Thermodynamic entropy as a Noether invariant

TL;DR

This paper demonstrates that thermodynamic entropy can be understood as a Noether invariant linked to a specific symmetry involving a non-uniform time translation in a classical many-particle system with external control parameters.

Contribution

It introduces a novel connection between thermodynamic entropy and Noether invariants through a symmetry of non-uniform time translation in classical systems.

Findings

Entropy is characterized as a Noether invariant.

The symmetry involves a non-uniform time translation with a specific shift.

Trajectories are restricted to quasi-static processes.

Abstract

We study a classical many-particle system with an external control represented by a time-dependent extensive parameter in a Lagrangian. We show that thermodynamic entropy of the system is uniquely characterized as the Noether invariant associated with a symmetry for an infinitesimal non-uniform time translation $t\to t+\eta\hbar \beta$, where $\eta$ is a small parameter, $\hbar$ is the Planck constant, $\beta$ is the inverse temperature that depends on the energy and control parameter, and trajectories in the phase space are restricted to those consistent with quasi-static processes in thermodynamics.