Daniele Belardinelli

Mauro Sbragaglia

Roberto Benzi

Sergio Ciliberto

Lattice Boltzmann Simulations of Non-Equilibrium Fluctuations in a Non-Ideal Binary Mixture

In the recent years the lattice Boltzmann (LB) methodology has been fruitfully extended to include the effects of thermal fluctuations. So far, all studied cases pertain equilibrium fluctuations, i.e. fluctuations with respect to an equilibrium background state. In this paper we take a step further and present results of fluctuating LB simulations of a binary mixture confined between two parallel walls in presence of a constant concentration gradient in the wall-to-wall direction. This is a paradigmatic set-up for the study of non-equilibrium (NE) fluctuations, i.e. fluctuations with respect to a nonequilibrium state. We analyze the dependence of the structure factors for the hydrodynamical fields on the wave vector q in both the directions parallel and perpendicular to the walls, highlighting the long-range (∼ |q| -4 ) nature of correlations in the NE framework. Results at the small scales (high wavenumbers) quantitatively agree with the predictions of fluctuating hydrodynamics without fitting parameters. At larger scales (low wavenumbers), however, results show finite-size effects induced by confinement and call for further studies aimed at controlling boundary conditions in the fluctuating LB framework as well as compressibility effects. Moreover, in presence of a non-ideal (NI) equation of state of the mixture, we also observe that the (spatially homogeneous) average pressure changes, due to a genuinely new contribution triggered by NE fluctuations. These NE pressure effects are studied at changing the system size and the concentration gradient. Taken all together, we argue that the results of this article are useful and instrumental to boost the applicability of the fluctuating LB methodology in the framework of NE fluctuations, possibly in conjunction with experiments.

I. INTRODUCTION

The equations of fluctuating hydrodynamics supplement the deterministic equations of hydrodynamics with the effect of thermal fluctuations [1]. In a nutshell, the key idea is that whenever scales of observations are small enough, thermal fluctuations cannot be ignored anymore and the non-equilibrium (NE) fluxes in the conservation equations (i.e. diffusion, viscous, etc) need to be promoted to stochastic variables. By linearizing with respect to a homogeneous background and applying the fluctuation dissipation theorem (FDT), one obtains the structure factors for the hydrodynamical fields in agreement with the corresponding statistical mechanics predictions [START_REF] Reichl | A modern course in statistical physics[END_REF]. Away from criticality, correlations of hydrodynamical fields such as mass density and velocity come out to be short-ranged, and the experimental observations with light scattering and neutron scattering techniques confirm such predictions [START_REF] Berne | Dynamic Light Scattering with Applications to Biology, Chemistry, and Physics[END_REF][START_REF] Earnshaw | [END_REF][START_REF] Earnshaw | Proc. R. Soc. Lond. A[END_REF]. The assumption of full (thermodynamic) equilibrium of the background system greatly simplifies the theoretical approach to the study of thermal fluctuations, but is actually inappropriate in many situations where we have mechanical equilibrium even in presence of temperature or concentration gradients. This may be the case of a Rayleigh-Bénard cell [START_REF] Law | [END_REF]7] or the case of a binary mixture under the effect of an external field [8][9][10][11][12]. For such systems, the theory of equilibrium thermal fluctuations can be extended [START_REF] Ortiz De Zárate | Hydrodynamic fluctuations in fluids and fluid mixtures[END_REF] to predict fluctuations with respect to a non-equilibrium steady state. In general, NE effects are promoted by two sources: one source is the "mode" coupling between the fluctuating velocity and the background inhomogeneous scalar field under consideration, the temperature in single-component fluids [START_REF] Kirkpatrick | [END_REF][15][16][17][18][19], and both the temperature and the concentration for mixtures [20][21][22]. Another source can be identified in the spatial inhomogeneity of the thermal noise, due to the proportionality of the noise correlations to the temperature [23][24][25][START_REF] Tremblay | Recent Developments in Nonequilibrium Thermodynamics[END_REF], as stated by FDT. Typically, the effects induced by inhomogeneity in the noise are negligible with respect to the mode coupling effect [START_REF] Ortiz De Zárate | [END_REF]. The mode coupling effect causes the small-scale behaviour of scalar fluctuations to be divergent as ∼ |q| -4 , with q being the Fourier wave vector. This was first obtained in [START_REF] Kirkpatrick | [END_REF] in the framework of non-equilibrium statistical mechanics and later assessed in the framework of fluctuating hydrodynamics [15,19]. Experimental confirmations followed [6-9, 12, 28]. We emphasize that NE fluctuations cause long-range correlation effects; similar long-range correlation effects are absent in equilibrium situations, except close to criticality [29]. Moreover, the long-range nature of the NE effects causes fluctuation-induced forces. This feature has been extensively discussed in the literature [30][31][32][33][34][35]. Similar NE pressure effects have also been studied in the non-linear Navier-Stokes equations with imposed shear rate [36][37][38][39], where the NE effects are triggered by the non linear coupling between the imposed shear rate and the flow itself. Setup for the numerical simulations. The wall-to-wall distance is L and we take the convention that z = 0 indicates the center of the channel. A linear concentration background profile c0(z) = 1/2 + z∇c0 is imposed, corresponding to a constant concentration gradient ∇c0 in the vertical direction. The vectors v1-8, together with v0 ≡ 0, act as lattice links in the D2Q9 LB simulations.

Thermal fluctuations become relevant at mesoscales, where many complex hydrodynamic phenomena occur, like for example the motion of non-ideal (NI) interfaces [40], the coupling between colloidal particles and the fluid [41,42], the rheology of vesicles and red blood cells [43][44][45], the acoustic-magnetic effect in magnetic fluids [46]. The need of understanding complex hydrodynamic phenomena at mesoscales naturally sets a compelling case for the development of suitably designed numerical methods. Beyond the numerical simulations based on the continuum equations of hydrodynamics [47,48], in the recent years mesoscale simulations based on the lattice Boltzmann (LB) [49,50] have been proposed [START_REF] Gross | Journal of Statistical Mechanics: Theory and Experiment[END_REF][START_REF] Dünweg | [END_REF][53]. The LB method stands out due to its remarkable capability of handling complex boundary conditions and NI fluids with phase transitions/segregation [START_REF] Dünweg | Advanced Computer Simulation Approaches for Soft Matter Sciences III[END_REF][START_REF] Aidun | [END_REF][56][57][58]; hence the LB coupled with thermal fluctuations is a promising pathway for realizing powerful mesoscale simulation methods. The idea of including noise in LB, in fact, has constituted an active research field of the recent years [START_REF] Gross | Journal of Statistical Mechanics: Theory and Experiment[END_REF][START_REF] Dünweg | [END_REF][53][59][60][61][62]. All these implementations, however, consider hydrodynamical systems fluctuating around a state in full equilibrium. The aim of the present paper is to explore the applicability of the fluctuating LB in the context of NE fluctuations. While none of the approaches proposed in the literature [START_REF] Gross | Journal of Statistical Mechanics: Theory and Experiment[END_REF][START_REF] Dünweg | [END_REF][53][59][60][61][62] can be trivially extended to the case with temperature gradients in the background, in [62] it is discussed how to formulate noise in multicomponent systems, even in presence of an inhomogeneous background concentration c 0 (r). Numerical simulations showed convincing agreement between the numerically evaluated equilibrium structure factors and the theoretical predictions. The latter, which can be obtained directly in the kinetic framework (see [62]), coincide with the predictions of fluctuating hydrodynamics. However, this is obviously not enough to prove convergence of fluc-tuating LB towards fluctuating hydrodynamics. Indeed, the stochastic noise terms break one of the basic assumptions of Chapman-Enskog theory (i.e. having fields slowly varying in space and time). Hence, the coincidence of theoretical results (kinetic framework vs. hydrodynamics framework) seems like a lucky case, possibly valid in homogeneous cases. Hence, investigating NE in LB simulations, is also a further way to highlight the convergence of fluctuating LB towards fluctuating hydrodynamics. The article is organized as follow. In section II the system and its governing equations are presented. The used methodology is described in section III. The numerical results, both in equilibrium and out of equilibrium, are discussed and compared with the theoretical predictions in section IV. We conclude in section V. The appendix recalls some relevant definitions and results on the structure factors.

II. SYSTEM

In this paper we study the problem of NE fluctuations by considering a two dimensional binary mixture confined between two walls in presence of a constant concentration gradient ∇c 0 in the wall-to-wall direction (see Figure 1). The reference fluctuating hydrodynamical equations for density δρ = ρ -ρ, concentration δc = c -c 0 and velocity δU = (U x , U z ) fluctuations are

∂ t δρ + ρ∇ • δU = 0, (1) ρ
(∂ t δc + U z ∇c 0 ) = ρD∇ 2 δc -∇ • J , (2)
ρ∂ t δU = -∇δP + ρν(∇ 2 δU + ∇∇ • δU ) -∇ • Π, ( 3
)
where δP is the pressure fluctuation, while ρ, D and ν are reference values for total mass density, mass diffusion coefficient and kinematic viscosity, respectively. Equations

a T al ma m eq a (ρ, U ) λa 0 1 ρ ρ λ0 1 (v l )x jx ρUx λd 2 (v l )z jz ρUz λd 3 3|v l | 2 -2 e 3ρ|U | 2 λe 4 (v l ) 2 x -(v l ) 2 z Σww ρ(U 2 x -U 2 z ) λs 5 (v l )x(v l )z Σxz ρUxUz λs 6 (3|v l | 2 -4)(v l )x Qx 0 λQ 7 (3|v l | 2 -4)(v l )z Qz 0 λQ 8 9|v l | 4 -15|v l | 2 + 2 0 λ Table I.
Moments set for the D2Q9 model used in the LB simulations. The index S of the species has been omitted. As the set of velocities is finite, the set of T al forms a basis. The moments ma are computed according to Eq. (7). They relax toward their respective asymptotic values according to a time scale 1/λa. The first three rows cover the conserved moments.

