Dynamical gauge invariance of statistical mechanics

We investigate gauge invariance against phase space shifting in nonequilibrium systems, as represented by time-dependent many-body Hamiltonians that drive an initial ensemble out of thermal equilibrium. The theory gives rise to gauge correlation functions that characterize spatial and temporal inhomogeneity with microscopic resolution on the one-body level. Analyzing the dynamical gauge invariance allows one to identify a specific localized shift gauge current as a fundamental nonequilibrium observable that characterizes particle-based dynamics. When averaged over the nonequilibrium ensemble, the shift current vanishes identically, which constitutes an exact nonequilibrium conservation law that generalizes the Yvon-Born-Green equilibrium balance of the vanishing sum of ideal, interparticle, and external forces. Any given observable is associated with a corresponding dynamical hyperforce density and hypercurrent correlation function. An exact nonequilibrium sum rule interrelates these one-body functions, in generalization of the recent hyperforce balance for equilibrium systems. We demonstrate the physical consequences of the dynamical gauge invariance using both harmonically confined ideal gas setups, for which we present analytical solutions, and molecular dynamics simulations of interacting systems, for which we demonstrate the shift current and hypercurrent correlation functions to be accessible both via finite-difference methods and via trajectory-based automatic differentiation. We show that the theory constitutes a starting point for developing nonequilibrium reduced-variance sampling algorithms and for investigating thermally-activated barrier crossing.