened at large scales due to the restoring forces induced by Coulomb interactions. This seemingly natural result, which can be derived by linearized sDDFT and confirmed by Brownian dynamics simulations, leads to non-trivial properties such as hyperuniformity and exotic frequency-dependency in the noise spectrum (Sec. 4.2.3 and Sec. 4.2.4). Moreover, we compared these predictions made in the implicit solvent framework, to explicit solvent simulations in Sec. 4.3, yielding both a confirmation of the electrostatic relaxation mechanisms and a regime of validity for implicit solvent models in terms of length and time scales. While meanfield theories are expected to fail at high concentrations, where short-range interactions are largely due to high packing fractions, they can be taken into account using simple modifications of DDFT. The most noticeable weakness of the implicit models used here comes from the solvent description, which underestimates ionic correlations and neglects hydrodynamic effects. With this caveat in mind, understanding bulk electrical fluctuation in electrolytes was an important first step for this thesis. We now go on to study the case of explicitly confined systems.

Chapter 5

Electrical response of confined electrolytes After studying the electrical fluctuations properties of bulk electrolytes, the next step is to address the remaining word in the title of this thesis: confined. The path from bulk to confined system is strewn with pitfalls. As indicated in Sec. 4.2, ionic fluctuations in electrolytes greatly depend on the studied length and time scales.

Recent advances in nanotechnologies demonstrate that properties of fluids can drastically change when confined under sub-micrometre scales [START_REF] Kavokine | Fluids at the nanoscale: From continuum to subcontinuum transport[END_REF]. Reducing the size of a system decreases its volume by a factor cubed, while the surface surrounding this volume only decreases by a factor squared. Interfacial effects prevail in nanotechnologies due to this so-called square-cube law, allowing a variety of designs and applications by tuning the interfacial effects. Conversely, the gain of variety and complexity in those technological applications is equally as high as in the theoretical tools required to model and understand their extent. Models of electrochemical systems are often divided into three parts, the electrodes, the electrolyte, and the interface [START_REF] Chassagne | Compensating for electrode polarization in dielectric spectroscopy studies of colloidal suspensions: Theoretical assessment of existing methods[END_REF][START_REF] Pireddu | Frequency-dependent impedance of nanocapacitors from electrode charge fluctuations as a probe of electrolyte dynamics[END_REF], and yet the separation between these three regions can never be fully disentangled from experimental measurements. The microscopic properties are heavily dependent on the choice of electrode electrolyte pair, nevertheless, experiments such as current measurements in single nanotubes (see Sec. 1.2) reveal universal low-frequency noise for a variety of substrates, indicating that its origins may lie on fundamental physical mechanisms.

To decouple this problem, as a direct following of the study of bulk electrolytes in the previous chapter, we study a generic system composed of an electrolyte confined by two insulating walls (see Fig. 5.1). The system now has a break of symmetry in the direction perpendicular to the plates. We will study the effects of the characteristics of the system on perpendicular electrical conductivity. After detailing methodological points on the modelling confined electrolytes from Brownian dynamics simulations in Sec. 5.1.1 and Sec. 5.1.2, the characterization of the electrical response of confined electrolytes in Sec 5.1.3 and Sec. 5.1.4, and the methodological path to obtain the frequency-dependent conductivity using nonequilibrium simulations, equilibrium simulations, and analytical methods for ideal electrolytes in Sec. 5.1.4 and Sec. 5.1.6, we follow by assessing the information contained in conductivity spectra in the 0.1 to 100 GHz range. In those spectra, we identify microscopic mechanisms in the dynamics of the ions, that are investigated in a gradually complex phenomenology, starting with ideal confined electrolytes (Sec. 5.1.4). After identifying the linear response regime we move on to specific effects, with the confinement distance (Sec. 5.2.1), adsorption (Sec. 5.2.2), and electrostatic relaxation (Sec. 5.2.3). Finally, we go out of the linear response regime and study the effect of the amplitude of the external field for ideal electrolytes (Sec. 5.2.4). Those various effects and their coupling arising at specific scales can be rationalized in terms of time scales, naturally probed by the conductivity spectrum. Similar systems of confined electrolytes have been extensively studied in the topic of nanocapacitors, in particular with the pursuit of understanding charging and discharging processes by modelling the charge relaxation driven by the dynamics of the ions, both with analytical and simulation methods [START_REF] Pireddu | Frequency-dependent impedance of nanocapacitors from electrode charge fluctuations as a probe of electrolyte dynamics[END_REF][START_REF] Bazant | Diffuse-charge dynamics in electrochemical systems[END_REF][START_REF] Janssen | Transient dynamics of electric double layer capacitors: Exact expressions within the Debye-Falkenhagen approximation[END_REF][START_REF] Adelchi | Lattice Boltzmann electrokinetics simulation of nanocapacitors[END_REF][START_REF] Limmer | Charge Fluctuations in Nanoscale Capacitors[END_REF][START_REF] Kondrat | Charging dynamics and optimization of nanoporous supercapacitors[END_REF][START_REF] Palaia | Charging dynamics of electric double layer nanocapacitors in mean-field[END_REF][START_REF] Lobaskin | Diffusive-convective transition in the non-equilibrium charging of an electric double layer[END_REF][START_REF] Netz | Conduction and diffusion in two-dimensional electrolytes[END_REF][START_REF] Jeanmairet | Microscopic Simulations of Electrochemical Double-Layer Capacitors[END_REF].

Modelling the dynamics of confined electrolytes

After detailing modelling methods, in this section we show the benefits of using linear response theory to predict the frequency-dependent conductivity from equilibrium simulations, using the Green-Kubo expression appropriate for overdamped Langevin dynamics [START_REF] Felderhof | Linear response theory of sedimentation and diffusion in a suspension of spherical particles[END_REF][START_REF] Felderhof | Linear response theory of the viscosity of suspensions of spherical brownian particles[END_REF][START_REF] Contreras | A unifying mode-coupling theory for transport properties of electrolyte solutions. I. General scheme and limiting laws[END_REF]. Even though the static limit of the latter has already been used in Brownian dynamics simulations to determine the effective diffusion coefficient or the static conductivity [START_REF] Jardat | Transport coefficients of electrolyte solutions from smart brownian dynamics simulations[END_REF][START_REF] Jardat | Brownian dynamics simulations of electrolyte mixtures: computation of transport coefficients and comparison with an analytical transport theory[END_REF][START_REF] Jardat | Brownian simulations contribution to the study of ionic dynamics in aqueous solutions[END_REF][START_REF] Dahirel | Two-scale Brownian dynamics of suspensions of charged nanoparticles including electrostatic and hydrodynamic interactions[END_REF], the frequency-dependent result (for which we provided a slightly different derivation [START_REF] Hoang | Frequency and field-dependent response of confined electrolytes from Brownian dynamics simulations[END_REF] inspired from Ref. [START_REF] Joubaud | Langevin dynamics with space-time periodic nonequilibrium forcing[END_REF] in Sec. 2.3.3) does not seem to have been much exploited in the literature. The methods shown in this section were developed in collaboration with Gabriel Stoltz.

To investigate the influence of confinement, adsorption on surfaces, and ion-ion interactions on the response of electrolyte solutions to an oscillatory electric field. We consider Brownian particles with the same characteristics as in Sec. 4.1, similar to aqueous NaCl solution, with q Na + = -Cl -= e and D Na + = 1.28×10 -9 m 2 /s and D Cl -= 1.77×10 -9 m 2 /s, both bathing in implicit water of relative permittivity ϵ water = 78.5 at T = 289K and whose ionic interactions are modelled by the force fields detailed in Sec. 2.1.2. This solution is confined between two parallel walls spawning over the simulation box, and separated by distance L in a slab geometry. This system is illustrated in Fig. 5.1. The equilibrated system is submitted to a sinusoidal external electric field in the z direction, starting at t = 0, which takes the form:

E = E(t) êz E(t) = E 0 sin (ωt) H 0 (t) , (5.1)
with the two parameters E 0 its amplitude, and ω its (angular) frequency, and H 0 (t) the Heaviside step function defined as:

H 0 (t) = 0 , t < 0 1 , t > 0 , (5.2)
modelling the start of the external field at t = 0. We define the studied electrical current Eq. (2.69) projected in that direction:

J el (t) = i z i β i D i F i,z (t), (5.3)
which is the sum of Brownian velocities of all particles, weighted by their charge. The relation between J el (t) and E(t) will be the main objective of this chapter.

Modelling solid walls

Solid walls can be described using various models at different levels of description. In line with the description of the solvent, we adopt a continuous coarse-grained picture. Instead of describing the full atomic structure of the interface, we model it using a force field that only depends on the distance between the ion with the walls located at z = ±L/2, and consequently its position z:

U (z) = V w (z + L/2) + V w (z -L/2).
(5.4)

The chosen form arises from the sum of LJ interactions with every atom in the solid wall which are disposed in a closed-packed face-centred cubic lattice, with lattice parameter σ w , cut along a (100) face (see Fig. 5.2). By performing the summation of single particle LJ potentials, which possess the characteristic exponents 12-6 for the attractive and repulsive parts respectively (see Sec. 2.1.2), one gets the wall potential

V ads w (z) = 2 π ϵ w 2 5 σ w z 10 - σ w z 4 - √ 2
3 z/σ w + 0.61/ √ 2 3 . (5.5)
The resulting potential has characteristics exponents 10-4-3, including short-range repulsion with the first term of exponent 10, and an attractive well modelled with the terms of exponents 4 and 3 which leads to adsorption [START_REF] Steele | The physical interaction of gases with crystalline solids: I. gassolid energies and properties of isolated adsorbed atoms[END_REF][START_REF] Steele | The interaction of rare gas atoms with graphitized carbon black[END_REF][START_REF] Magda | Molecular dynamics of narrow, liquid-filled pores[END_REF][START_REF] Magda | Erratum: Molecular dynamics of narrow, liquid-filled pores[END_REF]. The energy ϵ w tunes the strength of the ion-wall attraction. To model purely repulsive walls, we use the same truncation and shifting as for the short-range ion-ion interactions, i.e. :

V rep w (z) = V ads w (z) -V ads w (z * ) , z ≤ z * 0 , z > z * , (5.6)
with z * ≈ 0.987σ w corresponding to the minimum of V ads w . Fig. 5.3 illustrates these ion-wall potentials for the cases considered in the present work. V rep w , βε w = 0.17 V ads w , βε w = 0.17 V ads w , βε w = 0.50

Simulations details

Simulations were carried out using the same procedure as described in Sec. 2.1. The trajectories of the system are generated by simulating the Brownian motion of the ions with implicit solvent (see Sec. 2.1). Embedded in a simulation box of size L 3 box , the system again consists of N pairs of modelled Na + Cl -at concentration C s ∼ N s /(L box × L box × L) (the real value being C s = N/(L box × L box × L), due to an effective confinement distance L that will be discussed in Sec. 5.2.1). The ions are confined in a slab geometry by two walls, of separation distance L, modelled an external potential, defined by Eq. (5.5) and Eq. (5.6) for attractive walls and purely repulsive ones respectively. The parameters for the simulated systems are summarized in Tab. 5.1.

For equilibrium (EQ) simulations, we use a time step δt = 50 fs, except for the electrolyte with the largest salt concentration (C s = 0.8 M) for which we use δt = 25 fs. The total length of the trajectory is T tot = 50 µs, except for the same system (T tot = 25 µs) and in the case of ideal particles with the most attractive walls (T tot = 100 µs). For each simulation, the trajectory is divided into N blocks = 100 blocks (200 for the most attractive walls) on which the properties such as time-correlation functions or their Fourier transforms are computed. The reported results and uncertainties correspond to the average and the 95% confidence interval computed using the standard deviation among blocks, respectively. The length of each block (500 ns for most systems, 250 ns for ideal particles with the most attractive walls) is much longer than all the correlation times in the various systems so that the blocks can be considered statistically independent from each other. For each block, the real part of the frequency-dependent conductivity is computed using a non-uniform sampling of the current ACF and numerical integration of Eq. (2.79) using the trapezoidal rule for the considered frequencies. Specifically, the short-time behaviour of the ACF (for t < 10 ps) is estimated from the first 2000 steps of each block with the current evaluated at every step, while the rest of the ACF is estimated from samples of the current every 100 times steps over the whole block. Nonequilibrium (NEQ) simulations are performed with repulsive walls separated by the distance L = 20 Åwith 200 pairs of particles (R20 Tab. 5.1) and in the presence of a field E(t) = E 0 sin(ωt), using a time step δt = 50 fs, with a total trajectory length T tot = 1 µs. The results presented in Fig. 5.5 are obtained for field magnitudes E 0 ∈ {0.001, 0.01, 0.1} V/Å and frequencies f = ω/2π ∈ {2, 4, 8} × 10 {-1, 0, 1} GHz. While the largest field considered is beyond what could be sustained by real water, it remains interesting to include it in the discussion of the various regimes that may arise upon increasing the external driving. For the results presented in Fig. 5.6, the magnitudes and frequencies are E 0 = 2 p 10 -2 V/Å with p ∈ {-3, -2, ..., 9, 10} and f = ω/2π ∈ {1, 2, 4} × 10 {-1, 0, 1, 2} GHz. In such nonequilibrium simulations, we monitor the nonequilibrium current J el (t) in the stationary regime, which is reached after a transient regime that depends on E 0 and ω but is shorter than 10 periods T = 2π/ω of the field in all cases. We therefore discard the first 10 periods of the signal and the results are averaged over the T tot ω/2π -10 remaining periods (90 periods for the smallest frequency).

The treatment of long-range electrostatic interactions in periodic-periodic-finite systems requires some special attention. The efficient reciprocal space computation using Ewald summation that was described in Sec. 2.1.3, and applied until now for 3D periodic systems modelling bulk electrolytes is only not straightforwardly applicable anymore. While Ewald-type computation methods for electrostatics in slab geometry is an ongoing research topic [START_REF] Tyagi | Electrostatic layer correction with image charges: A linear scaling method to treat slab 2d+h systems with dielectric interfaces[END_REF][START_REF] Barros | Efficient and accurate simulation of dynamic dielectric objects[END_REF][START_REF] Arnold | Efficient Algorithms for Electrostatic Interactions Including Dielectric Contrasts[END_REF][START_REF] Ondrej Maxian | A fast spectral method for electrostatics in doubly periodic slit channels[END_REF], in particular for the treatment of dielectric jumps using image-charge techniques, we neglect those in order to use the PPPM algorithm (Sec. 2.1.3) with a slab correction (of 3 times the cell size in the confinement direction) to approximate finite electrostatics. This method is notably often used in classical molecular dynamics simulations (See Ref. [START_REF] Yeh | Ewald summation for systems with slab geometry[END_REF]). It is known that dielectric junctions between fluids and solids can lead to non-trivial changes in the description of ions close to interfaces [START_REF] Rotenberg | Electrokinetics: insights from simulation on the microscopic scale[END_REF][START_REF] Croxton | Ionic solution near an uncharged surface with image forces[END_REF][START_REF] Levin | Surface tension of strong electrolytes[END_REF][START_REF] Levin | Ions at hydrophobic interfaces[END_REF][START_REF] Hanne | Dielectric Modulation of Ion Transport near Interfaces[END_REF], both at in the static and dynamical picture and even in the absence of surface charges. With the same arguments which were used to neglect electrokinetic effects (Sec. 3.1.4), we neglect the influence of dielectric jumps to restrict the focus on the dynamics driven by ionic interactions and mechanisms in implicit solvent. The additional effects caused by explicit interfacial effects are left for future work. With these caveats in mind, the presented model provides a starting point for a systematic investigation of electrolyte response to oscillatory electric fields.

