On Boltzmann vs. Gibbs and the Equilibrium in Statistical Mechanics

TL;DR

This paper clarifies conceptual issues in the comparison of Boltzmann and Gibbs frameworks in statistical mechanics, emphasizing the conditions under which their equilibrium values agree and clarifying the role of the Khinchin condition.

Contribution

It provides a conceptual clarification of the Boltzmann-Gibbs relationship and demonstrates the derivation of the Khinchin condition from standard probability theory assumptions.

Findings

Clarifies the use of 'Boltzmann equilibrium'

Shows the Khinchin condition follows from probability theory assumptions

Addresses conceptual confusions in the Boltzmann-Gibbs comparison

Abstract

In a recent article, Werndl and Frigg discuss the relationship between the Boltzmannian and Gibbsian framework of statistical mechanics, addressing in particular the question when equilibrium values calculated in both frameworks agree. In this paper, I address conceptual confusions that could arise from their discussion, concerning in particular the authors' use of "Boltzmann equilibrium". I also clarify the status of the Khinchin condition for the equivalence of Boltzmannian and Gibbsian, and show that it follows under the assumptions proposed by Werndl and Frigg from standard arguments in probability theory.