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Philosophy of Science

Von Neumann's Entropy Does Not Correspond to Thermodynamic Entropy

Meir Hemmo, Orly Shenker

Published Apr 1, 2006• 45 citations
Publisher

Abstract

Von Neumann argued by means of a thought experiment involving measurements of spin observables that the quantum mechanical quantity $S_{VN}=-k\mathrm{Tr}\,(\rho \mathrm{log}\,\rho) $ is conceptually equivalent to thermodynamic entropy. We analyze Von Neumann's thought experiment and show that his argument fails.

DOI
10.1086/510816

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Von Neumann argued by means of a thought experiment involving measurements of spin observables that the quantum mechanical quantity SVN=−kTr (ρlog ρ)S_{VN}=-k\mathrm{Tr}\,(\rho \mathrm{log}\,\rho) SVN​=−kTr(ρlogρ) is conceptually equivalent to thermodynamic entropy. We analyze Von Neumann's thought experiment and show that his argument fails.
Conventional wisdom holds that the von Neumann entropy corresponds to thermodynamic entropy, but Meir Hemmo and Orly Shenker have recently argued against this view by attacking von Neumann’s argument. I argue that Hemmo and Shenker’s arguments fail because of several misunderstandings about statistical-mechanical and thermodynamic domains of applicability, about the nature of mixed states, and about the role of approximations in physics. As a result, their arguments fail in all cases: in the single-particle case, the finite-particles case, and the infinite-particles case.
Recently, attention has returned to the now-famous 1932 thought experiment in which John von Neumann establishes the form of the quantum mechanical von Neumann entropy -Tr $ ρ\ln ρ$ (SVNS_{VN}SVN​), supposedly by arguing for its correspondence with the phenomenological thermodynamic entropy (STDS_{TD}STD​). Hemmo and Shenker (2006) reconstruct von Neumann's thought experiment and argue that it fails to establish this desired correspondence. Prunkl (2019) and Chua (2019) challenge Hemmo and Shenker's result in turn. This paper aims to provide a new foundation for the current debate by revisiting the original text (von Neumann (1996, 2018)). A thorough exegesis of von Neumann's cyclical gas transformation is put forth, along with a reconstruction of two additional thought experiments from the text. This closer look reveals that von Neumann's goal is not to establish a link between SVNS_{VN}SVN​ and STDS_{TD}STD​, as is assumed throughout the current debate, but rather to establish a correspondence between SVNS_{VN}SVN​ and the Gibbs statistical mechanical entropy SGS_{G}SG​. On these grounds I argue that the existing literature misunderstands and misrepresents his goals. A revised understanding is required before the success of von Neumann's reversible gas transformation can be definitively granted or denied.
In 1932, John von Neumann argued for the equivalence of the thermodynamic entropy and −Trρlnρ, since known as the von Neumann entropy. Meir Hemmo and Orly R. Shenker 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.
Von Neumann proposed two quantum entropies, one in 1927 and the other in 1929. He intended to define thermodynamic entropy for quantum states in both cases. The 1927 entropy is the well-known von Neumann entropy, which is zero for all pure states. The 1929 entropy, which is nonzero for almost all pure states, has largely been ignored. After reviewing the history and properties of both entropies, we argue that the 1927 entropy is not thermodynamic entropy and should be used simply as a measure of entanglement, whereas the 1929 entropy is the true thermodynamic entropy for quantum states.
Statistical formulations of thermodynamic entropy, such as those by Boltzmann and Gibbs, were originally developed for classical systems and are well understood in that context. However, the foundational aspects of quantum statistical mechanics remain an area of active debate and are yet to be fully understood. This work is motivated by the need to develop a comprehensive understanding of the statistical measures of thermodynamic entropy in quantum systems - a topic intimately connected to the phenomenon of quantum thermalization. In particular, we investigate the conditions under which the von Neumann entropy can be regarded as a valid statistical measure of thermodynamic entropy in quantum systems. This paper demonstrates that the equivalence between the von Neumann and thermodynamic entropies is not universal, but instead depends on several subtle and often overlooked assumptions. In this context, we also briefly revisit key criticisms of von Neumann entropy - particularly its time-invariance and subadditivity - and argue that these concerns can be meaningfully addressed in the setting of thermodynamic systems. To substantiate some arguments and to clarify some issues, we provide suitable numerical results from our analysis of a spin-1/2 system.
On the Equivalence of −Trρln⁡ρ-\text{Tr}\rho\ln\rho−Trρlnρ and Thermodynamic Entropy.
In 1939, von Neumann argued for the equivalence of the thermodynamic entropy and −Trρln⁡ρ-\text{Tr}\rho\ln\rho−Trρlnρ, 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.