Why Noether's Theorem applies to Statistical Mechanics

TL;DR

This paper demonstrates how Noether's Theorem extends to statistical mechanics, linking symmetries to conservation laws in thermal systems, with applications to ideal sedimentation and many-body correlations.

Contribution

It provides a pedagogical framework applying Noether's Theorem to thermal systems using functional derivatives, expanding its relevance beyond classical mechanics.

Findings

Identifies conservation laws in thermal and out-of-equilibrium systems.

Derives identities relating forces and correlations in many-body systems.

Applies the framework explicitly to ideal sedimentation.

Abstract

Noether's Theorem is familiar to most physicists due its fundamental role in linking the existence of conservation laws to the underlying symmetries of a physical system. Typically the systems are described in the particle-based context of classical mechanics or on the basis of field theory. We have recently shown [Commun. Phys. $\textbf{4}$, 176 (2021)] that Noether's reasoning also applies to thermal systems, where fluctuations are paramount and one aims for a statistical mechanical description. Here we give a pedagogical introduction based on the canonical ensemble and apply it explicitly to ideal sedimentation. The relevant mathematical objects, such as the free energy, are viewed as functionals. This vantage point allows for systematic functional differentiation and the resulting identities express properties of both macroscopic average forces and molecularly resolved correlations in many-body systems, both in and out-of-equilibrium, and for active Brownian particles. To provide further background, we briefly describe the variational principles of classical density functional theory, of power functional theory, and of classical mechanics.