System

Wall Int. Ionic Int. Table 5.1: Summary tables on simulated systems in this chapter.

Ion pairs N

(real) Salt concentration C s (mol/L) L box (Å) L (Å) R20 Rep. LJ-1043 βϵ w = 0.17
First column is the reference of the system, second the type of wall interaction, LJ-1043 is the bare Steel potential with fixed parameter distance σ w = 3 Å, either in purely repulsive form with V rep w given by Eq. (5.6), or in bare form with V ads w given by Eq. (5.6) including the attractive part. The adsorption potential for the latter is tuned with the energy constant ϵ w . Third column indicates the presence or not of inter-ionic interactions potentials (see Sec 2.2.1), fourth column indicates the total number of ion pairs in the system that are simulated, fifth column shows the corresponding salt concentration (no inter-ionic interactions count as infinite dilution), two last columns indicate the (finite) perpendicular and (periodic) lateral lengths. While applying an oscillating electric field to a conducting material at equilibrium, an electrical current is created usually following these general rules (see Fig. 5.4):

Nonequilibrium electrical current

• For t < 0, the system being at equilibrium, there is no average current: ⟨J el (t < 0)⟩ 0 = 0.

• Shortly after the moment the oscillating external field is turned on at t = 0 + , an electrical current is induced: ⟨J el (t > 0)⟩ E ̸ = 0, which does not exhibit perfect periodic shape. The characteristics of this current depend on the initial state of the system, as this part of the response is not time translational invariant we call this the transient response.

• After a certain amount of time, characteristic of the relaxation time of the system to that particular oscillating field, the system reaches what we call a nonequilibrium steady state (NESS). In this regime, we observe that the electrical current follows the same repeating pattern, at the same periodicity as the electric field T = 2π/ω. The system is time statistically translation invariant (up to a phase that depends on the time origin).

Using the NESS time translation symmetry of the system, we shift the time reference such that we discard the transient regime, in this case, E(t) = E 0 sin (ωt). The electrical current repeated pattern can be averaged over the simulation trajectory to obtain a smooth time signal. We represent in Fig. 5.5 this stationary signal with an in-phase representation to highlight the variations of shape and amplitude, and a current-field relation, which is reminiscent of the potentiodynamic measurement of cyclic voltammetry techniques in electrochemistry, where an electrical current is measured in response of an oscillating applied tension. Fig. 5.5 shows the deviation from the bulk ideal behaviour of confined ideal electrolytes for different fields and frequencies. On the right panels, for every amplitude of the electric field shown, in the high-frequency regime above 10 GHz the current (blue tones), the signals coincide with the ideal bulk current predicting a sine wave with amplitude equal to V σ NE E 0 . By decreasing the frequency, the amplitude of the signal decreases and vanishes for low frequencies (red tones). We can also observe that for high fields (panel a) the current signal is strongly deformed from the sine form. This deformation can also be identified in the voltammograms (current against field) on the right. The ideal bulk response which was a proportionality relation, is not verified anymore. From these observations, one can identify an ideal bulk-like response at high frequencies and an insulating phase at low frequencies. The transitions and shape of the signals feature characteristics of the system and of the amplitude of the external field. In the next section, we will model this transition.

Conductivity under confinement

From Fig. 5.5, we observe that depending on the oscillation frequency and amplitude of the electric field, the current differs from ideal conductivity where:

J el (t) = V σ NE E(t) , (5.7)
with V is the volume of the system, σ NE = βe 2 C s (D Na + + D Cl -) the ideal Nernst-Einstein conductivity of bulk NaCl electrolyte (Eq. (2.71)). This proportionality constant is the charge-weighted sum of the mobility of each ion in the infinite dilution approximation (ideal) and is independent of the characteristics of the applied electric field. For a confined electrolyte under NESS, we can characterize the deviation from that ideal relation using Fourier decomposition. In order to extend this current/field relation to the general case, we define the generalized complex frequency and electric field dependent conductivity σ(E 0 , ω) taking the form of an Ohm's law:

J el (t) V E 0 = Im σ(E 0 , ω) e iωt . (5.8)
Here it is important to notice that we have assumed that the response is composed of a single mode identical to the external field, which was shown not to be the case in Sec. 5.1.3 for high fields, seen in the deformation of the sine signal. We will come back to this point in Sec. 5.2.4. For now, we focus on the fundamental model response that we define as the electrical current Fourier component at the same frequency as the external field:

J ω = 2 T ˆT 0 J el (t) sin ωt dt . (5.9)
Then the Ohm's law for the real electrical conductivity reads

Re [σ(E 0 , ω)] = σ(E 0 , ω) = J ω V E 0 . (5.10)
This formula is used to compute frequency-dependent electrical conductivity from nonequilibrium simulations, using direct current measurements. Coming back to the simulation results from Fig. 5.5, employing the definitions of Eq. ( 5.10), one can observe this general behaviours:

           σ(ω → ∞, E 0 ) = σ NE σ(ω → 0, E 0 ) = 0 σ(ω, E 0 → ∞) = 0 σ(ω, E 0 → 0) = σ NE (5.11)
We have seen that the deviation from the ideal bulk electrolyte behaviour depends on the frequency and the amplitude of the electrical field, where the conductivity goes from the Nernst-Einstein conductivity to zero for decreasing frequency as well as for increasing amplitude. In order to rationalize this conducting to insulating transition, one can propose the following model.

For an ideal electrolyte under an oscillating electric field, there are natural time scales that arise from the slab confinement, the diffusion, the electric field, and its oscillation:

τ diff = L2 π 2 D , τ E = L βeDE 0 , ω -1 .
(5.12)

The first one, τ diff , is the time an ion in the slab confinement takes to diffuse from one wall to the opposite, where L is the effective confinement distance that will be discussed in the Sec. 5.2.1 treating specificities of the walls influence. The two others are induced by the external field: τ E the time an athermal ion takes to migrate over that same distance L, due to a constant external field of amplitude E 0 , accelerating the particle to the constant speed βeDE 0 , and the last one, ω -1 , is the period of the external field. Due to the geometrical slab confinement, one can explain the conducting/insulating phases with the following arguments:

• When an electric field is applied, only the ions that are freely moving in the inter walls region can be set into motion and participate in the electrical conductivity. When all ions are in that situation, the system behaves as if it was not confined, consequently restoreing the full Nernst-Einstein ideal conductivity, σ NE .

• By moving in one direction due to a driving force, an ion can meet the wall, which blocks its motion by exactly compensating the driving force, with the force deriving from potential Eq. 5.4. Once all ions are saturated on the surface of the walls and respectively blocked, there is no conduction anymore.

• The saturation mechanism is only possible if the oscillation of the driving force is slow enough such that the ion has had time to persist in its motion until reaching a wall.

• The driving force can either be migration due to the external field, or thermal diffusion.

From these arguments, one can deduce, with the time scales ω -1 , τ E , τ diff an adimensionalized semi-quantitative law for the conducting insulation behaviour, which takes the following binary form:

σ(ωτ diff ≫ 1 and ωτ E ≫ 1) = σ NE σ(ωτ diff ≪ 1 or ωτ E ≪ 1) = 0 . (5.13)
Furthermore, having 3 independent time scales, we can build another couple of adimensionalized parameters. Here we have chosen:

ωτ diff and τ diff /τ E . σ(ωτ diff ≫ 1 and τ diff /τ E ≪ ωτ diff ) = σ NE σ(ωτ diff ≪ 1 or τ diff /τ E ≫ ωτ diff ) = 0 . (5.14)
A more detailed version of the time-scale composition is given in the nonlinear regime study in Sec. 5.2.4. We report the simulation results from the NEQ simulations in Fig. 5.6, which reveals the excellent agreement of the conducting/insulating model Eq. (5.14) with the Brownian dynamics simulations. The choice of ωτ diff and τ diff /τ E as scaling parameters allows to highlight the appearance of a regime τ diff /τ E ≪ 1 where the conducting law does not depend on the external field anymore but only the frequency ωτ diff . This is a characterization of frequency-dependent linear response, existing for a sufficiently small external field, such that E 0 ≪ π 2 k B T eL , where the motion of the ions is diffusion-dominated. Then in the linear response limit where E 0 → 0 and J el ∝ E 0 , one finds that the conductivity does not depend on E 0 : σ(ω, E 0 → 0) = σ(ω). Furthermore, the conductivity, which is in general a complex quantity, must obey the Kramers-Kronig relation, relating the real and imaginary parts. Therefore it is sufficient to restrain the study to the real part:

σ(ω) = Re [σ(ω)].
Nevertheless, as displayed in Fig. 5.5, for low external fields, the signalto-noise ratio increases significantly, rendering a precise study of that regime difficult to access using conventional NEQ simulations.

Frequency-dependent conductivity from equilibrium simulations

Here we show that noise issues for NEQ simulations can be turned into a powerful tool to compute the frequency-dependent conductivity, by harnessing the information contained in those fluctuations. We previously saw that the motion of the ions can be induced by either external fields or thermal fluctuations, and following Onsager's regression hypothesis, the system exhibits the same response to one or the other in the linear regime approximation. Taking advantage of that, one can exploit the Green-Kubo formula Eq. (2.79) derived in Sec. 2.3.3, where equilibrium thermal electrical current fluctuations are related to electrical conductivity. Furthermore, to validate this approach, we again focus on the case of ideal electrolytes. While this model is rather crude for real electrolytes, apart from the infinite dilution regime, this treatment will also serve as a tool to validate the method of Brownian dynamics simulations and of the Green-Kubo formula Eq. (2.79). Indeed, for the case of ideal confined particles, we present a way to directly (semi) analytically compute the stochastic dynamics by solving the (1-body) Fokker-Planck equation Eq. (2.25) through spectral propagation, giving direct access to the equilibrium electrical current fluctuation spectra appearing in Eq. (2.79). This method was developed with the helpful collaboration of Gabriel Stoltz.

For the case of ideal particles, hence independent, at equilibrium, the Fokker-Planck operator (see Eq. (2.24) in Sec. 2.2.1) is a sum of identical one-particle operators:

L 0 eq = i L 0 eq,i (5.15)
with L 0 eq,i only acting on functions of particle (i) as

L 0 eq . = D ∂ 2 z . -β(∂ z U (z))∂ z . , (5.16)
with D the diffusion coefficient of the ion (taken as averaged between anion and cations), β the inverse temperature, and U (z) the potential describing the walls (Eq. (5.4)). This operator only acts on functions of the position z due to the confinement (See Fig. 5.1), and is endowed with Neumann non-flux boundary conditions at the walls

∂ z f (±L/2) = 0.
(5.17)

The 2N-particle density can be factorized in a product of 1-particle density written ρ i (z i , t) and ions individually create 1-particle microscopic current J 0 el = -βDqU ′ (z) which are statistically independent. Therefore in this section, following Sec. 2.2.1, we will establish a semi-analytical resolution of the 1-particle FP equation

∂ t -L 0 eq • p(z, t) = 0 , (5.18)
for the probability density function p(z, t), in order to compute the spectral density of current appearing in the GK formula Eq. (2.79)

ˆ∞ 0 J 0 el (0) J 0 el (t) 0 e -iωt dt = ˆ∞ 0 J 0 el , e tL 0 eq J 0 el 0 e -iωt dt, (5.19)
where the current autocorrelation function is rewritten as a scalar product, under the Boltzmann metric, between the 1-particle current operator and its time-advanced counterpart. Explicitly:

⟨f (0) g(t)⟩ 0 = ⟨f, g⟩ 0 = ˆL/2 -L/2 f (z)g(z)p B (z) dz , (5.20)
where we recall the 1-particle equilibrium Boltzmann distribution (see Eq. (2.29) in Sec 2.2.1)

p B (z) = e -βU (z) ´L/2 -L/2 e -βU (z) dz .
(5.21)

In the Fokker-Planck equation (Eq. (5.18)), the linear Fokker-Planck operator (Eq. (5.16)) plays the role of propagating the system in time, and in the forward time signature convention is negative definite and symmetric with respect to the scalar product ⟨•, •⟩ 0 , therefore it can be diagonalized in an orthogonal basis set,

L 0 eq = - α |ϕ α ⟩ λ α ⟨ϕ α | , (5.22)
with {λ α , |ϕ α ⟩} α solutions of :

L 0 eq ϕ α = -λ α ϕ α , ⟨ϕ α , ϕ β ⟩ 0 = δ α,β . (5.23)
Using this decomposition, the spectral density can be decomposed as

ˆ∞ 0 J 0 el (0) J 0 el (t) e -iωt dt = α J 0 el , ϕ α 0 2 iω -λ α (5.24)
This formula allows us to explicitly compute the complex current spectrum as a discrete sum using numerical diagonalization of L 0 eq . But due to the divergent nature of the potential, the Boltzmann metric vanishes exponentially fast at the boundaries, so a direct numerical computation is incompatible with Neumann boundary conditions and prone to instabilities. We circumvent this issue by applying the following change of variable:

ϕ α (z) = 1 p B (z) ψ α (z) (5.25)
then by inserting Eq. (5.25) in Eq. (5. [START_REF] Hoogerheide | Probing surface charge fluctuations with solid-state nanopores[END_REF]) one shows that {λ α , |ψ α ⟩} α obey of the stationary Shrödinger equation, and form a normalized orthogonal basis of solutions under the flat functional metric:

Hψ α = λ α ψ α , ˆ+L/2 -L/2 ψ α (z)ψ * β (z) dz = δ α,β . (5.26)
for a Hamiltonian operator H with an effective potential:

H = D -∂ 2 z + βW , W (z) = β 4 U ′ (z) 2 - 1 2 U ′′ (z) (5.27)
Under this transformation, Neumann boundary conditions on ϕ α (z) translate to Robin boundary conditions for ψ α (z):

∂ z ψ α (±L/2) + ψ α (±L/2)βU ′ (±L/2)/2 = 0 , (5.28)
which simplify to Dirichlet boundary conditions for singular potentials:

ψ α (±L/2) = 0 . (5.29)
Numerically, to avoid instabilities in the dynamics due to the diverging potential at the boundaries, we consider the problem on a smaller closed interval [-L/2 + 0.7σ w , L/2 -0.7σ w ], divided in P + 2 equally spaced lattice points z i of spacing h = (L -1.4σ w )/(P + 1) where the eigenfunctions are set to zero at the extremal points. The P interior points are sought by solving the eigenvalue problem on the lattice. In this numerical scheme, H is represented by a P × P tridiagonal matrix using a central finite difference scheme for the one-dimensional Laplacian:

H i,j = -D δ i+1,j -2δ i,j + δ i-1,j h 2 + βDW (z i ) δ i,j , (5.30)
where δ i,j is the Kronecker function representing the identity operator. This symmetric diagonal matrix is numerically diagonalized using standard NumPy linear algebra library. The output is a finite set λ α , ψ i α 0≤α≤P -1 ordered in increasing eigenvalues and corresponding normalized orthogonal eigenfunctions. P = 3000 is chosen such that the difference between P and P + 1 lattice points is smaller than 1% over the whole spectrum, which is estimated around P ≥ 2000.

Using the diagonalization of the FP operator, we can efficiently and continuously propagate in time any arbitrary initial density p i (t = 0) on the lattice domain without the need for a numerical integration, by expressing the discrete density as p i (t) = P -1 α=0 e -tλα+βU (z i )/2 ψ i α h

P -1 j=0 ψ j * α e βU (z j )/2 p j (t = 0) . (5.31) This combination of spectral methods and finite difference has relatively strong stability given enough lattice points as shown in Fig. 5.7. Situations ranging from close to very far from equilibrium all converge to the Boltzmann probability density for the given wall potential. Remarkably, even the cases of discontinuous initial conditions are regularized. In addition, highly unstable situations where the initial density does vanish not close to the interface are rapidly expelled by the wall, due to incompatibility with the Boltzmann measure, can be computed with high time resolution. The high time resolution/stability of this method is equivalently what allows us to accurately compute the current fluctuation spectrum, which spans over a wide range of frequencies. The computation of the conductivity spectrum ranging from MHz to THz is done instantly (few seconds) compared to propagating the real dynamics with Brownian dynamics simulations. Without any further convergence study, we assume that the frequency-dependent conductivity for ideal electrolyte under linear response is accurately computed, and will serve as a reference for the particle simulation studies.

