Martin Mihelich
email: martin.mihelich@cea.fr

Bérengère Dubrulle
email: berengere.dubrulle@cea.fr

Didier Paillard
email: didier.paillard@lsce.ipsl.fr

Corentin Herbert
email: cherbert@ucar.edu

entropy

Keywords: maximum entropy production, Kolmogorov-Sinai Entropy, ASEP model

The asymmetric simple exclusion process (ASEP) has become a paradigmatic toy-model of a non-equilibrium system, and much effort has been made in the past decades to compute exactly its statistics for given dynamical rules. Here, a different approach is developed; analogously to the equilibrium situation, we consider that the dynamical rules are not exactly known. Allowing for the transition rate to vary, we show that the dynamical rules that maximize the entropy production and those that maximise the rate of variation of the dynamical entropy, known as the Kolmogorov-Sinai entropy coincide with good accuracy. We study the dependence of this agreement on the size of the system and the couplings with the reservoirs, for the original ASEP and a variant with Langmuir kinetics.

Introduction

The theory of dynamical systems at statistical equilibrium consists of well identified principles lying on firm mathematical foundations [START_REF] Khinchin | The Mathematical Foundations of Statistical Mechanics[END_REF][START_REF] Ruelle | Statistical Mechanics: Rigorous Results[END_REF][START_REF] Ellis | Large Deviations, and Statistical Mechanics[END_REF]. Although the representative point of the system in phase space (the microstate) can have a complicated dynamics, the statistics of any function on phase space (an observable) are well described, at least for large enough systems (in the thermodynamic limit) by simple invariant measures depending only on the invariants of the system, and not on the details of the dynamical equations. On the contrary, general theories describing the statistics of non-equilibrium systems are still in their infancy; although there have been attempts at formulating general principles governing non-equilibrium systems [START_REF] Prigogine | Introduction to Thermodynamics of Irreversible Processes[END_REF][START_REF] De Groot | Non-Equilibrium Thermodynamics[END_REF], they have not reached the level of mathematical rigor of their equilibrium counterpart, partly because there may not exist a universal way of being out-of-equilibrium, and remain valid essentially close to equilibrium.

Nevertheless, some progress have been achieved recently, in particular in the framework of "toy models". Similarly to the Ising model which played a central role in the theory of critical phenomena, the so-called asymmetric simple exclusion process (ASEP) [START_REF] Derrida | Non-equilibrium steady states: Fluctuations and large deviations of the density and of the current[END_REF][START_REF] Chou | Non-equilibrium statistical mechanics: From a paradigmatic model to biological transport[END_REF], which describes the transport of particles between two reservoirs, have become the dominant paradigm for non-equilibrium systems. Over the past few years, this system and others have benefited from the theory of large deviations, developed by Donsker and Varadhan [START_REF] Donsker | Asymptotic evaluation of certain Markov process expectations for large time[END_REF][START_REF] Donsker | Asymptotic evaluation of certain Markov process expectations for large time[END_REF][START_REF] Donsker | Asymptotic evaluation of certain Markov process expectations for large time-III[END_REF][START_REF] Donsker | Asymptotic evaluation of certain Markov process expectations for large time[END_REF]. This theory has introduced the large deviation function, which characterizes the statistics of the system in an asymptotic regime (large system, large time, low noise,...) and is a natural candidate for playing the out-of-equilibrium analogous role to the equilibrium free energy. In the long-time limit, the ASEP reaches a steady-state, and a macroscopic current is established between the two reservoirs. The average steady-state current, and the corresponding density profile, have been computed exactly [START_REF] Derrida | Non-equilibrium steady states: Fluctuations and large deviations of the density and of the current[END_REF][START_REF] Schütz | Phase Transitions and Critical Phenomena[END_REF][START_REF] Blythe | Nonequilibrium steady sstates of matrix product form: A solver's guide[END_REF], and more recently, the full large deviation function, which also describes the atypical fluctuations around these averages, has also been solved exactly [START_REF] Gorissen | Exact current statistics of the asymmetric simple exclusion process with open boundaries[END_REF], using a generalization of the matrix product ansatz.

Powerful as these techniques may be, they remain far from being applicable to more realistic systems. Indeed, if many systems in nature share some conceptual properties with the ASEP (establishment of a steady-state with fluctuations dominating the average behaviors and the resulting anomalous transport properties), not all of them are easily written as stochastic processes driven by a master equation. For instance, it has become clear over the years that deterministic systems with a very large number of degrees of freedom such as turbulent fluids, and in particular the atmosphere and the ocean of the Earth, should be thought of in terms of non-equilibrium statistical mechanics. However, up to now, attempts to apply such tools in the climate system have been restricted to phenomenological applications of the "Maximum Entropy Production" variational principle [START_REF] Paltridge | Global dynamics and climate-a system of minimum entropy exchange[END_REF][START_REF] Ozawa | The second law of thermodynamics and the global climate system: A review of the maximum entropy production principle[END_REF][START_REF] Herbert | Present and Last Glacial Maximum climates as states of maximum entropy production[END_REF]. The success of these approaches points at a deeper connection with the general properties of non-equilibrium systems. A justification has been attempted [START_REF] Dewar | Information theory explanation of the fluctuation theorem, maximum entropy production and self-organized criticality in non-equilibrium stationary states[END_REF], and dismissed [START_REF] Grinstein | Comments on a derivation and application of the 'maximum entropy production' principle[END_REF][START_REF] Bruers | A discussion on maximum entropy production and information theory[END_REF], following the ideas of Jaynes that non-equilibrium systems should be characterized by a probability distribution on the trajectories in phase space, instead of just the points in phase space at equilibrium. In the context of Markov chains, this idea has proved to provide a natural generalization of equilibrium statistical mechanics [START_REF] Monthus | Non-equilibrium steady states: Maximization of the Shannon entropy associated with the distribution of dynamical trajectories in the presence of constraints[END_REF], by considering the Kolmogorov-Sinai (KS) entropy. In this Letter, we establish a bridge between the phenomenological principle of maximum entropy production on the one hand, and the maximization of KS entropy on the other hand, in the context of the ASEP. We also consider a variant of the ASEP with Langmuir kinetics [START_REF] Parmeggiani | Phase coexistence in driven one-dimensional transport[END_REF], which mimics the phenomenological climate model designed by Paltridge [START_REF] Paltridge | Global dynamics and climate-a system of minimum entropy exchange[END_REF]. In both cases, rather than solving exactly the system for given dynamical rules, we show that the maxima location of the thermodynamic entropy production and the KS entropy as functions of the rate of transition coincide, and study the validity of this agreement when the system size and reservoir couplings vary.

Model Description

In this paper, we have considered a modified version to the ASEP model-hereafter named the CASEP model (for Changing ASEP model) or the LK-CASEP model when Langmuir dynamics is added-so as to be able to change the particle flux. A class of toy models for out-of-equilibrium systems consists of variants of the exclusion process (with open boundaries): let us consider a one dimensional lattice gas model composed of L sites between two reservoirs. Each site contains at most one particle. The particle undergoes a random walk on the lattice with the following rules: at each time step a site is chosen randomly. If this site belongs to the bulk and is occupied by a particle, the particle can jump right with a probability p and left with a probability q provided the neighboring sites are empty. At the boundaries, the system is coupled with the two reservoirs: at the left (resp. right) boundary, particles from the reservoir can enter the lattice with probability α and particles from the lattice can exit with probability γ (resp. δ and β). This coupling is equivalent to imposing the respective densities ρ a = α α+γ and ρ b = δ δ+β . When ρ a = ρ b , the system reaches an equilibrium state, while it becomes increasingly out-of-equilibrium as the difference between ρ a and ρ b increases. The numerous variants of the exclusion process depend on choices of constraints on the hopping rates p and q: the symmetric simple exclusion process (SSEP) imposes p = q. Imposing a preferred direction for particle transport (e.g., p > q) yields the asymmetric simple exclusion model (ASEP). An extreme case is the totally asymmetric simple exclusion model (TASEP), for which q = 0. Here, we shall consider a particular class of ASEP, by imposing that the particle has to hop to a neighboring site-provided it is empty; it cannot remain in place unless both the neighboring sites are occupied (see Figure 1).

