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

2004/05/21 by O. Costin, J. L. Lebowitz, Costin, O. +1 · 1 citation
Mathematics · Physics and Astronomy · #FOS: Physical sciences #Mathematical Physics (math-ph) #Statistical Mechanics (cond-mat.stat-mech) #cond-mat.stat-mech #math-ph #math.MP

paper · pdf · doi:10.48550/arxiv.cond-mat/0405519

arxiv created 2004/05/21 · arxiv updated 2009/12/01

Abstract

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(\bf r). In d=1 we give necessary and sufficient criteria on ρg(\bf 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, ρ_n(\bf r₁, ..., \bf r_n), are given explicitly as determinants. We give several examples.

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