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. 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.

Variance of fluctuations from Noether invariance

“…The Noetherian invariance against the displacement implies that the value of the grand potential remains unchanged upon shifting, and hence Ω½V ϵ ext ¼ Ω½V ext 23 . As a consequence, both the first and the second-order terms in the Taylor expansion (2) need to vanish identically, and this holds irrespectively of the value of ϵ; i.e.…”

“…Noether's theorem has recently been suggested to be applicable in a genuine statistical mechanical fashion [22][23][24] . Based on translational and rotational symmetries the theorem allows to derive exact identities ("sum rules") with relative ease for relevant manybody systems both in and out of equilibrium.…”

Dynamic decay and superadiabatic forces in the van Hove dynamics of bulk hard sphere fluids

“…(See Refs. [82,86] for exact force sum rules that stem from Noether's Theorem.) Geigenfeind et al [59] defined the differential force density G(r, t) as a linear combination of species-resolved force densities…”

Fundamental Cause of Bio-Chirality - Space-Time Symmetry: Concept Review

“…2018]. The symmetries determinants, critical in classical and quantum information processing [Penchev.2021], are recognized as the mechanisms underlying biological information processing [Higgins et al 2022;Hermann & Schmidt. 2022] According to current knowledge, "the mechanisms which ensure invariant left-right asymmetry of the heart, viscera, and brain represent a thread connecting biomolecular chirality to human cognition, along the way involving fundamental aspects of cell biology, biophysics, and evolutionary biology" [Levin.…”