Comparison of methods

Now we have three methods to compute the linear frequency-dependent conductivity of ideal electrolytes in our hands:

• Nonequilibrium simulations using direct current measurement (Eq. (5.10)):

σ(E 0 , ω) = J ω V E 0
• Equilibrium simulations using the frequency-dependent Green-Kubo formula (Eq. (2.79)):

σ(ω) = σ NE - βe 2 V ˆ∞ 0 ⟨J el (0) J el (t)⟩ 0 cos ωt dt
• Semi-analytical, also using the same Green-Kubo formula, but with the diagonalization of the Fokker-Planck evolution operator (Eq. (5.24)):

σ(ω) = σ NE - α J 0 el , ϕ α 0 2 λ α ω 2 + λ 2 α
Here we compare the three different methods and explain the advantages and disadvantages of each of them. Results are shown in Fig. 5.8. Panel (a) compares the real part of the frequency-dependent conductivity. First, from the discrete points (coloured and linked for a given electric field amplitude) indicating the spectrum computed from NEQ simulations (corresponding to horizontal slices of Fig. 5.6), one confirms the existence of the linear response regime. For decreasing amplitude of electric fields, the different spectra converge for τ diff /τ E = L e E 0 π 2 k B T ≪ 1 to a fixed curve, underlining a response that is independent of the electric field amplitude. But one also observes a significant increase in uncertainty. EQ results are in excellent agreement with the limit suggested by the NEQ method. Finally, the exact solution for the diagonalization of the Fokker-Planck operator confirms the two latter results for the conductivity spectrum at the linear regime. Panel (b) also demonstrates the perfect agreement between equilibrium Brownian dynamics simulations and analytical results on the real and imaginary parts of the complex conductivity Eq. (5.8). Lastly, one also observes that the insulating-to-conducting transition can be unambiguously located at the frequency ω = 1/τ diff from the maximum of the imaginary part. In principle both real and imaginary parts contain the same information about the linear response of the system, due to Kramer-Kronig relations, the latter is more convenient to extract the transition frequency which coincides with the maximum of the imaginary part. We now defined the transition frequency as:

ω * = arg max ω Im[σ(ω)] .
( 5.32) This quantity will serve as a precise characterization value for the conductivity spectrum in the next sections. As we will see, it reflects many characteristics of the system. The NEQ simulation method, being straightforward, allows to obtain the frequencydependent conductivity for any external field, but requires a full simulation for each field. With nonequilibrium current measurement, the signal-to-noise ratio diminishes with the amplitude of the field, consequently, the uncertainty increases with decreasing field, making linear response hardly accessible. Furthermore, one needs to estimate beforehand the duration of the transient regime to establish the steady state. These points make the NEQ approach not the most suitable method to study linear response in our case.

The EQ simulation method, while being numerically costly, due to the slow convergence of low-frequency phenomena, offers many advantages. By exploiting the fact that thermal fluctuations possess a flat white spectrum, the Green-Kubo formula allows us to compute the conductivity for the entire spectrum from a single (in principle infinitely long) numerical simulation. In practice, high and medium frequencies are well converged and determined with low variance, but the low frequencies are limited by the length of the trajectory.

Computing directly the conductivity spectrum using Eq. (5.24) and the Green-Kubo formula Eq. (2.79), is by far the fastest method. But while being formally exact for ideal electrolytes, this semi-analytical method is hardly extendable to real electrolytes due to ionic interactions, where the Fokker-Planck equation for the 2N -density distribution cannot be factorized. This method can only be applied to ideal electrolytes and will serve as a reference for EQ simulations.

Interpreting conductivity spectra

We previously saw that confined electrolyte conduction goes from zero at low frequencies to ideal ionic conductivity at high frequencies, we now focus on specific mechanisms that contribute to the motion of ions that are reflected in this transition. More precisely, Sec. 5.2.1 discuss the effects of confinement under purely repulsive walls given by Eq. (5.6), Sec. 5.2.2 that of adsorption with walls modelled by the Steele potential Eq. (5.5). While these two sections are dedicated to confined ideal electrolytes, aiming only at confinement effects, Sec. 5.2.3 treats the case of real electrolytes modelled with interionic interactions (Sec. 2.1.2), and in particular electrostatic relaxation effects. Finally in Sec. 5.2.4 we step back to ideal electrolytes to assess the effects of nonlinear couplings with the external field.

Ideal electrolytes: effect of confinement

Now we focus on one parameter of the model: the confinement distance L. Using equilibrium simulation and FP resolution methods (Sec. 5.1.5), we compare the conductivity spectra of ideal electrolytes using the same repulsive wall potential (R20, R16, and R12, see Tab. The equilibrium density profiles obtained in simulations (see Fig. 5.9a) follow the expected Boltzmann distribution Eq. (5.21) corresponding to the confining potential associated with the repulsive wall potentials V rep w (z) Eq. (5.6). For such hard repulsive walls, the density is mostly flat between the walls with a constant value ρ b , and rapidly decays to zero close to the interface within a characteristic distance of σ w . To define an accurate confining distance, one has to infer the position of an equivalent sharp interface. That interface can be identified as the Gibbs Dividing Surface (GDS), whose position z ± GDS is defined as:

z - GDS : ˆz- GDS -L/2 ρ(z) dz = ˆ0 z - GDS [ρ b -ρ(z)] dz (5.33)
at the left interface, and:

z + GDS : ˆz+ GDS 0 [ρ b -ρ(z)] dz = ˆ+L/2 z + GDS ρ(z) dz (5.34)
for the right interface. In this case, the GDS is equivalent to equating the depleted and excess density at a smooth interface, and can be understood as a geometrical construction as illustrated in Fig. 5.10. Then the effective confining distance is simply the separation between the two GDS:

L = z + GDS -z - GDS . (5.35)
In practice, we find that the position of the GDS for repulsive walls (Eq. (5.6)) of parameters σ w and βϵ w = 0.17, is located at z ± GDS ∼ ±L/2 ∓ 0.85σ w , so that L ∼ L -1.7σ w . (see Fig. 5.9) for the crossover frequency.

System L (Å) L (Å) τ ′-1 diff = π 2 D/L 2 (GHz) τ -1 diff = π 2 D/
Results for the complex frequency-dependent conductivity σ(ω) (Eq. (5.8)) are shown in Fig. 5.9b. We observe that for decreasing confining distances L, the transition frequency increases. This is both seen in the real and imaginary part of the conductivity from simulation results and is confirmed with the analytical predictions. The precise value of the transition frequency defined as ω * (Eq.(5.32)) is detailed in Tab. 5.2. While this trend is correctly captured by the naive inverse time pi 2 D/L 2 , Tab. 5.2 and Fig. 5.9c show that the exact transition is given by the effective confining distance π 2 D/ L2 . The rescaling of the frequency by the latter collapses the spectra to a signal transition function, while the naive one shown in the inset panel does not. From these, we can deduce that for separation distances much greater than atomic distances, the difference between τ ′ diff and τ diff would be only marginal, but for nanometric confinement, this correction can be substantial. Consequently, this implies that the frequency-dependent conductivity can serve as a probe of both the dynamics of the ions and the properties of interfacial confinement. We will see that this extends to other interfacial mechanisms such as adsorption in Sec. 5.2.2.

Ideal electrolytes: effect of adsorption

As suggested by the dependency of pink noise observed at low frequencies in single nanotubes [START_REF] Secchi | Scaling behavior for ionic transport and its fluctuations in individual carbon nanotubes[END_REF][START_REF] Stuart F Knowles | Current fluctuations in nanopores reveal the polymer-wall adsorption potential[END_REF][START_REF] Heerema | 1/f noise in graphene nanopores[END_REF][START_REF] Hoogerheide | Probing surface charge fluctuations with solid-state nanopores[END_REF] (see Sec. 1.2), surface properties may play a significant role in the electrical current fluctuations. In particular surface adsorption in nanopores, which is linked to surface charge, leads to trapping mechanisms that were described with stochastic processes in the literature [START_REF] Allen | Intermittent permeation of cylindrical nanopores by water[END_REF][START_REF] Piasecki | Kinetic models of ion transport through a nanopore[END_REF][START_REF] Levitz | Random flights in confining interfacial systems[END_REF], and recently in Ref. [START_REF] Gravelle | Adsorption Kinetics in Open Nanopores as a Source of Low-Frequency Noise[END_REF][START_REF] Sf Knowles | Noise properties of rectifying and non-rectifying nanopores[END_REF][START_REF] Stuart F Knowles | Current fluctuations in nanopores reveal the polymer-wall adsorption potential[END_REF][START_REF] Levitz | Molecular intermittent dynamics of interfacial water: probing adsorption and bulk confinement[END_REF] and very recently in Ref. [START_REF] Robin | Disentangling 1/f noise from confined ion dynamics[END_REF]. For us, in order to investigate adsorption effects from the point of view of the dynamics of the ions in the described slab geometry, we model the ions-surface interaction by considering the three different potentials illustrated in Fig. 5.3, for a fixed distance between the walls L = 20 Å: repulsive with βϵ w = 0.17, attractive with βϵ w = 0.17 and attractive with βϵ w = 0.50 (see Eqs.(5.5) and (5.6)). The first case corresponds to the one discussed in previous sections Sec. 5.1.4 and Sec. 5.2.1 and serves as a reference. Fig. 5.11a shows the normalized equilibrium density profiles, which follow the Boltzmann distribution Eq. (5.21), again as expected for ideal particles. Upon introducing short-range attraction to the wall, one observes the desired adsorption of particles with peaks in the density near the surfaces, which grow with increasing βϵ w . Fig. 5.11b shows the position of the corresponding GDS, which is not sufficient to describe the conductivity spectrum.

System

Wall Int.

L (Å) τ -1 diff = π 2 D L2 (GHz) Measured ω * (GHz)

R20

Repulsive βϵ w = 0.17 The simulation results for the frequency-dependent conductivity, shown in Fig 5 .11c and Tab. 5.3, are in good agreement with the spectral method. We observe that the conducting/insulating transition is modified. Contrary to the purely repulsive wall case, the determination of L and τ diff is not sufficient to characterize the crossover frequency, as shown in Tab. 5.3. When the confining potential is not purely repulsive there can be different time scales involved other than the diffusing time from one wall to the other. From the inset of Fig. 5.11c, one can clearly identify two peaks in the imaginary part, corresponding to two distinct frequencies. From Fig. 5.11a, one could interpret these two crossover frequencies with two diffusion processes, one at high frequencies where an ion diffuses in a well close to the wall (surface diffusion), the other at low frequencies, as the diffusion process moving an ion from one well to the other (surface hopping). This means that an intermittent process, such as adsorption, can create low-frequency signatures in the conductivity spectrum. This qualitative assessment could be quantitatively explored by studying the related Kramer's escape problem. This part illustrates again how frequencydependent conductivity can be used as a probe for interfacial mechanisms if proper models are established. V rep w , β w = 0.17 V ads w , β w = 0.17 V rep w , βε w = 0.17 V ads w , βε w = 0.17 , which depend on the potential used and serve as rescaling frequencies for each system respectively [START_REF] Hoang | Frequency and field-dependent response of confined electrolytes from Brownian dynamics simulations[END_REF].

Electrostatic interactions between ions

In the previous section, we limited ourselves to the oversimplified model of ideal electrolytes, in order to assess separately the effects of confinement and adsorption. We now turn to the more realistic case where ions interact between themselves by short-range repulsion and long-range electrostatics (see Sec. 2.1.2). In the previous chapter, we saw that electrostatic interactions play a substantial role in the charge fluctuations and electrical conductivity of bulk electrolytes. The main dynamical mechanism for equilibrium fluctuation dynamics being electrostatic screening, we recall the associated arising length and time scales (see Sec. 4.1 for 1:1 electrolyte assuming identical diffusion coefficients)

τ Debye = λ 2 D /D λ D = (8πl B C s ) -1/2 .
These quantities will serve as guides to follow the complex interplay between confinement and electrostatic relaxation. For the case of interacting ions, there is no direct way of diagonalizing the 2N -Fokker-Planck operator to compute the conductivity spectrum. We therefore only present results from equilibrium simulations. We start with the asymptotic limit L ≫ λ D , i.e. the situation where the confinement length is much larger than the Debye length. .36) This means that the electrostatic relaxation occurs at a much shorter timescale than diffusion over the confining distance. While the shape of the conducting/insulating transition looks qualitatively the same as in the case of ideal electrolytes, the crossover frequency has a different microscopic origin. Indeed, it is not the inverse diffusion time anymore (see inset of Fig. 5.12 ) but the inverse Debye time:

τ diff = L2 π 2 D ≫ τ Debye = λ 2 D D . ( 5
ω * = τ -1
Debye .

(5.37)

Electrostatic relaxation, which dominates at shorter times, suppresses slower charge fluctuations over scales larger than the Debye length as described in Sec. 4.1 and Sec. 4.2 (kills the fluctuations before reaching confinement regimes). Now that we have quantitatively studied the asymptotic regimes λ D ≪ L and the non-interacting confined case (Sec. 5.2.1), which was actually equivalent to λ D ≫ L (non-interacting ions corresponds to the infinite dilution case and infinite Debye length (Eq.(5.2.3))), which are electrostatic and confined dominated respectively, we consider the intermediate case. Insights from the last sections allow us to assert the phenomenology for the electrostatic/confinement competition in terms of time scale comparison. We, therefore, consider the ratio

τ Debye τ diff = πλ D L , (5.38)
to discuss the response of the systems with various confining length L and salt concentration C s . Fig. 5.13 shows the continuous evolution of the frequency-dependent conductivity spectrum with the dimensionless parameter πλ D / L. Various values of (C s , L) results in the similar frequency-dependency shape but shifted. The collapse of the curves in the thin double layer regime πλ D / L ≪ 1 (blue) under the Debye time rescaling, shows again that the system is dominated by electrostatic relaxation. The increase of that ratio monotonously shifts the curve away from the electrostatic transition, back to the ideal confined regime where πλ D / L ≫ 1 (red). The precise value of the transition frequency ω * defined by Eq. (5.32) is reported in Fig. 5.14. which receives it name from framework of equivalent circuits [START_REF] Chassagne | Compensating for electrode polarization in dielectric spectroscopy studies of colloidal suspensions: Theoretical assessment of existing methods[END_REF][START_REF] Pireddu | Frequency-dependent impedance of nanocapacitors from electrode charge fluctuations as a probe of electrolyte dynamics[END_REF][START_REF] Palaia | Charging dynamics of electric double layer nanocapacitors in mean-field[END_REF]. Here, we do not consider the response of an electrolyte confined between metallic electrodes to a voltage step, but rather the response to an applied electric field, which corresponds to boundary conditions of fixed surface charge rather than surface potential. In this work restricted to the linear response regime, the conductivity corresponds to the fluctuations of the current under conditions of vanishing applied field, hence vanishing surface charge. This situation can be recovered in Ref. [START_REF] Bazant | Diffuse-charge dynamics in electrochemical systems[END_REF] by considering the limit δ → ∞ in their result Eq. ( 50) for the time scale characterizing the total charge in a half-cell. Indeed, while the effective thickness λ S introduced in the electrostatic boundary conditions (Eq. ( 14) in Ref. [START_REF] Bazant | Diffuse-charge dynamics in electrochemical systems[END_REF]) has a different interpretation in their work, the limit λ S → ∞ amounts to imposing a zero electric field at the surface of the solid. Using the notation of the present work one obtains in this limit:

τ Bazant τ Debye = 1 - L 4λ D coth L 4λ D sech L 2λ D . (5.40)
The solid line in Fig. 5.14 corresponds to the assumption ω * = 1/τ Bazant , which is in excellent agreement with the simulation results. One recovers in particular the two limiting regimes, τ Bazant ∼ τ Debye and τ Bazant ∼ 48 5π 2 τ diff ≈ 0.97τ diff for small and large values of πλ D / L, respectively. In the intermediate region, τ Bazant is close to τ RC , but the corresponding range of πλ D / L is rather limited. The time scale in Eq. (5.40) corresponds to the dynamics of the total charge of the half-cell, but other time scales could be determined in Ref. [START_REF] Bazant | Diffuse-charge dynamics in electrochemical systems[END_REF], such as for the local charge density of the electrolyte at the surface of the electrodes. The good agreement between our results for the crossover frequency and the former time (instead of the latter or other measures of the charge dynamics) suggests that the evolution of the total charge of a half-pore better reflects the slower fluctuations of the total ionic current.

