The Gibbs "volume" entropy is incorrect

TL;DR

This paper critiques the Gibbs volume entropy, demonstrating it fails to meet thermodynamic postulates and proposing a surface integral approach that correctly accounts for negative temperatures as a valid concept.

Contribution

It introduces a surface integral formulation of entropy that corrects Gibbs volume entropy and supports the validity of negative temperatures in thermodynamics.

Findings

Gibbs volume entropy fails to satisfy thermodynamic postulates

Negative temperatures are consistent with the corrected entropy formulation

Surface integral approach provides accurate equilibrium entropy calculations

Abstract

In recent papers, several authors have claimed that a definition of the thermodynamic entropy in terms of the logarithm of a volume in phase space, originally suggested by Gibbs, is the only valid definition. Arguing from the Gibbs entropy, these authors claim that thermodynamics cannot be extended to include negative temperatures. However, the Gibbs entropy fails to satisfy the postulates of thermodynamics, leading to serious errors. In particular, predictions of the Gibbs entropy for systems with non-monotonic energy densities are incorrect. We show that the correct expression for the equilibrium entropy contains an integral over a surface in phase space, and negative temperature is a valid thermodynamic concept.