Broken ergodicity of right triangular billiard systems

TL;DR

This paper proves analytically that right triangular billiard systems exhibit broken ergodicity, challenging assumptions about their long-term statistical behavior despite their simple geometric setup.

Contribution

It provides the first analytical proof of broken ergodicity in right triangular billiard systems and discusses the underlying mechanism.

Findings

Analytical proof of broken ergodicity

Numerical evidence supporting the proof

Discussion of the ergodicity-breaking mechanism

Abstract

A right triangular billiard system is equivalent to the system of two colliding particles confined in a one-dimensional box. In spite of their seeming simplicity, no definite conclusion has been drawn so far concerning their ergodic properties. To answer this question, we transform the dynamics of the right triangular billiard system to a piecewise map and analytically prove the broken ergodicity. The mechanism leading to the broken ergodicity is discussed, and some numerical evidence corroborating our conclusion is provided.