Evolution of correlation functions in the hard sphere dynamics

TL;DR

This paper derives a series expansion for the evolution of correlation functions in a finite hard sphere system from the Liouville equation, introducing a graphical language for describing collision histories.

Contribution

It provides a direct derivation of the correlation function evolution and recovers the BBGKY hierarchy with minimal regularity assumptions, using a novel graphical approach.

Findings

Derived series expansion from Liouville equation

Recovered BBGKY hierarchy from the expansion

Developed a collision history graphical language

Abstract

The series expansion for the evolution of the correlation functions of a finite system of hard spheres is derived from direct integration of the solution of the Liouville equation, with minimal regularity assumptions on the density of the initial measure. The usual BBGKY hierarchy of equations is then recovered. A graphical language based on the notion of collision history originally introduced by Spohn is developed, as a useful tool for the description of the expansion and of the elimination of degrees of freedom.