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

## TL;DR

This paper derives exact analytical formulas for the mean times between hops, wall collisions, and disc collisions for two hard discs in a box, using a billiard model and ergodic theory, validated by simulations.

## Contribution

It provides the first exact analytical expressions for collision and hop times in a two-disc system, connecting ergodic theory with molecular dynamics.

## Key findings

- Analytical formulas match simulation results with high accuracy.
- Exact mean times for hops, wall collisions, and disc collisions are obtained.
- The approach links billiard models and ergodic theory to molecular dynamics.

## 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.