Exact hopping and collision times for two hard discs in a box

Abstract

We study the molecular dynamics of two discs undergoing Newtonian ("inertial") dynamics, with elastic collisions in a rectangular box. Using a mapping to a billiard model and a key result from ergodic theory, we obtain exact, analytical expressions for the mean times between the following events: hops, i.e.~horizontal or vertical interchanges of the particles; wall collisions; and disc collisions. To do so, we calculate volumes and cross-sectional areas in the four-dimensional configuration space. We compare the analytical results against Monte Carlo and molecular dynamics simulations, with excellent agreement.