arXiv subjects

Richard B. Lavine

Publications and source records attributed to Richard B. Lavine.

3 recordsLinked to original sources

arXiv2026

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

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.

physics.class-ph↗

arXiv2023

Entropy and Thermodynamic Temperature in Nonequilibrium Classical Thermodynamics as Immediate Consequences of the Hahn-Banach Theorem: I. Existence

The Kelvin-Planck statement of the Second Law of Thermodynamics is a stricture on the nature of heat receipt by any body suffering a cyclic process. It makes no mention of temperature or of entropy. Beginning with a Kelvin-Planck statement of the Second Law, we show that entropy and temperature -- in particular, existence of functions that relate the local specific entropy and thermodynamic temperature to the local state in a material body -- emerge immediately and simultaneously as consequences of the Hahn-Banach Theorem. Existence of such functions of state requires no stipulation that their domains be restricted to equilibrium states. Further properties, including uniqueness, are addressed in a companion paper.

math-ph↗

arXiv2023

Entropy and Thermodynamic Temperature in Nonequilibrium Classical Thermodynamics as Immediate Consequences of the Hahn-Banach Theorem: II. Properties

In a companion article it was shown in a certain precise sense that, for any thermodynamical theory that respects the Kelvin-Planck Second Law, the Hahn-Banach Theorem immediately ensures the existence of a pair of continuous functions of the local material state -- a specific entropy (entropy per mass) and a thermodynamic temperature -- that together satisfy the Clausius-Duhem inequality for every process. There was no requirement that the local states considered be states of equilibrium. This article addresses questions about properties of the entropy and thermodynamic temperature functions so obtained: To what extent do such temperature functions provide a faithful reflection of ``hotness"? In precisely which Kelvin-Planck theories is such a temperature function essentially unique, and, among those theories, for which is the entropy function also essentially unique? What is a thermometer for a Kelvin-Planck theory, and, for the theory, what properties does the existence of a thermometer confer? In all of these questions, the Hahn-Banach Theorem again plays a crucial role.

math-ph↗