"Stochastic Analysis on Large Scale Interacting Systems". October 26～29, 2015. edited by Ryoki Fukushima, Tadahisa Funaki, Yukio Nagahata, Makoto Nakashima, Hirofumi Osada and Yoshiki Otobe. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.We report recent rigorous and numerical results obtained on the derivation of Fick's law for deterministic dynamics in random environment

Chiffaudel, Yann

Lefevere, Raphaël

English

Kyoto University Research Information Repository

Title On the derivation of the laws of macroscopic diffusion(Stochastic Analysis on Large Scale Interacting Systems)Author(s)Chiffaudel, Yann; Lefevere, RaphaëlCitation数理解析研究所講究録別冊 = RIMS Kokyuroku Bessatsu(2016), B59: 15-21Issue Date2016-07URL http://hdl.handle.net/2433/243589Right© 2016 by the Research Institute for Mathematical Sciences,Kyoto University. All rights reserved.Type Departmental Bulletin PaperTextversionpublisherKyoto UniversityRIMS Kôkyûroku BessatsuB59 (2016), 15−21On the derivation of the laws of macroscopic diffusionByYann CHIFFAUDEL* and Raphaël LEFEVERE**AbstractWe report recent rigorous and numerical results obtained on the derivation of Fick’s lawfor deterministic dynamics in random environment.§1. IntroductionEver since the works of the founding fathers of statistical mechanics, the derivationof the macroscopic laws of physics as the result of the motion of the microscopic compo‐nents has been a major challenge which remains largely unsolved to this day. Fick’s lawis one of those central laws of macroscopic physics. It states that, after some transienttime, the current of particles crossing an extended macroscopic system of length  L de‐creases like the inverse power of  L . A paradigmatic model in this context is provided bythe Lorentz gas: it consists of tracer particles moving freely in a box and colliding withfixed obstacles. The only rigorous derivation of Fick’s law was achieved by Bunimovichand Sinai in [1] for a finite horizon Lorentz gas when the scatterers have a specificshape that gives rise to a strongly chaotic dynamics. However, it is unlikely that themicroscopic dynamics of a typical material possess the special properties of a stronglychaotic billiard. A more satisfactory result from a conceptual point of view would be toestablish diffusion in a random Lorentz gas. In that case, obstacles of arbitrary shapeReceived January 31, 2016.2010 Mathematics Subject Classification(s):  82C05Key Words: Mirrors model, Derivation of diffusion law.Supported by JAPAN SUPPORT *Laboratoire de Probabilités et Modèles Aléatoires (CNRS UMR 7599), Université Paris Diderot,UFR de Mathématiques, bâtiment Sophie Germain, 5 rue Thomas Mann, 75205 Paris CEDEX 13France e‐mail: chiffaudel@math. univ−paris−diderot. fr **Laboratoire de Probabilités et Modèles Aléatoires (CNRS UMR 7599), Université Paris Diderot,UFR de Mathématiques, bâtiment Sophie Germain, 5 rue Thomas Mann, 75205 Paris CEDEX 13France e‐mail: lefevere@math. univ−paris−diderot. fr© 2016 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.16 Yann Chiffaudel and  APHA\ddot{E}L Lefevereare thrown at random in a box. The goal is to show that, after a diffusive rescaling  0space and time is performed, macroscopic observables obey the laws of diffusion withvery large probability with respect to the distribution of the obstacles. If one looks atFick’s law for the macroscopic current, this requires to control not only its average butalso, at least, its variance. In contrast to the the Bunimovich‐Sinai case, the randomnessof the scatterers induce correlations between the trajectories and therefore also betweenoccupation numbers (or local empirical densities) at different points in space. We reporthere the results of [3] and [2] on the derivation of Fick’s law in the context of