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Hanon, D. (2000). Diffusion in stationary and non-stationary flows (Unpublished doctoral dissertation). Université libre de Bruxelles, Faculté des sciences, Bruxelles.

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Faculté des Sciences Service de Chimie Physique

Diffusion in Stationary and Non-Stationary Flows

Thèse présentée en vue

de l’obtention du titre

de docteur en Sciences

David Hanon

Faculté des Sciences Service de Chimie Physique

Diffusion in Stationary and Non-Stationary Flows

Thèse présentée en vue de l’obtention du titre de docteur en Sciences David Hanon

Juin 2000

Thanks and Acknowledgments

Unlike the cover of this work may suggest, it is by far not the resuit of one single person’s efforts, as many hâve influenced, helped and supported me throughout these years.

To Professor Jean-Pierre Boon, I must express ail my gratitude.

He has not only welcomed me into his group without any reserves, giving me more freedom and trust than I could ever expect, but has also been a teacher and a friend ail along; without his knowledge, his comments and help this thesis wouldn’t hâve corne into existence.

I would also like to thank Professor Grégoire Nicolis. He not only introduced me to the vast subject of non-linear physics, but it has been a pleasure to be part of his service; his remarks about my work hâve always been pertinent and hâve influenced some of my key decisions.

Among those who hâve also had a direct impact on my researches, I can not fail to mention Olivier Tribel and Eric Vanden Eijnden and must thank them warmly. The many discussions we had were invalu- able and determining as well as enjoyable; I should add that they both taught me more than I dare admit.

I also wish to do justice to Jean-Pierre’s group as a whole; Alberto Suarez, Jôrg Weimar, Patrick Grosfils and Christopher Bodenstein ail forced me to think more than I was willing to at varions times, for which I am thankful.

There are also the people who made sure I enjoyed my studies, es-

pecially those who hâve shared them. For their friendship and the good

times we shared I must thank the ever doubting Jérome Dethier, the

never doubting Marc Massai, Michael Theunissen whose doubts are subjected to more fluctuations and to Laurent Friob whose presence fluctuated along with his doubts. While I will not make an exhaustive list, Denis, Bonnie, Fabian, Stéphane, Pierre-François, Anne-Sophie, Bernard, Nicolas also contributed to these good moments, I hope they know to what extent I appreciate this.

After several years of résistance on my part, the people next door hâve turned me into a cofFee drinker, a major feat, for which I should probably hâte them. Thanks Philippe, José, Thierry, Jésus, Arnaud, Fabrice, Stam, Stéphanie and the others for their presence, their colîee and their humor. In the same vein, the FoScCuP has been an enjoyable place to relax ail these years, thanks to Bruno among others.

The Belgian Fond à la Recherche à l’Industrie et l’Agronomie de- cided that it was a good idea to fund my researches for the maximum of four years their régulation allows for. It goes without saying that this was one of the primary conditions for me to undertake this work and I wish to thank the F.R.I.A.’s jury for its trust.

My fifth and final year of research devoted to this Ph.D. thesis was made possible by the European School and by roughly six dozen kids who turned my first year of teaching into an enjoyable expérience while leaving me some time to end what had been started.

This is certainly the right time and place to write how grateful I am to those people who lured me into the study of physics and to those who ensured I could appreciate it — little did I know how much it would change and shape me. In chronological order, let me thank Herr Grabo, Professor Jean Doyen, Professor Jean Jeener and Professor Radu Balescu.

And since Pm already writing about the Old Times, I must also

thank Mr. Yao, a semi-mythical Chinese Emperor of the 23rd century

B.C., who according to legend invented WeiQi, the Game of Go, in

order to instruct his idle (and presumably stupid) son Dan Zhu. Even though the practice of this game bas taken its toll on the time I dis- posed of as a student for leisure or study, it is at last repaying its debt, furnishing me with a subject for my “thèse annexe”. Professer Raymond Devillers had the kindness to accept the supervision of this work, which earns him the immense gratitude of an addicted player.

I had the opportunity to visit several conférences and labs during the course of my thesis, which was certainly a chance.

In this context, I wish to thank Professer Michel Maréchal who enabled me to visit the CECAM in Lyon as well as Professer Grégoire Nicolis who made it possible for me to attend the seventh European Turbulence Conférence near Nice.

I am also in debt with Professer Dominique D’Humière, Professer Pierre Lallemand and their group for the welcoming atmosphère which reigned at Paris IX and at the École Normale Supérieure de Paris during my visit there, as well as for the time they gave me.

As I found out, there is also a darker side to research, consisting of doubts, unanswerable questions and in the end —despite the presence of ail those friends and colleges— a tremendous loneliness.

For the better or for the worst, the reader will be judge, but he

would not be reading this at this moment if Florence had not corne

into my life as a companion and friend. Nothing I care to write here

will express this well enough, I will therefore let it be.

Résumé

Cette thèse traite du phénomène de la diffusion au sein de systèmes basés sur les automates de gaz sur réseau et à périodicité spatiale. Les modèles que nous avons explorés permettent de simuler les caractéris­

tiques macroscopiques de fluides réalistes alors même que leur concep­

tion se fait à partir d’énormes simplifications de leur caractéristiques microscopiques, ce qui facilite considérablement leur étude théorique et expérimentale.

La première partie du travail porte sur l’étude des fluctuations de densité dans un gaz sur réseau à deux espèces à l’équilibre. Nous développons la théorie de Boltzmann sur réseau et calculons analy­

tiquement le facteur de structure dynamique. L’expression obtenue est évaluée explicitement dans la limite hydrodynamique par l’identi­

fication des modes hydrodynamique du gaz sur réseau, dont un mode purement diffusif (transport de couleur). L’expression du spectre est ensuite traitée numériquement, afin de quantifier l’étendue des échelles sur lesquelles l’hypothèse de Boltzmann reste valable.

La seconde partie de la dissertation concerne des systèmes de non- équilibre à l’état stationnaire. Nous développons d’abord la théorie des flots compressibles dont la moyenne et la covariance sont connus; en­

suite nous mettons en œuvre uin syst‘eme de mesures expérimentales basé sur les automates de gaz sur réseau produisant un flot périodique structuré bruité dans lequel des traceurs sont ensemencés. Nous mon­

trons à partir du développement multi-échelle de l’équation de Liou-

ville stochastique qui régit la dynamique des traceurs que ceux-ci sont

soumis à une dynamique asymptotique difFusive, ce qui est confirmé expérimentalement. Le traitement quasi-linéaire du développement multi-échelle de l’équation de Liouville stochastique permet de for­

muler des prédictions quantitatives sur les coefficients de transport en excellent accord avec les mesures effectuées dans le modèle d’automate de gaz sur réseau, et ce malgré les grandes variations d’échelles pou­

vant apparaître entre la diffusion microscopique provenant du chaos moléculaire et la diffusion macroscopique résultant de la présence de structures à grande échelle dans le flot moyen.