1 2 • • • L Reservoir Reservoir 1 -f f δ β α γ 1 2 • • • L Reservoir Reservoir 1 -f f δ β α γ wa w b
This amounts to setting p = f and q = 1 -f . We shall call this variant the Changing asymmetric simple exclusion process (CASEP).

The exclusion processes can be seen as microscopic models of transport. We are interested here in the connection with heat transport in turbulent flows at a macroscopic scale, like the atmosphere and the ocean. For such systems, phenomenological variational principles were suggested [START_REF] Paltridge | Global dynamics and climate-a system of minimum entropy exchange[END_REF] to compute the energy transport associated with the establishment of a temperature gradient, outside of the diffusive regime. In the atmosphere, radiative exchanges on the vertical coexist with this meridional transport. To mimic this effect, we shall also consider a variant of the CASEP which includes Langmuir Kinetic dynamics (LK-CASEP) [START_REF] Parmeggiani | Phase coexistence in driven one-dimensional transport[END_REF]: at each time step, particles can appear in an empty site with probability w a and disappear from an occupied site with probability w b .

The exclusion processes, and in particular the CASEP, are special cases of discrete Markov processes with 2 L states. It is therefore difficult to study numerically this system for L larger than 10. These processes are characterized by their transition matrix P = (p ij ) which is irreducible. Thus, the probability measure on the states converges to the stationary probability measure µ stat = (µ stat 1 , ..., µ stat 2 L ) which satisfies:

µ stat i = 2 L j=1 µ stat j .p ji ∀i ∈ [[1, 2 L ]] (1)
The states are labeled using the base 2. The p ij are equal to f.dt (resp. (1 -f ).dt) for a transition corresponding to a right jump of particule (resp. left) inside the bulk. For a transition corresponding to a jump at the boundary of the lattice, the p ij are equal to α.dt, β.dt, γ.dt, δ.dt. In the LK-CASEP Model, for a transition corresponding to the appearance or disappearance of a particle, p ij = w a .dt or p ij = w b .dt. Once the transition matrix has been defined through these simple rules, the dynamics can then be obtained by moving particules at each time steps, until a stationary states has been reached. The density of particules on the lattice ρ = (ρ 1 , ..., ρ L ) in such a stationary states can then be computed numerically.

Most of the studies of the ASEP model consider a fixed dynamics (given value for p and q) and try to solve exactly the resulting statistics for the steady-state current [START_REF] Gorissen | Exact current statistics of the asymmetric simple exclusion process with open boundaries[END_REF]. Here, we rather consider that the dynamics is not known exactly, and vary the parameter f in order to obtain different stationary states and calculate their entropy production σ function of f . Indeed, changing the parameter f is equivalent to change the flux of particles J. At fixed f , we can also use the dynamical properties of the CASEP model to compute the Kolmogorov-Sinai Entropy h KS as a function of f and compare it to σ(f ).

In the f = 1 2 case, the system reaches a steady state that is reminiscent of a "conductive state", with a linear density behaviour over the lattice, going from ρ a = α α+γ on the left side to ρ b = δ δ+β on the right side [START_REF] Derrida | Non-equilibrium steady states: Fluctuations and large deviations of the density and of the current[END_REF]. When f = 1 2 , the system reaches another stationary state, ressembling a "convective state", in which the density is mainly constant in the bulk, with steep transitions near the edges towards the left and the right side densities ρ a and ρ b .

Dynamical and Thermodynamical Entropy

We now define both the Macroscopic Entropy production and the Kolmogorov-Sinai Entropy for the CASEP model in order to compare them.

Komogorov-Sinai Entropy

There are many ways to estimate the Kolmogorov-Sinai entropy associated with a Markov chain [START_REF] Monthus | Non-equilibrium steady states: Maximization of the Shannon entropy associated with the distribution of dynamical trajectories in the presence of constraints[END_REF][START_REF] Billingsley | Ergodic Theory and Information[END_REF]. Here we compute Kolmogorov-Sinai Entropy as the time derivative of the Jaynes Entropy. To characterize the dynamics of the system during the time interval [0, t], one considers the possible dynamical trajectories Γ [0,t] and the associated probabilities p Γ [0,t] . The dynamical trajectories entropy-the Jaynes entropy-reads:

S Jaynes (t) = - Γ [0,t] p Γ [0,t] . log(p Γ [0,t] ) (2)
For a Markov chain we find that:

S Jaynes (t) -S Jaynes (t -1) = - (i,j) µ istat p ij log(p ij ) (3)
Thus, the Kolmogorov-Sinai Entropy for the Markov chain is:

h KS = - (i,j) µ istat p ij log(p ij ), (4)
where µ i is the stationary measure and p ij the transition matrix.

Entropy Production

For a macroscopic system subject to thermodynamic forces X i and fluxes J i , the thermodynamics entropy production is given by [START_REF] De Groot | Non-Equilibrium Thermodynamics[END_REF][START_REF] Balian | Physique Statistique et Themodynamique Hors équilibre[END_REF]:

σ thermo = i J i X i (5)
The fluxes to consider for a diffusive particules model are fluxes of particules and the thermodynamics forces can be written X = ∆(-ν T ) where T is the temperature and ν the chemical potential proportionnal to log(ρ) for an ideal gas [START_REF] Balian | Physique Statistique et Themodynamique Hors équilibre[END_REF]. So, as the temperature is here fixed, the Thermodynamics Entropy production of a given stationary state takes the form:

σ thermo ∝ L i=1 J.(log(ρ i ) -log(ρ i+1 )) = J.(log(ρ 1 ) -log(ρ L )) ( 6
)
where ρ is the stationary density distribution and J the particle fluxes, that are both (nonlinear) function of f . Note that we can find this result considering a continuous system. Indeed, the statistical entropy for an ideal gas confined between the walls A + and B -is:

S(t) = - B - A + ρ(x, t) log(ρ(x, t))dx (7)
Taking the times derivative of Equation ( 7) with the continuity equation div(J) + ∂ρ ∂t = 0 yields:

∂S ∂t = B - A + ∂J(x, t) ∂x (1 + log(ρ(x, t)))dx (8)
We can write Equation ( 8) to make entropy production appears:

∂S ∂t - B - A + div(J(x, t)(1 + log(ρ(x, t)))) = - B - A + J(x, t) ∂ log(ρ(x, t)) ∂x dx (9)
Thus, the entropy production reads:

σ thermo = - B - A + J(x, t) ∂ log(ρ(x, t)) ∂x dx (10)
This is nothing but the continuous limit of Equation ( 6).