(1), ( 2) and ( 3) describe respectively conservation of total mass, diffusion of one species into the other and momentum balance, in their linearized form (see Eqs. ( 17)-( 19) below). The terms J and Π are the stochastic contributions to the deterministic equations of hydrodynamics in a Langevin-like approach [63][64][65][START_REF] Zwanzig | Nonequilibrium statistical mechanics[END_REF]. Specifically, J is a stochastic flux and Π is a stochastic stress tensor satisfying FDT:

J i (r, t)J j (r , t ) = 2kBT ρDχδ ij δ(r -r )δ(t -t ), (4)
Π ij (r, t)Π kl (r , t ) = 2kBT ρν∆ ijkl δ(r-r )δ(t-t ), [START_REF] Earnshaw | Proc. R. Soc. Lond. A[END_REF] where δ ij is the Kronecker delta, ∆ ijkl = δ ik δ jl + δ il δ jk , kB the Boltzmann constant and T the temperature, while χ indicates the inverse osmotic susceptibility: χ -1 = (∂µ/∂c) P,T , with µ the chemical potential and P the fluid pressure. If Eqs. (1)-(3) are integrated in time, the construction of the noise ensures that the proper thermal stationary state, characterized by some average properties, is reached in the limit of large times. Such a characterization will be given in section IV. We remark that, in the context of Langevin-type equations, the term "average" refers to the ensemble average over all the possible trajectories and is equivalent to the canonical ensemble average, provided that Eqs. ( 4) and ( 5) are satisfied. The very same technique is used to introduce fluctuations in the LB algorithm, as explained in the next section.

III. METHODOLOGY

The basic idea behind the LB methodology is to derive the equations of hydrodynamics from the more fundamental kinetic theory [START_REF] Shan | [END_REF]. In this section we briefly review the fluctuating multicomponent LB model that we use. Extensive details are reported in [62]. The model does not consider directly the hydrodynamic fields, but considers a kinetic description of a multicomponent fluid with two species, say A and B, having mass densities ρ A and ρ B . The corresponding total mass density is indicated with ρ = ρ A + ρ B , while mass concentration is conventionally taken as c = ρ A /ρ.

A. General Framework

The LB method makes use of a set of Q distribution functions f S l (r, t) (l = 0, . . . , Q -1), representing the number of particles of the species S = A,B at time t in an elementary lattice cell of unit volume around the position r moving with velocity v l . Mass densities are recovered as ρ S = l f S l [START_REF]species by setting each of them equal to unit[END_REF]. One then introduces the (isotropic) lattice spacing ∆r and the time interval ∆t to rescale positions and times, respectively. Coherently, velocities are rescaled by ∆r/∆t. Dimensionless variables will be noted in the same way as the variables themselves. In this way, while t varies on the natural set, the velocities v l act as links connecting the lattice points r (see Figure 1). The LB evolution is described by the following algorithm:

f S l (r + v l , t + 1) = f S l (r, t) + R S l (r, t). (6)
Here, R S l is the responsible for the change of f S l when moving along the link v l in a time step. It is better written in terms of the moments m S a (a = 0, . . . , Q -1), which are defined by the following invertible transformation [62]:

m S a = l T al f S l , f S l = w l a T al N a m S a . (7)
In table I it is reported the chosen set of T al for the D2Q9 lattice used in the simulations. This is a 2-dimensional lattice with Q = 9 velocities (see Figure 1): v 0 = (0, 0), respectively. While the lattice mass densities coincide with their physical counterpart, the physical baricentric velocity U is constructed as [62] ρU = (ρU x , ρU z ) = j

v 1 = (1, 0) = -v 3 , v 2 = (0, 1) = -v 4 , v 5 = (1, 1) = -v 7 , v 6 = (-1, 1) = -v 8 ,
A + j B + 1 2 ρa. (8)
The additional term is a correction related to the time discretization and involves the effective body-force density ρa acting on the fluid. We can write ρa = ρ A a A + ρ B a B , and decompose each term in the sum of non-ideal (NI) and non-equilibrium (NE) contributions by writing for each species a S = a S NI +a S NE . The former is constructed on the lattice and takes the form [START_REF] Shan | [END_REF][70][71][72][73]

a A NI (r, t) = -G l w l v l ρ B (r + v l , t), (9)
and an analogous expression holds for a B NI , having B replaced by A on the rhs. The positive constant G is the same for both the species and is a tunable parameter in the model [74]. It regulates the intensity of interactions between the two fluids, which are assumed to be separately ideal (in the expression of a A NI only ρ B appears). This produces a NI contribution in the equation of state (see Eq. ( 16) below). The NE contribution is chosen in such a way that it imposes a concentration gradient ∇c 0 = (0, ∇c 0 ) in the steady state, which is important for the study of NE effects:

a A NE = 1 3 (0, ∇c 0 /c), (10)
where the prefactor has been conveniently chosen equal to the lattice speed of sound for the D2Q9 model, that is 1/3 [START_REF] Shan | [END_REF]. The analogous expression for a B NE is obtained by replacing ∇c 0 /c with -∇c 0 /(1 -c) = ∇c 0 /(c -1). Notice that the momentum balance and consequently the pressure are unaffected by the NE forcing, since

ρ A a A NE + ρ B a B NE = 0. (11)
In this way, the NE acceleration (10) gives a contribution in the diffusion current proportional to ρ∇c 0 (see Eq. ( 18) below), thus fixing ∇c = ∇c 0 in the stationary steady state [75]. Moments for a = 3, . . . , 5 are related to the viscous stress tensor, while higher order moments have no hydrodynamical counterpart and constitute the so-called "ghost" sector (see table I). Close to a local equilibrium state only the first moments contribute, as we can write m S a = m eq a (ρ S , U ), with the equilibrium hydrodynamical moments m eq a (ρ, U ) given in table I. With these ingredients, the last term in ( 6) can be written as

R S l = w l a T al N a (C S a + F S a + ξ S a ). ( 12
)
The first term in the round brackets models the relaxation towards the local equilibrium:

C S a = λ S a [m eq a (ρ S , U ) -m S a ]. (13)
The dimensionless constants λ S a are the lattice relaxation frequencies, 1/λ S a being the corresponding lattice relaxation times. This is the multiple relaxation times (MRT) generalization of the celebrated BGK (for Bhatnagar, Gross and Krook [76]) form of the Boltzmann collision integral. All the λ S a are tunable parameters of the model, with some restrictions imposed by the request of mass and momentum conservations. Since m eq 0 (ρ S , U ) = m S 0 = ρ S , mass conservation is ensured for each species separately, independently on the actual value of λ S 0 . Instead, the second argument of the equilibrium distribution in ( 13) is the baricentric velocity U of equation (8), allowing in this way the diffusion of one species into the other. Conservation of total momentum is then enforced by conveniently choosing λ S 1,2 = λd, where the lattice diffusion relaxation frequency λd is the same for both the species. Similarly, the lattice relaxation frequencies associated to the shear moments Σ ww and Σ xz (a = 4, 5, see table I) are chosen as λ S 4,5 = λs, with the lattice shear relaxation frequency λs being the same for both the species. The second term in the round brackets of equation ( 12) models the action of the interactions between the fluid particles. The first order moments (a = 1, 2) are given by

(F S 1 , F S 2 ) = ρ S a S .
We omit the expression of the moments of order higher than one, for shortness, by remarking that they must be included for a proper simulation of a non-homogeneous fluctuating system [62].