La troisième et dernière partie est consacrée à l’étude des systèmes

stationnaires aux systèmes non stationnaires. Nous présentons une

généralisation des méthodes analytiques et numériques développées

pour les flots stationnaires et nous montrons que si le flot évolue sur

des échelles de temps plus grandes que celle définie par la fonction

d’autocorrélation du champ de vitesse, les traceurs exhibent toujours

un comportement diffusif.

1 Introduction 1

1.1 Lattice Gas Automata... 1

1.2 Diffusion... 3

1.3 OverView... 4

2 Diffusion in a binary fiuid 7 2.1 Introduction... 7

2.2 Generalities... 8

2.2.1 The FHP Lattice Gas Automaton... 8

2.2.2 General formalism of LG As... 9

2.2.3 Units... 14

2.2.4 Golored models... 14

2.3 Boltzmann formalism... 20

2.3.1 Liouville équation... 20

2.3.2 The LGA’s Boltzmann équation... 21

2.3.3 Dynamic Structure factor... 23

2.4 The hydrodynamic limit... 26

2.4.1 The hydrodynamic modes... 26

2.4.2 The dynamic structure factor... 31

2.5 The power spectrum... 36

2.5.1 Hydrodynamic régime ... 38

2.5.2 Boltzmann régime ... 41

vii

2.5.3 Spurious Invariants... 44

3 Diffusion in stationary flows 47 3.1 Introduction... 47

3.2 Theoretical aspects... 49

3.2.1 General considérations... 49

3.2.2 Effective transport coefficients... 53

3.2.3 Solenoidal flows... 56

3.2.4 White noise... 57

3.2.5 Parallel flows... 57

3.2.6 Application : Kolmogorov flow... 60

3.3 Experimental aspects... 63

3.3.1 Introduction... 63

3.3.2 Kolmogorov and ABC flows... 63

3.3.3 Lattice gas automaton... 68

3.3.4 Tracer dynamics ... 70

3.3.5 Results... 74

4 Diffusion in non-stationary flows 83 4.1 Introduction... 83

4.2 Theoretical frame... 84

4.3 Experimental setup and procedure ... 86

4.3.1 Producing the flow... 86

4.3.2 Measuring the mean velocity fleld... 86

4.3.3 Numerical évaluation of Dap... 89

4.3.4 Characterizing the flows... 90

4.4 Results... 92

4.4.1 Observations ... 92

4.4.2 Numerical prédictions ... 92

5 Conclusions 97

A Multiscale expansion 101

B Quasilinear treatment 107

B.l Equation closure...107 B.2 Stationary flows... 114 B. 3 Non-stationary flows...116

C Numerical methods 119

C. l Introduction... 119 C.2 Measuring diffusivity...120 C.3 Intégration of PDEs ...122

Bibliography 126

List of figures 135

List of tables 141

Index 143

Introduction

1.1 Lattice Gas Automata

In order to understand complex phenomena arising in nature, one suc- cessful strategy has been to study model Systems with simpler char- acteristics than their actual physical counterparts, thus allowing for analysis in greater depth, while hopefully capturing the major compo- nents of complex Systems.

The application of this reasoning to the extreme to computer mod- els of fluids has given rise to the class of models known as Lattice Gas Automata (LGA), which were conceived to lead to very efficient sim­

ulations, a resuit obtained by the complété discrétisation of space and time. As early as 1973, Hardy, de Pazzis and Pomeau [41] introduced their ‘HPP’ model in order to study certain aspects of ergodicity. Due to a lack of symmetries in its construction, this model cannot produce the correct isotropie Navier-Stokes behavior, and it was not before 1986 that the ‘FHP’-model, an LGA-model without this major flaw, was introduced by Frisch, Hasslacher and Pomeau [36].

The LGA did not fulfill the dreams of engineers of providing a

1

cheap and efficient way to simulate the équations of hydrodynamics in general because of the intrinsic spontaneous fluctuations they ex- hibit^. However, these very fluctuations turn the LG A into interesting Systems to study per se, providing an interesting and relatively sim­

ple way to test the usual tools of Statistical Mechanics, as they are in many cases more convenient to treat mathematically than their realistic continuons counterparts.

Because the FHP lattice gas lacks an independent collisional in­

variant for energy, it is not suited for the modeling of thermal fluids.

An appropriate generalization was realized by the construction of the model proposed by Grosflls, Boon and Lallemand (GBL) [38]. Their study was motivated by the analysis of the corrélations of spontaneous fluctuations in lattice gas automata (LGA) in order to find whether the fluctuations’ power spectrum would be in accordance with those observed in actual fluids. Indeed the dynamical structure factor - the power spectrum of the density fluctuations corrélation function - gains its importance by providing insight to the dynamical behavior of the fluid [9], and the LGA was found to exhibit correct properties at global equilibrium: the spectra obtained by simulations of the GBL model présent the same characteristics as those obtained from neutron- and light-scattering experiments in real fluids.

Many other variations of the original FHP model hâve been con- structed in order to tackle Systems of fluid-mixtures, three-dimensional hydrodynamics in complex geometry or reactive Systems.

^Alternative methods were nonetheless derived from the techniques at the origin

of LGA, in particular, the direct simulation of an LGA’s continuons Boltzmann

équation has been shown to be compétitive in complex geometries.

1.2 Diffusion

From the movement of pollen grains observed by Brown in water to the flow of beat in a spoon dipped into coffee to the dispersion of a pollu- tant in the océan, the équations governing the density of the observed quantity are the same: The phenomenon of diffusion is présent in our daily life, arising in very different Systems, themselves characterized by very different scales.

As established by the laws of thermodynamics, a physical System left on its own for a sufficiently long amount of time will evolve towards a stable State of equilibrium. If it is then subjected to small pertur­

bations, these will eventually decay and disappear, and ail fiuxes (of mass, impulsion, energy...) eventually vanish.

The way in which these small perturbations evolve are themselves a subject of interest and are precisely at the center of our attention in the first part of this work, as their study can give us insight about the way a System not too far from equilibrium will strive to reach this State. In particular, one of the universal mechanisms leading to the State of equilibrium is diffusion.

A System subjected to an external forcing of weak intensity will not reach this equilibrium State, but will nonetheless evolve towards a steady state, in which fiuxes reach asymptotic, stable values. When the external forces are increased beyond certain limits, the System will not be able to respond linearly to the forcing; the non-linear nature of its reaction may then lead to a vast range of States, ranging from periodic behavior to chaos.

