First part of Clausius heat theorem in terms of Noether's theorem

TL;DR

This paper explores the connection between the invariance of entropy in thermodynamics and Noether's theorem, using Lagrangian and Hamiltonian frameworks to relate heat over temperature to a Noether charge.

Contribution

It establishes a novel link between thermodynamic invariance principles and Noether's theorem through a detailed theoretical framework.

Findings

Derived a Noether charge expression for heat over temperature.

Connected entropy invariance with Noether's theorem in thermodynamics.

Extended the approach to both Lagrangian and Hamiltonian formalisms.

Abstract

After Helmholtz, the mechanical foundation of thermodynamics included the First Law $d E = \delta Q + \delta W$, and the first part of the Clausius heat theorem $\delta Q^\text{rev}/T = dS$. The resulting invariance of the entropy $S$ for quasistatic changes in thermally isolated systems invites a connection with Noether's theorem (only established later). In this quest, we continue an idea, first brought up by Wald in black hole thermodynamics and by Sasa $\textit{et al.}$ in various contexts. We follow both Lagrangian and Hamiltonian frameworks, and emphasize the role of Killing equations for deriving a First Law for thermodynamically consistent trajectories, to end up with an expression of ``heat over temperature'' as an exact differential of a Noether charge.