Chaos and ergodicity of two hard disks within a circular billiard

Abstract

We investigate dynamical properties of the system of two interacting hard disks within a circular billiard numerically in the case of zero total angular momentum. Varying the radius of two identical disks, we examine chaotic irregularity and ergodicity of the system. Single-particle configuration and velocity distributions are obtained from dynamical trajectory calculations and compared with those in the microcanonical ensemble. We also analyze properties of trajectories by calculating the finite-time maximum Lyapunov exponent and clarify the existence of sticky motions around Kolmogorov-Arnold-Moser (KAM) tori even for small radii of disks. It is shown that the present system is almost ergodic in spite of the existence of tori for small radii of disks since the ratio of tori to the whole phase space is extremely small. On the other hand, a number of tori increase abruptly as the radius of disks increases beyond some value and tori prevent trajectories to run over the phase space uniformly, which makes the ergodicity of the system broken down.