Maximum Entropy Principle, Equal Probability a Priori and Gibbs Paradox

TL;DR

This paper demonstrates the equivalence of the maximum entropy approach to classical statistical mechanics and highlights its limitations in the grand canonical ensemble, resolving the Gibbs paradox without quantum mechanics.

Contribution

It shows the mathematical equivalence between MaxEnt and classical approaches and clarifies the correct application of MaxEnt in statistical mechanics, avoiding the Gibbs paradox.

Findings

MaxEnt approach is equivalent to classical statistical mechanics methods.

Applying MaxEnt to grand canonical ensemble requires careful prior selection.

The Gibbs paradox is resolved without quantum mechanics using classical assumptions.

Abstract

We prove that information-theoretic maximum entropy (MaxEnt) approach to canonical ensemble is mathematically equivalent to the classic approach of Boltzmann, Gibbs and Darwin-Fowler. The two approaches, however, "interpret" a same mathematical theorem differently; most notably observing mean-energy in the former and energy conservation in the latter. However, applying the same MaxEnt method to grand canonical ensemble fails; while carefully following the classic approach based on Boltzmann's microcanonical {\em equal probability a priori} produces the correct statistics: One does not need to invoke quantum mechanics; and there is no Gibbs paradox. MaxEnt and related minimum relative entropy principle are based on the mathematical theorem concerning large deviations of rare fluctuations. As a scientific method, it requires classic mechanics, or some other assumptions, to provide meaningful {\em prior distributions} for the expected-value based statistical inference. A naive assumption of uniform prior is not valid in statistical mechanics.