Realizability of Point Processes

Abstract

There are various situations in which it is natural to ask whether a given collection of k functions, ρ j (r 1 , . . . , r j ), j = 1, . . . , k, defined on a set X, are the first k correlation functions of a point process on X. Here we describe some necessary and sufficient conditions on the ρ j 's for this to be true. Our primary examples are X = R d , X = Z d , and X an arbitrary finite set. In particular, we extend a result by Ambartzumian and Sukiasian showing realizability at sufficiently small densities ρ 1 (r). Typically if any realizing process exists there will be many (even an uncountable number); in this case we prove, when X is a finite set, the existence of a realizing Gibbs measure with k body potentials which maximizes the entropy among all realizing measures. We also investigate in detail a simple example in which a uniform density ρ and translation invariant ρ 2 are specified on Z; there is a gap between our best upper bound on possible values of ρ and the largest ρ for which realizability can be established.

Convergence of Density Expansions of Correlation Functions and the Ornstein–Zernike Equation

“…For a graph g ∈ G n,n+k we define the activitỹ 22) as well as its version without the test function φ, but with dependence on a fixed configuration q 1 , . .…”

“…Note also that in one of these earlier works, Hiroike and Morita suggest that using more complex re-summations the theory of classical fluids may be constructed with the knowledge of the pair distribution function alone, even if a form of the pair interaction potential is not known. This is also closely related to the inverse or realizability problem, where one seeks to find a priori properties of the correlation function, see [22] and the references therein. All these considerations are purely formal; it is not even in the high-temperature and low-density regime a priori clear that these calculations can be made rigorous.…”

Well-Posedness of the Iterative Boltzmann Inversion

“…where we let 10) which is a positive number; this readily implies (4.8). The induction base n = 1 of (4.9) is obviously correct because ε < 1 according to (4.10).…”

“…The question of existence and uniqueness of a solution u † of (1.1) for a given g † is referred to as the inverse Henderson problem, because Henderson [8] was the first to investigate the identifiability problem associated with (1.1), i.e., whether the radial distribution function is enough data to uniquely recover the underlying pair potential; see Kuna, Lebowitz, and Speer [10] for a more rigorous mathematical treatment of the uniqueness problem.…”

“…The absolute value of ϕ N can be estimated by means of a tree-graph inequality 10) where T N is the set of trees with N vertices. Note that this inequality, which can be found in Ueltschi [16], makes use of the stability bound (5.4), which in turn requires |ζ| ≤ 1.…”