 04160 O. Costin and J.L. Lebowitz
 On the Construction of Particle Distributions with Specified Single
and Pair Densities
(31K, Latex)
May 21, 04

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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 $\rho$ and radial
distribution function $g({\bf r})$. In $d=1$ we give necessary and
sufficient criteria on $\rho g({\bf r})$ 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 $\rho D \leq e^{1}$: the maximum density
obtainable from diluting a Poisson process. We then describe briefly
necessary and sufficient conditions, valid in every dimension, for
$\rho g(r)$ to specify a determinantal point process for which all
$n$particle densities, $\rho_n({\bf r}_1, ..., {\bf r}_n)$, are given
explicitly as determinants. We give several examples.
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