Compositional Thermostatics

TL;DR

This paper introduces a formal framework for thermostatic systems using convex spaces and operads, unifying classical and quantum thermodynamics and allowing systematic combination of such systems based on entropy maximization.

Contribution

It defines thermostatic systems as convex spaces with entropy functions and constructs an operad formalism to rigorously combine these systems, encompassing classical, quantum, and generalized probabilistic theories.

Findings

Defines thermostatic systems as convex spaces with entropy.

Constructs an operad for combining thermostatic systems.

Proves thermostatic systems are algebras of this operad.

Abstract

We define a thermostatic system to be a convex space of states together with a concave function sending each state to its entropy, which is an extended real number. This definition applies to classical thermodynamics, classical statistical mechanics, quantum statistical mechanics, and also generalized probabilistic theories of the sort studied in quantum foundations. It also allows us to treat a heat bath as a thermostatic system on an equal footing with any other. We construct an operad whose operations are convex relations from a product of convex spaces to a single convex space, and prove that thermostatic systems are algebras of this operad. This gives a general, rigorous formalism for combining thermostatic systems, which captures the fact that such systems maximize entropy subject to whatever constraints are imposed upon them.