Diffusion approximation for billiards models

1. Presentation of billiards models2. The dynamic system approach

2.1 Notations and definitions2.2 Fundamental results2.3 Diffusion approximation for finite horizon billiards2.4 Infinite horizon billiards. Anomalous diffusion2.5 Numerical simulations

3. The PDE approach3.1 The Liouville equation3.2 A compacity lemma3.3 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

ra

Associated lattice  : LPhysical space : X = {xR2 d(x, L) r}

Associated periodized space : Y image of X by the canonical projection from R2 to R2/LVelocity space : V = S1

Poincaré section : ={(n,,) :representation of (x,v)XV 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 (xn=x(tn+0),vn=v(tn+0)) where tn is the nth collision instant.

Transition operator : T : (also T0 : 00 in the periodized domain) where T(n, ,)=(n1, 1,1) (see picture above)

2. 1 The dynamic system approach. Definitions

L = (a,0) (a

2,a 3

2)Z Z L = (a,0) (0,a)Z Z

1

1

ergodicity :

Theorem 1 (Sinaï, 1970) T0 is ergodic on 0 with respect to the

Liouville measure d0=Z-1cos()dd.

(moreover, T0 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 (1)

X .Xn 0 C F e a n( )

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)

0

1 1

2 nS e dun

u2

2

z

zn

2n 0

nX .X

S Xn k

k 1

n

ra 3

4

 Theorem 4 : (Bunimovitch, Sinaï, Chernov) : for any finite horizon billiard, if (x(0),v(0)) is distributed with respect to the measure d=Z-1dydv on an elementary cell, then there exists a

Gaussian distribution with density g(x) such that:

 

for any bounded and open set A of R2.

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

0 ( /x(0),v(0))x(t)

tA g(x)dx

At

D limx(t) x(0)

tv(0)

t

2

0.v(s) ds

2

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 n0.  

2.4 Anomalous diffusion (1)

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 :

 

2.4 Anomalous diffusion (2)

t

x x

t t

lim

( ) ( )

ln

t 0

n

x x

lim

( ) ( )T

nlnn

n

(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 (1)

Collision velocity autocorrelations <vx(tn).vx(0)> 

     

finite horizon billiard infinite horizon billiard

2.5 Numerical simulations (2)

Mean square displacement  <x2(t)> or <x2(tn)>

  

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) XV :

f(t,x,v)+v.xf = 0 (t,x,v)R+XV

f(t,x,v)=f(t,x,v*) if xX and v.nx >0 with v*=v2(v.nx)v

f(0,x,v)=f0(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)=f0(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 XS1S1 such that 

 

(example : k1/2 v’.nx : diffusive reflection)The following scaling of variables: rr, aa and of the unknown function: f(t,x,v)f(t/,x,v) is then realized.

3.1 Boundary conditions

f(t,x,v) f(t,x,v*) (1 ) k(x,v,v')f(t,x,v')dv'v'. xn 0

k(x, v, v' ) v. n dv = v' . n

k(x, v, v' ) dv' = 1

k(x, v, v' ) C v' . n

v.nx 0x x

v'.nx 0

x

v’?

v* v

 Proposition 1 (L.D.)

Let K and J two operators in H=L2(,0) :

 

 

Then, [(IJ)-1K]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

KF(x,v) k(x,v,v')F(T (x,v'*))dv'

JF(x,v) F(T (x,v))

v'.nx 0

-1

-1

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)1C. Moreover r()2/C if is such that:

vD(s,C).

3.2 idea of the proof

D(s,C) v S / v.kC

k k Z \ (0,0)1

s2

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]XS1) where F is the solution of the heat equation:

t F(t,x) DF=0 t0, xR2

F(0,x)=f0(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], L2(XS1))

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

D1

2(v (y,v) v (y,v))dydv1 1

Y S12 2

v. v (y,v) Y V

(y,v) v'.n (y,v')dv' y Y, v.n > 0

y

yv'.ny 0

y

1

2

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 

weak density limit (rr 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 

D r02