Dynamique d un gaz de sphères dures et équation de Boltzmann

Transcription

1 Dynamique d un gaz de sphères dures et équation de Boltzmann Thierry Bodineau Travaux avec Isabelle Gallagher, Laure Saint-Raymond

2 Outline. Linear Boltzmann equation & Brownian motion Linearized Boltzmann equation & acoustic equations Fluctuation field

3 " Microscopic scale N particles of size " Newtonian dynamics N" d 1 N" d 1 1 Macroscopic scale Fluid equations of hydrodynamics (Euler, Navier-Stokes)

4 " Microscopic scale N particles of size " Newtonian dynamics N" d 1 N" d 1 1 Stochastic perturbations: Olla,Varadhan,Yau. Macroscopic scale Fluid equations of hydrodynamics (Euler, Navier-Stokes)

5 " Microscopic scale N particles of size " Newtonian dynamics N" d 1 Stochastic perturbations: Olla,Varadhan,Yau. N" d 1 1 Mesoscopic scale Boltzmann equation Macroscopic scale Fluid equations of hydrodynamics (Euler, Navier-Stokes) Low density limit Fast relaxation limit: Bardos,Golse,Levermore Golse,Saint-Raymond

6 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 =

7 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

8 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 0 i + v0 j = v i + v j v 0 j 2 + v 0 j 2 = v i 2 + v j 2 v i x i v 0 i v j x j v 0 j

9 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) = " ( vi 0 + v0 j = v i + v j vj vj 0 2 = v i 2 + v j 2 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 ". v i x i v 0 i v j x j v 0 j

10 Initial Data Equilibrium distribution M N, (Z N )= 1 X Y NX exp v i 2 Y Z N, 2 Initial data : f 0 N, (Z N )= NY i=1 f 0 (z i )! i=1 M N, (Z N ) i6=j 1 xi x j >" Density of a particle at time t : Z f (1) N (t, z 1)= dz 2...dz N f N (t, z 1,z 2,...,z N ) Question. Convergence f (1) N (t, z 1)?! N!1 N" d 1 = f(t, z 1 )

11 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]

12 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!!!

13 Large time asymptotics Equilibrium distribution M N, (Z N )= 1 X NX Y exp v i 2 Y Z N, 2 i=1 i=1 Initial data for Lanford s theorem! NY fn, 0 (Z N )= f 0 X (z i ) M N, (Z N ) i6=j 1 xi x j >" ' exp(n) Perturbation of the equilibrium! distribution : Y Linear Boltzmann equation: perturbation of a tagged particle Linearized Boltzmann equation: f 0 (z) =1+ 1 N g 0(z)

14 The tagged particle 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 10

15 Tagged particle & Brownian motian R. Brown (1928) A. Einstein (1905) J. Perrin (1909) J. Perrin Les atomes

16 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 : f 0 N (Z N)=M N, (Z N ) 0 x 1 " Uniform bound: 0 x 1 apple µ 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 ))

17 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))

18 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, Rezakhanlou

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

20 [van Beijeren, Lanford, Lebowitz, Spohn] N particle system = N" d 1 N!1 Linear Boltzmann equation f (1) 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

21 [van Beijeren, Lanford, Lebowitz, Spohn] N particle system = N" d 1 N!1 Linear Boltzmann equation f (1) 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

22 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 Lorentz gas: [Bunimovich,Sinai], [Basile,Nota,Pezzotti,Pulvirenti]

23 Ornstein-Uhlenbeck process Gas particles: diameter : mass : " " Big tagged particle: " diameter : mass : 1 Boltzmann-Grad scaling N" d 1 =1 Ideal gas : [Holley], [Dürr, Goldstein, Lebowitz] [Szász, Tóth]

24 Ornstein-Uhlenbeck process Gas particles: diameter : mass : " " Big tagged particle: " diameter : mass : 1 Boltzmann-Grad scaling N" d 1 =1 Starting from equilibrium, limiting velocity process:

25 Ornstein-Uhlenbeck process Gas particles: " diameter : mass : Big tagged particle: diameter : mass : 1 " " Theorem [B.,Gallagher, Saint-Raymond] Starting from the equilibrium measure the velocity of the tagged particle converges to an Ornstein-Uhlenbeck process in the Boltzmann- Grad limit : N" d 1 =1

