Diffusion d une particule marquée dans un gaz dilué de sphères dures

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1 Diffusion d une particule marquée dans un gaz dilué de sphères dures Thierry Bodineau Isabelle Gallagher, Laure Saint-Raymond Journées MAS 2014

2 Outline. Particle Diffusion Introduction

3 Diluted Gas of hard spheres Gas of N hard spheres with deterministic Newtonian dynamics (elastic collisions). Dimension : d 2 Periodic domain: Td = [0,1]d Sphere radius = ε " T d Boltzmann-Grad scaling N" d 1 =

4 T Diluted Gas of hard spheres Gas of N hard spheres with deterministic Newtonian dynamics (elastic collisions). " T d Dimension : d 2 Periodic domain: Td = [0,1]d Sphere radius = ε P Solid Liquid Boltzmann-Grad scaling N" d 1 = Vapor

5 Boltzmann-Grad scaling tv " d 1 Volume covered by a particle = tv" d 1 N On average particles per unit volume On average, a particle has collisions per unit of time N " d 1

6 Diluted Gas of hard spheres Gas of N hard spheres with deterministic Newtonian dynamics (elastic collisions). Initial data at equilibrium and a tagged particle (x 1,v 1 ) Questions. In the Boltzmann-Grad scaling N " d 1 and N 1 1. Distribution of (x 1 (t),v 1 (t)) " 2. Position of the tagged particle x 1 ( t) when 1 5

7 Hard Sphere dynamics Gas of N hard spheres : Z N = {(x i (t),v i (t)} iapplen dx i dt = v i, dv i dt = 0 as long as x i(t) x j (t) >", and elastic collisions if x i (t) x j (t) = " ( v i (t + )=v i (t ) 1 " 2 (v i v j ) (x i x j )(x i x j )(t ) v j (t + )=v j (t )+ 1 " 2 (v i v j ) (x i x j )(x i x j )(t ) Liouville equation for the particle density f N (t, Z N ) t f N + v i r f xi N =0 in the phase space i=1 D N " := Z N 2 T dn R dn / 8i 6= j, x i x j >" with specular reflection on the D N ".

8 The tagged particle Equilibrium distribution M N, (Z N )= 1 Z N, exp 2 NX i=1 v i 2 Y i6=j 1 xi x j >" Particle Z 1 =(x 1, v 1 ) is tagged. Initial distribution : Uniform bound: 0 x 1 apple µ f 0 N (Z N)=M N, (Z N ) 0 x 1 Notation: Marginals t 0, 8s 1, f (s) N (t, Z s)= ZZ f N (t, Z N ) dz s+1...dz N Tagged particle distribution f (1) N (t, (x 1, v 1 ))

9 Limiting stochastic process single particle dynamics Position : x(t) = R t 0 v(u)du 1 Markov process on the velocities {v(t)} t 0 with generator L ZZ Lg(v) := [g(v) g(v 0 )] (v v 1 ) M (v + 1 ) dv 1 d v 0 = v +( (v 1 v)), v1 0 = v 1 ( (v 1 v))

10 Limiting stochastic process single particle dynamics Position : x(t) = R t 0 v(u)du 1 Markov process on the velocities {v(t)} t 0 with generator L Particle distribution M (v)' (x, v, t) follows the Linear Boltzmann t ' + v r x ' = L' Probabilist approaches : Tanaka, Sznitman, Méléard, Graham, Fournier...

11 [van Beijeren, Lanford, Lebowitz, Spohn] N particle system f (1) = N" d 1 N 1 Linear Boltzmann equation N (x 1, v 1,t) t>0 ' (x 1, v 1,t) M (v 1 )

12 [van Beijeren, Lanford, Lebowitz, Spohn] N particle system f (1) = N" d 1 N 1 Linear Boltzmann equation N (x 1, v 1,t) t>0 ' (x 1, v 1,t) M (v 1 ) 1 t = 1 Heat equation Large time asymptotic Brownian motion

13 [van Beijeren, Lanford, Lebowitz, Spohn] N particle system f (1) = N" d 1 N 1 Linear Boltzmann equation N (x 1, v 1,t) t>0 ' (x 1, v 1,t) M (v 1 ) N 1 t = 1 = p log log N Heat equation Large time asymptotic Brownian motion

14 Convergence to the Brownian motion Rescaled position of the tagged particle ( ) =x 1 with = p log log N Initial data f 0 N (Z N )=M N, (Z N ) 0 x 1 Theorem [B.,Gallagher, Saint-Raymond] converges weakly to a brownian motion with variance apple The distribution of the tagged particle converges as N 1to M (v 1 ) (x 1, ) f (1) N (x 1,v 1, = apple x on R + [0, 1] d, =0 = 0 Quantum brownian motion: [Erdös, Salmhofer, Yau] Lorentz gas

