A variational framework for the inverse Henderson problem of statistical mechanics

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.

Understanding and Modeling Polymers: The Challenge of Multiple Scales

“…This is known as the inverse Henderson problem. As proved in 1974 by Henderson for finite systems with fixed particle number and very recently by Frommer et al , for the thermodynamic limit, the problem has a unique solution for a rich class of interaction potentials which includes, among other the so-called Lennard-Jones type potentials . Nevertheless, the problem is ill-posed in the sense that a small noise in the RDF can lead to large changes in the potentials.…”

Stability, Speed, and Constraints for Structural Coarse-Graining in VOTCA

“…Examples for the latter are relative entropy minimization (REM) and force-matching. , Machine learning algorithms can be trained to yield potentials that reproduce, e.g., force-matching results . Two methods, inverse Monte Carlo (IMC) and iterative Boltzmann inversion (IBI), explicitly optimize pair potentials to match the radial distribution function (RDF) of the mapped atomic reference. , The RE method has been proven to be equivalent to IMC if pair potentials are optimized by a Newton scheme. , …”

“…9,10 The RE method has been proven to be equivalent to IMC if pair potentials are optimized by a Newton scheme. 11,12 The mathematical classification of the structural coarsegraining problem is that of an inverse problem. Only a complicated forward function from the potential to the RDF exists (the MD simulation), but the opposite direction is of interest.…”

“… 9 , 10 The RE method has been proven to be equivalent to IMC if pair potentials are optimized by a Newton scheme. 11 , 12 …”