  Noether invariance in statistical mechanics provides fundamental connections
between the symmetries of a physical system and its conservation laws and sum
rules. The latter are exact identities that involve statistically averaged
forces and force correlations and they are derived from statistical mechanical
functionals. However, the implications for more general observables and order
parameters are unclear. Here, we demonstrate that thermally averaged classical
phase space functions are associated with exact hyperforce sum rules that
follow from translational Noether invariance. Both global and locally resolved
identities hold and they relate the mean gradient of a phase-space function to
its negative mean product with the total force. Similar to Hirschfelder's
hypervirial theorem, the hyperforce sum rules apply to arbitrary observables in
equilibrium. Exact hierarchies of higher-order sum rules follow iteratively. As
applications we investigate via computer simulations the emerging one-body
force fluctuation profiles in confined liquids. These local correlators
quantify spatially inhomogeneous self-organization and their measurement allows
for the development of stringent convergence tests and enhanced sampling
schemes in complex systems.