Numerical Results

Comparison between KS Entropy and Macroscopic Entropy Production

Given a value of L, ρ a and ρ b , we have computed the stationary states of the CASEP model as a function of f and used them to compute numerically the entropy production in Equation ( 6) and the Kolmogorov-Sinai Entropy in Equation ( 4). An example is provided in Figure 2 for ρ a = 0.75, ρ b = 0.1 and L = 10. We see that both function are of approximate parabolic shape, reaching a maximum for f ≥ 1/2. In the sequel, we note f maxep (resp. f max ks the value of f at which the maximum of the entropy production (resp. of the Kolmogorov-Sinai entropy) is reached. The existence of a maximum of entropy production can be simply understood by considering the case L = 2, i.e., a model with only two boxes, where we assume for simplicity that ρ a > ρ b . Noting ρ 1 and ρ 2 the density of the box number 1 and number 2 respectively, it is easy to see that for J = 0, σ thermo = 0 whereas for J large enough ρ 1 = ρ 2 and σ thermo = 0. Thus, in between these two values of J, σ thermo has at least one maximum. We do not have such heuristic explanation for the existence of the maximum of the Kolmogorov-Sinai entropy. We note that although the entropy production and the Kolmogorov-Sinai entropy have similar parabolic shape, they do not coincide exactly: the entropy production and the Kolmogorov-Sinai entropy differ. For ρ a = ρ b (i.e., at equilibrium), their maxima take a common value f maxep = f max ks = 1/2, corresponding to the maximum of entropy: at equilibrium, all functionals are maximum for the same stationary state, corresponding to the "conductive state". As the difference between ρ a and ρ b increases, the two maxima f maxep and f max ks deviates from the equilibrium value 1 2 . This is the signature of a "convective dynamics". We would like now to explore the behaviour of the two entropy around this conductive state, by studying the behaviour of the maxima difference, ∆f max =|f maxep -f max ks |.

Behaviour of the Maxima Difference

Let us first fix the size of the box, L, and study the variation of ∆f max =|f maxep -f max ks | as function of ρ b and ρ a . This is shown in Figure 3 (Left). It can be seen that ∆f max is zero for at perfect equilibrium, and remains small near equilibrium, for ρ a ≈ ρ b . The difference then increases as the system deviates from equilibrium, reaching a maximal value of about 4% at L = 10. These general features are conserved when including the Langmuir-Dynamics as can be seen in Figure 3 (Right): for L = 8 the difference between the maximum of entropy production and the maximum of the Kolmogorov-Sinai entropy also of the order of 4%. This is still valid varying w a and w b in [0, 1 L ]. These shows that the coincidence between the maxima of the KS entropy and the entropy production is robust and independent of inclusion of vertical fluxes.

Influence of Box Size

The previous results have been obtained for a number of boxes comparable with the number of boxes used in climate models. However, it is interesting to investigate possible finite size effects by varying L and observe the orresponding variations of ∆f max . Due to limited computational power, we were not able to simulate systems with L larger than L = 10. We could however decrease L and observe the result. This is done in Figure 4, where both f maxep and f max ks are plotted as a function of ρ a and L for fixed ρ b . One sees that for fixed ρ a , the difference between the two values ∆f max decreases as L increases. So it is likely that for L → ∞, this quantity converges to 0 for any fixed ρ a and ρ b .

Discussion and Conclusions

In the present paper, we have built a bridge between stationary states that are maxima of Macroscopic Entropy production and maxima of Kolmogorov Sinai entropy, within a simple model of out-of-equilibrium physics. Both maxima coincide at equilibrium, and their difference increases with increasing out-of equilibrium. For fixed out-of-equilibrium conditions, however, the difference decreases with increasing system size. We may then postulate that for any fixed out-of-equilibrium conditions, both the Macroscopic Entropy production and the Kolmogorov Sinai entropy peak at the same stationary state, that thereby acquire a priviledged status. Whether such a stationary state is the natural one selected by the dynamics is actually an open question. Many studies point towards the relevence of the Kolmogorov Sinai entropy for out-of-equilibrium systems. For example, Latora and Baranger [START_REF] Latora | The rate of entropy increase at the edge of chaos[END_REF] established a link between the Kolmogorov-Sinai entropy, which is a microscopic quantity, with the derivative of the coarse grained entropy, which is a macroscopic quantity, in an out of equilibrium dynamical system. In the present paper, we have evidenced another connection between the Kolmogorov-Sinai entropy and a macroscopic quantity. This link, one hand, strengthens the hypothesis that Kolmogorov-Sinai entropy may play a key role in the study of out of equilibrium systems. On the other hand, its link with the Maximum Entropy Production may explain its rather successful role played in in many area of physics [START_REF] Martyushev | Maximum entropy production principle in physics, chemistry and biology[END_REF] such as solid physics [START_REF] Kirkaldy | Entropy criteria applied to pattern selection in systems with free boundaries[END_REF], electromagnetism, quantum physics or climate science [START_REF] Paltridge | Global dynamics and climate-a system of minimum entropy exchange[END_REF][START_REF] Herbert | Present and Last Glacial Maximum climates as states of maximum entropy production[END_REF]. At present time, the connection has only been established for a 1D discrete special system. Generalizing our result to continuous ASEP model is straightforward, using the expression of the Kolmogorov-Sinai entropy found by [START_REF] Lecomte | Thermodynamic formalism for systems with Markov dynamics[END_REF]. However, it would be very interesting to generalize the link between maxima of Maximum Entropy Production and maxima of Kolmogorov-Sinai entropy to more complicated systems, such as chaotic dynamical systems, lattice Bolzmann or turbulent flows.

Figure 1 .

 1 Figure 1. The dynamical rules of the asymmetric simple exclusion process (ASEP), without (top) and with (bottom, LK-ASEP) Langmuir kinetics.

Figure 2 .

 2 Figure 2. Macroscopic entropy production (top) and Kolmogorov-Sinai entropy (bottom) production function of f for ρ a = 0.75 and ρ b = 0.1 and L = 10. For these parameter values, we have limited the range of f from 0.4 to 1. Indeed f ≤ 0.4 corresponds to a state where the flux of particules is in the same direction as the gradient of ρ which is physically impossible.

Figure 3 .

 3 Figure 3. ∆f max = |f maxep -f max ks | as function of ρ a and ρ b for fixed box size L: left at L = 1) in the CASEP model. Right: at L = 8 for the LK-CASEP model with w a = 0.06 and w b = 0.03.

Figure 4 .

 4 Figure 4. Maxima difference ∆f max as function of ρ a at fixed ρ b = 0.1, for different system sizes L: blue: L = 6; red: L = 8; green: L = 10.

Acknowledgments

Martin Mihelich thanks IDEEX Paris-Saclay for financial support. We thank The National Center for Atmospheric Research which is sponsored by the National Science Foundation. The authors would like to thank the two anonymous reviewers whose comments help to clarify the manuscript.

Author Contributions

Each author contributed equally to this work.

Conflicts of Interest

The authors declare no conflicts of interest.


Nicolas Martin
email: nicolas.martin@b-com.com

Éric Jamet
email: eric.jamet@univ-rennes2.fr

Séverine Erhel
email: severine.erhel@univ-rennes2.fr

Géraldine Rouxel
email: geraldine.rouxel@univ-rennes2.fr

Evaluation of Technological Products in Mobility Context: First Steps Toward New Scales with Single Items

Keywords: Questionnaire, Échelles à Items Unique, UX, Facteurs Affectivo-Motivationnel Scale, Single Items, UX, Affective-Motivational Factors. ACM Classification H.5.2. User Interfaces: Evaluation/methodology

Récolter l'avis d'utilisateurs par questionnaire nécessite généralement l'utilisation de plusieurs items par dimension évaluée pour s'assurer de la fiabilité du questionnaire. Cela peut conduire à de longs questionnaires pouvant décourager les participants. Afin de diminuer cette difficulté, nous présentons ici le développement de deux échelles à items uniques. Ces échelles ont été construites à partir de données issues d'échelles à items multiples testées sur 305 participants évaluant une application mobile. D'après les résultats, l'utilisation d'échelles à items unique ne diminue pas la précision. Cette approche offre de nouvelles possibilités dans l'évaluation, notamment pour les études longitudinales.

