A note on the uniqueness result for the inverse Henderson problem

Abstract

The inverse Henderson problem of statistical mechanics concerns classical particles in continuous space which interact according to a pair potential depending on the distance of the particles. Roughly stated, it asks for the interaction potential given the equilibrium pair correlation function of the system. In 1974 Henderson proved that this potential is uniquely determined in a canonical ensemble and he claimed the same result for the thermodynamical limit of the physical system. Here we provide a rigorous proof of a slightly more general version of the latter statement using Georgii's version of the Gibbs variational principle. 1.Introduction. An important inverse problem in computational physics and computational chemistry is the determination of the interacting forces in a system of particles in continuous space, given structural information on the spatial distribution of the particles. In the simplest incarnation of this problem it is assumed that the potential energy of the particle ensemble is determined by a pair potential which only depends on the distance of the interacting particles. In an often cited paper Henderson [10] has claimed that under given conditions of temperature and density this pair potential is uniquely determined by the so-called radial distribution function, which -suitably normalized -assigns to each r > 0 the expected number of particles on a sphere of radius r around any given particle. Roughly speaking, the radial distribution function is obtained from the pair correlation function (called pair density function in the physical literature) associated with a canonical or grand canonical ensemble in a finite volume, when taking the limit of the volume to infinity, the socalled thermodynamical limit. To give credit to Henderson's contribution, this inverse problem of statistical mechanics is sometimes called the inverse Henderson problem.Henderson's argument makes use of a technique suggested by Hohenberg and Kohn [11], Mermin [16], and others, for studying a similar inverse problem for external potentials. The key idea is to apply a Gibbs variational principle, which states that in a system with given thermodynamic conditions the associated thermodynamic potential becomes minimal, if and only if the distribution of the particles is given by the probability measure associated with this system. The particular version of this principle used by Henderson is based on the free energy functional in a canonical ensemble, where the finite volume pair correlation function and not the radial distribution function is the relevant stochastic quantity. To extend the uniqueness result to the radial distribution function, Henderson subsequently turns to the thermodynamical limit, ignoring the possibility that the strict inequality of the variational principle * The research leading to this work has been funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) -Projektnummer 233630050 -TRR 146.

A variational framework for the inverse Henderson problem of statistical mechanics

“…But its simplicity offers a great opportunity for a mathematical analysis, which in turn may lead to a better understanding of other, more flexible coarse-graining techniques that are routinely being employed in practice. Still, only few rigorous mathematical results have yet been obtained, e.g., in [2,7,16,23,24,30], the reason being, again, the lack of explicit formulae to attack the problem.…”

A variational framework for the inverse Henderson problem of statistical mechanics

“…If P * satisfies a Ruelle condition (compare (A.1) in the appendix) then the specific entropy is finite; this is the case, e.g., when P * is a (µ * , u * )-Gibbs measure for some u * ∈ U and µ * ∈ R, compare [39,Corollary 5.3]. For this latter particular case it has further been shown in [7] that…”

“…But its simplicity offers a great opportunity for a mathematical analysis, which in turn may lead to a better understanding of other, more flexible coarse-graining techniques that are routinely being employed in practice. Still, only few rigorous mathematical results have yet been obtained, e.g., in [2,23,24,30,16,7], the reason being, again, the lack of explicit formulae to attack the problem.…”

“…Like Henderson's paper [18], the results in [29,36,41] lack mathematical rigor, because their arguments are restricted to bounded domains -whereas an unambiguous definition of a "translation invariant ensemble" is only possible in the full space. Concerning the Henderson theorem this shortcoming has recently been fixed in [7], building on fundamental work by Ruelle and by Georgii. Here we focus on the proposal by Shell and his colleagues, provide a rigorous justification of their results, and elaborate further on them.…”

Interpolation method for crystals with many-body interactions

“…Experimental techniques such as neutron and x-ray scattering [10][11][12][13][14] can yield pair distribution function (PDF) and hence do provide direct access to the atomic structure and interatomic correlations, and thus to the information on the nature of interatomic potentials. However, when PDF data alone are used in solving the corresponding inverse problem of extracting information about potentials, the well-known ambiguities arise [15][16][17] with distinctly different interatomic interactions resulting in similar (or identical within experimental errors) PDFs.…”