Unconditional Proof of the Boltzmann-Sinai Ergodic Hypothesis

TL;DR

This paper proves the Boltzmann-Sinai Ergodic Hypothesis for systems of elastically colliding hard balls on a torus, establishing hyperbolicity and ergodicity for all parameter choices using dynamical and geometric methods.

Contribution

It provides an unconditional proof of ergodicity and hyperbolicity for hard ball systems without relying on complex algebraic techniques, using only dynamical and geometric analysis.

Findings

Proves ergodicity for all parameters of hard ball systems.

Establishes full hyperbolicity in these systems.

Uses geometric analysis instead of algebraic methods.

Abstract

We consider the system of $N$ ($\ge2$) elastically colliding hard balls of masses $m_1,...,m_N$ and radius $r$ on the flat unit torus $\Bbb T^\nu$, $\nu\ge2$. We prove the so called Boltzmann-Sinai Ergodic Hypothesis, i. e. the full hyperbolicity and ergodicity of such systems for every selection $(m_1,...,m_N;r)$ of the external geometric parameters. The present proof does not use the formerly developed, rather involved algebraic techniques, instead it employs exclusively dynamical methods and tools from geometric analysis.