Gauge invariance and hyperforce correlation theory for equilibrium fluid mixtures

We formulate gauge invariance for the equilibrium statistical mechanics of classical multi-component systems. Species-resolved phase space shifting constitutes a gauge transformation, which we analyze using Noether's theorem and shifting differential operators that encapsulate the gauge invariance. The approach yields exact equilibrium sum rules for general mixtures. Species-resolved gauge correlation functions for the force-force and force-gradient pair correlation structure emerge on the two-body level. Exact 3g-sum rules relate these correlation functions to the spatial Hessian of the partial pair distribution functions. General observables are associated with hyperforce densities that measure the covariance of the given observable with the interparticle, external, and diffusive partial force density observables. Exact hyperforce and Lie algebra sum rules interrelate these correlation functions with each other. The practical accessibility of the framework is demonstrated for binary Lennard-Jones mixtures using both adaptive Brownian dynamics and grand canonical Monte Carlo simulations. In particular, we investigate the force-force pair correlation structure of the Kob-Andersen bulk liquid and we show results for representative hyperforce correlation functions in the symmetrical mixture of Wilding et al. confined between two asymmetric planar parallel walls.