The last term in the round brackets of ( 12) accounts for thermal fluctuations. These are modelled with zeromean Gaussian random variables, uncorrelated in time and with constant covariances (which can however depend on r). The derivation of the precise expression of the noise covariances has been achieved in [62]. It makes use of the FDT directly applied at the kinetic level. The covariance matrix appears to be diagonal in both moments [77] and space, as well as in time by construction, allowing us to write

ξ S a (r, t)ξ S a (r , t ) = ξ S ξ S a δ a,a δ r,r δ t,t . The quantities ξ S
a are arranged in the same way as the moments m S a . In particular, the stochastic injections of mass and momentum densities are ρ S ξ = ξ S 0 and j S ξ = (ξ S 1 , ξ S 2 ), respectively. As a direct consequence of mass conservation for each species it results that ξ S ξ S 0 = 0, coherently with an identically vanishing ρ S ξ . Momentum, instead, is not conserved separately for each species, due to diffusion effects. However, total momentum in conserved, so that j A ξ + j B ξ must be identically vanishing. Coherently, it is found that

ξ S ξ S 1,2 = -ξ A ξ B 1,2 = (2 -λd)λdkBT ρ A ρ B ρ . (14)
Higher order noise correlations have also to be taken into account. The only non vanishing are for S = S :

ξ S ξ S a = 3(2 -λ S a )λ S a N a kBT ρ S for a = 3, . . . , 8. ( 15
)
The factors ξ S ξ S a would depend on space through their dependence on the background fields ρ S . However, in order to focus on the mode coupling effects [START_REF] Ortiz De Zárate | [END_REF], we mainly performed simulations by keeping the ρ S in ( 14)-( 15) equal to their reference values. The effect of inhomogeneities in the ξ S ξ S a will also be highlighted in this study. Of central role in the derivation of the previous noise covariances are the properties of the noiseless (ξ S a ≡ 0) stationary state reached by the system. This is assumed to have the local equilibrium form m S a = m eq a (ρ S , 0) = ρ S δ a,0 , so that C S a = 0. Thus, by summing (6) over the species, using ( 11) and ( 9), and performing the continuum limit (formally, v l → 0), we get ∇ρ = -G∇(ρ A ρ B ). This can be written in the form ∇P = 0, allowing us to deduce the equation of state P = P (ρ, c) for the system at hand:

P = 1 3 (ρ A + ρ B ) + 1 3 Gρ A ρ B = 1 3 ρ + 1 3 Gρ 2 c(1 -c). (16)
The ideal equation of state P = ρ/3 is recovered by setting G = 0; recall that the factor 1/3 equals the D2Q9 lattice speed of sound [START_REF] Shan | [END_REF]. By applying the Chapman-Enskog procedure and treating the stochastic terms as generic external forces, one can prove [78] that the fluctuating hydrodynamic equations of a binary mixture with total mass density ρ, baricentric velocity U and mass concentration c are recovered [79] (the superscript denotes transposition):

∂ t ρ + ∇ • (ρU ) = 0, (17)
ρ(∂ t c + U • ∇c) = ∇ • [ρD∇(c -c 0 ) -J ], (18)
ρ(∂ t U + U • ∇U ) = -∇P +∇•[ρν(∇U + ∇U ) -Π], (19)
where J and Π are the noise fields whose variances are fixed by the Chapman-Enskog procedure, and thus satisfying FDT at kinetic level. The mass diffusion coefficient

D = 1 3 1 λd - 1 2 (20)
and the kinematic viscosity

ν = 1 3 1 λs - 1 2 (21)
respectively regulate the intensity of the diffusion fluxes and the viscous stresses [78] and are tunable in the model, by specifying λd and λs independently. Notice that the total mass density here is a dynamical variable. However, compressibility effects result to be small for the average profile, within an error of 1-2 %, hence by linearizing the equations around the background state one ends up with Eqs. ( 1)-( 3), with a reference total mass density ρ suitably chosen (see the end of next subsection). Summarizing, we use the LB solver described in [62] to simulate the hydrodynamical equations of a binary mixture in presence of a background stationary concentration gradient. If we trust the hydrodynamical limit of the LB model, we can then assess the properties of fluctuations by changing the background gradient ∇c 0 , the geometry used, the transport coefficients D and ν, and the interaction strength G that regulates the NI character of the mixture. We again remark that the fluctuating terms violate one of the basic assumptions of Chapman-Enskog theory (i.e. having fields slowly varying in space and time). We can only formally obtain Eqs. ( 17)- (19). Rather, the convergence towards the fluctuating hydrodynamical equations must be assessed via numerical simulations and a careful comparisons with the predictions of fluctuating hydrodynamics [32,80,81].

B. Set-up

We consider a two dimensional system with dimensions L x × L, with periodic boundary conditions in the stream-flow (x) direction and two solid walls located at z = ±L/2. The two dimensional choice is done to make the many computations affordable at changing L up to few tens of grid points. Indeed, the solutions of fluctuating hydrodynamics assume infinitely long parallel walls [START_REF] Ortiz De Zárate | Hydrodynamic fluctuations in fluids and fluid mixtures[END_REF]; hence, for a given L, the stream-flow lengthscale L x needs to be large enough to prevent spurious effects induced by periodicity. Regarding the boundary conditions, we choose the mid-way bounce back rule for the LB kinetic populations [START_REF] Succi | The Lattice-Boltzmann Equation[END_REF]: apart from small discrete effects, these provide a no-slip boundary condition for the tangential velocity (U x = 0) in absence of fluctuations. We also enforced exactly a zero normal velocity at the wall (U z = 0) at every time-step by properly readjusting the rest population at the wall. Regarding the boundary conditions on the concentration field, we impose that the densities of both components at the wall are equal to the neighboring fluid nodes [START_REF] Sbragaglia | [END_REF]. Both the noslip boundary condition and the conditions on the species densities (hence the boundary condition on concentration) are obviously changed by thermal fluctuations. To the best of the authors' knowledge there is no systematic study on the effects induced by thermal fluctuations on the LB boundary conditions and their hydrodynamic manifestations. A systematic study is surely warranted for the future. However, for the purposes of the present paper, we remark that boundary conditions affect the NE spectra only at large scales [32,80,81], while the smallscale behaviour is rather independent of the boundary conditions used. Moreover, regarding the large-scales, there are various solutions of fluctuating hydrodynamics that report the effects of hydrodynamic boundary conditions [32,80,81]. Thus, if from one side we can assess the universality in the small-scale behaviour, as a bonus we can also explore the importance of the boundary conditions by direct comparisons against analytical solutions available. From now on, when writing ρ we will mean that reference value for the total mass density such that, for given values of L x and L, the product ρL x L gives the total mass, which is exactly conserved by the algorithm. All the simulations are performed in such a way that c0 ≡ c 0 (z = 0) = 1 2 and we take c0 as the reference value for the concentration.

IV. RESULTS AND DISCUSSIONS

All the numerical results discussed in the following sections will be reported in LB units. In particular, in those units we set ρ = 1. We then set the other relevant physical parameters, such as concentration gradient, wall-towall separation, thermal energy and kinematic viscosity, to some "reference" base values in all the simulations, unless when explicitly varied. In both Sections IV A and IV B the reference values are ∇c 0 = 0.01, L = 32, kBT = 10 -4 , ν = 0.16667, D = 0.00427 as well as G = 0 (ideal binary mixture). The choice L x = 4L is enough to obtain negligible spurious effects for wavenumbers as small as 1/L. In Section IV C, to better highlight the NE effects induced to the average pressure, which increase with both the wall-to-wall distance and temperature (see Figure 10 below), we change L and kBT into L = 64 and kBT = 5×10 -4 . We also switch on non-ideal interactions, when needed, by setting G = 0.3; this guarantees that phase separation never occurs.

A. Equilibrium Fluctuations (∇c0 = 0)

The model that we use has already been extensively validated in unconfined homogeneous situations in [62]. Equilibrium fluctuations can be studied in Fourier space through the structure factors of the velocity and concentration fluctuations, respectively S Ux,z (q) and S c (q) (see Appendix), where q = (q x , q z ) is the wave vector. It is well known [START_REF] Ortiz De Zárate | Hydrodynamic fluctuations in fluids and fluid mixtures[END_REF] that for equilibrium fluctuations in unbounded domains the structure factors of both velocity and concentration fluctuations are independent of the wave vector. More quantitatively,

S Ux,z (q) = |U x,z (q)| 2 = kBT ρ , (22)
S c (q) = |δc(q)| 2 = 3 kBT ρ c0 (1 -c0 ). (23)
This corresponds to delta-like correlations in real space:

U x,z (z)U x,z (0) = kBT ρ δ z,0 , (24)
δc(z)δc(0) = 3 kBT ρ c0 (1 -c0 )δ z,0 . (25)
However, we will use confined simulations with wall boundary conditions for the NE fluctuations (∇c 0 = 0), it is therefore important to conduct a preliminary characterization of the equilibrium fluctuations (∇c 0 = 0) in such confined situations. We thus considered a homogeneous system (∇c 0 = 0) confined in a channel with fixed wall-to-wall separation L, and performed simulations at changing kBT in the range 10 -6 -10 -4 . The measured structure factors in both the streamflow, q = (q, 0), and the wall-to-wall, q = (0, q), directions are shown in Figure 2 on varying dimensionless wavenumbers q = qL,

whose smallest acceptable value for the wall-to-wall direction is 2π, by construction, while such a limitation is absent for the streamflow direction. The measured structure factors are close to the reference theory values in unconfined situations given in Eqs. ( 22)- (23). Notice that the velocity structure factors reported in Figure 2 do not exhibit the anisotropy typical of a divergenceless field. In particular, for a divergenceless velocity the scalar product q•δU (q) = q x U x (q x , q z )+q z U z (q x , q z ) identically vanishes, and consequently U x (q, 0) = U z (0, q) = 0 identically. This is not the case here, since the fluid at hand is not exactly incompressible. Some underestimates and anisotropy are observed, that we attribute to the presence of walls, which are not contemplated in Eqs. ( 22)- (23). Results reported in Figure 2 will serve as a reference case to quantify the importance of non equilibrium fluctuations. As shown in the following section, these properties are maintained by the velocities even in presence of a non zero concentration gradient, while the concentration itself exhibits long-range spatial correlations (see Figure 4).

