On the Construction of Particle Distributions with Specified Single and Pair Densities

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We discuss necessary conditions for the existence of probability distribution on particle configurations in d-dimensions i.e. a point process, compatible with a specified density ρ and radial distribution function g(r). In d=1 we give necessary and sufficient criteria on ρg(r) for the existence of such a point process of renewal (Markov) type. We prove that these conditions are satisfied for the case g(r) = 0, r < D and g(r) = 1, r > D, if and only if ρD ≤ e⁻¹: the maximum density obtainable from diluting a Poisson process. We then describe briefly necessary and sufficient conditions, valid in every dimension, for ρg(r) to specify a determinantal point process for which all n-particle densities, ρₙ(r₁,..., rₙ), are given explicitly as determinants. We give several examples.