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

TL;DR

This paper explores how the Hahn-Banach Theorem provides insights into the existence and uniqueness of entropy and temperature functions in classical thermodynamics, linking functional analysis with thermodynamic principles.

Contribution

It reveals the connection between the Hahn-Banach Theorem and the Second Law, showing how it guarantees existence and conditions for the uniqueness of thermodynamic functions.

Findings

Hahn-Banach Theorem ensures the existence of entropy and temperature functions satisfying the Clausius-Duhem inequality.

Uniqueness of these functions requires every state to be visited by a reversible process.

The interplay clarifies foundational 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.