To conclude this section, the shape of σ(ω), which is not exactly the same for all systems, can be understood through the functional form of the current ACF (see Eq. (2.79)): a mono-exponential decay results in a Lorentzian function for the frequency-dependent conductivity, but the current ACF may display more complex features, such as several exponential modes, meaning different time scales with different weights. This point was implicitly hinted by the spectral decomposition Eq. (5.24). The dimensional analysis would allow an infinite number of time scales of the form

τ ∝ λ u D Lv D , with u + v = 2 , (5.41)
also reflecting the various ones explored in Sec. 4.2.4. A full analysis from the Poisson-Nernst-Planck model for charging dynamics in electrical double-layer capacitors is found in Ref. [START_REF] Palaia | Charging dynamics of electric double layer nanocapacitors in mean-field[END_REF], which also discusses the case of binary electrolytes with unequal valencies and diffusion coefficients, leading to a rich phenomenology. Another somewhat more surprising observation is that the simulation results are well described by the mean-field PNP equation even for concentrations as high as 0.8 M. At such concentrations, this theory is not expected to be accurate, for example, the Debye screening length becomes comparable to the size of ions, and the influence of hydrodynamic effects (see underscreening in Sec. 1.3.3 and effects of hydrodynamics in Sec. 3.1.4 and Sec. 4.3), which questions the validity of continuous descriptions. Nevertheless, the above discussion further supports the relevance of the present approach to predict the frequency-dependent conductivity from current fluctuations in Brownian dynamics simulation.

Ideal electrolytes: effect of confinement in the nonlinear regime

We now come back to the current/field relation Eq. (5.8). This equation assumes that the current created has the same sinusoidal form as the external field. But the electrical response to a large monochromatic external field is actually multimodal in the general case, meaning that the current signal can be decomposed into a sum of modes:

J el (t) = Im[ n J νn e iνnt ] , (5.42)
where ν n = nω, with ω the fundamental frequency, that is equal to the forcing frequency.

In that case, the study of only J ω Eq. (5.9) is not sufficient to characterize the electrical response. One can instead study every mode present in the electrical current by analysing their power spectral density, which we define here as:

S [J el ] (ω ′ ) = ˆR J 2 ν δ(ν -ω ′ ) dν = ˆR 2 T ˆT 0 J el (t) sin νt dt 2 δ(ν -ω ′ ) dν .
(5.43)

In the following, we restrict this study of the nonlinear response to the simpler case of ideal electrolytes. ). When the frequency of the external field is high (blue), the power spectral density (PSD, Eq. (5.43)) displays a single peak at the same frequency. Upon decreasing the frequency (green and red), the PSD also displays peaks at harmonics of the fundamental frequency which is that of the applied field.

Fig. 5.15 reveals the harmonic generation for a fixed large field τ E ≫ τ diff . At high frequencies where ωτ E ≫ 1, one finds a single mode at ω ′ = ω, synonymous with a single sinusoidal component at the same frequency as the forcing. For a lower frequency forcing, one observes the appearance of regular equidistant peaks (shown in log scale) from ω ′ = ω with decreasing amplitude, characteristic of a harmonic generation: the current is a sum of different modes. One could continue that study using the power spectral distribution, but instead of considering the full harmonic generation, we restrict our analysis to the following characteristic physical quantities: 5.44) which are respectively the maximal current value, the Root Mean Square current (equivalently the amplitude of a constant signal which would dissipate the same power under resistive load), and lastly the Fourier component of the signal that is coinciding with the frequency of the external field (the one studied in the previous sections for linear response). These three characteristic quantities for electrical current respectively encoded useful information for signal detection, power application, and reactivity under a given sinusoidal drive. In the previously studied cases of electrical response at the linear regime, the three characteristics would be encoded in the single frequency-dependent electrical conductivity.

                 J max = max 0<t<T J el (t) J rms = 2 J 2 el (t) = 2 T ˆT 0 J 2 el (t) dt 1/2 J ω = 2 T ˆT 0 J el (t) sin(ωt) dt (
Indeed, at small fields, a generated sinusoidal current would verify the relation

J max = J rms = J ω = V σ NE E 0 . (5.45)
Deviation from this relation characterizes the nonlinear regime. We propose a saturation model to predict these values. In Fig. 5.5a, we observed that the electrical current, at high field and low frequencies, starts by following the ideal current, then saturates after a certain time and returns to zero. Then, on the opposite phase of the field, the same phenomenon appears when the external field changes sign. We model that signal by the following function defined on a half period:

J el (t) =    J max sin π 2t max t , t ∈ [0, 2t max ] , 0, t ∈ [2t max , T /2] , (5.46)
(and inversely in the second part of the period). This model relies on a value t max , which corresponds to the time the current reaches its maximum value. By the same arguments detailed in Sec. 5.1.4, it corresponds to the time ions take to move from one wall to the other. Under the approximation of high field, we neglect thermal effects, and compute τ max using a model for the position of the ion:

   dz dt = βDe E 0 sin ωt z(t = 0) = -L/2 . ( 5.47)
Then t max is the time such that z(t max ) = L/2. By integrating the equation of motion, one finds:

t max = arccos (1 -ωτ E ) ω , with τ E = L βDeE 0 (5.48)
under the condition ωτ E < 1, which coincides with the nonlinear phenomenologically explained in Sec. 5.1.4. It then follows

J max = V σ NE E 0 sin(ωt max ) . (5.49)
Then, we compute J max , J RMS and J ω in Eq. (5.44) by inserting Eq. (5.48) and Eq. (5.49) into the modelled signal Eq. (5.46). As final result, the model predicts the ideal conductivity Eq. (5.45) when E 0 ≤ k B T ω L De , and when Without going into a detailed study, this model of overdamped athermal ion can be extended to various electric field signals. For example, instead of a sinusoidal one, one could consider the response to an arbitrary signal E(t), and the model is generalized as:

E 0 > k B T ω L De :                    J max = V σ NE E 0 1 -(1 -ωτ E ) 2 J rms = J max 2 π arccos (1 -ωτ E ) J ω = J max 2 (1 -ωτ E ) 2 -(1 -ωτ E ) 4 arccos (1 -ωτ E ) (π/2) 2 -arccos 2 (1 -ωτ E ) . ( 5
J el (t) ∼ V σ NE E(t) , for t < t max with t max , such that ˆtmax 0 βDeE(t) dt = L (5.51)
where the underlying process is the saturation of the polarization of the electrolytes due to the confinement by the walls. While this model doesn't predict the exact shape of the current, it precisely estimates the value of t max , when the electrical current deviates from the ideal unconfined one.

Conclusion

In this last chapter, following the study of bulk electrolytes, we investigated the electrical response of confined electrolytes. In particular, we focused on electrolyte systems under various oscillating electric fields and perpendicular blocking confinements. Brownian dynamics simulations, coupled with descriptions of the electrical response, allowed us to extract the corresponding field-and frequency-dependent electrical conductivity. We demonstrated in Sec. 5.1.6 that the far from equilibrium regime i.e. with high amplitude of the external field, was well described using nonequilibrium simulations techniques, but the linear response regime was more efficiently studied with the adapted Green-Kubo formula Eq. (2.79). The simulation results were compared to relatively simple analytical models, yielding excellent agreements and allowing efficient phenomenological interpretations for the microscopic mechanisms, where the frequency-dependent conductivity always decays from bulk-like behaviour at high frequencies to vanishing conductivity at low frequencies, similar to capacitive systems. To understand the profound link between macroscopic electrical response and the microscopic dynamics of the charge carriers, we have decomposed the problem by addressing chosen physical mechanisms individually. Starting with ideal electrolytes, where we have studied the effects of the modelled wall confinement and the adsorption mechanisms, in Sec. 5.2.1 and Sec. 5.2.2 respectively. We followed by investigating the effects of coulombic interactions leading to electrostatic relaxation in Sec. 5.2.3, and finally nonlinear processes in confined ideal electrolytes under high electric fields in Sec. 5.2.4. While we did not address those effects coupled altogether, we quantitatively examined them through the scope of dynamical timescales, that are naturally probed by the conductivity spectrum. The nonlinear competition between migration and diffusion was addressed in Sec. 5.2.4 and the competition between confined diffusion and electrostatic relaxation in Sec. 5.2.3. The latter also confirmed linearized mean-field predictions from Ref. [START_REF] Bazant | Diffuse-charge dynamics in electrochemical systems[END_REF][START_REF] Palaia | Charging dynamics of electric double layer nanocapacitors in mean-field[END_REF], which were developed in other contexts. In conclusion, while the considered models suffer from a lack of important features, namely hydrodynamic effects and interfacial effects related to dielectric jump and polarization (see Sec. 5.1.2), the present work constitutes a first basis for the systematic investigation of electrical fluctuations in confined electrolytes and their transport properties, by analysing the frequency-dependent response in Brownian dynamics simulations.

General conclusion

The work presented in this manuscript had two complementary aims: firstly, to model ionic current fluctuations in nanopores, and secondly, to relate microscopic and macroscopic transport in electrolytes, by exploiting the intimate link between ionic transport and electrical fluctuations and shedding light on the relevant mechanisms. The first challenging task, due to the various time and length scales involved, was addressed by combining Brownian dynamics simulations, which allow for the exploration of longer times and larger system sizes than conventional explicit solvent molecular dynamics while incorporating the effect of thermal fluctuations into the dynamics of the ions, together with mesoscopic analytical theories at the mean-field level of description. The second aim was achieved by developing various methodological tools such as analytical models and response theory, whose scope outranges the phenomena studied in this thesis. Specifically:

• By gathering previously known results in the fields of electrolyte solutions, equilibrium statistical physics, response theory, and electrostatics of continuous media, we provided a unified framework for electrical response in electrolytes in Sec. 2.3.1. We related Maxwell constitutive relations in terms of generalized plane-wave linear response conductivity, to the charge-charge dynamical structure factor. The latter, while being often indirectly probed by experiments, is directly accessible from microscopic simulations.

• In the framework of implicit solvent models with overdamped Langevin dynamics, we tailored Green-Kubo formulae for the electrical conductivity in Sec. 2.3. With their emphasis on frequency-dependence, these results serve not only as modelling tools but also as a way to surgically probe the microscopic mechanisms, by exploiting the fact that correlations occur at specific time and length scales and are encoded in the electrical fluctuations.

• We developed numerical tools for the analysis of dynamical structure factors from microscopic simulations in Sec. 2.4, directly applicable to Brownian dynamics simulations, but also Langevin dynamics and molecular dynamics.

• In Chap. 3, we applied the framework of stochastic dynamical density functional theory (sDDFT) to aqueous electrolytes, arriving at the Poisson-Nernst-Planck model with stochastic flux at the mean-field level of description. While this analysis has been done numerous times in the literature and yields the usual description of electrostatic screening and relaxation, we have treated the general case of arbitrary electrolytes, and in particular the effects of unequal mobilities for the binary electrolytes, that is often approximated in the literature.

In Chap. 4, we started to exploit these methodological developments, which underlined the role of fluctuations in macroscopic transport, to bulk electrolytes, where we studied the mechanisms involved in charge density fluctuations.

• In Sec. 4.1, we recovered the electrostatic relaxation effect, but further gave new interpretations in terms of charge fluctuations in bulk electrolytes, compared to number fluctuations, and tested the theory for various concentrations. We used the density of modes of fluctuations encoded in dynamical structure factors, which were previously identified as an important object for transport.

• While the electrostatic relaxation for bulk electrolytes was unsurprising in terms of local modes of fluctuations, in Sec. 4.2, motivated by the Kirkwood-Buff theory (KB) which plays an important role in the thermodynamic of solutions, we showed that the effects of electrostatic relaxation are non-trivial when considering finite volumes. This was done by adapting the KB integral to reciprocal space and injecting results from sDDFT. In the static case, we recovered the property of charge hyperuniformity in aqueous electrolytes that was derived by different means in the literature. We also made an excursion to the abstract case of an electrolyte in arbitrary dimensions, and generalized hyperuniformity as a property arising from the competition between fluctuations in a volume, driven by entropy, and classes of interactions reducing them to surface fluctuations. We obtained new analytical results on that topic. Then, turning back to real electrolytes, we extended the KB integral to dynamical fluctuations and confirmed our predictions with Brownian dynamics simulations. We focused on the temporal fluctuation dynamics for the number and charges of ions in a closed volume. We found that the relaxation of the correlation in the number of ions behaves as a regular diffusion process in 3D, with scaling t -3/2 , whereas the total charge fluctuations, submitted to electrostatic relaxation, present exotic time/frequency scalings, which can be perceived as dynamical relaxation of hyperuniformity.

• As the precedent results were derived from sDDFT at the linearized mean-field level of description, then confirmed by BD simulations under the overdamped implicit solvent approximation, we investigated the limitations of these assumptions in Sec. 4.3.1. By comparing sDDFT to Brownian dynamics, Langevin dynamics, and molecular dynamics, we found that the breakdown of mean-field predictions and BD simulations occurs due to ionic correlations originating from steric effects, and hydrodynamic effects originating from the solvent respectively. We presented an efficient way to improve sDDFT beyond the mean-field approximation, making use of BD simulations, which encode ionic correlations in the static structure factor, and injecting it into the DDFT formulation. This procedure quantitatively improves dynamical predictions for high concentrations and could be in principle extended to hydrodynamic correlations as well.

• In Sec. 4.4, we gave preliminary results on the charge fluctuations in bulk binary electrolytes in the nonequilibrium regime, confirming predictions from sDDFT and opening the way to go beyond the linear response regime. Using our general theoretical developments from Chap. 3, we briefly addressed the effects of unequal mobilities on dynamical correlations.

The results obtained in the bulk simulations usually confirm the predictions that we derived through the sDDFT and agree with existing results in the literature, but Chap. 4 shows the need for numerical simulations in order to go beyond mean-field predictions.

After having addressed the case of bulk electrolytes, we turned to properties of confined ones in Chap. 5.

• We propose a way to model confined electrolytes in a slab geometry and study the response to a perpendicular oscillating electric field. We formalized the general field-and-frequency conductivity of the electrolytes, which constitutes a systematic analysis tool from numerical simulations. We demonstrated the efficiency of the adapted Green-Kubo formula for BD simulations, and obtained phenomenological insights into the microscopic mechanisms driving the dynamics of the ions, by interpreting the confined electrical conductivity spectrum through simple analytical models.