B. Non-equilibrium fluctuations (∇c0 = 0)

In this section we start by describing the NE fluctuations. In Figure 3 we report results for the structure factors for the velocity and concentration fluctuations. We observe that the structure factors for the velocities U x,z (top and central panels) do not show any substantial change with respect to the corresponding equilibrium situation reported in Figure 2. For the fluctuations in the concentration field δc (bottom panel), instead, the structure factors are strongly anisotropic and S Ux (q, 0) S Ux (q, 0) S Ux (q, 0) S Ux (0, q) S Ux (0, q) S Ux (0, q) theory theory theory 10 -7 S Uz (q, 0)

S Uz (q, 0)

S Uz (q, 0) S Uz (0, q) S Uz (0, q) S Uz (0, q) Sc(q, 0) Sc(q, 0) Sc(q, 0) Sc(0, q) Sc(0, q) Sc(0, q) theory theory theory Figure 2. Spectra of fluctuations of hydrodynamical fields around an equilibrium background (∇c0 = 0). Top panel: Velocity in the streamflow (x) direction; theoretical prediction in Eq. (22). Central panel: Velocity in the wall-to-wall (z) direction; theoretical prediction in Eq. (22). Bottom panel: Concentration fluctuations; theoretical prediction in Eq. ( 23).

All the theoretical predictions refer to unbounded fluids. In all cases, the viscosity and diffusivity have been fixed to their reference values (cfr. Section IV).

mode-dependent, which markedly contrasts the observations in the equilibrium situation. Correspondingly, the S Ux (q, 0) S Ux (0, q) eq. theory S Uz (q, 0) S Uz (0, q) eq. theory Sc(q, 0) Sc(0, q) eq. theory Figure 3. Spectra of fluctuations of hydrodynamical fields around a NE background (∇c0 = 0). Top Velocity in the streamflow (x) direction; equilibrium theoretical prediction in Eq. (22). Central panel: Velocity in the wall-to-wall (z) direction; equilibrium theoretical prediction in Eq. (22). Bottom panel: Concentration fluctuations; equilibrium theoretical prediction in Eq. ( 23). In all cases, the simulation parameters have been fixed to their reference values (cfr. Section IV). effect on the correlations in real space is highlighted in Figure 4: the two-point correlation function for the velocity (data shown only for the stream-flow velocity U x ) does not show any net NE contribution; in contrast, the NE contribution of the two-point correlation for the concentration highlights a correlation length that essentially spans the whole system size. To characterize such NE fluctuations on a more quantitative basis, we therefore continue our analysis for the concentration field c in a "parallel flow approximation", i.e. by taking the Fourier mode along the stream-flow direction, q = (q, 0). To facilitate a comparison with the existing literature on NE fluctuations we adopt the commonly used decomposition [START_REF] Ortiz De Zárate | Hydrodynamic fluctuations in fluids and fluid mixtures[END_REF] S c (q, 0) = 3

kBT ρ c0 (1 -c0 ) 1 + φ SNE(qL) , (27)
where

φ = 1 3 L 4 c0 (1 -c0 ) (∇c 0 ) 2 (ν + D)D . ( 28
)
Starting from the data reported in Figure 3 and the decomposition ( 27), we extracted the function φ SNE(q). The results are reported in Figure 5. At small scales (q 1) we observe the power-law scaling SNE(q) ∼ q-4 . This is perfectly in agreement with the expected powerlaw behaviour SNE(q) = q-4 predicted by the theory of NE fluctuations, which can be obtained from the equations of hydrodynamics linearized around a constant concentration gradient profile, i.e. Eqs. ( 1)-( 3). We emphasize that we just used the decomposition [START_REF] Ortiz De Zárate | [END_REF] and added no additional fitting parameters. We checked the goodness of the asymptotic behavior by separately changing the concentration gradient ∇c 0 , the wall-to-wall separation L and the kinematic viscosity ν, while keeping φ SNE(q)

φ SNE(q) bulk theory bulk theory bulk theory 27)-( 28)) as a function of the dimensionless wavenumber q (see Eq. ( 26)); the theoretical prediction, corresponding to SNE(q) = q-4 , is obtained from Eqs. the diffusion coefficient D unvaried. This can be done in the simulations thanks to the MRT generalization of the BGK model (see Eq. ( 13)), which allows to set different relaxation frequencies for different moments (see Eqs. ( 20)-( 21)). The plots reported in Figure 5 show changes in agreement with the corresponding change of φ in Eq. (28). We also checked that the product φ SNE(q) does not depend on kBT in the range 10 -6 -10 -4 , thus confirming the linear dependence of S c (q, 0) on temperature in the factorization [START_REF] Ortiz De Zárate | [END_REF]. These results provide a very strong indication that the fluctuating LB methodology is quantitatively able to reproduce the small scale behaviour of NE correlations obtained from fluctuating hydrodynamics in presence of NE fluctuations at the small scales [30][31][32][33][34][35].

At the small scales where the prediction of fluctuating hydrodynamics well matches the results of the simulations (cfr. Figure 5), we inspected the effect of the spatial dependence of the noise correlations in Eq. ( 14)- (15). Some authors before investigated the relative importance of mode-coupling effects and non-homogeneity in noise [START_REF] Ortiz De Zárate | [END_REF]84]. In particular, they show that for a case with temperature (with thermal diffusivity D T in place of diffusivity D) the importance of the mode coupling effect with respect to the inhomogeneity in noise scales inversely proportional to the quantity (ν + D T )D T . In other words, if the inhomogeneous noise is switched-on in the simulations, the diffussivity and the viscosity need to be properly chosen to guarantee that results of fluctuating hydrodynamics with homogeneous noise hold. To highlight this, we conducted various numerical simulations at changing both ν and D, from small values to larger values. In Figure 6 we reported the ratio between φ S(hom) NE (q) and φ S(loc) NE (q), i.e. the non-equilibrium structure factors computed by implementing the noise according to ( 14)-( 15) and using constant reference values for the mass densities (hom) and those obtained using the local (loc) values. We see that when both the transport coefficients (ν and D) are very small, the two simulations provide the same results. Instead, by increasing both transport coefficients different results are observed. This is just a qualitative statement, as a quantitative study requires further numerical, as well as theoretical analysis. We remark, however, that the effects induced by inhomogeneity in the noise are typically negligible with respect to the mode coupling effect for many practical purposes (details are given in [START_REF] Ortiz De Zárate | [END_REF]).

Going at smaller q we observe in Figure 5 that the power-law scaling ∼ q-4 becomes progressively underestimated by the numerically computed SNE(q). This is attributed to finite-size effects induced by confinement. Indeed, due to the long-range nature of NE spatial correlations, NE structure factors are necessarily affected by the boundary conditions. There are various papers aimed at the quantitative characterization of SNE(q) in presence of boundary conditions [32,80,81]. The results of these calculations share the common feature that the powerlaw behaviour ∼ q-4 is approached only at very small

| S(hom) NE (q)/ S(loc) NE (q)| | S(hom) NE (q)/ S(loc) NE (q)| | S(hom) NE (q)/ S(loc) NE (q)| | S(hom) NE (q)/ S(loc) NE (q)| | S(hom) NE (q)/ S(loc) NE (q)| Figure 6
. We report the ratio between the non-equilibrium structure factors computed by implementing the noise according to ( 14)-( 15) and using the constant reference value for the mass densities (hom) and local space-dependent (loc) density. Different transport coefficient are considered. The simulation parameters not given in the legends are detailed in Section IV.

scales, i.e. SNE(q) ∼ q-4 only for q → ∞. The smallq behaviour strongly depends on the boundary conditions used for both velocity and concentration. In what follows, we discuss three analytical (or semi-analytical) expressions for SNE(q) that can be gathered from the literature. All of them treat the wall as impenetrable:

U z | z=±L/2 = 0.
This condition is strictly imposed in all the simulations performed. One can then impose either no-slip (NS) or free-slip (FS) boundary conditions for U x , and independently either insulating (I) or conducting (C) boundary conditions for δc:

(NS,I) : (U x , ∂ z δc)| z=±L/2 = 0, (FS,C) : (∂ z U x , δc)| z=±L/2 = 0, (NS,C) : (U x , δc)| z=±L/2 = 0, (29)
Depending on the boundary conditions in (29), different expressions for SNE(q) can be found in literature, as reported explicitly in Appendix. The solution that better fits the data reported in Figure 7 is (NS,C). This is reasonable: we use a bounce-back for the kinetic population, thus reproducing the no-slip condition in the hydrodynamical limit; moreover, since fluctuations in the concentration are triggered by kBT 1, one may also say that the conducting boundary condition fits well in those conditions where the concentration fluctuations are much smaller than the average concentration. However, we hasten to remark that a quantification of the boundary conditions with LB in presence of noise is currently missing in the literature. This surely stimulates further work in the future. q φ SNE(q) bulk theory (q -4 ) no slip, insulating (NS,I) free slip, conducting (FS,C) no slip, conducting (NS,C)

Figure 7. Non-equilibrium structure factor contribution as a function of the dimensionless wavenumber q (see Eq. ( 26)). Different analytical formulas are checked, dependently on the boundary conditions (29). The choice of the simulation parameters is detailed in Section IV.

