Liouville ergodicity of linear multi-particle hamiltonian system with one marked particle velocity flips

TL;DR

This paper proves that a multi-particle Hamiltonian system with a velocity flip on one particle converges to the Liouville measure, despite having a continuum of invariant measures, by analyzing non-linear transformations on the energy surface.

Contribution

It introduces a novel approach to prove ergodicity in a multi-particle Hamiltonian system with velocity flips, addressing the challenge of multiple invariant measures.

Findings

Convergence to Liouville measure for any initial state

Velocity flip induces ergodicity in the system

Analysis of non-linear transformations on the energy surface

Abstract

We consider multi-particle systems with linear deterministic hamiltonian dynamics. Besides Liouville measure it has continuum of invariant tori and thus continuum of invariant measures. But if one specified particle is subjected to a simple linear deterministic transformation (velocity flip) in random time moments, we prove convergence to Liouville measure for any initial state. For the proof it appeared necessary to study non-linear transformations on the energy surface.