15 Boltzmann equation Theorem. For chaotic initial data fn 0 (Z N ) ' Q N i=1 f 0 (z i ) the density of the particle system converges up to a time t >0 to the solution of the Boltzmann equation when N 1, N" d 1 t f + v r x f ZZ = S d [f(v 0 )f(v1) 0 1 R d f(v)f(v 1 )] (v v 1 ) + dv 1 d with v 0 = v + (v 1 v), v1 0 = v 1 (v 1 v) [Lanford], [King], [Alexander], [Uchiyama], [Cercignani, Illner, Pulvirenti], [Simonella], [Gallagher, Saint-Raymond, Texier], [Pulvirenti, Saffirio, Simonella]

16 Boltzmann equation Theorem. For chaotic initial data fn 0 (Z N ) ' Q N i=1 f 0 (z i ) the density of the particle system converges up to a time t >0 to the solution of the Boltzmann equation when N 1, N" d 1 t f + v r x f ZZ = S d [f(v 0 )f(v1) 0 1 R d f(v)f(v 1 )] (v v 1 ) + dv 1 d with v 0 = v + (v 1 v), v1 0 = v 1 (v 1 v) Lanford's strategy leads to a short time convergence which depends on f. 0 The convergence time remains short even if initially the system starts from equilibrium

17 Equilibrium Properties P Boltzmann-Grad : N " d 1 Solid Liquid Low density : N " d = 1 Vapor High density : N " d = 1 T Open Problem : Vapor/Solid phase transition.

18 Equilibrium Properties P Boltzmann-Grad : N " d 1 Solid Liquid Low density : N " d = 1 Vapor High density : N " d = 1 T Open Problem : Vapor/Solid phase transition.

19 Equilibrium Properties P Boltzmann-Grad : N " d 1 Solid Liquid Low density : N " d = 1 Vapor High density : N " d = 1 T Open Problem : Vapor/Solid phase transition.

20 Liquid/Vapor phase transition Interaction potential H(X) = P i,j V xi x j " Lennard-Jones potential 400 V 200 Gibbs measure X = {x i 2 [0, 1] d ; i apple N} µ N,T (X) = 1 1 exp Z N,T T H(X)

21 Liquid/Vapor phase transition Interaction potential H(X) = P i,j V xi x j " Lennard-Jones potential 400 V 200 Gibbs measure X = {x i 2 [0, 1] d ; i apple N} µ N,T (X) = 1 1 exp Z N,T T H(X) There is a density = N" d and a temperature T µ N,T Phase transition N1 1 [Lebowitz,Mazel, Presutti]

22 Continuum percolation Disks of radius ε randomly distributed No interaction, No exclusion. density = N" d Phase transition : N 1 < c : small percolation clusters > c : percolation of a macroscopic cluster Many more models : R. Marchand

23 Continuum percolation Disks of radius ε randomly distributed No interaction, No exclusion. density = N" d Phase transition : N 1 < c : small percolation clusters > c : percolation of a macroscopic cluster Many more models : R. Marchand

24 Lattice gas models Simplifying the model: discrete variables i =1 i 2{0, 1} with i 2{1,..., 1 " }d nearest neighbor interaction H( ) = X i j i j h X i i " j =1 Gibbs measure µ N,T ( ) = 1 Z N,T exp 1 T H( )

25 Lattice gas models Simplifying the model: discrete variables i =1 i 2{0, 1} with i 2{1,..., 1 " }d nearest neighbor interaction H( ) = X i j i j h X i i " j =1 Gibbs measure µ N,T ( ) = 1 Z N,T exp 1 T H( ) P Solid Liquid Ising Model : i =2 i 1 Vapor B. de Tilière T

26 Equilibrium systems : Gibbs measure Parameters : (T, h); (T, P ) Critical curve : T = F h Non-equilibrium systems : Local dynamical rules Sandpile models; Driven particle systems Long range correlations; Self-organized criticality (in general no obvious Hamiltonian) M. Gorny : A mean field approach

Dynamique d un gaz de sphères dures et équation de Boltzmann

Dynamique d un gaz de sphères dures et équation de Boltzmann Thierry Bodineau Travaux avec Isabelle Gallagher, Laure Saint-Raymond Outline. Linear Boltzmann equation & Brownian motion Linearized Boltzmann