Diffusion approximation for billiards models 1 Presentation of

Diffusion approximation for billiards models 1. Presentation of billiards models 2. The dynamic system approach 2. 1 Notations and definitions 2. 2 Fundamental results 2. 3 Diffusion approximation for finite horizon billiards 2. 4 Infinite horizon billiards. Anomalous diffusion 2. 5 Numerical simulations 3. The PDE approach 3. 1 The Liouville equation 3. 2 A compacity lemma 3. 3 The diffusion approximation result 4. Conjectures. Future work

1. Presentation of billiards models Definition (Lorentz, 1905): array of circular obstacles randomly distributed with particles moving among them and specularly reflecting on them. Variants: periodic distribution of obstacles (Sinaï, 1970), partial accommodation reflection. r a Objective: study the long time and large range particle behavior

2. 1 The dynamic system approach. Notations Associated lattice : L Physical space : X = {x R 2 d(x, L) r} Associated periodized space : Y image of X by the canonical projection from R 2 to R 2/L Velocity space : V = S 1 Poincaré section : ={(n, , ) : representation of (x, v) X V with ingoing velocity}

2. 1 The dynamic system approach. Definitions 1 1 Continous dynamics: t (x(t), v(t)) with (x(0), v(0)) following a given distribution on X V. Discrete dynamics: n (xn=x(tn+0), vn=v(tn+0)) where tn is the nth collision instant. Transition operator : T : (also T 0 : 0 0 in the periodized domain) where T(n, , )=(n 1, 1) (see picture above)

2. 2 Fundamental results (1) ergodicity : Theorem 1 (Sinaï, 1970) T 0 is ergodic on 0 with respect to the Liouville measure d 0=Z-1 cos( )d d. (moreover, T 0 preserves this measure). mixing : Theorem 2 : (Bunimovitch, Sinaï, Chernov, 1991) : let F a function defined on 0 satisfying a Hölder condition and such that <F>=0 (average with respect to 0 ). Let Xn=F(Tn ). Then,

2. 2 Fundamental results (2) central limit theorem: Theorem 3 : (Bunimovitch, Sinaï, Chernov) : let and . If 0, then definition: a billiard is of finite horizon if the distance between two reflections is uniformly bounded. example: a triangular lattice billiard with

2. 3 Diffusion approximation Theorem 4 : (Bunimovitch, Sinaï, Chernov) : for any finite horizon billiard, if (x(0), v(0)) is distributed with respect to the measure dn 0=Z-1 dydv on an elementary cell, then there exists a Gaussian distribution with density g(x) such that: for any bounded and open set A of R 2. The Gaussian distribution g is a zero-average function and has a non singular covariance matrix. Moreover, the diffusion coefficient satisfies the Einstein-Green-Kubo formula:

2. 4 Anomalous diffusion (1) mean free path : Proposition 1 (Bleher, 1992): let r( )=x(T ) x( ). Then <r( )>=0 and < r( ) > = < (mean free path). Moreover, C/a < < C’/a. décorrelation: Proposition 2 (Bleher): < r( ) 2> = with logarithmic divergence and < r( ) r(Tn ) >< if n 0.

2. 4 Anomalous diffusion (2) Theorem (Bleher) : under 3 technical conjectures including a mixing hypothesis: <r( ). r (Tn )> <Cexp( n ) (under-exponential decay), one has: in the probability sense, where is a zero-average Gaussian random variable with a covariance matrix depending on the geometry of X. Moreover :

2. 5 Numerical simulations (1) (all these simulations were done by Garrido and Gallavoti) Velocity autocorrelations <vx(t). vx(0)> : finite horizon billiard infinite horizon billiard

2. 5 Numerical simulations (2) Collision velocity autocorrelations <vx(tn). vx(0)> finite horizon billiard infinite horizon billiard

2. 5 Numerical simulations (3) Mean square displacement <x 2(t)> or <x 2(tn)> finite horizon billiard infinite horizon billiard

3. 1 The PDE approach: the Liouville equation Let f(t, x, v) representing the density of particles at time t located at (x, v) X V : f(t, x, v)+v. xf = 0 (t, x, v) R+ X V f(t, x, v)=f(t, x, v*) if x X and v. nx >0 with v*=v 2(v. nx)v f(0, x, v)=f 0(x) The problem consists in studying f for large t (by introducing a small parameter ). N. B. the solution of the previous equation is given by f(t, x, v)=f 0(x(t)) with x(0)=x and v(0) v.

3. 1 Boundary conditions The specular reflection is replaced by the partial accommodation condition : where 0 <1 and k is a positive function défined on X S 1 such that v* v v’? (example : k 1/2 v’. nx : diffusive reflection) The following scaling of variables: r r, a a and of the unknown function: f(t, x, v) f(t/ , x, v) is then realized.

3. 2 A compacity result Proposition 1 (L. D. ) Let K and J two operators in H=L 2( , 0) : Then, [(I J)-1 K]2 is compact in H. Particular case : =0 , finite horizon billiard, diffusive reflection (C. Bardos, L. D. , F. Golse: J. S. P. 01/1997)

3. 2 idea of the proof In the case of infinite horizon billiards, an ergodization result of the torus by linear flows is used: Théorème (H. S. Dumas) Let: Then, for s>1, D(s, C) is non empty and for small enough C: m(D(s, C)c) 1 C. Moreover r( ) 2/C if is such that: v D(s, C).

3. 3 Diffusion approximation Theorem 1 (C. Bardos, L. D. , F. Golse: J. S. P. 01/1997) In the case of finite horizon billiards with diffusive reflections and assuming that the initial condition is smooth enough, then f converges to F in L ( [0, T] X S 1) where F is the solution of the heat equation: t F(t, x) D F=0 F(0, x)=f 0(x) t 0, x R 2 Theorem 2 (LD): the previous theorem can be extended to any infinite horizon billiards with partial accommodation reflection. The convergence is then achieved in L ( [0, T], L 2(X S 1))

3. 3 The diffusion coefficient is given by the following formula where =( 1, 2) is solution of the transport equation in the periodized domain:

3. 3 Sketch of the proof multi-scale asymptotic development «à la Benssoussan-Lions -Papanicolaou » . Fredholm alternative for the periodized problem (with the help of a compacity lemma) maximum principle for the transport equation.

4. Conjectures. Future works weak density limit (r r with 1 < < 2) partial result (Golse, 1992) : f converges to F in the weak consistency sense with an explicit diffusion coefficient. random distribution of obstacles (for instance Poisson). limit of D when tends to 1 (conjecture : value given by the Kubo formula) estimation of the diffusion coefficient : (conjecture : ) numerical simulation of billiards with partial accommodation. .