Abstract

Noether’s theorem is familiar to most physicists due its fundamental role in linking the existence of conservation laws to the underlying symmetries of a physical system. I will present how Noether’s reasoning also applies within statistical mechanics to thermal systems, where fluctuations are paramount. Exact identities (“sum rules”) follow thereby from functional symmetries. The obtained sum rules contain both, well-known relations, such as the first order term of the Yvon-Born-Green (YBG) hierarchy (i.e. the spatially resolved force balance), as well as previously unknown identities, relating different correlations in many-body systems. The identification of the underlying Noether concept enables their systematic derivation. Since Noether’s theorem is quite general it is possible to generalize to arbitrary thermodynamic observables. This generalization yields sum rules for hyperforces, i.e. the mean product between the considered observable and the relevant forces that act in the system. Simulations of a range of simple and complex liquids demonstrate the fundamental role of these correlation functions in the characterization of spatial structure, such as quantifying spatially inhomogeneous self-organization. Finally, we show that the considered phase-space-shifting is a gauge transformation in equilibrium statistical mechanics.