• In particular, due to the many mechanisms that are believed to play a role in the microscopic dynamics of ions and consequently, on ionic transport at the nanoscale, we separately investigated the influence of diffusion under confinement, migration under confinement, modelled interfacial properties such as the effective confinement distance and adsorption mechanisms, electrostatic relaxation and nonlinear coupling with the external field. The interpretation through timescale analysis, which was naturally probed by the conductivity spectrum, was demonstrated to be an efficient method to dissect the spectra. Notably, we addressed the coupling between confined diffusion and electrostatic relaxation, confirming analytical results derived in Ref. [START_REF] Bazant | Diffuse-charge dynamics in electrochemical systems[END_REF], and also studied the nonlinear coupling between confined migration and diffusion, arising at high electric fields.

These results show the efficiency of the methodology developed to describe electrical fluctuations. But the presented approaches, whose limitations arising from implicit solvent and semi-implicit interfaces modelling were discussed in this manuscript, could benefit from extensions and improvements. Firstly, one can note that in Chap. 4 we focused on spatio-temporal fluctuations of charges, but the description of local density fluctuation analysis in the confined case was lost in Chap. 5. Indeed, while in Sec. 2.3 we gave a Green-Kubo formula for the frequency and wavenumber-dependent conductivity, this result was not exploited. Therefore, the study of the electrical response of confined electrolytes under time and space-varying electrical fields both perpendicular and parallel to the confinement should be considered in the future. Secondly, the considered models should also include the treatment of interfacial effects arising from dielectric jumps, surface charge, surface polarization, surface chemistry, phonon-electrolytes interactions, quantum friction and all combined with hydrodynamics at the nanoscale, which were all neglected in our study. These effects are known to lead to important changes in the dynamics of ions and would be necessary to accurately model properties of ionic transport at the nanoscale.

One could then extend our approach to more complex systems, in particular, to relate to experimental setups and nanofluidic designs with specific interfaces and geometries. It is important to recall that the exact microscopic properties of materials are rarely directly probed. As a matter of fact, a recent work [START_REF] Robin | Disentangling 1/f noise from confined ion dynamics[END_REF] on the notorious 1/f electrical noise in nanopores deduces its origins not directly from what happens in the nanopore itself, but rather on diffusive transport mechanisms of ions (induced by hyperuniformity) happening in the reservoirs. Another important point to address is that most ionic current measurements in electrolytes are done with electrodes, therefore the modelling of the electrode-electrolyte interfaces is crucial. Lastly, one could also extend the methodology to different external drives than electrical plane waves, such as magnetic or pressure forcings, in order to get a more complete picture of ionic transport at the nanoscale.

In conclusion, to the question of predicting accurate transport properties of confined electrolytes at the nanoscale, while the studied models ostensibly possess limitations, our investigation yielded qualitative and quantitative results with fruitful interpretations. By shedding light on underlying mechanisms, which is arguably the key step to designing technological applications, the methodology and results presented in this manuscript constitute a short and efficient path between the microscopic dynamics of the ions and ionic transport properties through the analysis of electrical fluctuations in electrolytes. This partially open door can lead to improvements in the fundamental understanding of ionic transport at the nanoscale, motivated by promising new technological applications such as efficient energy storage, blue energy harvesting, water desalination, or nanopore sensing. More generally, the high specificity of the observed properties in these systems, arising from the choice of the solute, solvent, interfaces, geometry, and driving forces, if understood sufficiently well, offers new perspectives in the field of "iontronics". This emerging field attempts to harness the versatility of ionic transport to replace electrons in semiconductors with ions in nanofluidic systems, mimicking systems that naturally surround us in biological systems such as cellular transport or the human brain.

Résumé détaillé en français

La plupart des appareils électroniques et des cellules de stockage d'énergie, dans leurs formes modernes, reposent sur les propriétés électrochimiques de porteurs de charges microscopiques, à savoir les ions, pour transformer et transporter l'énergie électrique. Les dispositifs conventionnels, tels que les batteries, reposent sur des réactions chimiques pour stocker de l'énergie et utilisent le mouvement des porteurs de charge dans le processus de charge et de décharge. Au cours des dernières décennies, le développement technologique dans les sciences des matériaux, et en particulier les nanotechnologies, a conduit à une amélioration fulgurante de ces dispositifs, avec des matériaux et des géométries spécifiques, augmentant les performances des batteries par exemple. Mais avec l'extrême miniaturisation de ces systèmes, le passage à l'échelle sub-micrométrique a également conduit au développement de technologies disruptives dans le domaine du stockage de l'énergie, comme les supercondensateurs, reposant uniquement sur les propriétés des doubles couches électriques et l'adsorption d'ions dans des électrodes nanoporeuses. Bien qu'omniprésente dans les systèmes biologiques, le transport ionique à l'échelle nanométrique, à condition d'être maîtrisé, pourrait conduire à des avancées technologiques majeures. De plus, avec la polyvalence des nanotechnologies, de nouvelles applications peuvent être trouvées dans divers domaines. En particulier, le transport de solutions chargées à travers les nanopores offre des pistes prometteuses pour la récupération de l'énergie bleue, le dessalement de l'eau ou le séquençage ADN par nanopores. En ce qui concerne ce dernier, il est désormais possible de mesurer avec précision le courant électrique créé par le flux d'électrolyte à travers un nanotube unique, de sorte que le changement de courant lors de l'insertion d'une chaîne d'ADN dans le nanotube peut servir d'outil de séquençage performant et miniaturisé. Mais ces applications prometteuses sont basées sur des conceptions complexes, qui elles-mêmes reposent sur la compréhension du transport ionique à l'échelle nanométrique. De plus, du fait de la taille nanométrique de ces dispositifs et du faible nombre de porteurs de charge, les fluctuations électriques jouent un rôle clé dans les propriétés électriques de ces systèmes. En effet, de récentes mesures de courant ionique à travers des nanopores, biologiques ou artificiels, révèlent que les fluctuations de courant observées dans ces systèmes ne sont pas que du bruit parasite : la densité spectrale de ces fluctuations possède de manière quasi omniprésente un scaling en 1/f de type "bruit rose" à basse fréquence. Diverses études sur ce phénomène indiquent que l'amplitude de ces fluctuations est étroitement liée aux propriétés du pore avec son substrat, sa géométrie et son revêtement, de celles de l'électrolyte avec sa composition, sa concentration et son pH, ainsi que d'autres paramètres tels que la charge de surface ou la tension appliquée. Cependant l'origine et les propriétés de ces fluctuations, qui semblent refléter des propriétés fondamentales du transport ionique, ne sont pas totalement comprises à ce jour. Cette observation sur les nanopores n'est qu'une des multiples expériences impliquant les fluctuations électriques dans les électrolytes. Des expériences telles que la relaxation magnétique nucléaire , les mesures de forces de surfaces et l'électrochimie à l'échelle nanométrique, sans liens apparents, sondent en permanence les fluctuations de courant, de charge, de polarisation et de masse. Il est donc primordial de modéliser et de comprendre les fluctuations électriques des électrolytes à l'échelle nanométrique afin d'interpréter ces expériences. De plus, découvrir l'ensemble des mécanismes impliqués dans les fluctuations électriques dans les électrolytes pourrait être d'un grand intérêt comme outil fondamental pour sonder les propriétés du transport à l'échelle nanométrique [START_REF] Hoang | Electrical noise in electrolytes: a theoretical perspective[END_REF]. En outre, cette compréhension pourrait conduire à l'amélioration de nombreux concepts technologiques, avec l'augmentation du rapport signal-bruit dans la détection de molécules uniques pour la biophysique et les biotechnologies [START_REF] Dekker | Solid-state nanopores[END_REF]. Le modèle de dynamique Brownienne est décrit par une équation de Langevin suramortie. L'évolution de la position des ions R(t) = {r i (t)} dépend des forces conservatives -∇ R V ( R, t) qui s'exercent dans le système (internes et externes), et d'une force aléatoire décrite par le processus stochastique η, modélisant les fluctuations thermiques dues aux collisions ions-solvant. Dans cette équation β = 1/(k B T ) est la température inverse et D est la matrice de diffusion des ions.

d dt R = -βD • ∇ R V ( R, t) + (2D) 1/2 • η
Le travail présenté dans ce manuscrit a deux objectifs complémentaires : premièrement de modéliser les fluctuations du courant ionique dans les nanopores, et deuxièmement, de relier la dynamique microscopique des ions avec les propriétés macroscopiques du transport dans les électrolytes, en exploitant le lien complexe entre transport ionique et fluctuations électriques, et en analysant les mécanismes pertinents. Le premier défi, en raison des différentes échelles de temps et de longueur impliquées, a été résolu en combinant les simulations de dynamique brownienne, qui permettent d'explorer des temps plus longs et des tailles de système plus grandes que la dynamique moléculaire conventionnelle avec des solvants explicites, tout en incorporant l'effet des fluctuations thermiques sur la dynamique des ions, avec des méthodes de description analytique. Le deuxième objectif a été atteint par le développement de divers outils méthodologiques tels que des modèles analytiques et la théorie de la réponse, dont les applications respectives dépassent les phénomènes étudiés dans cette thèse.

En rassemblant des résultats déjà connus dans les domaines des solutions d'électrolytes, de la physique statistique, de la théorie de la réponse linéaire et de l'électrostatique des milieux continus, nous avons donné un cadre unifié pour la réponse électrique dans les électrolytes dans la Sec. 2.2.5. Nous avons relié les relations constitutives de Maxwell, formalisées en tant que réponses linéaires à des perturbations de forme d'onde-plane, caractérisées par la conductivité électrique σ(k, ω) et la permittivité ε(k, ω) généralisées, au facteur de structure dynamique de charge S qq (k, ω). Cette quantité, qui est souvent seulement indirectement sondée par des expériences, est directement accessible par des simulations microscopiques.

S qq (k, ω) = ˆ+∞ -∞ n,m
z n z m e -ik•(rn(t)-rm(t)) e -iωt dt Le facteur de structure dynamique de charge décrit les corrélations spatio-temporelles des charges. Dans cette expression, la position de l'ion n, de charge e z n , est r n (t). Ces corrélations de densités de charge sont analysées en termes de tranformées de Fourier.

Dans le cadre de modèles de solvants implicites avec dynamique de Langevin suramortie, nous avons redémontré une formule Green-Kubo pour la conductivité électrique dans la Sec. 2.3. Avec une importance mise sur la dépendance en fréquence, ce résultat sert à la fois d'outil de modélisation, mais aussi comme moyen de sonder les mécanismes microscopiques avec grande acuité, en exploitant le fait que les corrélations se produisent à des échelles de temps et de longueur spécifiques et sont encodées dans les fluctuations électriques.

La relation de Green-Kubo relie le spectre des fluctuations de courant électrique (Brownien) J el (t) = n ez n βD n F n (t) à l'équilibre et la conductivité dépendante de la fréquence dans le cadre de la réponse linéaire. Ici la décomposition issue du courant Brownien défini avec les forces conservatives offre directement la dépendance en fréquence comme une déviation de la conductivité idéale de Nernst-Einstein σ NE .

σ αβ (ω) = σ NE δ αβ - β V ˆ∞ 0 J α el (0) J β el (t)
eq e -iωt dt , De plus, nous avons développé des outils numériques pour l'analyse des facteurs de structure dynamiques à partir des simulations microscopiques en Sec. 2.4, directement applicable aux simulations de dynamique Brownienne, mais aussi dynamique de Langevin et dynamique moléculaire.

Au Chap. 3, nous avons appliqué l'outil de la Théorie Stochastique de la Fonctionnelle Densité (sDDFT) aux électrolytes aqueux, arrivant au modèle de Poisson-Nernst-Planck avec flux stochastique au niveau champ moyen. Alors que ce résultat a déjà été obtenu dans le passé, donnant la description connue de la relaxation électrostatique, nous avons traité le cas des électrolytes arbitraires, et en particulier avec les effets de mobilités inégales pour les électrolytes binaires proches de l'équilibre, cas souvent approximé dans la littérature. Au Chap. 4, nous avons commencé à exploiter ces développements méthodologiques autour du rôle des fluctuations dans le transport, dans le cas des électrolytes bulk, où nous avons étudié les mécanismes impliqués dans les fluctuations de densité de charge. Dans la Sec. 4.1, nous avons retrouvé l'effet de relaxation électrostatique, bien connu dans la littérature, mais nous en avons aussi donné de nouvelles interprétations en termes de mode de fluctuations de charge dans les électrolytes en bulk, par rapport aux modes de fluctuations de nombre d'ions. Nous avons utilisé la densité des modes de fluctuations encodée dans les facteurs de structure dynamiques, qui était précédemment identifiés comme des objets importants pour le transport ionique dans la Sec. 2.3. Alors que la relaxation électrostatique pour un électrolyte bulk n'était pas surprenante en termes de fluctuations locales, dans la Sec. 4.2, motivés par la théorie de Kirkwood-Buff (KB) qui joue un rôle important dans la thermodynamique des solutions, nous montrons que les effets de la relaxation électrostatique sont non triviaux lorsque l'on considère un volume fini. Ceci a été fait en adaptant l'intégrale de KB à l'espace réciproque et en injectant les résultats de sDDFT. Dans le cas statique, nous avons retrouvé la propriété de l'hyperuniformité de charge dans un électrolyte aqueux, qui a déjà été obtenue par différents moyens dans la littérature. Nous avons également fait une excursion vers le cas abstrait d'un électrolyte en dimensions arbitraires, et l'hyperuniformité a été généralisée comme une propriété découlant de la compétition entre les fluctuations en volume, de nature entropique, et des classes d'interactions les réduisant à des fluctuations de surface. Nous avons obtenu de nouveaux résultats exacts sur ce sujet. Puis, en revenant aux systèmes électrolytiques réels, nous avons étendu l'intégrale de KB aux fluctuations dynamiques. Bien que nous n'ayons pas trouvé explicitement le lien thermodynamique entre les intégrales de Kirkwood-Buff et de Green-Kubo, nous nous sommes concentrés sur la dynamique des fluctuations temporelles pour le nombre et les charges d'ions en volume fini. Nous avons constaté que la relaxation de la corrélation dans le nombre d'ions se comporte comme un processus de diffusion régulière en 3D, avec la décroissance temporelle en t -3/2 , alors que les fluctuations de charge totale, soumises à la relaxation électrostatique, présentent des lois en temps/fréquence plus exotiques, qui peuvent être perçues comme une relaxation dynamique de l'hyperuniformité. Comparaison entre simulations de dynamique moléculaire (MD, réalisées par J. Kim) et de dynamique Brownienne, prédictions analytiques du modèle de Poisson-Nernst-Planck et modèle balistique (aux très grands vecteurs d'onde), et prédiction semi-analytique de la DDFT permettant d'améliorer les prédictions dynamiques au-delà de l'approximation du champ moyen, en incorporant les corrélations statiques calculéess par simulations [START_REF] Hoang | Electrical noise in electrolytes: a theoretical perspective[END_REF].

Comme les résultats précédents ont été démontrés dans le cadre de la sDDFT, utilisant l'approximation du champ moyen, et ont été confirmés par des simulations BD sous l'approximation de solvant implicite, nous en avons cherché les limites dans la Sec. 4.3. En comparant la sDDFT à la BD ainsi qu'à la dynamique de Langevin et la dynamique moléculaire (MD), nous avons constaté que l'échec des prédictions de type champ moyen et des simulations de BD se produisent en raison des corrélations ioniques provenant respectivement d'effets stériques, et des effets hydrodynamiques dûs au solvant. Nous avons présenté un moyen efficace d'améliorer la sDDFT au-delà de l'approximation du champ moyen, en utilisant des simulations BD, qui encapsulent les corrélations ioniques dans le facteur de structure statique S qq (k), et en injectant ce dernier dans la formulation DDFT. Cette procédure améliore quantitativement les prédictions dynamiques pour les concentrations élevées et pourrait être étendue aux corrélations hydrodynamiques également.

