Two-point correlation properties of stochastic "cloud processes''

TL;DR

This paper derives exact formulas relating the two-point correlation functions of a particle distribution before and after stochastic subdivision, enabling the generation of highly uniform stochastic particle distributions from regular lattices.

Contribution

It provides a generalization of existing equations for stochastic displacement fields, including cases with correlated displacements among particles.

Findings

Derived exact relations between mother and daughter structure factors.

Established algorithms for creating highly uniform stochastic distributions.

Extended previous models to include correlated displacements.

Abstract

We study how the two-point density correlation properties of a point particle distribution are modified when each particle is divided, by a stochastic process, into an equal number of identical "daughter" particles. We consider generically that there may be non-trivial correlations in the displacement fields describing the positions of the different daughters of the same "mother" particle, and then treat separately the cases in which there are, or are not, correlations also between the displacements of daughters belonging to different mothers. For both cases exact formulae are derived relating the structure factor (power spectrum) of the daughter distribution to that of the mother. These results can be considered as a generalization of the analogous equations obtained in ref. [1] (cond-mat/0409594) for the case of stochastic displacement fields applied to particle distributions. An application of the present results is that they give explicit algorithms for generating, starting from regular lattice arrays, stochastic particle distributions with an arbitrarily high degree of large-scale uniformity.