Introduction

Evaluating products with users is a central aim in User-Centered Design approach (UCD) [START_REF]Ergonomics of humansystem interaction -Part 210: Human-centred design for interactive systems[END_REF]. It requires specific and reliable methods [START_REF] Law | Modelling user experience -An agenda for research and practice[END_REF] such as user testing, scales, experimental studies, etc. [START_REF] Abras | User-centered design. Bainbridge, W. Encyclopedia of Human-Computer Interaction[END_REF]. Thus, scales are often used because they can provide data quickly and they are easy to use. Nevertheless, scales have the disadvantages of requiring several items to ensure of their reliability [START_REF] Devellis | Scale development: theory and applications[END_REF][START_REF] Hinkin | Scale Construction: Developing Reliable and Valid Measurement Instruments[END_REF]. This can lead to very long scales for participants. Thus, to look after methods to reduce the number of items without decreasing the accuracy seems to be important for the Human Computer Interaction (HCI) field. Indeed, in real-life situations, especially when participants are recruited on social media, their participation is totally based on voluntariness and therefore the length of scales can lead them to give up. In this paper, we will present the application of a method to reduce the length of scales without compromising the precision, through the construction of single-items scales to evaluate factors of acceptability and user experience (UX).

Modeling with latent variables

Generally, in the HCI field, we evaluate theoretical constructs which are difficult to measure directly, for example perceived usability. For this purpose, we typically use scales to assess the observable outcomes of a theoretical variable [START_REF] Devellis | Scale development: theory and applications[END_REF]. From these observable variables, latent variables are often constructed to represent the theoretical variables [START_REF] Kenneth A Bollen | Latent variables in psychology and the social sciences[END_REF], especially in the Structural Equation Modeling [START_REF] Beaujean | Latent Variable Modeling Using R: A Step-by-Step Guide[END_REF][START_REF] Hoyle | Handbook of Structural Equation Modeling[END_REF][START_REF] Kline | Principles and Practice of Structural Equation Modeling[END_REF] (SEM) framework. For example, if you want to evaluate perceived usability, several items evaluating this theoretical construct should be developed (e.g., "Using this mobile app seems easy"). Based on data gathered with these items (i.e., observables variables), the latent variable is constructed statistically as the variable which influences the observed covariance between these items, in addition to errors of measure (δ) [START_REF] Beaujean | Latent Variable Modeling Using R: A Step-by-Step Guide[END_REF]. However, in order to use latent variables, multiple observable variables (e.g., items) are required. Indeed, the estimation of latent variables and related parameters requires that the model to be identified [START_REF] Kenny | Identification: A non-technical discussion of a technical issue[END_REF]: this means that the number of unique information (i.e., variances and covariances of observable variables) should be equal or greater than the number of parameters to be estimated (e.g., path between an observable variable and a latent variable or also error of variance of an item) [START_REF] Beaujean | Latent Variable Modeling Using R: A Step-by-Step Guide[END_REF]. For example, in the case of a model with one latent variable (as in figure 1) a minimum of three observable variables is required. Indeed, in this case, there are 6 pieces of unique information (variances of the 3 items and covariances between these 3 items) and 6 parameters to be estimated (2 paths between the latent variable and items1 ; measurement errors for the 3 items; variance of the latent variable). Thus, if one item is deleted, Latent variable " the number of unique information (i.e., variance of the 2 items and covariance between these 2 items = 3 pieces of information) is smaller than the number of parameters to be estimated (i.e., 1 path between the latent variable and the item 1 ; errors of variance for the 2 items; variance of the latent variable = 4 parameters to be estimated) and the model becomes not estimable. Thus, the constraint of multiple items can lead to very long questionnaires.

Item 2 1.0 δ2 = 0

Modeling with Single-Items

Decreasing the number of items so that scales contain only one item per evaluated construct offers practical advantages [START_REF] Bergkvist | The Predictive Validity of Multiple-Item Versus Single-Item Measures of the Same Constructs[END_REF]. Firstly, it can reduce the time spent by participants when testing a new application, and thus facilitates experiments [START_REF] Christophersen | Reliability, validity, and sensitivity of a single-item measure of online store usability[END_REF]. Moreover, it may reduce the number of missing data due to abandonment of studies. Consequently, it is possible to increase the number of studies, especially longitudinal ones. Indeed, if the participants in a study have to complete only a handful, as opposed to a large number of questions, motivation to participate can increase. Moreover, reducing the number of items provides more flexibility in modeling: the number of parameters to estimate would be then smaller [START_REF] Daniel | Measurement Error Models With Uncertainty About the Error Variance[END_REF]. Nevertheless, as presented previously, statistical models require several items for each evaluated theoretical variable (i.e., latent variable). Despite this requirement, the use of single-items scales (i.e., one item per evaluated theoretical construct) has been developed [e.g. ,1]. Generally, in this case, it can be assumed that the item perfectly measures the theoretical construct [START_REF] Hayduk | Should researchers use single indicators, best indicators, or multiple indicators in structural equation models[END_REF].

Technically, as presented above, reducing the number of items associated with a latent variable leads to a not estimable model. Therefore, to be able to calculate parameters, some modifications on modeling are carried out as presented in figure 2. First, the loading (λ) between the latent variable and the item is fixed to 1. Secondly, the measurement error (δ) of the item is generally fixed to 0. However, this hypothesis of a perfect measurement is difficult to sustain and can lead to errors in modeling [START_REF] Cole | Manifest variable path analysis: Potentially serious and misleading consequences due to uncorrected measurement error[END_REF]. So, one approach is to fix the measurement error (δ) to a specific non-zero value [START_REF] Beaujean | Latent Variable Modeling Using R: A Step-by-Step Guide[END_REF]. With this approach, measurement errors can be incorporated in the estimation of the model to reduce errors in modeling. Technically speaking, when only single-items are used from the creation of the scale, this value is generally chosen from an expected reliability and the variance of item [START_REF] Fuchs | Using single-item measures for construct measurement in management research[END_REF]. On the contrary, when the reduction of items is made after a factor analysis on a multiple-items scale, measurement errors of each item can be fixed from estimated measurement errors [START_REF] Hayduk | Should researchers use single indicators, best indicators, or multiple indicators in structural equation models[END_REF].

Context and current study

As previously mentioned, scales can be good tools to evaluate technological products. Nevertheless, statistical models require several items for each theoretical variable evaluated, meaning that it takes a long time for scales to be completed. To cope with the length of scales, we started the development of short scales including only single items because, to our knowledge, this type of scale can be found in the HCI field. Before going into detail about the creation of scales to evaluate technological products, a short overview of research on factors related to the perception of technological products is presented. Following that, first steps of the development of two short scales with single-items to evaluate technological products are described.

