Why Noether's Theorem applies to Statistical Mechanics

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. $\text{4}$, 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.

15 pages, 3 figures, J. Phys.: Condens. Matter (invited as Topical Review, revised version)

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• Hyper-density functional theory of soft matter
• Force balance in thermal quantum many-body systems from Noether's theorem
• Gauge Invariance of Equilibrium Statistical Mechanics
• Neural density functional theory of liquid-gas phase coexistence
• Why hyperdensity functionals describe any equilibrium observable
• Local measures of fluctuations in inhomogeneous liquids: Statistical mechanics and illustrative applications
• Noether invariance theory for the equilibrium force structure of soft matter
• Dynamic Decay and Superadiabatic Forces in the van Hove Dynamics of Bulk Hard Sphere Fluids
• Why gauge invariance applies to statistical mechanics
• Gauge invariance and hyperforce correlation theory for equilibrium fluid mixtures
• First part of Clausius heat theorem in terms of Noether's theorem
• Thermodynamics of an Empty Box