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. Objective: study the long time and large range particle behavior r a

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. Notations

Continous dynamics: t  (x(t),v(t)) with (x(0),v(0)) following a given distribution on X  V. Discrete dynamics: n  (x n =x(t n +0),v n =v(t n +0)) where t n is the n th collision instant. Transition operator : T :  (also T 0 :  0  0 in the periodized domain) where T(n, ,  )=(n 1,  1,  1 ) (see picture above) 2. 1 The dynamic system approach. Definitions 1 1   1 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 =0 (average with respect to  0 ). Let X n =F(T n ). Then, 2.2 Fundamental results (1)

 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 2.2 Fundamental results (2)

Theorem 4 : (Bunimovitch, Sinaï, Chernov) : for any finite horizon billiard, if (x(0),v(0)) is distributed with respect to the measure d  =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: 2.3 Diffusion approximation

 mean free path : Proposition 1 (Bleher, 1992): let r( )=x(T )  x( ). Then =0 and =  <  (mean free path). Moreover, C/a <  < C’/a.  décorrelation: Proposition 2 (Bleher): =  with logarithmic divergence and <  if n  Anomalous diffusion (1)

Theorem (Bleher) : under 3 technical conjectures including a mixing hypothesis:   <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 : 2.4 Anomalous diffusion (2)

(all these simulations were done by Garrido and Gallavoti) Velocity autocorrelations : finite horizon billiard infinite horizon billiard 2.5 Numerical simulations (1)

Collision velocity autocorrelations finite horizon billiard infinite horizon billiard 2.5 Numerical simulations (2)

Mean square displacement or finite horizon billiard infinite horizon billiard 2.5 Numerical simulations (3)

Let f(t,x,v) representing the density of particles at time t located at (x,v)  X  V : f(t,x,v)+v.  x f = 0 (t,x,v)  R +  X  V f(t,x,v)=f(t,x,v*) if x  X and v.n x >0 with v*=v  2(v.n x )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 The PDE approach: the Liouville equation

The specular reflection is replaced by the partial accommodation condition : where 0  <1 and k is a positive function défined on  X  S 1  S 1 such that (example : k  1/2  v’.n x  : 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. 3.1 Boundary conditions v’? v* v

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 A compacity result

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). 3.2 idea of the proof

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=0t  0,x  R 2 F(0,x)=f 0 (x) 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 Diffusion approximation

The diffusion coefficient is given by the following formula where  =(  1,  2 ) is solution of the transport equation in the periodized domain: 3.3 The diffusion coefficient

 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. 3.3 Sketch of the proof

s 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.. 4. Conjectures. Future works