Dans Sec. 4.4, nous avons donné des résultats préliminaires sur les fluctuations de charge en volume pour des électrolytes binaires dans le régime hors équilibre, confirmant les prédictions de la sDDFT et ouvrant la voie au-delà du régime de la réponse linéaire. Exploitant nos développements du Chap. 3, nous avons rapidement abordé les effets des mobilités inégales sur les corrélations dynamiques. Figure 5.23: Dynamique des ions confinés dans un pore fente: Le système étudié consiste en un électrolyte décrit par un modèle de solvant implicite, avec des parois bloquantes modélisées par des potentiels confinants. Le type de potentiel choisi permet de modéliser les interactions d'interface électrolyte-solide. Nous étudions la réponse de ce système à un champ électrique oscillant appliqué dans la direction perpendiculaire au confinement [START_REF] Hoang | Frequency and field-dependent response of confined electrolytes from Brownian dynamics simulations[END_REF].

Les résultats obtenus pour les électrolytes bulk confirment généralement les prédictions que nous avons dérivées par la Théorie Dynamique de la Fonctionnelle Densité Stochastique, les deux confirmant les résultats existants dans la littérature, mais aussi en les étendant à d'autres situations. En outre, le Chap. 4 montre le besoin de simulations numériques pour aller au-delà des prédictions du champ moyen. Après avoir abordé le cas des électrolytes bulk, nous nous sommes tournés vers les propriétés des électrolytes confinés dans le Chap. 5.

Le confinement et les interfaces peuvent drastiquement modifier les propriétés des électrolytes, ainsi nous proposons de modéliser des électrolytes confinés entre deux plans parallèles, et d'en étudier la réponse à un champ électrique oscillant perpendiculairement. Nous avons formalisé la conductivité dépendante de la fréquence (et de l'amplitude du champ électrique) des électrolytes σ(ω), ce qui constitue un outil d'analyse systématique à partir de simulations numériques. Nous avons démontré l'efficacité de la formule Green-Kubo adaptée pour les simulations de BD, et obtenu des aperçus phénoménologiques sur les mécanismes microscopiques conduisant à la dynamique des ions, en interprétant le spectre de conductivité électrique confinée à travers des modèles analytiques simples. Comparaison entre la méthode de simulation hors équilibre, à équilibre avec la formule de Green-Kubo, et la prédiction exacte obtenue par résolution de l'équation de Fokker-Planck, montrant la convergence des méthodes dans le régime linéaire. σ NE , τ diff , τ E sont repectivement la conductivité de Nernst-Einstein pour un électrolyte idéal bulk, le temps de diffusion entre les deux parois, et le temps de migration correspondant [START_REF] Hoang | Frequency and field-dependent response of confined electrolytes from Brownian dynamics simulations[END_REF].

En particulier, en raison des nombreux mécanismes qui sont supposés jouer un rôle sur la dynamique microscopique des ions et par conséquent le transport ionique à l'échelle nanométrique, nous avons étudié séparément l'influence de la diffusion en confinement, la migration en confinement, les propriétés interfaciales modélisées telles que la distance de confinement effective et les mécanismes d'adsorption, la relaxation électrostatique et le couplage non linéaire avec le champ externe. Bien que ces mécanismes aient surtout été étudiés individuellement et dans des cas simplifiés, l'interprétation par des échelles de temps, qui étaient naturellement sondées par le spectre de conductivité, s'est avérée être une méthode efficace. Nous avons notamment abordé le couplage entre la diffusion confinée et la relaxation électrostatique, confirmant les résultats analytiques déjà obtenus dans la littérature, et avons également étudié le couplage non linéaire entre migration confinée et diffusion confinée, survenant à des champs électriques élevés. Résulats de simulations en points colorés, prédiction analytique en orange en identifiant ω * = 1/τ Bazant [START_REF] Bazant | Diffuse-charge dynamics in electrochemical systems[END_REF]. Les lignes en pointillés montrent les régimes asymptotiques dominés par la relaxation électrostatique pour les doubles couches électriques minces / faibles confinements (λ D ≪ L en bleu), et par le confinement géométrique pour les doubles couches électriques épaisses / forts confinements (λ D ≫ L en rouge) [START_REF] Hoang | Frequency and field-dependent response of confined electrolytes from Brownian dynamics simulations[END_REF].

Ces résultats montrent l'efficacité de la méthodologie développée pour décrire les fluctuations électriques. Mais les approches présentées, dont les limites découlant de la description implicite du solvant et des interfaces ont été discutées tout au long de ce manuscrit, pourraient bénéficier de nombreuses extensions et améliorations. Tout d'abord, on peut noter que dans le Chap. 4 nous nous sommes concentrés sur les fluctuations de densité de charges en temps et en espace, mais la description des fluctuations de densité locale dans le cas confiné n'a pas été appliquée dans le Chap. 5. En effet, alors que dans la Sec. 2.4.3 nous avons donné une nouvelle formule de Green-Kubo pour la conductivité dépendante de la fréquence et du nombre d'onde, ce résultat n'a pas été exploité. Par conséquent, l'étude de la réponse électrique des électrolytes confinés à des champs électriques variables en temps et en espace, perpendiculaires et parallèles au confinement, devrait être envisagée à l'avenir. Dans ce dernier cas, les modèles envisagés devraient inclure le traitement des effets interfaciaux résultant des sauts diélectriques, de la charge, polarisation et chimie de surface, des interactions phonon-électrolytes, de la friction quantique -le tout combiné avec l'hydrodynamique à l'échelle nanométrique. Ces effets sont notamment connus pour conduire à des changements importants dans la dynamique des ions, et leur prise en compte serait nécessaire pour modéliser avec précision les propriétés du transport ionique à l'échelle nanométrique.

En conclusion, en ce qui concerne la prédiction précise des propriétés de transport des électrolytes confinés à l'échelle nanométrique, les modèles étudiés possèdent un certain nombre de limitations, mais notre étude a donné des résultats qualitatifs et quantitatifs avec des interprétations fructueuses. En mettant en lumière les mécanismes sous-jacents, ce qui constituent sans doute l'étape clé de la conception d'applications technologiques, la méthodologie et les résultats présentés dans ce manuscrit constituent un chemin court et efficace entre la dynamique microscopique des ions et les propriétés de transport ionique à travers l'analyse des fluctuations électriques dans les électrolytes. Cette porte partiellement ouverte peut conduire à des améliorations dans la compréhension fondamentale du transport ionique à l'échelle nanométrique, motivé par de nouvelles applications technologiques prometteuses telles qu'un stockage efficace de l'énergie, l'extraction de l'énergie bleue, le dessalement de l'eau ou la détection moléculaire par nanopores. Plus généralement, la forte spécificité des propriétés observées dans ces systèmes découlant du choix du soluté, du solvant, des interfaces, de la géométrie et des différents forçages externes, si suffisamment bien maîtrisée, offrirait de nouvelles perspectives dans le domaine de "l'iontronique". Ce champ de recherche émergent tente d'exploiter la polyvalence du transport ionique pour remplacer les électrons dans les semi-conducteurs par des ions dans les systèmes nanofluidiques, imitant ainsi les systèmes biologiques qui nous entourent tels que les cellules ou le cerveau humain.

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Abstract et résumé

Abstract: Ionic transport at the nanoscale plays a key role in fields as diverse as biological systems, DNA sensing through nanopore, electricity storage, or blue energy harvesting and desalination. In nanopores, the small number of charge carriers and their thermal motion result in the predominance of electrical current fluctuations compared to bulk electrolytes. These fluctuations encode information on the microscopic processes resulting in ionic transport. Interpreting these fluctuations remains however a great challenge for modelling, due to the variety of coupled phenomena to be considered and the corresponding length and time scales, such as thermal motion, electrostatic and hydrodynamic interactions, and interactions with the solid walls. In this work, we use Brownian dynamics simulations and stochastic density functional theory to model electrical fluctuations in electrolytes. We analyze the combined effects of salt concentration, confining distance, applied electric field, and adsorption at the wall surfaces on the field-and frequency-dependent conductivity. To achieve this goal, we develop and use dedicated theoretical approaches to compute transport coefficients and the dynamical structure factors. The comparison between implicit and explicit solvent simulations generally supports the predictions of the simpler descriptions, but also highlights their limitations and provides insights on how to improve them. Overall, this thesis contributes to a comprehensive understanding of electrical fluctuations and ionic transport at the nanoscale.

Résumé: Le transport ionique à l'échelle nanométrique joue un rôle clé dans des domaines aussi divers que les systèmes biologiques, la détection de l'ADN à travers les nanopores, le stockage de l'électricité ou la récupération d'énergie bleue et le dessalement de l'eau. Dans les nanopores, le petit nombre de porteurs de charge et leur mouvement thermique entraînent la prédominance des fluctuations du courant électrique par rapport aux électrolytes bulk. Ces fluctuations encodent des informations sur les processus microscopiques qui conduisent au transport ionique. L'interprétation de ces fluctuations reste cependant un grand défi pour la modélisation, en raison de la variété des phénomènes couplés à considérer et des échelles de longueur et de temps correspondantes, tels que le mouvement thermique, les interactions électrostatiques et hydrodynamiques, et les interactions avec les parois solides. Dans ce travail, nous utilisons des simulations de dynamique brownienne et la théorie fonctionnelle de la densité stochastique pour modéliser les fluctuations électriques dans les électrolytes. Nous analysons les effets combinés de la concentration en sel, de la distance de confinement, du champ électrique appliqué et de l'adsorption aux surfaces des parois sur la conductivité dépendante du champ et de la fréquence. Pour atteindre cet objectif, nous développons et utilisons des approches théoriques dédiées pour calculer les coefficients de transport et les facteurs de structure dynamiques. La comparaison entre les simulations de solvants implicites et explicites confirme généralement les prédictions des descriptions les plus simples, mais met également en évidence leurs limites et fournit des informations sur la manière d'améliorer ces descriptions. Dans l'ensemble, cette thèse contribue à une compréhension globale des fluctuations électriques et du transport ionique à l'échelle nanométrique.

Figure 1 . 1 :

 11 Figure 1.1: Experimental realization of nanotube nanofluidic set-up from Ref. [12]: a) (1-4): Insertion, sealing, retraction, finalized transmembrane boron nitride nanotube (BNNT) device (left sketch, corresponding SEM image). Embedding of BNNT nanotube (grey) through a drilled SiN membrane (green), and carbon sealing of the hole using electro-beam-induced deposition (orange). b) Top: schematic of the full experimental set-up. Bottom: sketch of the final BNNT transmembrane on the left, its experimental realization from TEM imagining on the right.

Figure 1 . 2 :

 12 Figure 1.2: 1/f noise in graphene and silicon nitride pores: Power spectral densities of electrical current across graphene and silicon nitride nanopores showing a different 1/f scaling behaviour at low frequencies[START_REF] Heerema | 1/f noise in graphene nanopores[END_REF]

Figure 1 . 3 :

 13 Figure 1.3: Effect of salt concentration and polymer adsorption on pink noise in nanopores: Power Spectral Density of electrical current fluctuation for KCl electrolyte at various salt concentrations, and PEG-coated nanopores with different molecular weights. Comparison between experimental measurements and Brownian dynamics simulations for modelled adsorption rates. [26].

Figure 1 . 4 :

 14 Figure 1.4: Breakdown of continuum descriptions at the nanoscale: overview of relevant physical features arising at various length scales (Ref. [5]).

  1.6).

Figure 1 . 6 :

 16 Figure 1.6: Deviations from the Debye screening length: Screening length λ S divided by the Debye length λ D Eq. (1.12) as a function of the particle radius σ also normalized by the Debye length for a 1:1 electrolyte. Flat solid line representing the Debye-Hückel prediction λ S = λ D , dashed line Landau fluctuation theory combined with MSA results for thermodynamic functions. Symbols are also computed from pair correlation function theory using the MSA approximations but as exact poles of the structure factors, with various parameters of the primitive model (Ref. [47]).

Figure 1 . 7 :

 17 Figure 1.7: Overview of microscopic simulation methods represented at the relevant time scale, along with corresponding experimental techniques, from smaller to larger scales from left to right. Courtesy of M. Jardat

Figure 1 . 8 :

 18 Figure 1.8: Implicit solvent modelling: Sketch of the implicit/explicit solvent coarse-graining process. From explicit water molecules (red and white) on the left, to implicit solvent model with only the anions and the cations (blue and green) to the right. Courtesy of G. Pireddu

Figure 1 . 9 :

 19 Figure 1.9: Self-diffusion coefficient of K + in KCl aqueous solution at 298 K, as a function of the salt concentration. Stars: experimental data from Ref. [127], disks: Brownian dynamics simulation without hydrodynamic interactions, triangles: Brownian dynamics simulation with hydrodynamic interactions taken as an effective mobility tensor (Rotne-Prager).Figure from Ref. [121].

Figure 1 . 10 :

 110 Figure 1.10: Equivalent molar conductivity of KCl aqueous solution at 298 K, as a function of the salt concentration. Stars, experimental data Ref. [128, 129], disks: Brownian dynamics simulation without hydrodynamic interactions, triangles: Brownian dynamics simulation with hydrodynamic interactions taken as an effective mobility tensor (Rotne-Prager).Figure from Ref. [121].

  ) numerical values are shown in Tab 2.1.

Figure 2 . 1 :

 21 Figure 2.1: Interionic potentials: a) Illustration of the Lennard-Jones Eq. (2.14), Weeks-Chandler-Andersen Eq. (2.16) and Coulomb Eq. (2.18) potentials, for T= 300 K, ε w = 78.5, ϵ ++ = ϵ --= ϵ +-= ϵ -+ = 0.5 k B T , σ ++ = σ --= σ -+ = σ +-= 3 Å and q + = -q -= 1.6 × 10 -19 C. b)Illustration of interionic potentials for a fully symmetric electrolyte, as combinations of the latter potentials. XXX -Coul. for cations/anions pairs and XXX + Coul. for equal sign charges pairs.

Figure 2 . 2 :

 22 Figure 2.2: Simulation Performance: Simulation speed for a few electrolyte systems. Simulations were done on a desktop computer with: AMD Ryzen Threadripper 1950X 16-Core Processor 3.4 GHz, 32 CPU ram 32 GB. On the left the simulation's parameters and resulting speed, on the right plot of the simulation speed as a function of the number of particles for 4 and 8 CPU cores (red squares and blue triangles respectively), showing a quasi-optimal linear scaling.

Figure 2 . 4 :

 24 Figure 2.4: Multiscale signal: on the left a representation of a multi-scale signal composed of two frequencies, characterized by periods τ 1 (red) ≪ τ 2 (blue) and noise (τ r ≪ τ 1 , τ 2 ). Due to the mixing of frequencies, the separation of time scales in the total signal is hard to achieve. On the right panel, a typical representation of a time correlation function with two characteristic relaxation processes, both the peaked short-time (blue) and slow decaying long-time (red) parts are relevant in this case. Double time sampling allows for a better convergence of a quadrature, which is used to compute the Power Density Spectrum, as displayed by the coloured rectangles.

× 4

 4 .1) where the reduced variables (K, Ω) are defined with Eq. (3.36). Eq. (4.1) are plotted in Fig. 4.1. Dκ 2 (b) Charge dynamical structure factor

Figure 4 . 1 :

 41 Figure 4.1: Dynamical structure factors in the (k, ω) plane for a 1:1 symmetric electrolyte. (a): Mass dynamical structure factor (unbounded), (b): Charge dynamical structure factor (bounded). The colourmap represents the amplitude of the fluctuation spectrum of mass and charge from Eq. (4.1), normalized by 4Dκ 2 .

