Billiards correlation functions ?

We discuss various experiments on the time decay of velocity autocorrelation functions in billiards. We perform new experiments and nd results which are compatible with an exponential mixing hypothesis, rst put forward by Friedman and Martin, FM]: they do not seem compatible with the stretched exponentials believed, in spite of FM] and more recently of Chernov, C], to describe the mixing. The analysis led us to several byproducts: we obtain information about the normal diiusive nature of the motion and we consider the probability distribution of the number of collisions in time t m (as t m ! 1) nding a strong dependence on some geometric characteristics of the locus of the billiards obstacles. ? This paper is archived in the electronic archive mp arc: send an empty e-mail message tomp [email protected] to get the instructions on how to copy a postscript version of this preprint. x1 Billiards correlation functions. The rst results on the ergodicity of the billiards go back about thirty years ago, S]. But, as it is well known, ergodicity is a very weak, and in some sense, not too interesting, property. More direct physical meaning is attached to the correlation functions and to their decay speed. Let be the system phase space, S t the evolution map, (dx) the Liouville measure. In the case of the billiards is three dimensional: a point x 2 is a point q 2 M, where M is the billiards table, see Fig. 1, x3, i.e. a periodic box of side 1 with a few circles of radii R 1 ; R 2 ; : : : taken out and regarded as obstacles, and an angle ' 2 0; 2] so that (q; ') represents the position and the velocity direction (with respect to the x-axis, say) of a point mass moving with unit velocity and direction ' until a collision takes place with the obstacles. Upon collision ' changes according to the elastic collision rules (equal incidence and reeection angles). The measure (dx) is simply (dx) = d 2 q d'=(2jMj), where jMj = area of M. The dynamical system ((; S t ;) will be called the continous or 3d system (because it has a 3 dimensional phase space). The average with respect to over the phase space will be denoted by the symbol h:i. An associated dynamical system is the collision system or 2d system: its (2 dimensional) …