Hyperforce balance via thermal Noether invariance of any observable

Abstract

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.

Noether invariance theory for the equilibrium force structure of soft matter

“…The Noether approach captures genuinely the mechanical fluctuations, as opposed to local chemical (particle number) [53,[84][85][86][87] and thermal (energy) fluctuations [53,86,87]. We lastly point to the recent hyperforce generalization [31] that arises from thermal Noether invariance and that applies to arbitrary observables.…”

“…Lastly, equation ( 35) can be viewed as a special case of the recent more general hyperforce correlation theory by Robitschko et al [31]. Their theory applies to general phase space functions Â(r N , p N ) and for configuration-dependent cases, Â(r N ), it ascertains the validity of the identity ⟨β F(r) Â(r…”

“…Noether's theorem of invariant variations [16,17] was put in a statistical physics setting in a number of different ways [18][19][20][21][22][23][24]. The recent thermal Noether invariance theory [25][26][27][28][29][30][31] is based on the identification of the symmetry of the statistical many-body systems against specific shifting and rotation operations on phase space. Corresponding force and torque correlation functions are generated and exact statistical mechanical identities ('sum rules') play the role that in more conventional uses of the Noether theorem are played by the resulting conservation laws.…”

“…Corresponding force and torque correlation functions are generated and exact statistical mechanical identities ('sum rules') play the role that in more conventional uses of the Noether theorem are played by the resulting conservation laws. The recent hyperforce approach by Robitschko et al [31] carries the thermal Noether concept further in that it allows to address the behaviour of arbitrary observables in equilibribum.…”