Given certain conditions, these driven Systems may still give rise

to the phenomenon of diffusion, even if the scales involved are strongly

dépendent on the nature of the flows resulting from the applied con-

straints. We then speak of effective diffusion coefficients, since these

reflect behaviors on large scales, while the précisé phenomena tak-

ing place on smaller scales may be vastly more complex and beyond

possible mathematical analysis.

The general object of this work is to contribute to the under- standing of the mechanisms through which diffusion arises in different régimes. We try to predict under which conditions and on which scales diffusion will take place in certain classes of Systems and test our re- sults on experimental data obtained from Systems based on Lattice Gas Automata.

1.3 OverView

The first part of our study, presented in Chapter 2, is an analysis of diffusion in terms of the spontaneous density fluctuations around the equilibrium State in a non-thermal two-species fluid modeled by a lat­

tice gas automaton. The LGA model is first presented in detail, while the following sections are devoted to the study of the power spectrum of the density fluctuations. This function is computed with statisti- cal mechanical methods, analytically in the hydrodynamic limit, and numerically from a Boltzmann expression for shorter time and space scales.

In Chapter 3 we focus on analytical results concerning a broad class of non-equilibrium stationary Systems, which are shown to exhibit asymptotic difîusive behaviour. Appendix A contains the explicit de­

tails of multiscale developments leading to this resuit and Appendix B shows how to evaluate the transport coefficients under the quasilinear approximation.

On the experimental side, in order to explore diffusive phenomena

in setups which are far from the thermodynamic equilibrium State, we

hâve been led to construct new LGA models. We présent them in the

second part of Chapter 3 and then proceed to confront our analytical

and numerical prédictions to the experimental observations obtained

from the automaton world.

Chapter 4 deals with non-stationary Systems, for which the theo- retical results presented in Chapter 3 can be generalized to a certain extent, as shown through confrontation with experimental data.

We close with Chapter 5 where we summarize our contributions

and outline the possibilities for extension to future research.

Diffusion in a binary fluid

2.1 Introduction

In this chapter, we investigate the phenomenon of diffusion in an equi- librium System, a non-thermal Lattice Gas Automaton with two types of colored particles for which we develop the lattice Boltzmann theory and perform automaton simulations.

The emphasis is on the fluctuation corrélations in order to obtain a microscopie analysis of diffusion dynamics in contrast to earlier studies based on macroscopie approaches.

On large space- and time-scales we find spectral features of the dy- namic structure factor in accordance with those of real fluids described by the Landau-Placzek theory. Because of the intrinsic simplicity of the lattice gas model, the varions wavelength régimes can be easily identified, and the propagator spectrum can be used to to compute the power spectrum over the full wavenumber domain, and to test the the validity of the Boltzmann hypothesis.

The présent study based on the analysis of spontaneous fluctu­

ations offers a microscopie approach to diffusion, and, through the

7

identification of a purely diffusive mode associated to color transport, compléments and supports earlier macroscopie investigations.

2.2 Generalities

2.2.1 The FHP Lattice Gas Automaton

The basic FHP model is based on a regular two dimensional lattice with hexagonal symmetry whose nodes are the only positions accessi­

ble to the fluid particles. Thus space is discretised, and in practice it is also finite. The boundary conditions may be chosen periodic, bounce- back, wind-tunnel-like or whatever may be convenient. In addition, momenta are also distributed in a discrète fashion: ail particles hâve the same mass (equal to one mass-unit) and equal velocity modulus

—equal to one speed-unit, i.e. they will fly from one lattice node to one of its nearest neighbors in one unit of time (time is also discrète).

Furthermore, an exclusion principle is imposed: no two particles may résidé simultaneously on the same node if their direction is identical.

On the FHP triangular lattice, this implies that there can be at most six particles per node, unless a rest-channel is added, as in most ver­

sions of the model, finally resulting in a maximum of seven particles per node. Let us notice that this allows to describe the State of each node by a word of seven bits The exclusion principle, introduced in order to permit such a simple description on a computer, has the direct conséquence that the equilibrium-distribution of the particles is of the type of a Fermi-Dirac distribution, a feature consistent with

^The HPP model, which is based upon a square tessellation of the plane can

be described by 4-bit words and is thus simpler, but its low symmetry makes it

impossible to yield large scale isotropy.

general statistical physics

Interactions between partiales are simple ; they may only take place on nodes with several particles, taking the form of local instan- taneous collisions. The collision rules are chosen in order to conserve both mass and momentum. In this model there is no additional con­

servation law associated to energy since energy is directly proportional to mass. However if an internai degree of freedom is excited on rest- particles, energy is conserved and the System acquires a bulk viscosity, since internai energy and kinetic energy may be exchanged.

The évolution of the System from one time-step to the next takes place in two successive stages:

• propagation : the particles move from their node to the nearest neighbor in the direction of their velocity vector,

• collision : particles on the same node may exchange momentum if it is compatible with the imposed invariance-rules.

One distinguishes traditionally five different FHP-models which differ by the number and the type of collisions included in their colli­

sion tables [52].

2.2.2 General formalism of LG As

A convenient formai description of the LGA can be given in terms of occupation channels. We associate to each node a number b of channels indexed with i = 1,..., 5; each channel is also characterized by a velocity vector Cj, taken from the ensemble of allowed velocities. The State of a channel is defined by a Boolean variable rij, represented by a bit which takes the value 0 or 1, corresponding to a free or occupied State respectively.

^One should however not interpret this as a quantum effect, ail the models we

consider axe built through classical reasoning.

Figure 2.1: A physical image of the standard FHP model. Arrows in-

dicate the presence and direction of moving particles, circles represent

resting particles. Particles may cross each-other’s paths to jump to

the nearest neighboring sites if necessary.

-.J...- -J...- -J ...■■■■y--...

k yk. X X

>)< >

>k^' >

<>k ^

Table 2.1; The 7-bits FHP models allow for fourteen types of non- transparent collisions. The input States of the nodes are shown in the first row of both tables, over the output state(s). In particular, the FHP-III includes ail of these and maximizes the number of combina­

tions for the transfer of momentum.

One may so describe the simultaneous propagation of particles as

rii{T,t + t) <= rii{T - CiT,t).

In this manner, the content of bit on node r is simply transferred to bit rii on node r + CiT. From now on, we will use LGA units, setting

T

= 1.

If the general state of a node is written s, the collision step consists in replacing state s with s'. Most LGA models are constructed proba- bilistically allowing for more than one output s' for a given input s the choice is then generated randomly with a probability A{s —>■ s') > 0.

A non-transparent or efficient collision will ensure that the number of particles présent on some or ail channels varies on the node on which it takes place.

