A Man Misunderstood: Von Neumann did not claim that his entropy corresponds to the phenomenological thermodynamic entropy

Abstract

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$ ($S_{VN}$), supposedly by arguing for its correspondence with the phenomenological thermodynamic entropy ($S_{TD}$). 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 $S_{VN}$ and $S_{TD}$, as is assumed throughout the current debate, but rather to establish a correspondence between $S_{VN}$ and the Gibbs statistical mechanical entropy $S_{G}$. 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.