26 Linearized Boltzmann equation Response to a small perturbation (@ t + v r x )g = L g, Z L g(v) := M (v 1 ) g(v)+g(v 1 ) g(v 0 ) g(v1) 0 (v 1 v) + d dv 1

27 Linearized Boltzmann equation Response to a small perturbation (@ t + v r x )g = L g, Z L g(v) := M (v 1 ) g(v)+g(v 1 ) g(v 0 ) g(v1) 0 (v 1 v) + d dv 1 Background Linear Boltzmann equation Z Lg(v) := [g(v) g(v 0 )] (v v 1 ) + M (v 1 ) dv 1 d

28 Linearized Boltzmann equation Response to a small perturbation (@ t + v r x )g = L g, Z L g(v) := M (v 1 ) g(v)+g(v 1 ) g(v 0 ) g(v1) 0 (v 1 v) + d dv 1 Background Linear Boltzmann equation Z Lg(v) := [g(v) g(v 0 )] (v v 1 ) + M (v 1 ) dv 1 d Probabilist interpretation : Norris

29 Linearized Boltzmann equation Response to a small perturbation (@ t + v r x )g = L g, Z L g(v) := M (v 1 ) g(v)+g(v 1 ) g(v 0 ) g(v1) 0 (v 1 v) + d dv 1 Tagged particle perturbation of the tagged particle

30 Linearized Boltzmann equation Response to a small perturbation (@ t + v r x )g = L g, Z L g(v) := M (v 1 ) g(v)+g(v 1 ) g(v 0 ) g(v1) 0 (v 1 v) + d dv 1 Tagged particle perturbation of the tagged particle perturbation of the background

31 Linearized Boltzmann equation Response to a small perturbation (@ t + v r x )g = L g, Z L g(v) := M (v 1 ) g(v)+g(v 1 ) g(v 0 ) g(v1) 0 (v 1 v) + d dv 1 Tagged particle A cloud of particles is modified. On averaged the distribution of each background particle changes by an order : t O N

32 Linearized Boltzmann equation Response to a small perturbation (@ t + v r x )g = L g, Z L g(v) := M (v 1 ) g(v)+g(v 1 ) g(v 0 ) g(v1) 0 (v 1 v) + d dv 1 Tagged particle A cloud of particles is modified. On averaged the distribution of each background particle changes by an order : t O N Goal: Capture corrections ' 1 N

33 Linearized Boltzmann equation Perturbation of order 1 f 0 N (Z N )=M N, (Z N ) g 0 z 1 corrections of order ' 1 N Perturbation of order N (symmetric version) fn 0 X N (Z N )=M N, (Z N ) g 0 z i Z with M (v)g 0 (z)dz =0 i=1 corrections of order ' 1 Question. Large time behavior of f (1) N (t, z 1)

34 N particle system f (1) N (x 1, v 1,t) = N" d 1 N!1 Linearized Boltzmann equation g (x 1, v 1,t) [van Beijeren, Lanford, Lebowitz, Spohn] (short time)

35 N particle system f (1) N (x 1, v 1,t) = N" d 1 N!1 Linearized Boltzmann equation g (x 1, v 1,t)!1 [Bardos,Golse,Levermore] Initially : g(0,x,v):= 0 (x)+u 0 (x) v + v 2 d 2 0 (x) density fluctuation momentum fluctuation energy fluctuation

36 N particle system f (1) N (x 1, v 1,t) = N" d 1 N!1 Linearized Boltzmann equation g (x 1, v 1,t)!1 [Bardos, Golse, Levermore] Initially : Acoustic equations g(0,x,v):= 0 (x)+u 0 (x) v + v 2 d 2 0 (x) g(t, x, v) := (t, x)+u(t, x) v + v 2 d (t, x) 2 8 t + r x u t u + r x ( + ) =0 t + r x u =0

