On the Mathematics of Thermodynamics

TL;DR

This paper develops a mathematical framework for thermodynamics based on measure spaces with orderings, deriving fundamental identities and potentials, and exploring informational and systematic aspects of thermodynamic equations.

Contribution

It introduces a measure-theoretic structure for thermodynamics and derives core identities like Maxwell relations from simple axioms, offering new insights into thermodynamic potentials and information requirements.

Findings

Derived Maxwell relations from measure space axioms

Established the existence of thermodynamic potentials

Explored informational aspects of equations of state

Abstract

We show that the mathematical structure of Gibbsian thermodynamics flows from the following simple elements: the state space of a thermodynamical substance is a measure space together with two orderings (corresponding to "warmer than" and "adiabatically accessible from") which satisfy certain plausible physical axioms and an area condition which was introduced by Paul Samuelson. We show how the basic identities of thermodynamics, in particular the Maxwell relations, follow and so the existence of energy, free energy, enthalpy and the Gibbs potential function. We also discuss some questions which we have not found dealt with in the literature, such as the amount of information required to reconstruct the equations of state of a substance and a systematic approach to thermodynamical identities.