randomLorentz gas on a lattice also called the mirrors model. In [3], the model consists of asimplification of the original mirrors model. One direction of the lattice is special in thesense that the motion of every particle in that direction takes place at constant speedin that direction.We recall now briefly the set‐up of the original mirrors model. Particles travel onthe edges of  \mathbb{Z}^{2} with unit speed. Mirrors are located at some vertices of the lattice andtake two possible angular orientations :   \{ \frac{\pi}{4}, \frac{3\pi}{4}\} . When a particle hits a mirror, it getsdeflected according to the laws of specular reflection, see Figure 2 for sample trajectoriesof particles. It is convenient to think that every particle starts at time zero with a givenvelocity at a vertex of the lattice  \mathcal{Q} that is obtained by taking the middle point  0every edge of  \mathbb{Z}^{2} . As all particles move with unit velocity, one can simply observe theevolution of the system at discrete times  t  \in N. At those times, the particles will bealways located at one of the vertices of the new lattice  \mathcal{Q} with a well‐defined velocity.In general, the orientation of the mirrors is picked randomly. It is obvious that themotion of a single particle can not be described as a Markov process. When a particlehits a mirror for the second time, no matter how far back in the past the first visitoccurred, its reflection is strongly affected by the way its was reflected at the first visit.For instance in Figure 2, the two orientations of the mirrors are picked at random, andin that case, at the second visit the reflection is always deterministic.We come now to a more general definition of the dynamics in  d dimensions. Wedenote by  z  =  (z1, :::, z_{d}) a generic element of  \mathbb{Z}^{d} . As for  \mathbb{Z}^{2} , we consider the set  0midpoints of edges of an hypercube of  \mathbb{Z}^{d} of side  N and with periodic conditions in allbut the first direction. We call this set  \mathcal{Q} . It may be described as follows :   \mathcal{Q}=\bigcup_{i=1}^{d}L_{i}where  L_{i}  =   \{z+\frac{1}{2}e_{i} : 0\leq z_{1} \leq N-1, (z2, :::, z_{d})\in (\mathbb{Z}/N\mathbb{Z})^{d-1}\} . Let (e1, :::,  e_{d} )the canonical basis of  \mathbb{R}^{d} , the space of possible velocities is  \mathcal{P}  =   \{\pm\frac{e_{1}}{2}, :::, \pm\frac{e_{d}}{2}\} andthe phase space of the dynamics is \mathcal{M}= {  (q, p) :  q\in  \mathcal{Q},  p\in \mathcal{P}s.  t . if  q\in L_{i} then   p=\pm\frac{e_{i}}{2} }.We denote a generic point of  \mathcal{M} by  (q, p) . The set of points in  \mathcal{M} whose spatial O N the derivation of the laws of macroscopic diffusion 17coordinate belongs to the boundaries of the system is  B=B_{-}\cup B_{+} , with B_{-}= \{x=(q, p)\in \mathcal{M} :q=(q_{1}, \ldots, q_{d})\in L_{1}, q_{1} =\frac{1}{2}\} B_{+}= \{x=(q, p)\in \mathcal{M} :q=(q_{1}, \ldots, q_{d})\in L_{1}, q_{1} =N-\frac{1}{2}\}.See Figure 1.Figure 1. The spatial component of the phase space  \mathcal{M} and of the sets  B_{-} and  B_{+} in 2D.For each  z\in \mathbb{Z}^{d} , we define the action of a “mirror” on the velocity of an incomingparticle by  \pi(z;\cdot) which is a bijection of  \mathcal{P} into itself. It satisfies the following conditions:(1.1)  \{\begin{array}{l}\pi(z;-\pi(z;p))=-p, \forall z\in \mathbb{Z}^{d}, \forall p\in \mathcal{P}\pi(0, z_{2}, \ldots, z_{d};-\frac{e_{1}}{2})= \frac{e_{1}}{2}\pi (N, z2, . . . , z_{d};\frac{e_{1}}{2})=-\frac{e_{1}}{2}, (z2, . . . , z_{d})\in (\mathbb{Z}/N\mathbb{Z})^{d-1}\end{array}The dynamics is defined on  \mathcal{M} in the following way. For any  (q, p)  \in \mathcal{M} : F(q, p)=(q+p+\pi(q+p;p), \pi(q+p;p)) .The map  F is a bijection on  \mathcal{M}.As we are interested in the transport of particles, we define occupation variables \sigma(q, p;t)  \in  \{0 , 1  \} that record the absence or presence of a particle at position  q withvelocity  p at time  t\in \mathbb{N} . When connecting the system to external particles reservoirs,we obtain the following evolution rule : given  \sigma(\cdot;t-1) , we define  \sigma(\cdot;t) for all  t\in \mathbb{N}^{*}18 YanN Chiffaudel and  APHA\ddot{E}L Lefevereby \sigma(x;t)=  \{\begin{array}{l}\sigma(F^{-1}(x);t-1) if x\not\in B_{-}\cup B_{+}\sigma_{\overline{x}}(t-1) if x\in B_{-}\sigma_{x}^{+}(t-1) if x\in B_{+}.