Variance of fluctuations from Noether invariance

TL;DR

This paper reveals how Noether invariance imposes exact sum rules on fluctuations in classical many-body systems, linking force variance and potential energy curvature, thereby aiding theoretical and simulation consistency.

Contribution

It demonstrates that second-order Noether invariance leads to exact sum rules relating fluctuations and energy curvature in classical systems, providing new theoretical tools.

Findings

Derivation of exact sum rules from Noether invariance.

Connection between force variance and potential energy curvature.

Guidance for theory development and simulation validation.

Abstract

The strength of fluctuations, as measured by their variance, is paramount in the quantitative description of a large class of physical systems, ranging from simple and complex liquids to active fluids and solids. Fluctuations originate from the irregular motion of thermal degrees of freedom and statistical mechanics facilitates their description. Here we demonstrate that fluctuations are constrained by the inherent symmetries of the given system. For particle-based classical many-body systems, Noether invariance at second order in the symmetry parameter leads to exact sum rules. These identities interrelate the global force variance with the mean potential energy curvature. Noether invariance is restored by an exact balance between these distinct mechanisms. The sum rules provide a practical guide for assessing and constructing theories, for ensuring self-consistency in simulation work, and for providing a systematic pathway to the theoretical quantification of fluctuations.