Acceptability and its components

According to previous research, acceptability (i.e., user's judgement toward a system before use [START_REF] Schuitema | Explaining differences in acceptability before and IHM'16 Travaux en Cours 25-28 oct[END_REF]) of a product is based on several factors. Firstly, meta-analyses [e.g. ,26] have shown an influence of usability (i.e., degree in which the user expects the target system to be free of effort [START_REF] Davis | Perceived Usefulness, Perceived Ease of Use, and User Acceptance of Information Technology[END_REF])

• Usefulness: I think that this mobile app might be useful and usefulness (i.e., degree in which the user expects that using a specific application system will increase his/her performance [START_REF] Davis | Perceived Usefulness, Perceived Ease of Use, and User Acceptance of Information Technology[END_REF]) on acceptability. More recently, research highlighted that users perceive a product beyond these functional qualities (i.e., beyond usability and usefulness) [START_REF] Bargas-Avila | Old Wine in New Bottles or Novel Challenges? A Critical Analysis of Empirical Studies of User Experience[END_REF]. Thus, two main variables have been studied: stimulation and trust. Stimulation (i.e., need of novelty [START_REF] Hassenzahl | The Interplay of Beauty, Goodness, and Usability in Interactive Products[END_REF]) has a positive effect on behavioral intention [START_REF] Van Schaik | An integrated model of interaction experience for information retrieval in a Web-based encyclopaedia[END_REF] and the influence of trust (i.e., positive belief about the perceived reliability in a system [START_REF] Fogg | The Elements of Computer Credibility[END_REF]) has been demonstrated in some papers [e.g. ,18]. These 4 variables are related to product qualities according to user experience (UX) theories. Lastly, in research, the perception toward a technological product is generally considered to be based on a rational evaluation of quality criterions [START_REF] Venkatesh | Why Don't Men Ever Stop to Ask for Directions? Gender, Social Influence, and Their Role in Technology Acceptance and Usage Behavior[END_REF]. Nevertheless, the human functioning is more complex and it is necessary to take into account some affective and motivational variables. Thereby, identification (i.e., self-image that the product returns of itself [START_REF] Hassenzahl | The Interplay of Beauty, Goodness, and Usability in Interactive Products[END_REF]) has been demonstrated as an important predictor of acceptability [START_REF] Van Schaik | An integrated model of interaction experience for information retrieval in a Web-based encyclopaedia[END_REF]. Moreover, an effect of intrinsic motivation (i.e., doing an activity for its inherent satisfactions [START_REF] Ryan | Intrinsic and Extrinsic Motivations: Classic Definitions and New Directions[END_REF]) on acceptability was demonstrated [START_REF] Matthew | Acceptance of Internet-based learning medium: the role of extrinsic and intrinsic motivation[END_REF]. Lastly, the social influence (i.e., degree in which an individual perceives how important others believe he or she should use the new system [START_REF] Venkatesh | User Acceptance of Information Technology: Toward a Unified View[END_REF]) was added and shows a positive effect on behavioral intention [START_REF] King | A meta-analysis of the technology acceptance model[END_REF]. These 3 variables are related to affective-motivational factors. Thus, we developed two short scales to evaluate the theoretical construct presented below: one to evaluate the product qualities (i.e., UX) and one to evaluate affective-motivational factors.

Method

Material

In order to evaluate the psychometrics qualities of our scales, it was necessary to collect data with these scales. Thereby, participants evaluated the sleeping tracker mobile application "Sleep Better ©". This application was selected because it is simple, free, designed for repeated use and relatively distributed.

Measures

To evaluate the users' perceived qualities about the mobile application, the following dimensions were measured: Some items were constructed specifically for this study and others were based on existing questionnaires as UTAUT [START_REF] Venkatesh | User Acceptance of Information Technology: Toward a Unified View[END_REF] and Attrackdiff [START_REF] Hassenzahl | The Interplay of Beauty, Goodness, and Usability in Interactive Products[END_REF]. For each item, participants had to answer on an 11 points Likert scale (from 0 to 10). Position of each item in the questionnaire was randomly delivered to avoid effects related to order of the items [START_REF] Benton | A Question Order Effect in a Local Government Survey[END_REF].

Participants

Using a web form, 305 participants (236 women), who had never used the app, completed evaluations before use (i.e., 32 items). They were recruited using social networks: Facebook, Twitter and LinkedIn. The average age was 28.83 years (SD = 9.35).

Procedure

The participants completed the scales after reading the presentation of the application proposed by Runtastic© on market app (AppStore and Google Play).

Results

To ensure of the quality of the scales, exploratory and confirmatory factor analyses were conducted. R [START_REF] Core | R: A Language and Environment for Statistical Computing[END_REF] with libraries lavaan [START_REF] Rosseel | lavaan: An R Package for Structural Equation Modeling[END_REF] and psych [START_REF] Revelle | psych: Procedures for Psychological, Psychometric, and Personality Research[END_REF] were used.

Exploratory factor analyses

To determine the number of factors to extract, Eigenvalues and an oblique rotation (Promax) [START_REF] Abdi | Factor Rotations in Factor Analyses[END_REF] were used. The number of extracted factors was consistent with the number of predefined factors: 4 for UX scale and 3 for A-M Scale. Exploratory factor analyses confirmed the uni-dimensionality of each dimension in UX and A-M Scales: items saturate above .30 their a priori defined factors (i.e., theoretical factors).

Confirmatory factor analyses

After the exploratory factor analyses, confirmatory factor analyses were computed. To deal with violation of multivariate normality, robust standard errors (Huber-White) and a scaled test statistic were used (Yuan-Bentler) [START_REF] Beaujean | Latent Variable Modeling Using R: A Step-by-Step Guide[END_REF]. Moreover, for handling with missing values, a full information maximum likelihood estimation was used [START_REF] Enders | The Relative Performance of Full Information Maximum Likelihood Estimation for Missing Data in Structural Equation Models[END_REF]. According to cutoff values by Hu and Bentler [START_REF] Hu | Cutoff criteria for fit indexes in covariance structure analysis: Conventional criteria versus new alternatives[END_REF], results indicated a good fit for UX and A-M scales with respectively

Construction of single-items scales

As explained previously, to reduce the length of scales without decreasing their accuracy, one approach is to take into account the measurement errors of items. For this purpose, values of measurement errors should be determined. Thus, confirmatory factor analyses were bootstrapped (10000 draws) to ensure of the stability of models' parameters [START_REF] Fan | Using Commonly Available Software For Bootstrapping In Both Substantive And Measurement Analyses[END_REF]. This method allows data to be resampled (random sampling with replacement) [START_REF] Fitzgerald Bone | A Bootstrap Procedure for Evaluating Goodness-of-Fit Indices of Structural Equation and Confirmatory Factor Models[END_REF]. Moreover, the observable variables (i.e., items) were standardized before being entered in the model. To develop short versions of the scales, we started by selecting one item per evaluated latent variable. This selection was based on statistical indexes (largest factor loadings and smaller estimated measurement errors) and wording of the items.

Selected items are shown in figure 3. Then the estimated measurement errors for these selected items were extracted from bootstrapped factor analyses. Estimated values are presented on figure 4 and5. With these values, it is possible to correct data collected with items, taking into account the errors of measure.

Multiple-items scales vs. single-items scales

To test the accuracy of single-items scales compared to multiple-items scales, we computed two models which explained Behavioral Intention (BI) using SEM. In both of these models, we estimated the influence of UX and A-M factors on BI. In the first model, scales composed of multiple items for each evaluated construct were used: 16 items for UX scale and 12 items for A-M scale. In the second model, scales composed of single items for each evaluated construct were used: 4 items for UX scale and 3 items for A-M scale. In this model, estimated measurement errors extracted from bootstrapped confirmatory factorial analyses were included to take into account the measurement error. Lastly, to be able to compare the models, the latent variable corresponding to BI was estimated using 4 items. Results indicated good and similar fit with respectively

Discussion and conclusion

In the current study, we presented the first steps of the development of short scales including single items. Before creating our short scales, a first version with multiple items for each evaluated dimension was developed. After, one item for each dimension was selected and measurement errors for each item were defined from bootstrapped confirmatory factor analyses. According to the results of SEM, the model which explained BI using single-items scales provides very close results to a model using multiple-items scales. Indeed, reducing the number of items (28 vs. 7 items) has a very little effect on paths between factors of acceptability and behavioral intention. Moreover, the explained variances are close between the both models.