\end{array}The families of random variables  \{\sigma_{\overline{x}}(t) : x\in B_{-}, t\in \mathbb{N}\} and  \{\sigma_{x}^{+}(t) : x\in B_{+}, t\in \mathbb{N}\}consist of independent Bernoulli variables with respective parameters  \rho_{-} and  \rho+\cdot Ione chooses  \{\sigma(x;0) : x \in \mathcal{M}\} to be a collection of independent random variables,then it is easy to see by induction that at any  t  \geq  0,  \{\sigma(x;t) : x \in \mathcal{M}\} is a collectionof i.i.  d Bernoulli random variables. To simplify a bit the discussion, we choose anhomogeneous initial distribution, i.e. all Bernoulli random variables have a commonparameter  \rho_{I} . We define the average current of particles that crosses the hyperplane \mathcal{Q}^{l}  = \{q\in \mathcal{Q} : q_{1} =l+\frac{1}{2}\},  l\in\{1, :::, N-2\} during a diffusive time interval  N^{2} :(1.2)  J(l, t)=  \frac{1}{N^{d+1}}\sum_{s=t+1}^{t+N^{2}}\sum_{x\in \mathcal{M}}\sigma(x;s)\triangle(x, l)where  \triangle(x, l)  =2(p\cdot e_{1})1_{q\in \mathcal{Q}^{l}} ,withx  =  (q, p) . Thus  \triangle(x, l) takes the value  +1 (resp. -1) if  x crosses the slice  \mathcal{Q}^{l} from left to right (resp. from right to left). We denote by \mathcal{N}_{\pm} the numbers of crossings from   B\pm to  B induced by  F , i.e.  \mathcal{N}_{\pm}  =  |s_{\pm}| where  s_{\pm}is given by s_{\pm}=\{x\in B\pm : \exists s>0, \forall 0< <s, F^{j}(x) \not\in B\pm, F^{s}(x) \in B \} :One notes that  \mathcal{N}_{+}  =  \mathcal{N}_{-} . Indeed, since every orbit is closed, it must containas many left‐to‐right than right‐to‐left crossings. Thus, we set  \mathcal{N}  =  \mathcal{N}_{+}  =  \mathcal{N}_{-}.  Acompletely analogous definition for the current can be given for the current of particlesfor the rings model [3]. In both the mirrors model and the rings model, it is possible toobtain theTheorem 1.1. For any   t\geq  |\mathcal{M}|,  E[J(l, t)]  =   \frac{\mathcal{N}}{N^{d-1}}(\rho_{-}-\rho_{+}) .Moreover, for every  \delta>0 , any   t\geq  |\mathcal{M}| and   l\in  \{1, :::, N-2\},(1.3)   \mathbb{P}[|J(l, t)-\frac{\mathcal{N}}{N^{d-1}}(\rho_{-}-\rho_{+})| \geq\delta] \leq 2\exp(-\delta^{2}N^{d+1}) .We take now random configurations of reflectors  \{\pi(z;\cdot) : z\in \mathbb{Z}^{d}\} . The law of thereflectors is denoted by  \mathbb{Q} . The map  F becomes now a random map.Definition 1.2. The model satisfies Fick’s law if and only if there exists some \kappa>0 (the conductivity) such that  \forall\delta>0,(1.4)   \lim_{Narrow\infty}\lim_{tarrow\infty}\mathbb{P}\cross \mathbb{Q}[|NJ(l, t)-\kappa(\rho_{-}-\rho_{+})| >\delta]=0. O N the derivation of the laws of macroscopic diffusion 19Figure 2.  