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.

Why Noether's Theorem applies to Statistical Mechanics

“…In the present contribution we demonstrate that the concepts of Ref. [14] apply to the canonical ensemble, as is relevant for confined systems [50][51][52] and for the dynamics [53][54][55]. Hence having an open system with respect to particle exchange is not necessary for the Noetherian arguments to apply.…”

“…In Ref. [14] we also apply Noether's thinking to a very recent variational approach for dynamics, called power functional theory [28], which propels the functional concepts from equilibrium to nonequilibrium [28][29][30][31][32][33][34][35][36][37][38][39][40][41][42], including the recently popular active Browian particles [43][44][45][46][47][48][49]. The generalization is important, as it shows that not only a dead Munchausen cannot bootstrap himself out of his misery, but that being alive does not help (in this particular case).…”

Dynamic Decay and Superadiabatic Forces in the van Hove Dynamics of Bulk Hard Sphere Fluids

“…The phenomenological equation of motion (23) of DDFT does not capture the full non-equilibrium dynamics of many-particle systems. Important physical effects such as drag [59], viscosity [58,75] and structural non-equilibrium forces [59,[76][77][78][79][80][81][82]82] are absent. PFT provides a formally exact method for including such effects and for calculating the full current in a non-equilibrium system [52]; see Ref.…”

“…(See Refs. [82,86] for exact force sum rules that stem from Noether's Theorem.) Geigenfeind et al [59] defined the differential force density G(r, t) as a linear combination of species-resolved force densities…”

Accumulation of Particles and Formation of a Dissipative Structure in a Nonequilibrium Bath

“…The Kullback–Leibler divergence between the steady state distribution and another distribution could be a good candidate. With tools like Noether’s Theorem, alternative formulations of active-particle statistical mechanics and of the Fractional Fokker–Planck Equation have been derived [ 25 , 26 ], with work in this direction appearing to be promising.…”