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

TL;DR

This paper demonstrates that entropy and temperature functions in nonequilibrium classical thermodynamics naturally arise from the Hahn-Banach Theorem, without assuming equilibrium conditions, highlighting their fundamental mathematical basis.

Contribution

It shows that entropy and temperature functions in nonequilibrium states emerge directly from the Hahn-Banach Theorem, without requiring equilibrium assumptions.

Findings

Entropy and temperature functions exist for nonequilibrium states.

These functions are derived as immediate consequences of the Hahn-Banach Theorem.

Existence of such functions does not depend on equilibrium restrictions.

Abstract

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.