On the Equivalence of von Neumann and Thermodynamic Entropy

TL;DR

This paper defends the classical equivalence between thermodynamic and von Neumann entropy, addressing recent claims of discrepancy by clarifying the correct calculation and implications for the second law of thermodynamics.

Contribution

The paper refutes recent challenges to von Neumann and thermodynamic entropy equivalence by correcting the calculation and demonstrating consistency with thermodynamic principles.

Findings

The alleged discrepancy is based on an incorrect entropy calculation.

Correcting the calculation restores the equivalence.

The challenge violates the second law of thermodynamics.

Abstract

In 1939, von Neumann argued for the equivalence of the thermodynamic entropy and $-\text{Tr}\rho\ln\rho$, since known as the von Neumann entropy. Hemmo and Shenker (2006) recently challenged this argument by pointing out an alleged discrepancy between the two entropies in the single particle case, concluding that they must be distinct. In this article, their argument is shown to be problematic as it a) allows for a violation of the second law of thermodynamics and b) is based on an incorrect calculation of the von Neumann entropy.