Figure 4 . 2 :

 42 Figure 4.2: Charge dynamical structure factors for various systems (Sec. 2.1, Tab. 4.1), normalized by the Debye time and plotted as a function of the frequency normalized by the inverse Debye time. Results are shown for different wavelengths λ = 2π/k normalized by the Debye length, computed by Brownian dynamics simulations (coloured continuous lines), and sDDFT predictions Eq. (4.1) (black dashed lines) for C s = 0.01, 0.04, 0.1 and 0.4M in panels (a,b,c,d) respectively.

Figure 4 . 3 :

 43 Figure 4.3: Ionic fluctuations in finite observation volumes. (a) Brownian dynamics simulation, here for a C s = 52 mM salt solution in a cubic, triply periodic simulation domain. Yellow (resp. blue) particles represent cations (resp. anions). The dark grey cube is a finite observation volume, here, one of the 64 boxes of size L obs = L box /4. (b) Within the observation box, the number of particles N = n + + n -and the charge Q = q(n +n -) fluctuate, where n + (resp. n -) is the number of positive (resp. negative) charges[START_REF] Hoang | Ionic fluctuations in finite volumes: fractional noise and hyperuniformity[END_REF].

L

  obs ≫ λ D L obs ≪ λ D

Figure 4 . 4 :

 44 Figure 4.4: Hyperuniformity of charge (a) Static charge fluctuations with observation box

Figure 4 . 5 :

 45 Figure 4.5: Algebraic decay of particle number fluctuations. (a) Correlation function of the number of ions in a cubic volume of side L box , ranging from light blue to dark blue for increasing observation box size. Brownian dynamics results are shown as triangles, with shaded areas indicating one standard deviation around the mean, while lines are predictions from Eq. (4.21).(b) Rescaled (a) plot showing algebraic decay at long times as 1/t 3/2 . (c) Associated frequency spectrum with the 1/f 3/2 signature of fractional noise[START_REF] Benoit | Fractional brownian motions, fractional noises and applications[END_REF][START_REF] Marbach | Intrinsic fractional noise in nanopores: The effect of reservoirs[END_REF]. Here, C s = 104 mM; coloured legends are shared across (a-c)[START_REF] Hoang | Ionic fluctuations in finite volumes: fractional noise and hyperuniformity[END_REF].

  4.6-b). Inserting Eq. (4.8) and Eq. (4.2) into Eq. (4.7) and taking d = 3 the charge equation takes the form:

  8.0 nm L obs = 4.0 nm L obs = 2.0 nm L obs = 1.0 nm L obs = 0

Figure 4 . 6 :

 46 Figure 4.6: Charge fluctuations decay exponentially with time for large observation volumes. (a) Charge fluctuations rescaled by ⟨N ⟩, with time, for increasing L box from yellow to dark red. Dots: results from BD simulations with shaded areas indicating one standard deviation around the mean; lines: Eq. (4.22). (b) Same as (a) in lin-log scale to highlight the exponential decay for large L box . Here, C s = 104 mM and λ D = 0.95 nm [192].

Figure 4 . 7 :

 47 Figure 4.7: A variety of timescales emerges in charge correlation relaxation. (a) Charge correlations rescaled by e t/τ Debye for L obs ≪ λ D . Vertical dotted orange and blue lines indicate scaling law intersections, as t = τ diff /π and t = 3τ Debye /2, respectively. (b) Similar as (a) but for L obs ≫ λ D . The vertical dotted orange and blue lines indicate t = √ 3τ diff /(2 7/6 π 5/12) and t = 4τ Debye /π 2 , respectively. (c) Rescaled charge correlations at long times, with a scaling law as C qq,V ∼ L obs . In all panels: dots: results from BD simulations with shaded areas (or error bars) indicating one standard deviation around the mean; lines: Eq. (4.22)[START_REF] Hoang | Ionic fluctuations in finite volumes: fractional noise and hyperuniformity[END_REF].

Figure 4 . 8 :

 48 Figure 4.8: Static charge-charge structure factor S qq (k) (see Eq (2.50)). The figure displaysresults from molecular dynamics (MD, blue), Langevin dynamics (LD, yellow), and Brownian dynamics (BD, red) simulations, for wave vectors ranging from the minimal value for the box size of the MD simulations, k min = 2π/L box , to 256k min , corresponding to wavelengths between 45.6 and 0.2 Å, as well as for k min /2 in the LD and BD case for which a larger simulation box was used. The results are also compared with the prediction of Debye-Hückel theory (see Eq. (2.51), dotted line) and for an ideal gas (dashed-dotted line). The vertical dotted lines indicate k = k min and k = κ D , the inverse Debye screening length[START_REF] Hoang | Electrical noise in electrolytes: a theoretical perspective[END_REF].

  MD qq (k) DDFT + S BD qq (k) PNP

Figure 4 . 9 :

 49 Figure 4.9: (a) Dynamic charge-charge structure factor, normalized by its initial value, which is reported in panel (b). Both panels show results from molecular dynamics (MD, blue), Langevin dynamics (LD, yellow, in panel (a) only), and Brownian dynamics (BD, red) simulations, for wave vectors ranging from the minimal value for the box size of the MD simulations, k min = 2π/L box , to 256k min , corresponding to wavelengths between 45.6 and 0.2 Å, as well as for k min /2 in the LD and BD case for which a larger simulation box was used. (c) Dynamic charge-charge structure factor for k = 4k min , from MD and BD. The simulation results are also compared with the prediction of Dynamical Density Functional Theory (DDFT, dashed lines), using the static structure factor S qq (k) obtained in the corresponding simulations in panels (b) and (c), or from Debye-Hückel theory (i.e. PNP, black dotted line) in all panels. Panels (a) and (b) also show the prediction for the ballistic regime, dashed-dotted line). The vertical dotted lines in panel (b) indicate k = k min and k = κ [7].

(4. 29 )

 29 Predictions from linearized DDFT only depend on the static structure factor and the diffusion coefficient. The results Eq. (4.1) and Eq. (4.2) for the concentration and charge correlations can be recovered by injecting the static structure factors from Debye-Hückel theory Eq. (2.51) into Eq. (4.29), coinciding with the sPNP model. Then, instead of the charge-charge static structure factor derived in the mean-field approximation, we can make use of the simulation results, displayed in Fig.4.8, to improve DDFT predictions, as demonstrated in Fig.4.9b and Fig.4.9c.

  4.1), couples with the transport mode related to migration (k = βeE ext , ω = iκβDeE ext ).

Figure 4 . 10 :

 410 Figure 4.10: Nonequilibrium dynamical structure factors at C = 0.1 M (see Tab. 4.1) in the presence of an external electric field in the z direction, E ext = 0.00817 V/Å. Simulation results are shown as solid coloured lines, with shaded areas indicating error bars, and analytical predictions Eq. (4.31) as black dashed lines. On the left the concentration (top) and charge (bottom) dynamical structure factors (Eq. (2.53)) in the k ⊥ E ext directions, normalized by the Debye time as a function of the frequency normalized by the Debye frequency, for various wavevectors (colour bar). On the right, the same plots but for k ∥ E ext , where the anisotropic nonequilibrium effects are observed.

Figure 4 . 11 :

 411 Figure 4.11: Effect of unequal diffusion coefficients on the nonequilibrium charge dynamical structure factors at C = 0.1 M (see Tab. 4.1) in a presence of an external electric field in the z direction, E ext = 0.00817 V/Å, with D Na + = 1.28 × 10 -9 m 2 /s and D Cl -= 1.77 × 10 -9 m 2 /s. Simulation results are shown as solid coloured lines, with shaded areas indicating error bars. Black lines are analytical predictions from the general sDDFT model for binary electrolytes Eq. (3.39). Dashed black lines indicate predictions with the approximation that both anion and cation diffusion coefficients are equal to the pair averaged value D = 1.53 × 10 -9 m 2 /s, consequently γ = 0 (Eq. (3.32)). This simplifies the prediction to Eq. (4.31). In solid black lines the general prediction from Eq. (3.39), with explicit accounting for the unequal diffusion coefficients: γ = -0.161. On the top panel, axis represented in log-log scale, and on the bottom panel in lin-log (without the lowest wavelength), highlighting the quantitative change in the resonance frequency.

Figure 5 . 1 :

 51 Figure 5.1: Ionic dynamics under slab confinement: The simulated systems consist of a 1:1 electrolyte, described by ions (red/blue spheres for the cations/anions respectively) in an implicit solvent, confined between parallel walls (grey plates) separated by a distance L. The box is cubic with dimensions L x = L y = L z = L box and periodic boundary conditions are used in the directions along the planes. The red arrow indicates the direction of the external field and the studied electrical current.[START_REF] Hoang | Frequency and field-dependent response of confined electrolytes from Brownian dynamics simulations[END_REF]

Figure 5 . 2 :

 52 Figure 5.2: Solid wall structure modelled by LJ particles (yellow spheres) in an FCC lattice of parameter σ w , with a drawn cut along a (100) crystallographic face in green indicating the solid interface. Courtesy of Laura Scalfi

Figure 5 . 3 :

 53 Figure 5.3: Illustration of the ion-wall potential for σ w /L = 0.3, with L the distance between the two walls and σ w the characteristic length entering in the potential. The orange and blue lines show the full (V ads w Eq. (5.5)) and truncated-shifted (V rep wEq. (5.6)) potential for βϵ w = 0.17, while the green line corresponds to V ads w for βϵ w = 0.50.[START_REF] Hoang | Frequency and field-dependent response of confined electrolytes from Brownian dynamics simulations[END_REF]

Figure 5 . 4 :

 54 Figure 5.4: Electrical current response: Electrical current Eq. (5.3) induced in the confined ideal electrolyte R20 (see Tab.5.1) by an external oscillating electric field Eq. (5.1) of amplitude and frequency E 0 = 0.1 V/Å , f = 16 GHz. The approximate separation of the transient and stationary states corresponds to the moment when the signal pattern becomes periodic.

Figure 5 . 5 :

 55 Figure 5.5: Averaged steady-state electric current: Electrical current Eq. (5.3) from nonequilibrium simulations with various sinusoidal external electric field parameters (Eq. (5.1)): E 0 = 0.1, 0.01, 0.001 V/Å for panels a, b and c respectively. Frequencies ranging from 0.2 GHz to 80 GHz are indicated by colours. Coloured shaded areas indicate associated error bars. Currents are shown normalized by the maximum ideal response V σ NE E 0 where V is the volume of the system, σ NE (Eq. (2.71)) the ideal (volumic) conductivity. The time is rescaled by the frequency of the field, resulting in in-phase signals. On the right, panels a', b', and c' show in-phase currentfield relations for the corresponding currents shown on the left. The ideal bulk response Eq. (5.7) is represented as black dashed lines, on the left as a normalized sine wave for current signals, and on the right as straight diagonal lines for current-field relation.[START_REF] Hoang | Frequency and field-dependent response of confined electrolytes from Brownian dynamics simulations[END_REF]

Figure 5 . 6 :

 56 Figure 5.6: Frequency and field dependent conductivity of an ideal confined electrolyte σ(E 0 , ω) (see Eq. (5.8)) determined from NEQ simulations of system SR (see Tab. 5.1) with a distance between walls of L = 20 Å. The conductivity is shown normalized by the ideal conductivity σ NE as a colourmap in the adimensionalized parameter plane (ωτ diff , τ diff /τ E ). The dashed lines represent the conducting/insulating transitions predicted by model Eq. (5.14) [208].

Figure 5 . 7 :

 57 Figure 5.7: Spectral time evolution: Solutions of the 1-particle Fokker-Planck equation with a confining/adsorbing potential (Eq. (5.5), shown in Fig. 5.3) for three different initial conditions (dashed lines): (a) Close to equilibrium, (b) Far from equilibrium, (c) Very far from equilibrium with discontinuities and density close to the left wall. Continuous coloured lines ranging from red to blue indicate the qualitative forward time propagation ranging in the nanosecond scale. All cases relax to the same Boltzmann equilibrium distribution (Eq. (2.29)) (dotted lines).

Figure 5 . 8 :

 58 Figure 5.8: Frequency-dependent conductivity of a confined ideal electrolyte (R20, see Tab. 5.1) from three different methods: (a) Normalized real conductivity σ(ω)/σ NE , where σ NE is the bulk ideal conductivity (Eq. (2.71)). NEQ simulations results are shown as linked coloured dots for different values of amplitude E 0 . EQ simulation results are represented with solid blue lines. Semi-analytical results are shown as a dashed black line. (b) Real (blue) and imaginary (red) parts of the complex conductivity, EQ simulation in solid lines, semi-analytical predictions in dashed black line, the vertical dotted line indicates the crossover frequency ω = 1/τ diff = π 2 D/ L2 [208].

  5.1) for various confinement distances L = 20, 16 and 12 Å.

Figure 5 . 9 :

 59 Figure 5.9: Equilibrium density profiles and conductivity spectrum: (a) Normalized equilibrium density profiles for three confining distances L (see systems R20, R16 and R12 in Tab. 5.1): simulation results in solid lines and Boltzmann distributions in dashed lines; vertical dotted lines indicate the position of the corresponding Gibbs dividing surface (see Eq. (5.33), Eq. (5.34) and Fig. 5.10). (b) Normalized complex frequency-dependent conductivity σ(ω)/σ NE (see Eq.(5.8) and Eq. (2.71)): simulation results in solid lines and semi-analytical results in dashed lines; vertical dotted lines indicate the measured crossover frequency ω * Eq. (5.32). (c) Real frequency-dependent conductivity rescaled by the predicted crossover frequency τ -1 diff = π 2 D/ L2 . Inset shows the results using τ ′-1 diff = π 2 D/L 2 [208].

Figure 5 . 10 :

 510 Figure 5.10: Gibbs dividing surface: Geometrical construction of the Gibbs dividing surface for purely repulsive walls. The density profile equal to the Boltzmann probability distribution Eq. (5.21) (continuous line) is approximated by a sharp distribution (dashed lines) with a jump positioned at z GDS Eq. (5.33). The geometrical construction relies on matching the excess density in red with the depleted one in blue with respect to the density at the wall (0), and the one in the bulk (ρ b ) respectively.

Figure 5 . 11 :

 511 Figure 5.11: Effects of surface interactions on ideal electrolyte: In blue the system with purely repulsive walls defined by Eq. (5.6) with (σ w , ϵ w ) = (3.0 Å, 0.1 kcal/mol), in orange attractive walls with Eq. (5.5) with (σ w , ϵ w ) = (3.0 Å, 0.1 kcal/mol), and in green highly attractive walls with (σ w , ϵ w ) = (3.0 Å, 0.3 kcal/mol) (systems S20-A01-A03 respectively, see Tab. 5.1). (a) Density profiles in solid lines, Boltzmann distributions (Eq. (5.21)). (b) Construction of the equivalent GDS (Eq. (5.33)) in dashed lines; dotted lines indicate z - GDS . (c) Normalized frequencydependent conductivity extracted from simulations using the Green-Kubo formula (Eq. (2.79)) in solid lines, semi-analytical predictions on dashed lines, and imaginary parts in the inset. Dotted lines indicate the associated inverse confined diffusion times τ -1 diff = π 2 D L2, which depend on the potential used and serve as rescaling frequencies for each system respectively[START_REF] Hoang | Frequency and field-dependent response of confined electrolytes from Brownian dynamics simulations[END_REF].

