Hyperforce balance via thermal Noether invariance of any observable

TL;DR

This paper establishes that classical phase space functions obey exact hyperforce sum rules derived from Noether invariance, enabling new insights into force correlations and local self-organization in equilibrium systems.

Contribution

It introduces hyperforce sum rules for arbitrary observables based on translational Noether invariance, extending traditional sum rules to local and higher-order identities.

Findings

Hyperforce sum rules relate mean gradients to force correlations.

Local force fluctuation profiles reveal inhomogeneous self-organization.

Sum rules enable improved sampling and convergence in simulations.

Abstract

Noether invariance in statistical mechanics provides fundamental connections between the symmetries of a physical system and its conservation laws and sum rules. The latter are exact identities that involve statistically averaged forces and force correlations and they are derived from statistical mechanical functionals. However, the implications for more general observables and order parameters are unclear. Here, we demonstrate that thermally averaged classical phase space functions are associated with exact hyperforce sum rules that follow from translational Noether invariance. Both global and locally resolved identities hold and they relate the mean gradient of a phase-space function to its negative mean product with the total force. Similar to Hirschfelder's hypervirial theorem, the hyperforce sum rules apply to arbitrary observables in equilibrium. Exact hierarchies of higher-order sum rules follow iteratively. As applications we investigate via computer simulations the emerging one-body force fluctuation profiles in confined liquids. These local correlators quantify spatially inhomogeneous self-organization and their measurement allows for the development of stringent convergence tests and enhanced sampling schemes in complex systems.