Variance of fluctuations from Noether invariance

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.

4 pages, 1 figure

References in corpus (9)

• Extending Noether's theorem by quantifying the asymmetry of quantum states
• Symmetries in Fluctuations Far from Equilibrium
• Phase separation and multibody effects in three-dimensional active Brownian particles
• Enhanced wavelength-dependent surface tension of liquid-vapour interfaces
• Density depletion and enhanced fluctuations in water near hydrophobic solutes: identifying the underlying physics
• Isometric fluctuation relations for equilibrium states with broken symmetry
• Fundamental measure theory of inhomogeneous two-body correlation functions
• Thermodynamical path integral and emergent symmetry
• Thermodynamic entropy as a Noether invariant in a Langevin equation

Cited by in corpus (14)

• Perspective: How to overcome dynamical density functional theory
• Noether-Constrained Correlations in Equilibrium Liquids
• Perspective: New directions in dynamical density functional theory
• Why neural functionals suit statistical mechanics
• Force balance in thermal quantum many-body systems from Noether's theorem
• Gauge Invariance of Equilibrium Statistical Mechanics
• Hyperforce balance via thermal Noether invariance of any observable
• Why hyperdensity functionals describe any equilibrium observable
• Local measures of fluctuations in inhomogeneous liquids: Statistical mechanics and illustrative applications
• Noether invariance theory for the equilibrium force structure of soft matter
• Reduced-variance orientational distribution functions from torque sampling
• 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