Noether's Theorem in Statistical Mechanics

Abstract

We apply Noether's calculus of invariant variations to equilibrium and driven many-body systems. Generating functionals, such as the free energy, yield mechanical laws under continuous translational and rotational symmetry operations. The resulting global theorems express vanishing of total internal and total external forces and torques. Local sum rules interrelate density correlators, as well as static and time direct correlation functions via infinite hierarchies, including memory. For anisotropic particles, systematic coupling of orbital and spin motion is identified. The theory allows to shed new light on the spatio-temporal coupling of correlations in complex systems. When applied to active Brownian particles, the theorem clarifies the role of interfacial forces in motility-induced phase separation. For active sedimentation under gravity the global internal Noether sum rule constrains the motion of the center of mass.