The Classical Inverse Problem for Multi-Particle Densities in the Canonical Ensemble Formulation

Abstract

We provide sufficient conditions for the solution of the classical inverse problem in the canonical distribution for multi-particle densities. Specifically, we show that there exists a unique potential in the form of a sum of m-particle (m greater then 1) interactions producing a given m-particle density. The existence and uniqueness of the solution to the multi-particle inverse problem is essential for the numerical simulations of matter using effective potentials derived from structural data. Such potentials are often employed in coarse- grained modeling. The validity of the multi-particle inverse conjecture also has implications for liquid state theory. For example, it provides the first step in proving the existence of the hierarchy of generalized Ornstein-Zernike relations. For the grand canonical distribution, the multi-particle inverse problem has been solved by Chayes and Chayes [J. Stat. Physics 36, 471-488 (1984)]. However, the setting of the canonical ensemble presents unique challenges arising from the impossibility of uncoupling interactions when the number of particles is fixed.