A simple mathematical proof of Boltzmann's equal a priori probability hypothesis

TL;DR

This paper provides a rigorous, first-principles mathematical proof of Boltzmann's equal a priori probability hypothesis, demonstrating relaxation to equilibrium and uniqueness of the microcanonical distribution in ergodic systems.

Contribution

It offers a novel, simple mathematical derivation of Boltzmann's postulate using the Dissipation and Fluctuation Theorems, applicable to dense fluids and nonmonotonic relaxation.

Findings

Initial distributions relax to uniform distribution over energy hypersurface.

In ergodic systems, the microcanonical distribution is the only stationary, dissipationless state.

The proof applies to both dilute gases and dense fluids, extending Boltzmann's ideas.

Abstract

Using the Dissipation Theorem and a corollary of the Fluctuation Theorem, namely the Second Law Inequality, we give a first-principles derivation of Boltzmann's postulate of equal a priori probability in phase space for the microcanonical ensemble. We show that if the initial distribution differs from the uniform distribution over the energy hypersurface, then under very wide and commonly satisfied conditions, the initial distribution will relax to that uniform distribution. This result is somewhat analogous to the Boltzmann H-theorem but unlike that theorem, applies to dense fluids as well as dilute gases and also permits a nonmonotonic relaxation to equilibrium. We also prove that in ergodic systems the uniform (microcanonical) distribution is the only stationary, dissipationless distribution for the constant energy ensemble.