Combining the propagation and collision steps into a single expres­

sion also known as the microdynamical équation, we write the évolution of a channel occupation as

ni{r -I-

t -I-1) =

+ Aj(r,

(2.1) The collision operator Ai(r, t) is evaluated for each time step by means of a Boolean variable ^s,s' equal to zéro for ail configurations s' except for the one chosen as the post-collisional state of the node considered, for which ^s,s> = 1- It should go without saying that the average of ^s,s' is independent of space and is given by

< Cs,s' >t = s') (2-2)

Since every initial state s must lead to a final state s', we also may Write

s'

1, Vs (2.3)

and thus, combining (2.2) and (2.3) we obtain the following conditions upon A{s —>• s') :

^ yl(s —)■ s') = 1, Vs (2.4)

s'

Likewise, every final State must be engendered by exactly one initial State and so we also hâve

E^s,s' = 1, Vs' (2.5)

$

which implies

^A(s-)■ s') = 1, Vs' (2.6)

S

Combining (2.4) and (2.6) gives

^A(s->s') = ^^(s'^s), Vs (2.7)

s' s'

which garanties that for any given State s the population’s growth and decay hâve the same rate. This very important condition is called semi-detailed balance, while it is not a very strong condition on A(s —¥ s'), it has a fundamental conséquence, as it has been shown [37] to be sufficient for any LGA to prove the existence of a H-theorem.

This in turn warrants the existence of an equilibrium distribution func- tion towards which the System will evolve asymptotically. The model we are interested in benefits from a much stronger requirement, the detailed balance

A(s —>■ s') = A(s'—> s), V(s,s'), (2.8) which of course implies (2.7) and présents further advantages as we will see.

We now proceed to write Ai(r, t) explicitly as a function of the col­

lision rules. The following identity is verified by the Boolean variables nj and Sj :

-«,)<*-•<) =^,7“*- (2^9)

It implies

-rii = 0, (2.10) s' j

and allows us to write équation (2.1) as

rii{T + Ci,t + 1) = (2-11)

s,s' j

in which the rhs is to be evaluated at r and at time t. Subtracting 0 as written in (2.10) from the rhs of (2.11) results in

Ai=- Si) (2-12)

s,s' j

by identification with équation (2.1).

2.2.3 Units

Ail OUI calculations are performed in the unit System defined by the Lattice Gas Automaton. The unit for lengths is the distance between two nearest-neighboring nodes. The time unit is defined to be equal to the intégration time between two propagation steps. The unit for mass is set to be equal to the mass of each automaton particle, regardless of other properties, such as color.

AU numerical values appearing in this work —whether in calcula­

tions or graphical présentations of results— are therefore expressed in these units.

2.2.4 Colored models

We now consider two-species Systems with particles that can be iden-

tified, say as ‘red’ particles and ‘blue’ particles.

The dynamic structure factor^ is a measure of the intensity and évolution of a system’s intrinsic density fluctuations. It is a tool which we use extensively throughout this chapter. In simple monomolecular Systems, it exhibits in the limit of long wavelengths three Lorentzian peaks whose origins can be traced back rather straightforwardly to the expression of the conservation laws in the équations governing hydrodynamic behaviour:

The first of these peaks (the Rayleigh peak) originates in the law of energy conservation and provides a measure of the intensity of entropy fluctuations as well as of their diffusivity.

The remaining two peaks (the Brillouin peaks) resuit from the combination of the laws of mass and momentum conservation and provide a measure of the speed and dissipation of sound waves.

The mixture of two real fluids exhibits a power spectrum in which the central peak is not a simple Lorentzian, even in the long wavelength limit [9]. It has a spectral structure where it is difficult to separate the contributions from entropy fluctuations and from concentration fluctuations which are not decoupled in general (unless one of the two components is in trace amounts, in which case the two modes can be identifled as they produce two independent central Lorentzians).

The FHP model can be generalized to study diffusive phenomena in binary fluids using ‘macroscopie’ experiments [6, 52, 51]. Typically the observer would be interested in the évolution of the density profile of ‘red’ particles in a System composed of ‘red’ and ‘blue’ particles where the color is a passive property used to distinguish species which otherwise do not differ from one another (it is necessary that they do in other circumstances [47]). The diffusion coefficient is then evaluated by fitting the ‘experimental’ profile to the solution of the diffusion équation subject to the appropriate boundary conditions [52, 51].

^also frequently called “power spectrum of density fluctuations” and which we

will define in more detail in subsection 2.3.3

channel’s occupation mass-bit co\oT-bit

zéro partiale: 0 0

one red partiale: 1 0

one blue partiale; 1 1

Table 2.2: Description of a two-species LGA in ternis of the properties associated to the partiale which occupies a channel.

Prom the above considérations the idea emerged to analyze and measure the fluctuation corrélations in a non-thermal two-species LGA fluid (in which the Rayleigh peak is absent) in order to study diffusion dynamics from a microscopie approach. Our particular choice of model is the two-species FHP-III model, which we now présent in more detail.

The two-species FHP-III can be described in term of seven mass- occupation channels to which one must furthermore also associate a ftitindicating whether the occupying partiale is ‘red’ (color 0) or ‘blue’

(color 1). The state of this bit is irrelevant if there is no partiale on the given channel and can be set to a default value of zéro. A physical image of this représentation is given page 17 in figure 2.2a.

This description has been used [52], but it is not the one we hâve chosen, as the simultaneous knowledge of mass-occupancy and color brings unnecessary complications in the LGA’s formai description.

For our purposes, the following description turns out to be much more natural: instead of associating a new bit to each channel to take into account the occupying particle’s color, we double the number of channels, half of which are now dedicated to each color’s presence and transport. In order to keep the model unchanged, the exclusion prin- ciple is now applied to pairs of channels, making sure no two partiales can ever hâve the same position and velocity^. We number the red

^While this is not specifically required, it has the advantage of allowing to

Figure 2.2: Microscopie représentations of the two species FHP-III model

a: The State of each velocity channel is given by two bits, one for occupancy, one for the extra property we call ‘color’. There axe therefore seven channels, each having three possible States.

b: The State of each velocity channel is given by two bits, but both represent

the presence or absence of a particle, one bit counting the ‘red’ particle’,

the other one counting the ‘blue’ particles. There are therefore fourteen

channels, each having only two possible States.

occupation red blue of the channel pair mass-bit mass-bit

zéro particle: 0 0

one red particle: 1 0

one blue particle: 0 1

Table 2.3: Description of a two-species LGA in terms of the properties associated to the channel which a partiale may occupy.

channels from one to eight and the remaining blue channels from nine to fourteen (see figure 2.2b) with the following définition:

Our model’s microscopie invariants can now be written in terms of the microscopie occupancies. We hâve, Vr and Vt :

7 7

^ni(r + Ci,t + 1) t=i

14

=

t=l 14

(2.13)

^nj(r + Ci,t + 1) 1=8

14

= ^77t(r,t) i=8

14

(2.14)

^Cini(r + Ci,t + 1) i=l

II

(2.15)

which implies for the collision operator Aj the following relations

7

=

14

(2.16)

=

i=8

14

(2.17)

O

î>

O (2.18)

compare prédictions and data with previous work at no further cost.

for ail configurations s = {sj = = 1,..., 14} which satisfy the exclusion principle applied to channel pairs^.

