Quantum corrections to the entropy and its application in the study of quantum Carnot engines

TL;DR

This paper investigates quantum corrections to entropy using phase-space methods, clarifies the relation between von Neumann and Gibbs entropy, and applies findings to quantum Carnot engines, enhancing understanding of quantum thermodynamics.

Contribution

It introduces a phase-space expansion of von Neumann entropy, explicitly derives quantum corrections, and applies these to analyze quantum Carnot cycle work extraction.

Findings

Quantum entropy corrections are expanded in powers of ℏ.

Odd ℏ terms vanish at thermodynamic equilibrium.

Quantum corrections influence net work in quantum Carnot engines.

Abstract

Entropy is one of the most basic concepts in thermodynamics and statistical mechanics. The most widely used definition of statistical mechanical entropy for a quantum system is introduced by von Neumann. While in classical systems, the statistical mechanical entropy is defined by Gibbs. The relation between these two definitions of entropy is still not fully explored. In this work, we study this problem by employing the phase-space formulation of quantum mechanics. For those quantum states having well-defined classical counterparts, we study the quantum-classical correspondence and quantum corrections of the entropy. We expand the von Neumann entropy in powers of ${\hbar}$ by using the phase-space formulation, and the zeroth order term reproduces the Gibbs entropy. We also obtain the explicit expression of the quantum corrections of the entropy. Moreover, we find that for the thermodynamic equilibrium state, all terms odd in ${\hbar}$ are exactly zero. As an application, we derive quantum corrections for the net work extraction during a quantum Carnot cycle. Our results bring important insights to the understanding of quantum entropy and may have potential applications in the study of quantum heat engines.