Before closing this section, some remarks on the issue of compressibility are needed. All the reported analytical results [32,80,81] are derived in the incompressible regime. The fluid velocity in our system is not divergenceless (see the discussion on the velocity structure factors reported in Figure 2). Thus, one could wonder if compressibility plays a role. To this aim, recall that in the incompressible case the equation for U x can be decoupled by taking the double curl of Eq. ( 3) and using ∇ • δU = 0. The resulting equations for U z and c are the starting point for the computations performed in [32,80,81]. By following the same procedure, in our case we would get additional contributions in the momentum balance involving ∇(∇ • δU ), whose z-component in Fourier space reads -q z q • δU (q). These contributions are identically vanishing for q = (q, 0), thus giving the same bulk equations as in the incompressible case. Notice, however, that the incompressibility condition is routinely used in formulating a "closed" boundary condition in terms of the velocity component U z [32,80,81] and to the best of our knowledge, there is no analytical result for confined flows involving compressibility effects in the imposition of the boundary conditions. Therefore, at this stage, it is not clear how much of the observed behaviour at large scales depends on compressibility effects. This point surely stimulates further research for the future. Summarizing, the LB solver described in [62] generates a fluctuating hydrodynamical system that under the presence of a constant concentration gradient and homogeneous noise develops the typical long-range correlations characterizing NE fluctuations. Remarkably, neither fitting parameters nor corrective factors are needed to match numerics and analytical results at the sall scales. Confinement effects also seem captured, although a more systematic study of the boundary conditions emerging in the simulations is needed.

C. Non-Equilibrium Pressures

Recent papers of the literature [30][31][32][33][34][35] supported the fact that the long-range effects deriving from NE fluctuations (see Figure 4) cause a NE "Casimir" pressure. The rationale behind this effect hinges on the connection between the pressure and concentration fluctuations. In a nutshell, the local equilibrium assumption relates mass density and concentration to pressure through an equation of state P = P (ρ, c) satisfying ∇P = 0 (see equation (16) for the case at hand), which is expected to be still valid in average. Fluctuations of ρ and c are then accompanied by fluctuations of P that one expects to be vanishing at linear order [33,35,85]. By keeping the first non vanishing terms, one gets [33,35]

PNE(z) = 1 2 A c |δc(z)| 2 NE , (30)
where the vanishing of the linear order is used to express δρ in terms of δc. The constant A c plays the role of a second order coefficient, and is a function of the background fields computed at their respective reference values. Two important comments are in order. First, based on the prediction for the NE pressure in (30), one would expect NE Casimir pressures to be triggered by the nonideality of the mixture (see Eq. ( 16)): an ideal equation of state (G = 0) would just deliver A c = 0 and hence PNE = 0. Second, the NE correlation |δc(z)| 2 NE may be non homogeneous in space, depending on the choice of the boundary conditions [32,33,35]. Thus, the resulting NE pressure in (30) is space-dependent and one may wonder how this could be reconciled with an average mechanical balance. Indeed, the mode coupling effect triggers NE effects only in the concentration fluctuations, while velocity fluctuations are unchanged (see Figure 3); thus, one would expect the equilibrium condition of a constant (average) pressure to be recovered. As already pointed out [35], the mechanism of compensation is a NE renormalization of the background profile which provides a zero derivative of the total pressure. In other words, the pressure may be seen as the sum of an equilibrium contribution and the NE contribution of Eq. ( 30); z-dependency in the latter causes the former to be z-dependent in such a way that the total pressure has zero derivative. Based on the numerical model that we used, we are in a condition to test directly these properties. The setup to measure the NE "Casimir" pressure is the same used for the measure of the NE structure factors with the same boundary conditions. In practice, the total pressure is evaluated by its mechanical definition, that is as half of the trace of the pressure tensor [73], whose bulk behaviour is expected to coincide with (16) in the hydrodynamical limit. Results are reported in Figures 89and fully confirm the above views. Specifically, we fixed a non-zero concentration gradient ∇c 0 = 0.01, and we performed simulations with kBT > 0 and kBT = 0 for an ideal mixture (G = 0, Figure 8, top panel) and for a non-ideal mixture (G > 0, Figure 8, bottom panel). We observed that the total pressure profiles are homogeneous in z and that the pressure receives a correction by thermal fluctuations only when G > 0. Only when the pressure receives a correction, the average density profile slightly changes with thermal fluctuations. This can be seen in Figure 9, where we reported the deviation ∆ρ(z) of the average density profile with respect to its values in the noiseless case. Note that the effect on the density is quite small, and to point it out in the simulations we had to use large L and k B T to maximize it (see later discussion on the scaling laws of NE pressure). These facts said, we wanted to further characterize the NE Casimir pressure from our simulations, hence we sticked with a non-ideal mixture with fixed G > 0. The spatial average pressure will then depend on kBT , L and ∇c 0 , i.e. P = P (kBT, L, ∇c 0 ). To make progress we wanted to study the scaling properties of the NE Casimir pressure as a function of the system size L, concentration gradient ∇c 0 and kBT . We emphasize that fluctuations are expected to induce pressure effects also in equilibrium conditions (∇c 0 = 0), and that the latter effects are particularly large close to the critical point (critical "Casimir" pressure) and decay to zero at large L [29]. For the parameters chosen [86] the critical point corresponds to G = 2, while we kept G = 0.3 in all the non-ideal simulations. In such conditions thermal fluctuations only trigger some small effects in equilibrium conditions, that we detect only at the smallest L considered; however, aiming at characterizing the NE pressure at changing L, we needed to remove such small contributions. We proceeded as follows. For a given system size L, we first performed a numerical simulation in equilibrium conditions (∇c 0 = 0) without thermal fluctuation (kBT = 0); we then repeated the simulation with the desired kBT .

In both simulations we have computed the average pressure and we estimated the pressure difference induced by thermal fluctuations as [87] ∆P eq (kBT, L) = P (kBT, L, 0) -P (0, L, 0).

Then, for the desired ∇c 0 > 0, we performed two other simulations without thermal fluctuations (kBT = 0) and with the desired kBT . The NE contribution to the spatial average pressure has been identified as P NE(kBT , L, ∇c 0 ) = P (kBT, L, ∇c 0 ) -P (0, L, ∇c 0 )

-∆P eq (kBT, L).

In Figure 10 we plot the measured P NE as a function of ∇c 0 , L and kBT . While the scalings P NE ∼ (∇c 0 ) 2 and P NE ∼ kBT are in agreement with the theoretical predictions [30][31][32][33][34][35], the behavior of P NE ∼ L 2 reflects the two dimensional character of the system. This can be understood by looking at the unbounded behavior ∼ |q| -4 of |δc(q)| 2 NE in Fourier space. Indeed, the computation of the NE average pressure from Eq. ( 30) requires the integration of |q| -3 in a two dimensional system, in contrast with the integration of |q| -2 for a three dimensional system, as those considered in [30][31][32][33][34][35]. Consequently, if an infrared cutoff proportional to L -1 is introduced, then a two dimensional system furnishes ∼ L 2 , while a three dimensional system gives ∼ L. Predicting the offsets requires the complete control of the boundary conditions.

V. CONCLUSIONS

We applied the fluctuating lattice Boltzmann (LB) methodology described in [62] to a system out of thermodynamic equilibrium. Specifically, we considered a binary mixture confined between two parallel walls in presence of a constant concentration gradient in the wall-to-wall direction. We studied structure factors and spatial correlations of the velocity and concentration fluctuations, and found good agreement with the theoretical predictions of fluctuating hydrodynamics [START_REF] Ortiz De Zárate | Hydrodynamic fluctuations in fluids and fluid mixtures[END_REF]. We further inspected the behavior of the resulting NE pressure as a function of both the concentration gradient and the wall-to-wall distance, and verified the correctness of the corresponding expected scaling laws [32,33,35], in agreement with a constant average total pressure. The results here reported naturally warrant other future quantitative studies, both numerical and theoretical. In the context of the LB methodology, the analysis of the structure factors revealed the necessity of a better control in implementing the boundary conditions in presence of thermal fluctuations. Furthermore, the extension of the Chapman-Enskog procedure to the fluctuating case is missing. In this sense, the results of this paper support the convergence of fluctuating LB towards fluctuating hydrodynamics. In the context of finite size effects in the NE fluctuations, it would be interesting to further inspect the importance of compressibility effects for analytical solutions in presence of confinement, with the aim of comparing with the LB simulations. On a more general perspective, we remark that NE effects are continuously invoked in a variety of situations of experimental interest involving complex hydrodynamics. These include studies with colloidal suspensions [88][89][90][91], transient and enhanced diffusion effects [92][93][94][95], driven active matter [96], complex polymeric fluids [97], finite Reynolds numbers fluids [98]. In particular, for the future, it could be insightful to design experiments involving colloidal particles exhibiting a mechanical-chemical coupling with the fluid [99], in such a way that NE fluctuations effects can be indirectly reconstructed and studied from the particles trajectories. The LB methodology has proven capable of remarkable versatility in the simulation of colloidal particles [START_REF] Dünweg | Advanced Computer Simulation Approaches for Soft Matter Sciences III[END_REF][START_REF] Aidun | [END_REF][56][57][58], hence results of the present paper are instrumental for the use of LB as a validated methodology to support and complement experimental studies in the aforementioned direction.

