Noether's Theorem in Statistical Mechanics

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.

arXiv comment17 pages, 6 figures

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• Why Noether's Theorem applies to Statistical Mechanics
• Perspective: How to overcome dynamical density functional theory
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• Noether-Constrained Correlations in Equilibrium Liquids
• Perspective: New directions in dynamical density functional theory
• Why neural functionals suit statistical mechanics
• Hyper-density functional theory of soft matter
• Force balance in thermal quantum many-body systems from Noether's theorem
• Variance of fluctuations from Noether invariance
• Gauge Invariance of Equilibrium Statistical Mechanics
• Hyperforce balance via thermal Noether invariance of any observable
• Neural density functional theory of liquid-gas phase coexistence
• Neural density functionals: Local learning and pair-correlation matching
• Neural force functional for non-equilibrium many-body colloidal systems
• Why hyperdensity functionals describe any equilibrium observable
• Noether invariance theory for the equilibrium force structure of soft matter
• Reduced-variance orientational distribution functions from torque sampling
• Inhomogeneous steady shear dynamics of a three-body colloidal gel former
• Learning the bulk and interfacial physics of liquid-liquid phase separation with neural density functionals
• Why gauge invariance applies to statistical mechanics
• Dynamic Decay and Superadiabatic Forces in the van Hove Dynamics of Bulk Hard Sphere Fluids
• Gauge invariance and hyperforce correlation theory for equilibrium fluid mixtures
• Active crystallization from power functional theory
• Non-equilibrium Onsager-Machlup theory
• Metadensity functional learning for classical fluids: Regularizing with pair correlations
• First part of Clausius heat theorem in terms of Noether's theorem
• Adiabatic and irreversible classical discrete time crystals
• Thermodynamics of an Empty Box