Limits and future studies

Several limits and improvements can be identified. Firstly, the approach presented to take into account the error of measurement has a major disadvantage. Indeed, before being able to use only single items, it is necessary to collect data from participants with multiple items scales to estimate measurement errors. Moreover, as measurement errors are estimated on only one sample, estimated parameters can be biased. To assess this potential bias, future works should strive to collect data with multiple-items scales on several samples or technological products, as well as determining the difference between estimated errors on the first sample and new data. In addition, these data can be used to estimate an average error measurement for each item. Secondly, actual versions of scales include only items for a judgement before use. In order to have a complete tool to evaluate a technological product, it is essential that it can be evaluated after use. Lastly, as scales have been tested in French, it is necessary for use in an international context to translate it into English and revalidate it with Englishspeaking people.

To conclude this preliminary work, reducing the number of items should decrease the time of completion for participants without decreasing the possibility for statistical modeling and the accuracy of models. We can obtain data from users about a product more easily and then facilitate a user-centered design approach. Moreover, understanding factors leading to adoption and long term use is essential for companies who develop products. For this purpose, it is necessary to acquire data at several stages: before use, just after first use, and over use (e.g. after one week, one month, etc.) [START_REF] Martin | From Acceptability to Acceptance: Does Experience with the Product Influence User Initial Representations?[END_REF]. This requires participants to complete scales regularly. Thus, having scales with a minimum of items can facilitate the collection of data for researchers and simplify responses for participants.

Figure 2 . 3 Figure 1 .

 231 Figure 2. Single item scale: measurement error is fixed to 0 -assumes that item measured the latent variable perfectly

Figure 3 .

 3 Figure 3. Items selected for the single-items scales

Figure 4 .

 4 Figure 4. Results of bootstrapped factor analysis for UX scale. Note: Bold line indicate the selected item for single item scale

  99 & .98; RMSEA = .03 & .05 and SRMR = .03 & .04

  49 & 1.6; CFI = .97 & .99; RMSEA = .04 & .05 and SRMR = .04 & .01. Moreover, the paths between BI and UX or Affective-Motivational factors were very similar between scales as presented in figure 6 and 7. Lastly, the explained variances with multiple-items and

Figure 5 .

 5 Figure 5. Results of bootstrapped factor analysis for AM scale. Note: Bold line indicate the selected item for single item scale

Figure 6 .Figure 7 .

 67 Figure 6. Modeling with multiple-items scales

For standardization reasons[START_REF] Beaujean | Latent Variable Modeling Using R: A Step-by-Step Guide[END_REF], the variance of the latent variable or the loading of one item needs to be fixed to 1. Here, the loading between the latent variable and the item 1 is fixed.

Acknowledgements

We would like to thank all those who participated in any way in this research. This work was carried out within the IRT b<>com (ANR-07-A0-AIRT).


Sandrine Ngo

Anton Peshkov

Igor S Aranson

Eric Bertin

Francesco Ginelli

Hugues Chaté

Large-scale chaos and fluctuations in active nematics

Keywords: numbers: 05.65.+b, 45.70.Vn, 87.18.Gh

We show that "dry" active nematics, e.g. collections of shaken elongated granular particles, exhibit large-scale spatiotemporal chaos made of interacting dense, ordered, band-like structures in a parameter region including the linear onset of nematic order. These results are obtained from the study of the relatively simple and well-known (deterministic) hydrodynamic equations describing these systems in a dilute limit, and of a self-propelled particle Vicsek-like model for this class of active matter. In this last case, revisiting the status of the strong fluctuations and long-range correlations now considered as landmarks of orientationally-ordered active phases, we show that the giant number fluctuations observed in the chaotic phase are a trivial consequence of density segregation. However anomalous density fluctuations are present in the homogeneous quasi-ordered nematic phase and characterized by a non-trivial scaling exponent.

Many of the recent studies on active suspensions and active gels have reported the existence of instabilities leading to spontaneous flows [1][2][3][4]. At the nonlinear level, the term "bacterial turbulence" has been used to describe the fast, collective but chaotic motion of swimmers evolving in what remains a very low Reynolds number, inertialess, world [5][6][7][8]. In remarkable in vitro experiments on actomyosin motility assays, the fluid in which filaments and motors evolve seems to play a key role in the emergence of local order leading in turn to erratic large-scale flows [9]. Other actomyosin systems, like the active nematics suspensions studied by Dogic et al., display spontaneous large-scale dynamics mediated by the nucleation and motion of topological defects [10][11][12][13].

In contrast, most of the recent studies of "dry" active matter (where the fluid in which active particles move can be safely neglected), have not reported widespread occurence of large-scale chaos or turbulence [14]. Following the seminal papers of Vicsek et al. [15] and Toner and Tu [16], a lot of attention has been paid to the nature of the onset of orientational order/collective motion and to the existence of generic long-range correlations and anomalous fluctuations in spatially-homogeneous ordered phases [17,18,21]. The existence of long-wavelength instabilities of homogeneous ordered states leading, in dilute systems, to some phase separation between highdensity high-order structures (bands, waves) is now recognized as a generic feature [18][19][20][22][23][24][25][26][27][28][29], but the stability and large-scale dynamics of these structures remain largely unknown.

In this Letter, we show that large-scale spatiotemporal chaos arises generically in dry active nematics. We first demonstrate that the solutions of the relatively simple and well-known (deterministic) hydrodynamic equations describing these systems are chaotic in a region of parameter space including the linear onset of nematic order. We show in particular that the nonlinear ordered band solution found before [30] is unstable, leading to a disordered phase in which elongated dense and ordered structures curve, extend, merge, and split on very large time-and length-scales. Returning to the Vicsek-style model for dry active nematics introduced in [32] to investigate further the effect of fluctuations, we provide evidence that its segregated regimes where dense, ordered structures form are also actually chaotic on large scales, rendering this phase asymptotically disordered. The giant number fluctuations reported in [32] to be in agreement with the predictions of Ramaswamy et al. [33] are thus a trivial consequence of phase separation. Non-trivial fluctuations are nevertheless present in the homogeneous quasiordered phase overlooked in [32] but found here at larger densities. Their scaling exponent, though, differs from the simple one derived in [33] using a linearized theory. Interestingly, we find that it takes a value similar to that calculated by Toner and Tu for polar ordered phases [16].

We start by recalling the Vicsek-style model for active nematics defined in [32], where point particles carrying a (uniaxial) nematic degree of freedom align locally and are forced to move randomly along one of the two directions defined by their axis. In two spatial dimensions the positions x t j and directors n t j ≡ cos θ t j , sin θ t j T with

θ t j ∈ [-π 2 , π 2
] of particles j = 1, . . . , N are updated synchronously at discrete timesteps according to:

θ t+1 j = 1 2 Arg   k∈Vj e i2θ t k   + ψ t j ; x t+1 j = x t j ± v 0 nt j (1)
where V j is the set of neighbors of particle j within unit distance, the sign in the second equation is chosen randomly with equal probability, and the random angle

ψ t j ∈ [-η π 2 , η π 2 ] (with η ∈ [0, 1]
) is drawn from a uniform distribution. As a matter of fact, this model has not been much studied beyond the initial paper [32] where numerical simulations performed on square domains of linear size L at global density ρ 0 = N/L = 1 2 concluded to an isotropic/nematic Berezinskii-Kosterlitz-Thoulesslike transition [34] as η is decreased, with the quasi-longrange ordered phase consisting of a single dense ordered band and supporting giant number fluctuations.