Figure 1 .

 1 Figure1. Setup for the numerical simulations. The wall-to-wall distance is L and we take the convention that z = 0 indicates the center of the channel. A linear concentration background profile c0(z) = 1/2 + z∇c0 is imposed, corresponding to a constant concentration gradient ∇c0 in the vertical direction. The vectors v1-8, together with v0 ≡ 0, act as lattice links in the D2Q9 LB simulations.

  the associated weights being w 0 = 4/9, w 1-4 = 1/9 and w 5-8 = 1/36. The normalization constants are obtained as N a = l w l T 2 al . In particular, lattice mass and momentum densities are given by ρ

0 5 × 10 - 6 1 × 10 - 5 1. 5 × 10 - 5 2 × 10 - 5 2. 5 × 10 - 5 3 × 10 - 5 3. 5 × 10 - 5 -Figure 4 .

 51061055105105510510551054 Figure 4. Real space NE correlations of velocity and concentration fluctuations. They are obtained by subtracting to the measured correlations the equilibrium values reported in Eqs. (24)-(25). The simulation parameters have been fixed to their reference values (cfr. Section IV).

Figure 5 .

 5 Figure 5. Non-equilibrium structure factor contribution (see Eqs. (27)-(28)) as a function of the dimensionless wavenumber q (see Eq. (26)); the theoretical prediction, corresponding to SNE(q) = q-4 , is obtained from Eqs. (1)-(3) for unbounded systems. Top panel: NE structure factor contribution for different concentration gradients ∇c0. Central panel: NE structure factor contribution for different wall-to-wall separations L. Bottom panel: NE structure factor contribution for different kinematic viscosities ν. The simulation parameters not given in the legends are detailed in Section IV.

  Figure 5. Non-equilibrium structure factor contribution (see Eqs. (27)-(28)) as a function of the dimensionless wavenumber q (see Eq. (26)); the theoretical prediction, corresponding to SNE(q) = q-4 , is obtained from Eqs. (1)-(3) for unbounded systems. Top panel: NE structure factor contribution for different concentration gradients ∇c0. Central panel: NE structure factor contribution for different wall-to-wall separations L. Bottom panel: NE structure factor contribution for different kinematic viscosities ν. The simulation parameters not given in the legends are detailed in Section IV.

D

  = 0.00427, ν = 0.00433 : D = 0.00427, ν = 0.16667 : D = 0.01852, ν = 0.03333 : D = 0.5, ν = 0.33333 : D = 3.16667, ν = 0.66667 :

Figure 8 .Figure 9 .

 89 Figure 8. Average total pressure. Top Average total pressure for an ideal binary mixture (G = 0). Bottom panel: Average total pressure for a non-ideal binary mixture (G > 0). The choice of the simulation parameters is detailed in Section IV.

Figure 10 .

 10 Figure 10. Scaling laws for the NE contribution to the spatial average pressure computed according to Eq.(31). Top panel: NE pressure contribution as a function of the concentration gradient ∇c0. Central panel: NE pressure contribution as a function of the wall-to-wall separation L. Bottom panel: NE pressure contribution as a function of the thermal energy kBT . The choice of the simulation parameters not given in the legends is detailed in Section IV.

Appendix: Structure factors

In this appendix we report the essential technical details for the computations of the structure factors of a generic scalar field ϕ = U x , U z , c. Given the wave vector q = (q x , q z ), we started from the partial Fourier transform around a z-dependent background ϕ 0 (z):

Based on Eq. (6.30) in [START_REF] Ortiz De Zárate | Hydrodynamic fluctuations in fluids and fluid mixtures[END_REF], we defined the quantity C ϕ (q x , z, z ) through the equal-time mixed correlation:

where . . . indicates the ensemble average computed via the equal time average in the statistically stationary state. We then Fourier-transformed in z and z to define the structure factor (see Eq. ( 31) in [80]):

We can also write δϕ(q, t) * δϕ(q , t) = S ϕ (q)(2π) 2 δ(q -q ), which gives S ϕ (q) = |δϕ(q, t)| 2 on a two dimensional lattice, where (2π) 2 δ(q -q ) is replaced by δ q,q .

The decomposition [START_REF] Ortiz De Zárate | [END_REF] for the structure factor of the concentration field ϕ = c, computed in the stream-flow direction, q = (q, 0), can then be used to extract the NE contribution SNE(q), where q = qL. Depending on the boundary conditions at z = ±L/2 (see Eq. ( 29)), different expressions for SNE(q) are obtained. The (NS,I) solution found in Eq. ( 30) of [32] is valid for ν D. It predicts S(NS,I)

For the (FS,C) boundary conditions one can get an exact solution (see Eq. ( 35) in [80]):

where H(q, 0) = 1 -cosh q 4q sinh q 15 -7q sinh q + q2 1 -cosh q sinh 2 q .

(A.3) Details for (NS,C) are found in Eq. ( 20) of [81] (see also (7.36) in [START_REF] Ortiz De Zárate | Hydrodynamic fluctuations in fluids and fluid mixtures[END_REF]). In particular, this solution comes from a Galerkin truncation of exact equations. Hence it is a semi-analytical estimate [START_REF]This semi-analytical result underestimates by 20% the exact large q behavior[END_REF]. Quantitatively, it reads S(NS,C) NE (q) = 30 36

Sc + 1 Sc + Ã(q) 27q 2 28(q 2 + 10) [(q 2 + 12) 2 + 360] ,

(A.4) where Ã(q) = (q 2 + 12)(q 2 + 10) q4 + 24q 2 + 504 . (A.5)


Marion Real¹
email: marion@fablabbcn.org

Milena Juarez Calvo¹

Boosting co-creation practices in makespaces

Keywords: Fab Labs, Co-creation, Community Engagement, Sustainable Education, Eco-design

Introduction Makespaces as enabling spaces for local circular transformations

Circularity is one core value present in the manifesto of the Fab City network [START_REF] Diez | Fab City Whitepaper Locally productive, globally connected self-sufcient cities[END_REF]. The key guiding principles of Fab City are aligned with the primary directives of the Circular Economy Action Plan by contributing to the reduction of municipal waste and packaging by 2030 [START_REF] Diez | [END_REF]. Barcelona is one of the pioneers of the Fab City network. In July 2014, the city hosted the Fab10 annual conference in which Fab City Global Initiative was officially launched. In this occasion, the Barcelona City Council committed to the goal of producing locally half of everything consumed in the city by 2054. This initiative proposes a model for cities to be resilient, productive and self-sufficient in order to respond to the current societal challenges. It also aims at recovering the knowledge and the capacity to make things and empower citizens to become the leading agents of their own destiny.

At a local scale, Fab City initiative boost the use of community production spaces as hubs to connect citizens with existing products and services that foster new sustainable practices in consumption and production. According to Stewart and Tooze's definition (Stewart and Tooze, 2018), 'makespace' can be used as a catch-all term for an open access community fabrication workshop regrouping Fab Labs, Hackerspaces, (Re)Makespaces and other facilities with a suite of fabrication tools and technologies openly accessible for use by a community.

Meeting local needs through local inventiveness, these spaces have a great potential to foster synergies between different actors in a territory using open source and redistributed fabrication. [START_REF] Dewberry | Developing scenarios for product longevity and sufficiency[END_REF] asserts that a form of redistributed manufacturing composed of new, localised structures of design and manufacturing, could enable large reductions in resource consumption by limiting waste in a supply chain, and through addressing the flows of resources at critical times in the lifecycle of products.

The maker network has been growing empowering people and offering enabling opportunities through digital social innovation strategies. By developing research programs, maker communities are helping to create blueprints for a new generation of prototypes that are tailored to local circumstances. Following the idea of bringing opportunities for people to develop innovative solutions, the present study describes the combination of two methodologies to foster eco-innovative practices enabling citizens to develop skills and knowledge for territorial circular transformations.

Responsible Research Innovation and co-creation practices to support local community transformations

Responsible research and innovation (RRI) and co-creation have been considered as potential approaches to engage citizens and different levels of decision making authorities in collaborative processes. Responsible research and innovation (RRI) is a concept that has recently gained currency, particularly in Europe. According to the European Commission (2014), RRI means that societal actors work together during the whole research and innovation process in order to better align both the process and its outcomes, with the values, needs and expectations of European society. Von [START_REF] Schomberg | A Vision of Responsible Research and Innovation[END_REF] defines RRI as a transparent, interactive process by which societal actors and innovators become mutually responsive to each other with a view to the (ethical) acceptability, sustainability and societal desirability of the innovation process and its marketable products.

