Finding Exact Forms on Thermodynamic Manifolds

TL;DR

This paper presents an algorithm to automatically identify exact 1-forms on thermodynamic manifolds, aiding in the discovery of fundamental thermodynamic quantities like entropy from data.

Contribution

It introduces a simple algorithm leveraging differential forms to find exact 1-forms on thermodynamic manifolds, extending previous vector calculus approaches.

Findings

Algorithm successfully finds exact 1-forms in thermodynamics

Potential to re-discover entropy expressions from diverse data

Enhances machine learning methods with differential forms

Abstract

Because only two variables are needed to characterize a simple thermodynamic system in equilibrium, any such system is constrained on a 2D manifold. Of particular interest are the exact 1-forms on the cotangent space of that manifold, since the integral of exact 1-forms is path-independent, a crucial property satisfied by state variables such as the internal energy dE and the entropy dS. Our prior work shows that given an appropriate language of vector calculus, a machine can re-discover the Maxwell equations and the incompressible Navier-Stokes equations from simulation. We speculate that we can enhance this language by including differential forms. In this paper, we use the example of classical thermodynamics to show that there exists a simple algorithm to automate the process of finding exact 1-forms on a thermodynamic manifold. Since entropy appears in various fields of science in different guises, a potential extension of this work is to use the machinery developed in this paper to re-discover the expressions for entropy from data in fields other than classical thermodynamics.