How the Hahn-Banach Theorem Sheds Bright Light on Fundamental Questions in Classical Thermodynamics
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.
Keywords
Cite
@article{arxiv.2604.22717,
title = {How the Hahn-Banach Theorem Sheds Bright Light on Fundamental Questions in Classical Thermodynamics},
author = {Martin Feinberg and Richard B. Lavine},
journal= {arXiv preprint arXiv:2604.22717},
year = {2026}
}