Recently RRI literature have introduced the perspective of co-creation, with the aim to associate these two areas of knowledge and cross potential connections between them. Co-creation methodologies, when applied in RRI, have the potential to efficiently engage citizens and civil society organizations, bringing on board the widest diversity of actors normally excluded from making decisions, on matters of science and innovation. Co-creation is defined as the participation of end-users in the process of innovation (von Hippel, 1987), in which knowledge and solutions are co-produced by different actors working in synergy, in an inclusive process for civil society and citizens [START_REF] Regeer | Knowledge co-creation: Interaction between science and society[END_REF].

Adapting the use of co-creation in RRI as a bottom-up approach to address local solutions, Fab Lab Barcelona has been developing a pilot within SISCODE EU project applying co-design tools to find a common language among different participants. SISCODE is an H2020-EU funded project aimed at stimulating the use of cocreation methodologies in policy design, using bottom-up design-driven methodologies to pollinate Responsible Research and Innovation (RRI), and Science Technology and Innovation (STI) Policies. The project looks for new ways to reach the gap between grassroots, citizen initiatives and policy design when facing societal challenges. The assumption is that "Labs" (Fab Labs, Living Labs, and Science Galleries) could play an important role in creating bridges and activating the desired societal changes. 10 pilots are being conducted in 10 different EU places through the implementation of co-creation journeys that last around 18 months. In these experiments, each lab aims to tackle a specific societal challenge, (meaningful for the context in which the lab is located but at the same time transversal and relevant at a wider scale) and engage with it a set of stakeholders in a co-creation process from the stage of co-design where stakeholders are analysing the context, reframing the problem and envision alternatives, to that of co-production of prototypes within an iterative process. (Real et al., 2019;[START_REF] Rizzo | Deliverable 1.2: co-creation in RRI practices and STI policies[END_REF] El Barri Circular: from co-creation to the emergence of a new synergistic system for local food cycles

The Fab Lab Barcelona's pilot has been exploring how co-creation and RRI can be approached to locally develop solutions to improve the circularity of surplus food and bio-waste at a neighbourhood scale. The main challenge started by exploring the following question: How to endeavour a circular economy in the neighbourhood by co-producing projects that connect local resources and territorial needs?

"El Barri Circular", which means "the circular neighbourhood", by mixing Spanish and Catalan languages, has been engaging local stakeholders at different levels to create synergies between existing initiatives related to the food system. The pilot aims to stimulate the use of co-creation methodologies, through an active learning process, to support a transition towards re-valuing surplus food and bio-waste at the neighbourhood scale.

After six months of exploration and learning, the core network of stakeholders has designed a "synergistic system for local food cycles. The system is acting at the neighbourhood level in collaboration with local stakeholders, and include ongoing food waste generators in the co-development of three circular community projects proposing food waste redistribution, bio-waste-based material development and collective composting activities. To activate and enhance the community empowerment and circular knowledge capacity building, three types of community services have been imagined. These include: a logistical service for food waste collection and community engagement, a set of learning practices to support the local design and production of dedicated tools, and an environmental plan that measures the flow of materials, energy and resources in the local food system. The system model will be developed and tested in Poblenou, the maker district of Barcelona in further steps. This case study presents the "El Barri Circular" journey as an example of how to use co-creation and RRI into the innovation process to locally develop solutions to improve the circularity of a territory. The paper will first describe the co-creation process and then give an overview and discuss the emerging synergistic system for supporting the development of local food cycles.

Method: The co-creation process

Co-creation combined with Responsible Research Innovation was adopted for this study to develop deeper understanding of community engagement and its role in finding solutions to address local issues regarding surplus food and bio-waste. The research strategy is based on a specific neighbourhood, known as the maker and creative area of Barcelona, the district of Poblenou. Fab City Global Initiative has already started a first exploration in this territory, in which a prototype of a productive and scalable city has been tested. The initial idea aimed to contribute to the city's reindustrialisation through activities, services and projects that promote interaction between local communities and citizen initiatives. Following the same principles of people-centred, inclusive and locally productive approaches, El Barri Circular has been applying participatory design and learning by doing experiences to co-create and empower citizens as drivers for change.

'The experimentation was based on a methodology divided into 4 phases (Real, 2019;[START_REF] Rizzo | Deliverable 1.2: co-creation in RRI practices and STI policies[END_REF] in which social innovation was fostered through the use of co-creation to combat local challenges: 1. Analysing the context, 2.

Reframing the problem, 3. Envision alternatives, 4. Selecting the idea. The main stakeholders involved in this process were represented by local markets and km0 restaurants, cooperatives, local association's urban gardens, composting initiatives, projects with food redistribution, makers and material designers.

 The first phase consisted of 'analysing the context' by having a better understanding of the existing instruments for circular economy, identifying the policies about food cycles at the different local scales and analysing the dynamism of Poblenou neighbourhood and Barcelona. For this phase, the Fab Lab's team participated in 5 policy making events, 35 informal interviews with 50 local actors in order to identify the ongoing policies and resources that are participating in the local circular transition. Moreover, an effort has been done to develop a consistent mapping of current initiatives related to circular economy. The mapping phase focused on spreading awareness about ongoing activities and projects developed at Fab Lab Barcelona that are related to co-creation activities and empowerment of citizens through sustainable and regenerative cities.

 The second phase consisted of 'reframing the problem' through a shaping of the set of data collected to better structure the future interventions with local stakeholders. A first co-creation workshop named "Synergy Soup" was proposed (called "Sopa de Sinergias) destined to a core group of local stakeholders identified through the different interviews, events and participative observations. The event aimed at identifying synergies among the actors by matching local resources with local needs from each stakeholder. The workshop was built upon existing systemic design methodologies regrouping industrial symbiosis identification tools [START_REF] Van Capelleveen | A literature survey of information systems facilitating the identification of industrial symbiosis[END_REF][START_REF] Makinen | NISP Canada -Waste Reduction through Re-Purposing[END_REF], resource mapping, input-output system modelling [START_REF] Bistagnino | Systemic Design to Develop Local Distributed Economies[END_REF][START_REF] Barbero | Systemic Design Method Guide for Policymaking[END_REF]2017). It was composed by four activities: (1) a presentation of the context and key aspects of circular design; (2) an icebreaker where participants were engaged in the preparation of the soup presenting themselves, picking some pre-cut vegetables and commenting their subjectivity by associating shape, colour to their emotion and personal inspirations. (3) Then, they were divided in three subgroups and were asked to first identify local needs and resources at individual, organisational and neighbourhood level and second to create projects that are matching identified resources and needs. They were playing on cardboards with three types of cards to fill (needs, resources, and projects) and advertisement paper collected and reused to be cut into lines to help them build and show the different connexions. Finally each group made a restitution while note-taking and soup distribution were offered by the two facilitators. The workshop permits to identify 58 needs, 36 resources and 31 ideas of projects were generated by participants. It allowed getting to know each other and start creating shared value and a sense of community. After this event, the team developed and proposed a new plan of co-creation activities to guarantee an effective engagement and collaboration of the local community through an action learning process. The communication plan was redefined according to the target identified. An identity for the pilot was created and named "El Barri Circular #Poblenou, episod: Food, waste and local crafts." Social media channels and personal invitations have permitted to maintain the engagement of the core group of stakeholders while opening the challenge more broadly to the community.

 The third phase which consists in "envisioning alternatives", was composed of five events during one month, in which the participants of El Barri Circular were actively involved to foster ideation and participate to learning experiences. The first event was a community ideation workshop which took place at an historic community place, "El Ateneu de la Flor de Maig"2 where 5 pre-identified projects were challenged in groups through three short activities: First, participants used a customised version of the 6W tools to define the ideal solution of each concept and cross the different visions; Secondly, the facilitators created a new specific onion diagram that can be described as a back casting-value opportunity mapping challenging "how" to reach the solution, and identifying needs and opportunities in terms of materials, tools, resources and skills, and other project dimensions. After this problem definition activity, each group was asked to design idea cards adapted from the n to describe from 1 to 3 concepts framing diverse actions that could redefine and facilitate the development of the project. Finally a restitution of each group were realised. The community ideation workshop was followed by two learning by doing experiences, which were proposed and co-organised with local participants to raise knowledge and answer the need to "make things together". The Fab Textiles3 ran a workshop for realising biomaterials and bio-composites from local waste collected in restaurants. Three techniques were explored by the participants: 3D extrusion, bioplastic sheets and bowls mold design. The second experience was facilitated by MACUS Cooperative4 , based on digital fabrication tools and machine design. The fourth event was composed by a session of co-design and scenario building, together with three stakeholders. The workshop was proposed during a local event about circular economy which took place in Palo Alto Foundation5 , another historical place of Poblenou. The workshop aims at understanding better the impact of food waste valorisation processes and designing new scenarios for local valorisation and supports.