®The application of an exclusion principle to pairs of channels is not required

in the theory developed in Secs. 2.3 and 2.4, but it does not provide any com-

putational inconvenience and makes easy connection with the single-species FHP

lattice gas.

2.3 Boltzmann formalism

In order to develop further arguments and calculation, we now summa- rize in the next two sections the foundations of Lattice Gas Automata’s Statistical Physics. We will then proceed to show how the two-species FHP-III can be treated in accordance with methods which were first applied in classical Boltzmann theory.

2.3.1 Liouville équation

indexphase space The key concept is ‘phase space’, denoted by® P, which is the ensemble of ail possible microscopie configurations for the set of the LGA’s nodes, denoted by £. A single point in this space corresponds to one single possible configuration s.,

s. = {sj(r) ; i = 1,14 ; Vr G £} ,

Next, we write the System évolution operator as £ and décomposé it into a collision operator C and a streaming operator <S, so that

£ — SoC. If we now write the system’s probability to be in state s. at time t as P(s.,t), we may aiso write the system’s Liouville équation, which describes this probability évolution

P{£s.,t + l) = P{s.,t) (2.19) and expresses that the probability of a given configuration is conserved by the System évolution. The décomposition of £ also allows us to Write this in the form of a Chapman-Kolmogorov équation

P{Ss.\ t + 1) = ^ B ^ (^(r) W) s.er re£

®‘phase space’ is also denoted by W in several sources

( 2 . 20 )

which shows that the probability of a given configuration s.' at time t + 1 is equal to the sum of ail configurations s. which can produce s' weighed by the product of the micro-states transition probabilities (since the Systems we consider are Markovian).

2.3.2 The LGA’s Boltzmann équation

We define the average Q of an observable q expressed in terms of a con­

figuration s by taking the ensemble average over F of the microscopie function q{r,t), i.e.

Q{ t , t) =< q{r, t) >= g(s(r))P(s-, t). (2.21)

s.er

Making direct use of this ensemble average on the micro-dynamic équation (2.1) and making the highly non-trivial hypothesis^ that par- ticles on a same node are uncorrelated before collision, thereby allow- ing us to factorise < Aj >, we obtain

/i(r + Cj,t + l) = /,(r,() + Af(S) (2.22) Af(S) =

S,s'

j

where fi is the ensemble average of nj.

This is the lattice Boltzmann équation. It can be shown that it yields (at least) one stable equilibrium solution [37]. The equilibrium populations are of the form

/r = --- 7^---T (2-23) 1 -I- exp(o!i -t- 7i • Ci)

^the validity of this hypothesis needs to be corroborated a posteriori. It is how-

ever quite intuitive that it ought to be correct for low-density Systems. Actually,

it turns out that experiments and theoretical results based on this assumption are

in very good agreement, even for densities which cannot honestly be described as

very low.

where «j and

i are functions of the conserved quantities such as den- sity, colored density and momentum.

Equation (2.23) can be further reduced for Systems with zéro global velocity. This leads to the considération that the populations must be independent of the Cj (consider the system’s spatial invariance when subjected to mirroring), meaning they are constants.

The red mass density t) is the number of red particles at node r at time t and the fluctuations t) are defined in terms of the red channel occupations rij (i G {1,..., 6}, with 2b the total number of channels per node)

^ [ni(r,t)-< nj(r,t) >], (2.24)

where <> dénotés the equilibrium ensemble average; in basic equilib- rium

fe

< nj(r,t) >= fi = <

/(I

for i = 1, ... ,b (red channels), 9) ÏOT i = b + 1, ..., 2b

(blue channels),

(2.25)

with / the average density per pair of channels, and 9 the concentra­

tion of red particles. Note that, in accordance with the usual descrip­

tion of the FHP-III model 2b

< ni{T, t) >= bf9 + 6/(1 -9) = p^^^ -t- = p. (2.26)

i=l

This also deflnes the respective average densities per node.

2.3.3 Dynamic Structure factor

The Ted mass’ dynamic structure factor a»), defined as the space and time Fourier transform of the van Hove corrélation function

>, (2.27) is given by

00

pre<i5’-erf(k,a;) = ^ { t , \t\) (2.28)

rec t=-oo

1

= 17 E e-“‘<6p"''(k,|t|) V-(k,0) >,(2.29) ' t——oo

where 5p"‘^(k, t) is the spatial Fourier transform of Sp^^‘^{r,t), and V is the total number of nodes, also interpreted as the volume of the lattice universe (here the lattice C is finite and has periodic bound- ary conditions). 5’'®‘^(k, a;) is also expressed in terms of the kinetic propagator [39] defined by

Tij{k,t) Kj — < (5ni(k,t)(5n*(k,0) >, i,j = (2.30) where ôrii{k,t) is the spatial Fourier transform of ôni{r,t), and kj = fj{l — fj). Using (2.30), we write the dynamic structure factor (2.28) as

OO 6 6

p^“‘S™‘'(k,a>) = e-“‘i:2:r«(k.t)Ki, (2.31) t=—oo

j=l

and the static structure factor (the Fourier transform of the equal-time van Hove function) as

p"" S'“‘(k) = î; E (k, 0) Kj (2.32) i=l j=l

b b 6

— ^ ^ij ^3 ~ >

i—l3=1 3=\

(2.33)

or

= i-fe. ^(2.34)

We now evaluate the kinetic propagator in the Boltzmann approx­

imation The lattice gas équation for the single particle distribution fi{r,t) reads [14]

/i(r + Ci,t-|-l) = /i(r,t) + A({rij}). (2.35) Here A({nj}) is the collision term, which is expanded around the sta- tionary equilibrium distribution < Ui > to yield

26

A({< Ui> + Jnj}) = '^üijôuj + '^0{6nj6nk), (2.36)

1=1 j,k

where we hâve used the property A({<nt>}) = 0 which follows from mass conservation. The explicit form of Qÿ is given in terms of the transition matrix A(s s') between pre- and post-collisional States s and s' respectively

