A variational framework for the inverse Henderson problem of statistical mechanics

TL;DR

This paper establishes a rigorous variational framework for the inverse Henderson problem in statistical mechanics, connecting relative entropy minimization with iterative schemes and analyzing convexity properties of the pressure and entropy functions.

Contribution

It extends the variational approach to the inverse Henderson problem in the thermodynamic limit, including Lennard-Jones potentials, and analyzes the convexity and Hessian of the relative entropy.

Findings

Pressure is strictly convex in pair potential and chemical potential.

Specific relative entropy is strictly convex as a function of pair potential.

Hessian of the relative entropy extends to a positive semidefinite quadratic form.

Abstract

The inverse Henderson problem refers to the determination of the pair potential which specifies the interactions in an ensemble of classical particles in continuous space, given the density and the equilibrium pair correlation function of these particles as data. For a canonical ensemble in a bounded domain it has been observed that this pair potential minimizes a corresponding convex relative entropy functional, and that the Newton iteration for minimizing this functional coincides with the so-called inverse Monte Carlo (IMC) iterative scheme. In this paper we show that in the thermodynamic limit analogous connections exist between the specific relative entropy introduced by Georgii and Zessin and a proper formulation of the IMC iteration in the full space. This provides a rigorous variational framework for the inverse Henderson problem, valid within a large class of pair potentials, including, for example, Lennard-Jones type potentials. It is further shown that the pressure is strictly convex as a function of the pair potential and the chemical potential, and that the specific relative entropy at fixed density is a strictly convex function of the pair potential. At a given reference potential and a corresponding density in the gas phase we determine the gradient and the Hessian of the specific relative entropy, and we prove that the Hessian extends to a symmetric positive semidefinite quadratic functional in the space of square integrable perturbations of this potential.