Noether's Theorem in Statistical Mechanics

TL;DR

This paper extends Noether's theorem to statistical mechanics, deriving local sum rules and hierarchies of correlation functions from symmetries in generating functionals, with applications to active matter systems.

Contribution

It introduces a novel application of Noether's theorem to generate local sum rules and correlation hierarchies in many-body statistical systems, including active particles.

Findings

Derived identities relate forces and torques to symmetries.

Established hierarchies of correlation functions and memory effects.

Clarified the role of interfacial forces in active phase separation.

Abstract

Noether's calculus of invariant variations yields exact identities from functional symmetries. The standard application to an action integral allows to identify conservation laws. Here we rather consider generating functionals, such as the free energy and the power functional, for equilibrium and driven many-body systems. Translational and rotational symmetry operations yield mechanical laws. These global identities express vanishing of total internal and total external forces and torques. We show that functional differentiation then leads to hierarchies of local sum rules that interrelate density correlators as well as static and time direct correlation functions, including memory. For anisotropic particles, orbital and spin motion become systematically coupled. The theory allows us to shed new light on the spatio-temporal coupling of correlations in complex systems. As applications we consider active Brownian particles, where the theory clarifies the role of interfacial forces in motility-induced phase separation. For active sedimentation, the center-of-mass motion is constrained by an internal Noether sum rule.