It was divided in three steps starting by the presentation and exhibition of three different initiatives using bio waste and food surplus. Then, each process was screened onto table where people, in groups used specific stickers for identifying the environmental impact at different stages of the process. This was inspired by simplified eco-designed tools such as LIDs eco-design wheel [START_REF] Brezet | EcoDesign Strategy Wheel. WikID, The Industrial Design Engineering wiki[END_REF]) or Material, Energy, Toxicity (MET) matrix (Gertsakis, 1997). The last part consisted in creating emotional narratives so to support new visions and services in the neighbourhoods to envision local logistic, educational services and new bank-time systems. Specific co-creation tools (Personas and moodboards) were given to participants to support them in scenario design. Finally, a last event was organised -the "convivial agora" in collaboration with the 40 students of the summer school of Degrowth 6 to go beyond environmental issues and discussing about the conviviality of the emerging projects. The event was designed with one context and theoretical presentation, one co-creation workshop and one larger public debate. The co-creation workshop has been designed within the inspirational frame of the design tools for conviviality [START_REF] Lizarralde | A framework for the integration of the conviviality concept in the design process[END_REF][START_REF] Real | Practices, Places, Projects: Enrolling Stakeholders for Circular Fashion[END_REF][START_REF] Vetter | The Matrix of Convivial Technology-Assessing technologies for degrowth[END_REF]. Through all these events, the internal team in collaboration with external stakeholders defined and refined a synergistic model for supporting local food waste transformation in the neighbourhood. The model is described below.

Results and Discussion: Toward a new synergistic system for local food waste transformation

The model is aimed at synthesizing the vision shared within the first three phases of the project to support circular food transformations at a neighbourhood scale. By identifying and co-designing with the local stakeholders as well as observing and raising knowledge about the food cycles, four community services have been identified, will be designed and experimented in further steps. 4) food cooperatives, particularly active in Barcelona and Catalonia (Espelt, 2013) where part of local residents are building self-organised groups that connects local producers and consumers with a short loop and historical Catalan cooperativism principles. Seeing as well generator of waste and actors of changes, these stakeholders are weaving an important amount of the local food waste.

Three circular activities have been selected, simplified from the food waste hierarchy [START_REF] Papargyropoulou | The food waste hierarchy as a framework for the management of food surplus and food waste[END_REF] and adapted for the specificities of the neighbourhood scale (Poblenou in that case).

 First, the activity of cooking and redistributing food surplus has been found through diverse practices: internally into families or communities, through sharing applications or within social organisations. One specific initiative caught up our attention as a promising social initiatives. The "Bacuinetes from BarcelonaActua" 7 is a collective of grandmothers and local people (often in social difficulties) who collect and cook once a week the food surplus collected in local shops and distribute them in a collective meals organised for people in needs each week. Social associations from the city and neighbourhoods are supporting the initiative by connecting people and providers as well as booking a kitchen place each week.

 Another activity concerns the design of materials and products based on food waste with craft and innovative production techniques. This activity is generally under-explored at such a small scale. However, an opportunity has been identified with the strong presence of artisans and makers locally and the wish to tend to enhance the autonomy of citizens by allowing them to be aware of how things are made, to design products and to have access to the means of production. The Fab Textiles and others makers are yet experimenting in material innovations from food waste and look after new modes of production more distributed. Within the first workshop, several techniques have been tested to create biomaterials and bio composites from coffee, orange, yucca peels as well as eggs and mussels shells.

 The third circular activity is the experimentation of effective compost techniques at individual, building and neighbourhood scale. Several possibilities have been already provided by the city like municipal organic waste collection point, the distribution of collective compost bins in building. Another technique, the worm compost system is being experimented in a local urban garden, using waste from residents, local restaurants and markets with a semi-industrial worm composting to create compost distributed for local gardens and others uses to turn green the local area.

To maintain, support and scale up this grassroots initiatives at the neighbourhood level, four types of community services have been imagined from the emerging needs of local stakeholders i.e. services for community building, logistic, making education and environmental monitoring.

 First, stakeholders identified a necessity for re-connecting people, informing them as well as opening or promoting existing spaces and events for encouraging community synergy building. The service will be built upon collaborations with the stakeholders to access to the diversity of places present in the neighbourhood that could be private or public, to communicate about and participate in local events, to create new moments for synergy emergence, where co-creation tools can be shared and used in an open way.

 Another need is about logistics questioning how to integrate in the actual system and habits of stakeholders the possibility for them to be active in the local food waste transformations. The logistic system consists of collecting waste in restaurants and markets, processing them and finally re-distributing meals, products and composts. This requires a preparation phase for organising people and tasks while providing necessary tools to make it possible. Three actions are proposed before launching the service:

-Designing a shared soft mobility device (a cargo-bike) that permits to transport bio-wastes, products and materials from one point to another.

-Gathering bio-waste providers (restaurants and markets) around a series of meetings aimed to discover and discuss material innovation possibilities, to best select the type of waste, the disposal devices and storage of resources and finally to agree on a schedule.

-Devices or organisational mechanisms could be developed to gather the list of community tasks to be done and connect them to local stakeholders, making them accessible through micro-volunteering platforms or other forms of collaborations that ensure and diversify their effective engagement.

 Local stakeholders want experiences to learn by doing, so to improve their internal capacity and go beyond cognitive gaps and individual learning limits. Educational services are composed by a set of learning and codesigned activities that support the local design and co-production of dedicated tools needed to feed local food waste transformations. In the case of cooking, material innovation or composting, specific workshops could be realised in cooperation with the local expertise, from practicing bio-material recipes from existing database, to the design of new ideas via co-creation tools, to building moulds with digital fabrication tools.

Collective co-development could be organised like the realisation of a cargo-bike system that is made with self-helped bike workshop.

 The last service is about environmental monitoring and suppose the exploration on how to measure the flow of material, energy and resources for the emerging local food system. An essential step towards achieving a relevant model of circularity at the neighbourhood scale is gaining a better understanding of the evolution of the environmental impacts from the existing to the new emerging multi-loop systems. First measures from material analysis flow can be done at the prototype level and scenario of upscaling could be then debated in collaboration with other studies and emergent work from other scales.

Conclusions

The case-study shows that co-creation methodologies have permitted to engage local stakeholders, support them to adopt new practices and co-design a contextual set of services for encouraging local food waste at the neighbourhood level. It highlights the complexity of local transformations and underlies the hypothesis that makespaces can participate in supporting the design and development of local circular ecosystems, thanks to an activist mindset, a knowledge-making culture and global connexions. In the project El Barri Circular, the makespace was piloting the co-creation process connecting local initiatives in a global societal challenge.

An important and emergent aspect for such initiative remains the stakeholder engagement process and respective design activities that supported it. Indeed, the evolution of the process and its results in this case was highly dependent of the synergies established with local stakeholders combined with the tools and methods shared within the EU SISCODE project. Starting with an understanding of the actual ecosystem was crucial to strengthen connection between actors and promote collaboratively actions, supporting existing initiatives. From the identification of stakeholders and initiatives, the immersive visits to the synergy mapping and ideation workshops, the stakeholder engagement process, was like the divergence and convergence phases of design processes, enlarging and consolidating the active network. Through discussions and events, the initial concepts have been transformed, the emerging model has gained in legitimacy.

The synergistic model for local food waste transformations is proposing to co-produce with local stakeholders services for community building, logistic, making education and environmental monitoring to optimise how food waste are redistributed and transformed in meals, materials, products and compost. A new playground is now being developed to make the proposal tangible, and define more precisely what could be the role of makespaces in the system and how to revisit the interactions between the quintuple helix of stakeholders, meaning public bodies, the civil society, residents, economic actors and academics. According to the specific context, answers could be different from makespace-leaded initiatives, public-leaded actions to self-organized ecosystems in neighbourhoods.

The project outputs will be discussed within broader networks and feed a collective handbook that will contribute to envision the design of new circular practices in makespaces and thus, shape new forms of learning in local areas. There is an interest to adapt such processes in other sectors or even to let it open to transversal applications.

This weak signals will be also reinforced by replicating it in other neighbourhoods considering the specificities of the context (cultural, political, economic, and demographic), as well as creating connexions with other scales, specifically at the city and regional levels. Those interactions can help to avoid new forms of disconnexions, distances and dispersions between people in territories.

Creating new forms of values for territories through co-creation is a constant engagement for reaching RRI values.

It consists of making more efforts towards inclusion, transparency and openness while adopting new postures to be aware of the potential barriers and conflicts behind system development, and looking for overcoming emergent frustrations that can create more convivial and sustainable environments. Encouraging local stakeholders to learn about design, as well as revisiting the deep roots, principles and practices of design for transforming territories is compulsory to build effective systemic changes toward circularity.

Figure 1 .

 1 Figure 1. Synergistic system for local food waste transformation Four type of stakeholders have been categorised as potential enactors of circular food shifts in the neighbourhood of Poblenou: (1) Urban Gardens as mixed and heterogeneous places where people are reconnecting with nature, experiments techniques, organise events for both creating social bonds and raise knowledge and discussions about environmental and health issues; (2) restaurants and (3) markets in the dynamic of building sustainable alternatives (with km O and slow food networks), (4) food cooperatives, particularly active in Barcelona and Catalonia(Espelt,

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