Figure 5 . 12 :

 512 Figure 5.12: Conductivity spectrum for thin double layer / thick slab: Frequencydependent conductivity σ(ω) normalized by the ideal conductivity σ NE for a fixed distance L = 100 Å ( L = 94.9 Å) between the walls and 5 salt concentrations C s (see EI in Tab. 5.1). These conditions correspond to the limit of thin electric double layers (λ D ≪ L). The frequency is scaled by the Debye time τ Debye (with the dotted vertical line indicating ωτ Debye = 1). The inset shows the same results as a function of the frequency scaled by the diffusion time τ diff (with the dotted vertical line indicating ωτ diff = 1) [208].

Fig. 5 .

 5 Fig.5.12 shows the frequency-dependent conductivity for five concentrations (0.05, 0.1, 0.2, 0.4, 0.8 M), for the fixed inter-wall L = 100 Å, and the same type of repulsive walls set by the potential given in Eq. (5.6), resulting the effective confinement length L = 94.5 Å. The concentrations studied with the systems E1 (Tab. 5.1), ranging from

Figure 5 . 13 :Figure 5 . 14 :

 513514 Figure 5.13: Conductivity spectrum of confined electrolytes: Frequency-dependent conductivity σ(ω) normalized by the ideal conductivity σ NE for a wide range of concentrations and several distances between the walls (see Tab. 5.1 for the definition of the systems labeled EI, EII, and EIII). The results are shown as a function of the frequency scaled by the Debye time τ Debye and coloured according to the ratio πλ D / L, with λ D the Debye length. The vertical dotted line indicates ωτ Debye = 1, which corresponds to the crossover between low-and high-frequency regimes in the limit πλ D / L ≪ 1 [208].

Fig. 5 .

 5 Fig. 5.14 shows the value of ω * Eq. (5.32) for the corresponding spectra in Fig. 5.13. Interestingly, all our numerical results are very well described by an analytical prediction (also shown in Fig. 5.14) following from the work of Bazant et al. who solved the Poisson-Nernst-Planck (PNP) equation to analyze the charge dynamics in electrical double layer capacitors [76], and deduced characteristic relaxation time. This work generalizes the RC time τ RC = λ D L D (5.39)

Figure 5 . 15 :

 515 Figure 5.15: Harmonic generation for large-amplitude and low-frequency electric fields. Normalized spectral density of the electrical current for various frequencies and constant amplitude forcings. Simulations are carried out for system R20 (see Tab. 5.1) with external electric field of amplitude E 0 = 0.1 V /Å and frequency f ∈ {0.4, 8.0, 80} GHz (see Sec. 5.1). When the frequency of the external field is high (blue), the power spectral density (PSD, Eq. (5.43)) displays a single peak at the same frequency. Upon decreasing the frequency (green and red), the PSD also displays peaks at harmonics of the fundamental frequency which is that of the applied field.

. 50 )

 50 These results are in very good agreement with the simulation results presented in Fig.5.16.

Figure 5 . 16 :

 516 Figure 5.16: Characterization of the nonlinear regime: J max (disks), J rms (squares) and J ω (triangles) (see Eq. (5.44)) computed from NEQ simulations for system R20 (see table Tab. 5.1, L = 20 Å) and various parameters E 0 , ω of the electric field (Eq. (5.1)). Values are plotted as a function of the inverse of the frequency scaled by the migration time τ E (see Eq. (5.48)). The three currents are normalized by the maximal current for ideal particles in the absence of confinement by the walls (bulk case), V σ NE E 0 (see Eq. (5.45)), as in Fig. 5.5, for all the simulations corresponding to Fig. 5.6 where only the frequency limited to the range ωτ diff ≳ 2 are shown here. Dashed lines indicate predictions from the athermal saturation model Eq. (5.50) [208].

  . 5.1, L = 20 Å) and various parameters E 0 , ω of the electric field (Eq. (5.1)). Values are plotted as a function of the inverse of the frequency scaled by the migration time τ E (see Eq. (5.48)). The three currents are normalized by the maximal current for ideal particles in the absence of confinement by the walls (bulk case), V σ NE E 0 (see Eq. (5.45)), as in Fig. 5.5, for all the simulations corresponding to Fig. 5.6 where only the frequency limited to the range ωτ diff ≳ 2 are shown here. Dashed lines indicate predictions from the athermal saturation model Eq. (5.50) [208].

Figure 5 . 17 :

 517 Figure 5.17: Réalisation expérimentale de mesure de courant ionique à travers un nanotube unique en nanofluidique: a) Fabrication du dispositif: incorporation du nanotube de nitrure de bore (BNNT, gris) dans une membrane de nitrure de silicium (SiN, vert) et colmatage en carbone (orange) par déposition induite par faisceau d'electrons. (1-4) : Insertion, scellage, rétractation, et nanotube transmembrane finalisée (gauche : schéma, droite : image correspondante par microscopie électronique en transmission (MET)). b) Haut : schéma du dispositif de mesure de courant. Bas : schéma du nanotube transmembrane SiN-BNNT sur la gauche, image correspondante par MET à droite.

  Cependant, la modélisation des mécanismes sous-jacents est d'une grande complexité, notamment à cause d'une grande variété de phénomènes se manifestant sur des échelles de temps et d'espace extrêmement étendues. De plus, à l'échelle nanométrique, les descriptions continues traditionnellement employées pour les fluides cessent d'être valides, pour laisser place à des descriptions discrètes de types atomiques et/ou moléculaires.

Figure 5 . 18 :

 518 Figure 5.18: Limite des descriptions continues des fluides a l'échelle nanométrique : aperçu des différents éléments et mécanismes selon l'échelle à considérer [5].

Figure 5 . 19 :

 519 Figure 5.19: Modèles d'électrolyte: illustration de la description tout-atome à gauche, décrivant explicitement les ions commes des particules chargées (en vert et bleu) ainsi que les molécules d'eau (rouge et blanc), dit de solvant explicite. A droite le modèle dit de solvant implicite, où nous ne considérons que les ions explicitement, le solvant étant décrit par un milieu continu. Image réalisée par G. Pireddu.

× 4 Figure 5 . 20 :

 4520 Figure 5.20: Facteurs de structure dynamiques: pour un électrolyte symétrique 1:1. (a) facteur de structure de la densité d'ions. (b) Facteur de structure de la densité de charge. La palette de couleurs représente l'amplitude du facteur de structure dynamique en fonction du vecteur d'onde et de la fréquence (k, ω), normalisé (4×) le temps de Debye 4Dκ 2 .

Figure 5 . 21 :

 521 Figure 5.21: Fluctuations ioniques en volume fini : Illustration des observables considérées dans la section 4.2. Pour un électrolyte bulk dans un volume infini, on propose d'étudier dans un sous-volume fini, les variations du nombre d'ions N et de charge totale Q contenus dans ce volume. Les corrélations statistiques temporelles et spatiales de ces observables présentent des propriétés exotiques qui proviennent la dynamique microscopique des ions[START_REF] Hoang | Ionic fluctuations in finite volumes: fractional noise and hyperuniformity[END_REF]. Image réalisée par S. Marbach.

Figure 5 . 22 :

 522 Figure 5.22: Limite basse fréquence du facteur de structure de charge dynamique en fonction du vecteur d'onde :Comparaison entre simulations de dynamique moléculaire (MD, réalisées par J. Kim) et de dynamique Brownienne, prédictions analytiques du modèle de Poisson-Nernst-Planck et modèle balistique (aux très grands vecteurs d'onde), et prédiction semi-analytique de la DDFT permettant d'améliorer les prédictions dynamiques au-delà de l'approximation du champ moyen, en incorporant les corrélations statiques calculéess par simulations[START_REF] Hoang | Electrical noise in electrolytes: a theoretical perspective[END_REF].

19 Figure 5 . 24 :

 19524 Figure 5.24: Conductivité dépendante de la fréquence : transition isolant / conducteur pour un électrolyte idéal confiné. Comparaison entre la méthode de simulation hors équilibre, à équilibre avec la formule de Green-Kubo, et la prédiction exacte obtenue par résolution de l'équation de Fokker-Planck, montrant la convergence des méthodes dans le régime linéaire. σ NE , τ diff , τ E sont repectivement la conductivité de Nernst-Einstein pour un électrolyte idéal bulk, le temps de diffusion entre les deux parois, et le temps de migration correspondant[START_REF] Hoang | Frequency and field-dependent response of confined electrolytes from Brownian dynamics simulations[END_REF].

Figure 5 . 25 :

 525 Figure 5.25: Valeur de la fréquence de transition isolant / conducteur pour un électrolyte confiné : fréquence de transition ω * normalisée par le temps de Debye τ Debye , en fonction du rapport entre la longueur de Debye et la distance de confinement. Résulats de simulations en points colorés, prédiction analytique en orange en identifiant ω * = 1/τ Bazant[START_REF] Bazant | Diffuse-charge dynamics in electrochemical systems[END_REF]. Les lignes en pointillés montrent les régimes asymptotiques dominés par la relaxation électrostatique pour les doubles couches électriques minces / faibles confinements (λ D ≪ L en bleu), et par le confinement géométrique pour les doubles couches électriques épaisses / forts confinements (λ D ≫ L en rouge)[START_REF] Hoang | Frequency and field-dependent response of confined electrolytes from Brownian dynamics simulations[END_REF].

  Ionic transport at the nanoscale . . . . . . . . . . . . . . . . . . . 2 1.2 Electrical current noise in nanopores . . . . . . . . . . . . . . . . . 2 1.3 Modelling electrical current noise in electrolytes . . . . . . . . . 5 1.3.1 Historical approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.4 Main objectives and plan . . . . . . . . . . . . . . . . . . . . . . . .

						Chapter 1
					Introduction
	Contents
	1.1 14
	1.3.6 Fluctuations:	a	link	between	microscopic	dynamics
	and macroscopic response

6 1.3.2 A challenge of size . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3.3 Electrostatic screening in electrolytes . . . . . . . . . . . . . . . . . . 9 1.3.4 Transport in electrolytes . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.3.5 Simulation methods . . . . . . . . . . . . . . . . . . . . . . . . . . .

  2.2.

	Pair coefficients : (ϵ ij , σ ij , D ij )	Na +	Cl -
		0.35264 kcal/mol	0.06715 kcal/mol
	Na +	2.1595 Å	3.4950 Å
		1.28 m 2 /s	0 m 2 /s
		0.06715 kcal/mol	0.01279 kcal/mol
	Cl -	3.4950 Å	4.8305 Å
		0 m 2 /s	1.54 m 2 /s
	Table 2.2: Pair potential coefficients and diffusion coefficients for Sec.4.3: Lennard-
	Jones (Eq. (2.14)) parameters taken Ref. [175], and diffusion coefficients are tailored from MD
	simulation, with SPC/E water model, extracted from infinite dilution mean-square-displacement
	(MSD) Ref. [7]. The null value of inter-diffusion coefficients indicates the lack of hydrodynamic
	mediated interactions.

  Derivation from the Langevin equation . . . . . . . . . . . . . . . . . 50

	Chapter 3
	Dynamical density functional theory
	for electrolytes
	Contents
	3.1 Stochastic dynamical density functional theory . . . . . . . . . .	50
	3.1.1

3.1.2 Mean-field approximation . . . . . . . . . . . . . . . . . . . . . . . . 52 3.1.3 Phenomenological interpretation . . . . . . . . . . . . . . . . . . . . 52 3.1.4 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.2 Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.2.1 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.2.2 Binary electrolytes and normal modes . . . . . . . . . . . . . . . . . 56

  Contents 4.1 Diffusion and electrostatic relaxation . . . . . . . . . . . . . . . . 60 4.1.1 Predictions for a 1:1 symmetric electrolyte . . . . . . . . . . . . . . . 60 4.1.2 Simulation details . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 salt concentration . . . . . . . . . . . . . . . . . 62 4.2 Ionic fluctuations in finite volumes . . . . . . . . . . . . . . . . . . 63

	4.1.3 Comparison	between	sDDFT	and	BD	simulations,
	the effect of

4.2.1 Simulation details . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.2.2 Number and charge fluctuations in arbitrary dimensions . . . . . . . 65 4.2.3 Hyperuniformity and arbitrary spherical volume . . . . . . . . . . . 67 4.2.4 Time-correlations and dynamics in electrolytes . . . . . . . . . . . . 70 4.3 Implicit vs explicit solvent . . . . . . . . . . . . . . . . . . . . . . . 74 4.3.1 Molecular, Langevin and Brownian Dynamics . . . . . . . . . . . . . 74 4.3.2 Simulation details . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.3.3 Static properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.3.4 Dynamical properties . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.3.5 Extensions of DDFT . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.4 Driven regime and asymmetric diffusion coefficients . . . . . . . 80

Table 4 . 1 :

 41 Simulated salt concentrations and associated parameters. The relevant parameters for the electrostatic relaxation: the Debye length and the Debye time (Sec. (4.1.1)). The packing fraction ϕ = Ntotvions

	3 πr 3 ions C s

Vsim

=

Table 4 . 2 :

 42 Main physical parameters for all simulated systems: number of ion pairs, salt concentration, Debye screening length and packing fraction.

				packing fraction
	32	13	2.7	0.17 %
	64	26	1.9	0.35 %
	128	52	1.3	0.71 %
	256	104	0.95	1.4 %
	512	207	0.67	2.8 %

  Contents 5.1 Modelling the dynamics of confined electrolytes . . . . . . . . . . 86 5.1.1 Modelling solid walls . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.1.2 Simulations details . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.1.3 Nonequilibrium electrical current . . . . . . . . . . . . . . . . . . . . 92 5.1.4 Conductivity under confinement . . . . . . . . . . . . . . . . . . . . 94 5.1.5 Frequency-dependent conductivity from equilibrium simulations . . . 96 5.1.6 Comparison of methods . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.2 Interpreting conductivity spectra . . . . . . . . . . . . . . . . . . . 103 5.2.1 Ideal electrolytes: effect of confinement . . . . . . . . . . . . . . . . 103 5.2.2 Ideal electrolytes: effect of adsorption . . . . . . . . . . . . . . . . . 107 5.2.3 Electrostatic interactions between ions . . . . . . . . . . . . . . . . . 109 5.2.4 Ideal electrolytes: effect of confinement in the nonlinear regime . . . 113

Table 5 . 2 :

 52 Crossover frequencies: Parameters (see Tab. 5.1), predictions and simulation results

	L2 (rad/ns)	Measured ω * (rad/ns)

Table 5 . 3 :

 53 Crossover frequencies Parameters (see Tab. 5.1) with fixed distance L = 20 Å, naive predictions and simulation results (see Fig. 5.11) for the crossover frequency.

Remerciements

Publications from this work [START_REF] Hoang | Frequency and field-dependent response of confined electrolytes from Brownian dynamics simulations[END_REF] Thê Hoang Ngoc Minh, Gabriel Stoltz, and Benjamin Rotenberg. Frequency-and field-dependent response of confined electrolytes from Brownian dynamics simulations. The Journal of Chemical Physics, 158 [START_REF] Feng | Single-layer MoS2 nanopores as nanopower generators[END_REF], 03 104103, 2023 [START_REF] Hoang | Electrical noise in electrolytes: a theoretical perspective[END_REF] Thê Hoang Ngoc Minh * , Jeongmin Kim * , Giovanni Pireddu, Iurii Chubak, Swetha Nair, and Benjamin Rotenberg. Electrical noise in electrolytes: a theoretical perspective.

To appear in Faraday Discussions (2023) [START_REF] Hoang | Ionic fluctuations in finite volumes: fractional noise and hyperuniformity[END_REF] Thê Hoang Ngoc Minh, Benjamin Rotenberg, and Sophie Marbach. Ionic fluctuations in finite volumes: fractional noise and hyperuniformity. To appear in Faraday Discussions (2023)