How the Hahn-Banach Theorem Sheds Bright Light on Fundamental Questions in Classical Thermodynamics

Martin Feinberg; Richard B. Lavine

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arXiv · 2604.22717

How the Hahn-Banach Theorem Sheds Bright Light on Fundamental Questions in Classical Thermodynamics

Martin Feinberg·Richard B. Lavine

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Abstract

The Hahn-Banach Theorem, a cornerstone of modern functional analysis, is a natural companion of the Second Law of Thermodynamics. From a Kelvin-Planck version of the Second Law, the Hahn-Banach Theorem delivers, immediately and simultaneously, entropy and thermodynamic-temperature functions of the local material state such that the Clausius-Duhem inequality is satisfied for every process a particular material might admit. For \emph{existence} of such functions there is no need at all to require that their domain be restricted to states of equilibrium. However, the Hahn-Banach Theorem also indicates that for \emph{uniqueness} of such a pair of functions across the entire state-space domain, every state must be visited by a reversible process. This review is intended to help make accessible to both thermodynamics scholars and mathematicians the remarkable interplay of the Hahn-Banach Theorem and the Second Law.

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Martin Feinberg, Richard B. Lavine. 2026-04-24. How the Hahn-Banach Theorem Sheds Bright Light on Fundamental Questions in Classical Thermodynamics. https://arxiv.org/abs/2604.22717

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