37 N particle system f (1) N (x 1, v 1,t) = N" N!1 Linearized Boltzmann equation g (x 1, v 1,t) Theorem [BGSR] For d = 2, convergence for any t>0 Initially : Acoustic equations g(0,x,v):= 0 (x)+u 0 (x) v + v 2 d 2 0 (x) g(t, x, v) := (t, x)+u(t, x) v + v 2 d (t, x) 2 8 t + r x u t u + r x ( + ) =0 t + r x u =0

38 N particle system f (1) N (x 1, v 1,t) = N" N!1 Linearized Boltzmann equation g (x 1, v 1,t) p log log log N!1 Initially : Acoustic equations Theorem [BGSR] For d = 2, convergence for any t>0 g(0,x,v):= 0 (x)+u 0 (x) v + v 2 d 2 0 (x) g(t, x, v) := (t, x)+u(t, x) v + v 2 d (t, x) 2 8 t + r x u t u + r x ( + ) =0 t + r x u =0

39 Goal. Fluctuating Boltzmann equation Microscopic scale : Newtonian dynamics " Z N (t) =(x i (t),v i (t)) iapplen

40 Goal. Fluctuating Boltzmann equation Microscopic scale : Newtonian dynamics " Z N (t) =(x i (t),v i (t)) iapplen Starting at equilibrium 1 N NX i=1 h z i (t)! N!1 E(h)

41 Goal. Fluctuating Boltzmann equation Microscopic scale : Newtonian dynamics " Z N (t) =(x i (t),v i (t)) iapplen Starting at equilibrium 1 N NX i=1 h z i (t)! N!1 E(h) Fluctuation field. N (h, Z N (t)) = 1 p N N X i=1 h(z i (t)) with Z h(x, v)m (v)dxdv =0 Question. Dynamical fluctuations at equilibrium?

42 Fluctuation field. N (h, Z N (t)) = 1 p N N X i=1 h(z i (t)) with Z h(x, v)m (v)dxdv =0 Covariance of the fluctuation field : E N g, Z N (0) N h, Z N (t) = 1 N Z M N, (Z N ) NX g z i! NX h z i (t)! i=1 i=1

43 Fluctuation field. N (h, Z N (t)) = 1 p N N X i=1 h(z i (t)) with Z h(x, v)m (v)dxdv =0 Covariance of the fluctuation field : E N g, Z N (0) N h, Z N (t) Symmetry = 1 N = Z Z M N, (Z N ) M N, (Z N ) NX i=1 NX i=1 g z i! g z i! NX i=1 h z 1 (t) h z i (t)!

44 Fluctuation field. N (h, Z N (t)) = 1 p N N X i=1 h(z i (t)) with Z h(x, v)m (v)dxdv =0 Covariance of the fluctuation field : E N g, Z N (0) N h, Z N (t) Symmetry = 1 N = Z Z M N, (Z N ) M N, (Z N ) NX i=1 NX i=1 g z i! g z i! NX i=1 h z 1 (t) h z i (t)! New initial data

45 Fluctuation field. N (h, Z N (t)) = 1 p N N X i=1 h(z i (t)) with Z h(x, v)m (v)dxdv =0 Covariance of the fluctuation field : Consequence of the previous Theorem lim E M N, N (h, Z N (0)) N ( h, Z N (t)) N!1 Z = dz M (v)exp t(v r x + L) h(z) h(z)

46 Fluctuation field. N (h, Z N (t)) = 1 p N N X i=1 h(z i (t)) with Z h(x, v)m (v)dxdv =0 Covariance of the fluctuation field : Consequence of the previous Theorem lim E M N, N (h, Z N (0)) N ( h, Z N (t)) N!1 Z = dz M (v)exp t(v r x + L) h(z) h(z) Question. Convergence of the field to the Ornstein-Uhlenbeck process? [Rezakhanlou]

47 Conclusion Deterministic dynamics of a diluted gaz of hard sphere: Brownian motion Ornstein-Uhlenbeck process Linearized Boltzmann equation & acoustic equations

48 Conclusion Deterministic dynamics of a diluted gaz of hard sphere: Brownian motion Ornstein-Uhlenbeck process Linearized Boltzmann equation & acoustic equations Open problems. Linearized Boltzmann equation in dimension 3 Stochastic fluctuations [Spohn] Understanding the dissipation Boltzmann equation for large times