As shown in [30], the rather well known hydrodynamic equations for dry active nematics [31] can be derived in a simple and controlled way from this model, with all transport coefficients depending explicitly on ρ 0 and the noise strength, the only two parameters remaining after rescaling. In this approach, one assumes a dilute limit and a molecular chaos hypothesis, which allows to write a Boltzmann equation for the one-body distribution function f (x, θ, t). By expanding f in Fourier series of θ, f (x, θ, t) = 1 π k=∞ k=-∞ fk (x, t)e -i2kθ , the kinetic equation becomes a hierarchy which can be truncated and closed, assuming a diffusive scaling ansatz and the proximity of the onset of nematic order. The first non-trivial order yields a nonlinear equation governing the nematic complex field Q ≡ f1 , together with the continuity equation governing the density field ρ ≡ f0 [40]:

∂ t ρ = 1 2 ∆ρ + 1 2 Re ∇ * 2 Q (2) ∂ t Q = µ(ρ) -ξ |Q| 2 Q + 1 4 ∇ 2 ρ + 1 2 ∆Q (3)
where µ(ρ) = µ ′ (ρ -ρ t ) and we have used the complex operators 2) and ( 3) is given in Fig. 1a. The condition ρ = ρ t , defining the line σ t , marks the linear instability of the disordered solution Q = 0 and the emergence of the homogeneous ordered solution |Q| = µ/ξ (for µ > 0). But this ordered solution is itself linearly unstable to long-wavelength perturbations transversal to nematic order in a region bordering the basic line ρ t , σ t . Deeper in the ordered phase, below the line ρ s , σ s , the homogeneous ordered solution is linearly stable.

∇ ≡ ∂ x +i∂ y , ∇ * ≡ ∂ x -
It was shown in [30] that Eqs.( 2) and ( 3) support an inhomogeneous solution in the form of a band of nematic order with density ρ band > ρ s surrounded by a disordered gas with ρ gas < ρ t . Supposing the nematic order is along x, for this solution we obtain ρ = R 0 (y) ≡ Q 0 (y) + ρ gas and

Q 0 (y) = 3(ρ t -ρ gas ) 1 + a cosh 4µ ′ (ρ t -ρ gas )y (4)
with a = 1 -9ξ(ρ t -ρ gas )/2µ ′ , ρ gas being a constant fixed by density conservation [30]. Its existence domain (σ min , σ max ) actually extends beyond the region of linear instability of the homogeneous ordered solution (σ min < σ s and σ max > σ t ). Its ordered part occupies a fraction of the y dimension of the system going continuously from zero (near σ max ) to one (near σ min ).

If considered only as a one-dimensional function of y, the band solution Eq. ( 4) is linearly stable. We now show that it is always unstable with respect to long-wavelength undulations along the x-axis. To study the linear stability with respect to wavenumber k, we seek the perturbative solution in the form (Q, ρ) = (Q 0 (y), R 0 (y)) + (q(y), r(y)) exp(λt + ikx). Substituting the growth rate λ into Eqs. ( 2),(3), we obtain a linear system for q(y), r(y). Noting that for k = 0 the solution to this system is the translational mode r = q = ∂ y Q 0 , we can further simplify the problem in the long-wave limit k → 0 by expanding the perturbative solution in k and employing the following anzatz: λ = λ 1 k 2 , q(y) = u(y) + iv(y) and

  r(y) u(y) v(y)   =   ∂ y Q 0 (y) ∂ y Q 0 (y) 0   +   k 2 r 1 (y) k 2 u 1 (y) ikv 1 (y)   (5)
We then obtain linear inhomogeneous self-adjoint equations for the functions r 1 , u 1 , and v 1 . The solvability condition yields an explicit expression for the growth rate λ in terms of integrals of Q 0 and ∂ y Q 0 . The analysis, detailed in [35], shows that λ > 0, implying that the band solution is always unstable. However, the instability can be suppressed in small systems. At the nonlinear level, for large enough systems, the instability of the band first manifests itself as some periodic modulation in space and time localized along its borders. This then turns into localized chaotic behavior (Fig. 1b), which eventually develops into full-blown spatiotemporal chaos for large enough system sizes and integration times. There, distorted band-like structures evolve on very long timescales and large lengthscales, elongate, split, merge, without ever forming the original macroscopic band again (Fig. 1c and[35]). We have observed this spectacular dynamics all along the (σ min , σ max ) interval [42]. As σ is varied from σ max to σ min , the largest structures observed have increasing sizes. We measured the global nematic order parameter S(t) = | Q x,y | and the two-point spatial correlation function of the density field for different square systems of linear size L. For large enough systems, the time-averaged order parameter S decreases like 1/L, indicating the existence of a finite, L-independent correlation length (Fig. 1d) [43].

The segregated phase of the hydrodynamic equations for active nematics is thus asymptotically disordered. We now investigate the robustness of the above results with respect to fluctuations by coming back to the Vicsek-like model defined by Eq. (1). This also provides an opportunity to gauge the faithfulness -at a qualitative level-of the hydrodynamic equations ( 2)-(3) to the model they were derived from. We performed extensive simulations of Eq. ( 1) at various global densities ρ 0 , varying the noise strength η and the system size L.

At large enough ρ 0 , we do observe, for low enough η, a spatially-homogeneous, non-segregated, quasi-ordered phase (not shown). Because it possesses unusually strong density fluctuations (see below) which may be hard to distinguish from the fluctuating structures of the segregated phase, the location of η low , the noise value marking the lower extent of the inhomogeneous phase, is difficult to characterize beyond visual inspection provided by movies and snapshots such as in Fig. 2 and[35]. We used scaling and fluctuation properties of the global nematic order parameter S, whose full probability distribution P (S) can be measured with good statistics only for moderate system sizes (up to L = 256). In the homogeneous phase, its mean S decreases algebraically with L with an exponent ζ(η) < 1 8 (quasi-long-range order), and P (S) quickly converges, as L is increased, to the Bramwell-Holdsworth-Pinton (BHP) distribution, well-known to describe almost perfectly the quasi-ordered, vortex-free phase of the equilibrium XY model [36] (Fig. 2a). The inhomogeneous segregated phase, by contrast, is characterized by a departure of P (S) from the BHP distribution which becomes more important as the system size is increased (Fig. 2c). During the corresponding events, the main dense ordered band typically observed at such moderate system sizes reorganizes itself. We used the finite-size behavior of P (S) to define -admittedly rather roughly-the threshold value η low : for η < η low , P (S) falls on the BHP distribution, whereas for η > η low P (S) deviates from BHP, and these deviations eventually become so important that the initial algebraic decay of S with L accelerates at large L values (inset of Fig. 2c).

Using the above approach, we find that η low , decreases with ρ 0 (Fig. 2b). For ρ 0 = 1 2 , the density used in [32], we estimate η low ≃ 0.02, relegating the homogeneous regime to numerically difficult, very small noise strengths regimes not probed in [32]. For even smaller ρ 0 values the homogeneous phase is practically unobservable.

For the η high line of the phase diagram separating the segregated from the homogeneous disordered phases, we used the location of the peak of the susceptibility of the nematic order parameter. Even though both phases are disordered, the segregated one still shows sizeable S values at the finite sizes we can probe numerically.