= Y. Ms ^ s')(s- - Si)sj n

{5,s'} fc=l

This resuit is obtained with the assumption that particles on different channels of the same node are uncorrelated before collision (Boltzmann Ansatz), i.e. by factorising the averages < nj nj > {i ^ j)- Combining (2.35) and (2.36), we obtain the linearized Boltzmann équation which reads in k-space

26

ôm{k,t + l) = fly)5nj(k,t). (2.38)

1=1

®We merely sketch the dérivation which proceeds essentially along the Unes of the dérivation given in [39]

Uk < 1 - Uk

Equation (2.38) is straightforwardly solved by itération; inserting its solution into (2.30) yields

rÿ(k,t) Kj = • {ô + n)Y.Kj , (t > 0). (2.39)

L J tj

Here is a diagonal matrix. From (2.31) and (2.39) we obtain

^red5red(k^^) F"‘^(k,o;)

b b r l

1

SjÇi Le^+»k-‘: - S+ n ' 2ïij J

)

(2.40) (2.41) where IZe dénotés the real part. This expression for the dynamic structure factor is exact within the Boltzmann approximation, but the explicit analytical inversion of the 6x6 matrix in (2.41) cannot be performed in ail generality.

In section 2.4 we use perturbation methods to compute S'''®‘^(k, u)

analytically in the hydrodynamic limit. Beyond the long wavelength

- long time domain, it is the recourse to the numerical évaluation of

(2.41) which allows us to compute the ‘Boltzmann’ power spectrum,

we discuss this in section 2.5.

2.4 The hydrodynamic lirait

The hydrodynamic scales are those for which our intuition is best developed, as we can relate to it in our daily lives. Mathematically speaking, these scales are so much larger than those defined by molecu- lar movement that we may idealize them through the limits |k| —> 0 and Lü C?(|k|), 0(|kp). Keeping track of inverse times of 0(|k|) is necessary in order describe propagative phenomena, while keeping track of inverse times of O(lkp) also allows us to describe diffusive behaviour, the major theme of this work.

2.4.1 The hydrodynamic modes

We first notice that the linearized collision operator Cl is not symmet- rical, with the conséquence that its left and right eigenvectors are not each other’s transpose. However when the detailed balance condition is satisfied, the matrix product DjjKj is symmetrical [32], and the left and right eigenvectors of fi are related by \<f>)i = /Cj it can also be shown that to each of the N collisional invariants corresponds an eigenvector (4n|(n = 1,N) of fi, with zéro eigenvalue. The compo- nents are given by the conserved quantities carried by each channel i :

red mass : (^li = 1, if Z = 1,..., b ; (i2|i = 0. if i = 6 + 1,..., 2b ; blue mass : if Z = 1, ...,6;

(5|i = l, if Z = 6 + 1,..., 2b ; x-momentum : (■fxji ~ Cj Il )

y-momentum : {By\i — Cj ■ _{' ly •}

(2.42)

From these considérations and by analogy with the thermal scalar

product introduced in [39], we define the colored scalar product

(A\B) = j2Mci)i^iB{Ci), (2.43)

i=l

where the weight K{ dépends on density and concentration. Since ^ijKj is a symmetrical matrix, the colored scalar product has the symmetry

26 26

(A|n|B) = {B\n\A) = Mci) b { cj ) . (2.44)

i=l j=l

Following a method introduced by Résibois [55], we consider, as the starting point, the propagator (2.39) which is the t-th power of the non-symmetrical matrix e"**^'*^ • (5 + fi), and we use the eigenvalue problem formulation

e-“-'■ (« + fi) |^,(k)) = e-'W |V>„(k)>, (2.45)

<^„(k)| e-*" ■ (<S + fJ) = e’«M(ÿ„(k)|. (2.46) The eigenmodes of the propagator may be separated into two groups;

the slow modes, corresponding to eigenvalues

|i(k) close to zéro when k{= |k|) tends to zéro, and the fast modes corresponding to eigenvalues TZeZfiÇk) < 0 leading to exponentially fast decay. The latter are the kinetic modes; the slow modes which decay infinitely slowly when k 0 will be identified as the hydrodynamic modes. They are the dominant modes in the hydrodynamic régime where the kinetic modes can be neglected.

For jkj = 0 the matrix ' • (5 + f2) reduces to 5 + fî, whose null-

space spanned by the eigenvectors (2.42) has the dimension given by

the number of collisional invariants (here 4). For jk| ^ 0 but small, we

can express the eigenvectors of e“**^ '^-(5+f2) as a linear combination of

the collisional invariants, and we can expand 6“**^ *^, |'0^(k)), (0^(k)|,

and respectively as

e-tkc = .5 - {ik) +

\M^)) = i<^r>i + {ik) + (<^M(k)|

+ {ik) {(f)W\ +

z^{k) = {ik) z^ +

with c, • • = ^ij b .? ^ > ...,2b, where

\2 _2

liikf (ik)^

{ikf {0\

(ikf

+ ,

(2.47)

Cj onto k. Substitution of the first and second expressions of (2.47) into (2.45) and identification of the successive powers of k yields the hierarchy

0{k°) : = 0, (2.48)

0{k^) : (2.49)

0(fc2) : = (c, +

+ +^ (c,+4^)(5)^| . (2.50)

The solution to zero-th order is straightforward; one has

l’/'D = (2.51)

n=l

where the coefficients are to be determined subsequently.

The first order solution is obtained by taking the scalar product of {Am\ with (2.49) where the previous order solution is substituted; the resuit has the form as an A/'-dimensional eigenvalue problem:

H (^m| (c, + I bn , =

n=l

0, (2.52)

mode eigenvector eigenvalue

shear i4"’) = i^’i) = 0

acoustic |4“’> = \P,)-c.\M) =

IV-W) = \P,) + c,\M)

color diffusion -

O

--

Table 2.4: The propagator’s four slow modes expressed in terms of the LGA’s four collisional invariants

which yields the four eigenvectors and eigenvalues shown in table 2.4.1 In this table \M) is the sum of \R) and \B), and P±

are the projections of the momentum onto k and perpendicular to k respectively, kj = fj{l — fj) with j = 1,..., b for «r, and j = 6+1,..., 26 for Kb, and

C,

= {{P,\P,)/{M\M))'/^ (2.53)

will be identified as the speed of sound. Note that, for ail FHP models with a zero-velocity channel, we hâve Cs = ^3/7, independently of the modulus of k and —more importantly— of its orientation.

