Ergodicity of non-Hamiltonian equilibrium systems

TL;DR

This paper extends ergodic theory proofs of relaxation to equilibrium from Hamiltonian systems to more general dynamics that preserve a mixing distribution, focusing on averages of physical properties.

Contribution

It introduces a proof of ergodicity for non-Hamiltonian systems that preserve a mixing equilibrium distribution, broadening the scope of ergodic theory applications.

Findings

Weak relaxation applies to physical property averages

Requires ergodic consistency of initial and final distributions

Does not address distribution relaxation or relaxation times

Abstract

It is well known that ergodic theory can be used to formally prove a weak form of relaxation to equilibrium for finite, mixing, Hamiltonian systems. In this Letter we extend this proof to any dynamics that preserves a mixing equilibrium distribution. The proof uses an approach similar to that used in umbrella sampling, and demonstrates the need for a form of ergodic consistency of the initial and final distribution. This weak relaxation only applies to averages of physical properties. It says nothing about whether the distribution of states relaxes towards the equilibrium distribution or how long the relaxation of physical averages takes.