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

2026/04/24 by Martin Feinberg, Richard B. Lavine · 1 voice
Physics and Astronomy · Mathematics · #physics.class-ph #math-ph #math.DS #math.FA #physics.chem-ph

paper · pdf

arxiv published 2026/04/24 · arxiv updated 2026/06/14

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 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 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.

Citations

• Entropy and Thermodynamic Temperature in Nonequilibrium Classical Thermodynamics as Immediate Consequences of the Hahn-Banach Theorem: II. Properties
• Entropy and Thermodynamic Temperature in Nonequilibrium Classical Thermodynamics as Immediate Consequences of the Hahn-Banach Theorem: I. Existence
• Entropy and Convexity for Nonlinear Partial Differential Equations: Introduction
• The entropy concept for non-equilibrium states
• The physics and mathematics of the second law of thermodynamics
• The Hahn-Banach theorem: the life and times
• The thermodynamics of elastic materials with heat conduction and viscosity
• On the equilibrium of heterogeneous substances

Discussions

• Martin Feinberg, Richard B. Lavine: How the Hahn-Banach Theorem Sheds Bright Light on Fundamental Questions in Classical Thermodynamics https://arxiv.org/abs/2604.22717 https://arxiv.org/pdf/2604.2271 [bsky, 1 points, 0 comments]