We define the currents \jn) as

y,} = (c, ++«) Wi“’>. (2-54) and we note, by multiplication of (2.49) by that the currents are orthogonal to As a conséquence the currents do not belong to the fi kernel, and we may write the formai solution to the first order équation (2.49) as

“ “ i'=i

(2.55)

The coefficients bfj,^ are determined by substituting by (2.55) in the second order équation (2.50), and multiplying the resuit by

{u

^ jj) to obtain

= -0.|i + fli>. (2.56) The expression for follows from the évaluation of the product of équation (2.50) with which yields

0.1 (n +1) \ù)

(2.57) We anticipate that is the kinematic viscosity {i'), that .f = =?>

is the Sound dumping (T), and that z^^]{ is the color-diffusivity (D), as will be justified subsequently by the analysis of the power spectrum.

Explicit évaluation of (2.56) shows that the only non-zero off- diagonal éléments of the matrix formed by the are the two coef­

ficients

= - 6_,+ = A, ^(2.58)

Z C

The diagonal éléments 6^^ remain unknown, but this is unimportant because, as will be seen, they do not contribute to the power spectrum (see section 2.5).

We hâve now identified the four hydrodynamic modes in the LGA.

The shear mode and the acoustic modes are independent of color re-

lated properties, and the mode |'0diff) describes color diffusion only. As

will be shown below, the density power spectrum reflects this prop-

erty. Notice that the purely diffusive behavior of color is related to an

observable (defined below) which is neither the concentration of one

of the components nor the différence between the two concentrations

[52, 51].

2.4.2 The dynamic structure factor

The left and right eigenvectors of matrix • (d + fi) which are defined by équations (2.45-2.46) can be related simply. Transposing Eq.(2.46) which defines the left eigenvectors of 6"**^*^ • (5 -h fi), and multiplying the resuit by on the left, we find that and \ip^) are related by

|,^„(k)> = — e+*"|V-,(k)>. (2.59)

where is a normalization constant. If the eigenvectors ((/»^| and form a complété bi-orthonormal set, i.e.

2b

y : = ô, and = ô,, , (2.60)

M=i

we may write • (ô -H fi) as

26

e-^^-^.{ô + n) = (2-61)

,1=1

We will use this expression to recast the spectral function F'‘®‘^(k, a;) We first rewrite équation (2.41) as

b b r

EE t=l j=l

1 1^

giu+ik-c _ 5 _ fi 2 .

Ki ij

= {^1-

1 1

+ 7^à\R) I gùj+ik c _ ^ _ fi 2

= {R\--- ^---n--- {ô + ny'' + lô\R) ' (e-**"-‘= • (5 -f fi)) ^ - 5 2

= {R\--- --- + U\R), (2.62)

' (e-*‘‘-'= • (5 + fi)) - 5 2

where the last equality is obtained by noticing that the action of (d + fi) ^ upon \R) is the identity operation since \R) belongs to the kernel of fi. Then making use of (2.61), we find

We observe that each mode /x contributes a spectral line whose am­

plitude dépends on This factor becomes large for {iu — z^) 0, that is for small in the limit of small u>. The modes for which

^(k) tends to zéro at long wavelength are precisely the slow modes identi- fied in (2.4.1). So we may approximate (2.63) by neglecting the fast kinetic modes in the sum over /x. It is then consistent to make use of the approximation (e® — 1)“^ +

~ ^(^) ^^r a: 1 in the évaluation of V^; with (2.47), we obtain

The final step is the évaluation of in terms of the fc-expansion of using (2.47) in (2.59), expanding in powers of k, and iden- tifying the successive orders. To orders 0{k^) and 0{k^) respectively, we find

+ i](ÿ„(k)|B>

b

(2.63)

with Aff, = {R\i}f,{]s.)){(l)f,{k)\R), and T>^

(2.64) (2.65)

( 2 . 66 )

{^n\ and \ipfi); this is accomplished by expressing in terms of

(2.67)

-

, ,(0)|\(0)„ (wric/l'^’r) + 2(V>ri’/’i'’)) (’/'ri (2,68) {iPn \iPn )

which results are inserted into (2.63) to yield

K ^{1 + 2 ih}

+ ik {R\ c M'>) {R\i>f')

— 2ik

— ik

(2.69)

We now discuss the évaluation of for each hydrodynamic mode separately:

• M =-L •

As the vectors {R\ and |'0i ) are orthogonal, we hâve

A/^i = 0 (2.70)

and the shear mode will not show up in the density fluctuations power spectrum.

• /i = ±.

To order 0{k°), A4 = + '^b)- The computation of the next order requires, in principle, complété knowledge of but in fact this is unnecessary because the terms including the unknown 4 m

cancel each other; thus we obtain

4-(l=F‘*:r/(2c,)+0((:")). (2.71)

2 Kr + Kb ^ '

• Il = diff.

The contributions to order 0{k^) ail vanish identically, so the ex­

pression to 0{k°) is correct up to corrections of 0{k'^). We thus are

left with

= b-^ (2,72)

Combining with the explicit expression of (using (2.4.1), (2.54) and (2.57) in (2.66)), we obtain F’’®‘^(k, w), and therefrom the analytical expression of the dynamic structure factor which reads

^red5red(k,^) ^

+ b

K,r + Kf, \{u) ± CgkY + (rA;2)2^ E

Kr + «6 ±

TA: c.k ± üj

+ h KrK,b

\Cg (w ± Csky + (rA;2)2 2DP

Kr + KbUJ^ + {Dk'^y (2.73)

At fixed value of k, the spectrum consists of a Brillouin-doublet cen- tered around ±A:Cs, and of a central peak characterizing color diffusion.

To first order in A:, Xm(z^) yields the frequency shift of the spectral peak corresponding to the propagating modes (// = ±), and to second order in A:, 'R,e{z^{k)) yields the dissipation coefficients which déter­

mine the line-widths of the spectral components. Note that (2.73) and (2.34) yield do;S'''®‘^(k,a;)/27rS'''®‘^(k) = 1; the spectra shown in the figures are normalized accordingly.

Considering the fluctuations

SpéAr, t) = pdiff(r, t)- < pdiff(r, t) >

of the observable Pdi» defined as

P... = - ^-p“”,

Kr + Kb Kr + Kb the corresponding spectral density

OO

p,„S‘“"(k,a,) = E E re£ t=-oo

(2.74)

(2.75)

e-"‘-‘k r < t) 5p„„(0,0) > . (2.76)

can be computed straightforwardly along the Unes of the évaluation of the power-spectrum w); since l'i/’aur) is orthogonal to the other eigenvectors, there is no coupling with the other modes, and the dynamic structure factor is a single Lorentzian

KrKb 2Dk^

Kr + K,b + {Dk'^y ' (2.77)

with similar normalization as for The power spectrum

(2.77) characterizes color diffusion alone, a feature which will be used

in the analysis of the simulation data in the next section.

