A Leibniz rule of distributional pairing and hyperforce sum rule

TL;DR

This paper generalizes the hyperforce sum rule, a key concept in statistical mechanics, by employing distribution theory and Leibniz rule, extending its applicability to various boundary conditions.

Contribution

It introduces a reformulation of the hyperforce sum rule using Schwartz space and distribution pairing, broadening its theoretical foundation and applicability.

Findings

Derived hyperforce sum rule for Euclidean space

Extended the sum rule to systems with periodic boundary conditions

Unified the hierarchy with distributional calculus

Abstract

We reformulate and generalize the equilibrium hyperforce sum rule, a generalization of the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy, by employing the Schwartz space and its dual. We show that the hyperforce sum rule for the Euclidean space and the equilibrium BBGKY hierarchy at arbitrary level are derived through the Leibniz rule of the derivative for the pairing of tempered distributions and Schwartz functions. We also apply the Leibniz rule to obtain the hyperforce sum rule for systems with periodic boundary conditions.