On the stationary BBGKY hierarchy for equilibrium states

TL;DR

This paper introduces a new direct integration method for solving the stationary BBGKY hierarchy in equilibrium states of infinite classical particle systems, establishing existence, uniqueness, and equivalence with Kirkwood--Salsburg equations.

Contribution

It presents a novel direct integration approach for the stationary BBGKY hierarchy and proves its equivalence with Kirkwood--Salsburg equations, addressing existence and uniqueness at low densities.

Findings

Established a new integration method for the hierarchy.

Proved equivalence with Kirkwood--Salsburg equations.

Solved existence and uniqueness for low-density systems.

Abstract

A new direct integration method is established to construct the solutions of the stationary BBGKY hierarchy, assuming the usual form of the equilibrium correlation functions, for infinite classical systems of particles interacting via a smooth, stable and regular two body potential. The equivalence between the corresponding infinite hierarchy and the Kirkwood--Salsburg equations is proved. A problem of existence and uniqueness of the solutions of the hierarchy with appropriate boundary conditions is thus solved for low densities. The result is extended in a milder sense to systems with a hard core interaction. Comparisons are provided with different integration techniques.