Well-posedness of the Iterative Boltzmann Inversion

TL;DR

This paper provides initial theoretical analysis demonstrating the well-posedness of the iterative Boltzmann inversion method for determining pair potentials from radial distribution functions, under general assumptions.

Contribution

It offers the first rigorous convergence analysis of the iterative Boltzmann inversion, establishing conditions for its well-definedness near the true pair potential.

Findings

The algorithm is well-defined near the true potential under broad assumptions.

Proved decay properties of the Ursell function for Lennard-Jones type potentials.

Established key properties of the cavity distribution function.

Abstract

The iterative Boltzmann inversion is an iterative scheme to determine an effective pair potential for an ensemble of identical particles in thermal equilibrium from the corresponding radial distribution function. Although the method is reported to work reasonably well in practice, it still lacks a rigorous convergence analysis. In this paper we provide some first steps towards such an analysis, and we show under quite general assumptions that the algorithm is well-defined in a neighborhood of the true pair potential, assuming that such a potential exists. On our way we establish important properties of the cavity distribution function and provide a proof of a statement formulated by Groeneveld concerning the rate of decay at infinity of the Ursell function associated with a Lennard-Jones type potential.