Hyperforce balance via thermal Noether invariance of any observable

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.

15 pages, 6 figures

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Cited by in corpus (8)

• Hyper-density functional theory of soft matter
• Gauge Invariance of Equilibrium Statistical Mechanics
• Neural density functional theory of liquid-gas phase coexistence
• Why hyperdensity functionals describe any equilibrium observable
• Noether invariance theory for the equilibrium force structure of soft matter
• 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