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

TL;DR

This paper explores the properties and uniqueness of entropy and temperature functions derived from the Hahn-Banach Theorem in nonequilibrium thermodynamics, examining their physical interpretation and the concept of thermometers.

Contribution

It investigates the conditions under which entropy and temperature functions are unique and meaningful in nonequilibrium thermodynamics using the Hahn-Banach Theorem.

Findings

Temperature functions can faithfully reflect hotness under certain theories.

Uniqueness of entropy and temperature functions depends on specific conditions.

The concept of a thermometer is formalized within the theoretical framework.

Abstract

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.