2.5 The power spectrum

The above results obtained in the hydrodynamic limit are in accor­

dance with the Landau-Placzek theory for continuons fluids [9]. Now the hydrodynamic theory breaks down at short wavelengths, but the Boltzmann theory should remain valid down to k values where the kinetic domain starts. Lattice gas automata are appropriate model Systems to investigate quantitatively the varions régimes covering a wide range of wavelengths. In order to characterize the wavelength domain of the varions régimes, we consider the quantities k£f, where l/p) is the mean free path, and / = p/Pmaxi the reduced density (or the average density per channel). Accordingly the hydrodynamic régime is defined by kif <C 1, the generalized hydrodynamic régime (Boltzmann régime) by kif < 1, and the kinetic régime by ~ 1.

We now discuss the power spectra obtained from automaton sim­

ulation data and we compare the results to the analytical Landau- Placzek expressions and to the prédictions of the lattice Boltzmann theory. For the latter, we use the eigenvalue spectrum of the propaga- tor r which can be evaluated numerically over the complété /c-domain, so extending the computation of the power spectrum to the région of k values where analytical évaluation can no longer be performed. In Fig.

2.3 we show a typical eigenvalue spectrum as computed numerically, where the fourteen modes of the 14-bit model described in Sec.2.2.4 can be distinguished.

The numerical experiments are performed using the ‘color’ FHP-

III lattice gas (see Sec.2.2.4) at equilibrium. The lattice size is 256 x

256 nodes, and the simulation duration is 40.000 time steps. Spatial

Fourier transforms are computed at every time-step, and time-Fourier

transforms are taken over intervals of 16384 time-steps shifted by 20

time-steps for averaging; data shown in the figures are smoothed by

low-pass frequency filtering.

0.0 0.5 1.0 1.5 2.0 k

Figure 2.3: Eigenvalue spectrum of the 14-bit model propagator:

Boltzmann computation (full Unes) and hydrodynamic limit (dashed

Unes). The reduced density and concentration are / = 0.15 and

Gred = 30% respectively. The wave-number k = |k| is given in re-

ciprocal lattice units

2.5.1 Hydrodynamic régime

In Fig. 2.3 we observe that in the range 0 < k < 0.4, corresponding to wavelengths A > 15^o (with io the lattice unit length), the four slow modes are well separated from the kinetic modes, and their behavior is correctly given by the hydrodynamic expressions. We therefore ex- pect that in this domain, the Landau-Placzek theory should provide a correct description of the dynamic structure factor. This is indeed confirmed by the comparison between the results obtained from simu­

lation data and the theoretical prédictions as shown in Fig. 2.4, where we also notice that the Landau-Placzek spectrum is indistinguishable from the Boltzmann spectrum.

The experimental measure for the diffusion coefficient D{f,6) is obtained straightforwardly from the diflFusive mode power spectrum:

(k, w) is a single Lorentzian (2.77) with half-width Au = Dk“^\

so plotting Au as a function of k"^ (see Fig. 2.5.a) yields, by least- squares fit, a slope whose value provides an experimental measure of D. In Figure 2.5.b, we show the diffusion coefficient as a function of density: we observe that the exact Boltzmann resuit for the diffusion coefficient of the présent model —which can also be cast in a power sériés expansion in terms of the density

— is in good agreement with the lattice gas simulation data up to / « 0.25; for larger / the theoretical prédiction deviates progressively from the measured values indicating that the molecular chaos assumption becomes invalid at high densities.

47 147

+ _ 1)2 + 11(0 _ l)4l / + .... (2.78)

49^ 2^ 49^ 2W ^ ’

Q)

(û

Figure 2.4: Power spectra of ‘red’ density fluctuations (a) and of Pdis fluctuations (b) at low density and small k. Comparison of experi­

mental data (full Unes) with theoretical prédictions: the Boltzmann results and the Landau-Placzek spectra coïncide (dashed Unes). Den­

sity / = 0.15; concentration dred = 30%; wave number |k| = 0.098 re-

ciprocal lattice units; u is given in reciprocal time units (27 t /T, where

T is total number of time-steps); the spectral functions are given in

reciprocal u units.

Figure 2.5: Spectral measurement of the diffusion coefficient: (a) Au = Dk^, f = 0.3 and 6red = 30% (open squares), 50% (black dots);

the least-squares fits (solid line) coincide. (b) D = D{f,6red), simula­

tion data (black dots) and Boltzmann prédiction (solid curve); the size

of the black dots corresponds to the largest error bar {\AD/D\ < 2%).

2.5.2 Boltzmann régime

As k increases from 0.4 to 1.4, there is still a distinct scale séparation between slow and fast modes (see Fig. 2.3), but the eigenvalues of the slow modes départ significantly from the hydrodynamic prédiction, indicating the breakdown of the local response hypothesis: the trans­

port coefficients become /c-dependent. As a resuit, the Landau-Placzek theory does no longer describe the power spectrum correctly - for in­

stance, there is a noticeable spectral line broadening in u>) (see Fig. 2.6a) -, but the complété Boltzmann spectrum is in good agree- ment with the simulation results, as seen in Figs. 2.6a and 2.6b. We hâve also observed that even at rather short wavelength (A ~ lO^o) the experimental data can still be approximated with a Landau-Placzek spectral function if the transport coefficients are parameterized, show- ing that a hydrodynamic type description holds qualitatively down to quite short wavelengths. For k > 1.5, ail modes exhibit compa­

rable decay rates (see Fig. 2.3.a), and ail modes with even parity in c± contribute significantly to the power spectrum. In this domain, the Landau-Placzek theory is invalid and the Boltzmann computation provides good agreement with the experimental spectrum (down to A « 4^o) as shown in Fig. 2.6.b.

From the agreement between the experimental data and the Boltz­

mann results, we found that the Boltzmann theory remains valid up to reduced densities of / w 0.25. At higher densities the discrepancy between the Boltzmann spectral density and the experimental power spectrum (see Fig. 2.7.a) refiects the failure of the Boltzmann theory to evaluate correctly the transport coefficients. Here the contributions of ‘ring-collisions’ [19] should be included in the évaluation of the dif­

fusion coefficient. Finally we note that at very high densities (/ = .9),

we observe a slight coupling between the color diffusion mode and the

Sound propagation modes as illustrated in Fig. 2.7.b.

(û

Figure 2.6: Power spectra at low density and high k: (a) |k| = 0.49, i.e.

A w 12^o; (b) |k| = 1.57, i.e. A w AI q . Experimental data (full Unes),

Boltzmann spectrum (dashed Unes), Landau-Placzek theory (doted

Unes). / = 0.15 and 6red — 30%.

CO

(0

Figure 2.7: Spectra at high density: / = 0.5 (a) and / = 0.9 (b).

Experimental data (full Unes) and Boltzmann prédiction (dashed line).

Concentration ôred = 30%; wave number |k| = 0.098.