N  =  6 . Crossing orbits are coloured in purple, internal loops in black andnon‐crossing orbits are coloured in green. The travel direction given by the arrows isarbitrary. Each edge of the crossing orbits will be used twice in a given orbit : once ineach direction. For this configuration of mirrors  \mathcal{N}=2.It is easy to infer from (1.3) that the following theorem holds.Theorem 1.3. Sufficient and Necessary Condition for Fick’s law : (1.4)holds if and only if there exists  \kappa>0 such that for any  \delta>0,(1.5)   \lim_{Narrow\infty}\mathbb{Q} [|\frac{\mathcal{N}}{N^{d-2}}-\kappa| >\delta] =0.To go further, it is important to note that, using the notationsO  =(( \frac{1}{2},0, \ldots, 0), \frac{e_{1}}{2})and  S=S_{-} :(1.6)   E[\frac{\mathcal{N}}{N^{d-2}}] =N\mathbb{Q}[O\in S]andVar  [ \frac{\mathcal{N}}{N^{d-2}}]  =   \frac{1}{N^{2d-4}}\sum_{\in x,B-}\delta(x, y)=   \frac{1}{N^{d-3}}\sum_{x\in B-}\delta(O, x)(1.7)with(1.8)  \delta(x, y) =\mathbb{Q}[x\in S, y\in S]-\mathbb{Q}[x\in S]\mathbb{Q}[y\in S].20 Yann Chiffaudel and  APHA\ddot{E}L LefevereNext observe that if the law of an orbit with respect to  \mathbb{Q} was similar to the law  0a simple random walk, then there would be a  \kappa>0 such that   \lim_{Narrow\infty}N\mathbb{Q}[0\in S]  =\kappa,this follows from the gambler’s ruin argument. Similarly, if the orbits were independentobjects, then the RHS of (1.7) would go to zero because the only non‐zero term wouldbe the one with  x=O and  \mathbb{Q}[O\in S]  \sim\kappa/N.In [3], the equality (1.5) is proven for the rings model in  d  \geq  7 . This is doneby comparing the crossing probability  \mathbb{Q}[O \in S] to the crossing probability of a lazyrandom walk. The correction due to the memory effect in the real orbits give rise toa correction that decays in the size of the system faster than  1/N . Correlations aretreated in the same way.In the mirrors model, we have tested the probability  \mathbb{Q}[O \in S] numerically [2].The log‐log plot of this probability versus the size of the system  N is given in Figure3. In order to show that the variance (1.7) goes to zero, we have also studied theFigure 3.  \mathbb{Q}(O\in S) for  N from 5 to 420. The 95% confidence interval is about half thesize of a dot.correlation functions  \delta(O, x) in  d=3 . For all but a few points, the correlations  \delta(O, x)for  x  \neq  O are not only small but negative within confidence intervals, see Figure 4.The only exceptions are points  ((1/2,1,0),  \frac{e_{1}}{2}) ,  ((1/2,0,1),  \frac{e_{1}}{2}) ,  ((1/2, N-1,0),  \frac{e_{1}}{2}) and ((1/2,0, N-1),  \frac{e_{1}}{2}) which give clearly positive correlations. However, we checked that for N=70,   \sum_{=1}^{N-1}\delta(O, ((1/2, y, 0), \frac{e_{1}}{2}))=-1.360\cross 10^{-04}\pm 1.47\cross 10^{-05} , i.e. it is negativewith a margin of more than  9\sigma.   \sum_{z=1}^{N-1}\delta(O, ((1/2,0, z), \frac{e_{1}}{2})) must be equal by symmetry.Increasing values of  N do not modify this behaviour. In particular, the number of pointswith positive correlations do not increase. Since we know already that  \mathbb{Q}[O\in S]  \sim\kappa/N,as  N  arrow  1 , we conclude with the same margin that   \sum_{x\in B-}\delta(O, x)  \leq  \kappa/N  arrow  0 , as Narrow 1 . We expect the same behaviour in  d>3. O N the derivation of the laws of macroscopic diffusion 21Figure 4.  \delta(O, x) for  x=((1/2, y, 0),  \frac{e_{1}}{2}) .  N=70 . We draw the 95% confidence interval.References[1] Bunimovich, L. A. and Sinai, Ya. G., Statistical properties of the Lorentz gas with periodicconfiguration of scatterers. Commun. Math. Phys., 78, (1980), 479‐497[2] Chiffaudel, Y. and Lefevere, R., The Mirrors Model : Macroscopic Diffusion WithoutNoise or Chaos, J. Phys. A: Math. Theor. .: 49 (2016), 10LT02[3] Lefevere, R., Fick’s law in a random lattice Lorentz gas Arch. Rat. Mech. and Anal. : 216,Issue 3 (2015), 983−1008[4] Ruijgrok, T. W. and Cohen,E. G. D., Deterministic lattice gas models. Phys. Lett. A 133(1988) 415

On the derivation of the laws of macroscopic diffusion (Stochastic Analysis on Large Scale Interacting Systems)

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On the derivation of the laws of macroscopic diffusion (Stochastic Analysis on Large Scale Interacting Systems)

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