On the solutions of the nonlinear Liouville hierarchy

TL;DR

This paper develops an explicit solution for the nonlinear Liouville hierarchy's initial-value problem using cluster expansions and cumulants, proving existence for integrable initial data.

Contribution

It introduces a novel explicit solution construction for the nonlinear Liouville hierarchy applicable to general interaction potentials.

Findings

Explicit solution expressed via particle cluster expansions

Proved existence of strong solutions for integrable initial data

Applicable to systems with general interaction potentials

Abstract

We investigate the initial-value problem of the non-linear Liouville hierarchy. For the general form of the interaction potential we construct an explicit solution in terms of an expansion over particle clusters whose evolution is described by the corresponding-order cumulant of evolution operators of a system of finitely many particles. For the initial data from the space of integrable functions the existence of a strong solution of the Cauchy problem is proved.