Entropy in Themodynamics: from Foliation to Categorization

TL;DR

This paper provides an overview of entropy in thermodynamics, exploring its mathematical foundations from differential forms to axiomatic and categorical perspectives, highlighting the conceptual and structural aspects.

Contribution

It introduces a unified view of entropy using differential geometry, axiomatic set theory, and category theory, offering a comprehensive theoretical framework.

Findings

Entropy can be represented using differential forms on manifolds.

Axiomatic definition of entropy as an ordering induced by adiabatic processes.

Category theory offers a new interpretation of the ordering structure in thermodynamics.

Abstract

We overview the notion of entropy in thermodynamics. We start from the smooth case using differential forms on the manifold, which is the natural language for thermodynamics. Then the axiomatic definition of entropy as ordering on a set that is induced by adiabatic processes will be outlined. Finally, the viewpoint of category theory is provided, which reinterprets the ordering structure as a category of pre-ordered sets.