Our numerical results obtained for η low < η < η high show that the decay of S accelerates at large system sizes as P (S) gradually departs from the BHP distribution. We thus conjecture that our system is then, asymptotically, in a disordered, not a quasi-ordered phase. This occurs even when the initial decay of S displays an exponent ζ < 1 8 , its value at the Berezinskii-Kosterlitz-Thouless transition used in [32] to define the isotropic/nematic transition (inset of Fig. 2c). In particular, at the onset of density segregation (η ≃ η low ), the decay exponent ζ is typically rather small (e.g. ζ ≃ 0.03 for ρ 0 = 0.5).

Observing full-blown disordered inhomogeneous regimes for η values such that the early decay of S occurs with an exponent ζ < 1 8 is however numerically very difficult, requiring huge system sizes and simulation times. The snapshots shown in Fig. 2e-f are actually taken in such a case: several dense, curved bands of various orientations are present. They evolve on very long timescales, elongating, merging, splitting, in a manner reminiscent of the band chaos reported in Fig. 1 (see [35]). Only the largest system sizes (Fig. 2f) reveal the existence of a finite characteristic length, in agreement with the nonlinear analysis of the equations ( 2)- (3).

Next, we investigated anomalous, "giant" number fluctuations, which we characterize by the scaling of ∆n the variance of the number n of particles contained in a square sub-system. Even though this anomalous scaling is a landmark of fluctuating active ordered phases, it can be probed in the fluctuation-free context of the fully chaotic regime of the hydrodynamic equations ( 2) and (3). We find ∆n 2 ∼ n 2 for boxes of sizes up to the characteristic lengthscale of chaos, followed by a crossover to normal fluctuations (∆n 2 ∼ n) for larger boxes (not shown). For the inhomogeneous disordered phase of the Vicsek-like model (1), as already reported in [32], we also find ∆n 2 ∼ n 2 but are numerically unable to see the expected crossover to normal fluctuations. These observations are the trivial result of the segregation of density in dense bands [37] and not due to the sophisticated mechanism put forward by Toner, Ramaswamy, et al. as a typical feature of ordered, non-segregated, active phases. However, the homogeneous regimes found for η < η low in the microscopic model do constitute such a phase. Measuring number fluctuations in this case, we find anomalous fluctuations, ∆n 2 ∼ n α , but with a characteristic scaling exponent α close to 1.6, not 2 (Fig. 2d). In other words, we do not find the value α = 2 derived by Ramaswamy et al. from a linearized theory, but, surprisingly, a value close to that calculated by Toner and Tu for active polar ordered phases.

We finally mention results obtained on the Vicseklike model defined by Eq. ( 1), but where neighbors are chosen to be those forming the first shell of Voronoi polygons around a given particle. In this "metric-free" model, the basic instability of the homogeneous ordered state leading to the segregated phase is suppressed [28]. Accordingly, our simulations reveal only two phases, the homogeneous disordered one, and the homogeneous quasi-ordered one, separated by a Berezinskii-Kosterlitz-Thouless-like transition. The quasi-ordered phase also exhibits anomalous number fluctuations, also with a nontrivial scaling exponent α ≃ 1.6 (not shown).

To summarize, we showed that dry active nematics exhibit large-scale spatiotemporal chaos consisting of moving, elongating, splitting, merging high-density highorder band-like structures. At the level of the (deterministic) hydrodynamic equations ( 2) and ( 3), this chaos is the outcome of the linear instability, in two dimensions, of the band solution (4), and is observed in the whole region of existence of the solution, i.e. between the σ min and σ max lines in Fig. 1a, a region which encompasses the linear onset of nematic order. At the level of the Vicsek-like microscopic model defined by (1), the corresponding lines are the η low and η high lines of Fig. 2b. The phase diagram of dry active nematics thus comprises three different phases, a homogeneous disordered phase at strong noise, an inhomogeneous disordered chaotic phase at intermediate noise values, and a homogeneous ordered phase. In the microscopic model, this last phase is only quasi-ordered and displays anomalous number fluctuations. The order-disorder transition is thus located between the ordered phase and the chaotic inhomogeneous phase. Its nature, even at the "mean-field" level of the hydrodynamic equations, remains unclear: if the correlation scales of chaos were found to diverge when σ → σ min , then the transition would probably be continuous. Unfortunately, the numerical determination of these scales in this limit is a very difficult task beyond the scope of this Letter [44].

Our results should eventually be completed by a study of the mesoscopic theory derived in [30], either by numerical integration of these Langevin equations or by some renormalization group analysis. (This could in particular provide some explanation to the anomalous scaling expo-nent for number fluctuations found here similar to that of polar ordered phases.) This difficult task is left for future studies. At the experimental level, we believe that microtubule motility assays of the type studied in [38] should be able to demonstrate the phenomena described here [39]. , where the Pk are the coefficients of the Fourier series of the noise distribution. In the following, we used a Gaussian distribution of variance σ 2 so that Pk = e -2k 2 σ 2 .

[42] This was done at a global density ρ0 = 1, but we do not have any reason to believe that a qualitative change of behavior occurs as ρ0 is varied.

[43] This is corroborated by the two-point correlation function of the density field: it is axisymmetric for long enough averaging times and its radial average decreases exponentially, with a system-size independent cutoff for large enough L (not shown).

[44] Note that this would be an even harder task with the microscopic model.

  i∂ y , and ∆ ≡ ∇∇ * . The transport coefficients µ ′ , ρ t , and ξ are positive constants depending on the noise strength σ [41]. The phase diagram of Eqs.(

FIG. 1 :

 1 FIG. 1: (color online) Mesoscopic dynamics: (a) phase diagram of hydrodynamic equations for active nematics. (b) global order parameter vs system size at different noise values in chaotic regime (the solid line has slope -1). (c,d) snapshots of density field in chaotic regime for L = 1600 and ρ0 = 1; (c): localized chaos (σ = 0.26); (d) fully-developed chaos (σ = 0.28)

FIG. 2 :

 2 FIG. 2: (color online) Microscopic dynamics: (a,c) Rescaled distribution of S at various system sizes in the homogeneous ordered (a) and inhomogeneous (c) phases. Insets of (a,c): log log plots of S × L -ζ vs L, where ζ is estimated from a fit of the initial decay of S . (b) Phase diagram in the (η, ρ0) plane. (d) Variance ∆n 2 as a function of mean number of particle n , in the homogeneous ordered phase. Dashed lines: power laws with exponents 2 (magenta) and 1.6 (cyan). (e-f) Snapshots of coarse-grained density in the chaotic phase of panel (c) for system sizes L = 8192 (e) and L = 16384 (f). Parameters: in all cases v0 = 0.3, for (a,d): ρ0 = 2, η = 0.1, ζ = 0.007 and for (c,e,f): ρ0 = 1 8 , η = 0.038, ζ = 0.103.

  miliar symmetric traceless tensor fieldQ since ρ[Q]xx = -ρ[Q]yy = 1 2 Re f1 and ρ[Q]xy = ρ[Q]yx = 1 2Im f1 but the complex notations are very convenient and we keep them in the following.[41] They read µ(ρ)

We thank the Max Planck Institute for the Physics of Complex Systems, Dresden, for providing the framework of the Advanced Study Group "Statistical Physics of Collective Motion" within which much of this work was conducted. FG and SN acknowledge support from grant EPSRC First Grant EP/K018450/1. The work of ISA was supported by the US Department of Energy, Office of Basic Energy Sciences, Division of Materials Science and Engineering, under contract no. DEAC02-06CH11357.