Two-point correlation properties of stochastic splitting processes.

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 nontrivial 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 formulas are derived relating the structure factor (power spectrum) of the daughter distribution to that of the mothers. An application of these 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. Such distributions are of interest, in particular, in the context of studies of self-gravitating systems in cosmology.

[1]  A. Gabrielli,et al.  Gravitational dynamics of an infinite shuffled lattice of particles. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[2]  J. Stadel,et al.  An Alternative to Grids and Glasses: Quaquaversal Pre-Initial Conditions for N-Body Simulations , 2006, astro-ph/0606148.

[3]  Carlos S. Frenk,et al.  The large-scale structure of the Universe , 2006, Nature.

[4]  F. S. Labini.  Statistical Physics for Cosmic Structures , 2006 .

[5]  D. Lévesque,et al.  A Method of generating initial conditions for cosmological N body simulations , 2004, astro-ph/0411607.

[6]  B. Marcos,et al.  Quantification of discreteness effects in cosmological N -body simulations: Initial conditions , 2004, astro-ph/0410451.

[7]  A. Gabrielli.  Point processes and stochastic displacement fields. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[8]  Salvatore Torquato,et al.  Local density fluctuations, hyperuniformity, and order metrics. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[9]  A. Gabrielli,et al.  Causality constraints on fluctuations in cosmology: A study with exactly solvable one-dimensional models , 2003, astro-ph/0303169.

[10]  L. Pietronero,et al.  Generation of primordial cosmological perturbations from statistical mechanical models , 2002, astro-ph/0210033.

[11]  Eric Renshaw,et al.  Two-dimensional spectral analysis for marked point processes , 2002 .

[12]  D. Heggie.  The gravitational million-body problem , 2001, astro-ph/0111045.

[13]  A. Gabrielli,et al.  Glass-like universe: Real-space correlation properties of standard cosmological models , 2001, astro-ph/0110451.

[14]  M. Kerscher.  Constructing, characterizing, and simulating Gaussian and higher-order point distributions. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[15]  Thomas M Truskett,et al.  Density fluctuations in many-body systems , 1998 .

[16]  I. R. Mcdonald,et al.  Theory of simple liquids , 1998 .

[17]  István Groma,et al.  Link between the microscopic and mesoscopic length-scale description of the collective behavior of dislocations , 1997 .

[18]  B. Wandelt,et al.  Causality and the power spectrum. , 1995, Physical review. D, Particles and fields.

[19]  Aggelos K. Katsaggelos,et al.  Iterative maximum likelihood displacement field estimation in quantum-limited image sequences , 1995, IEEE Trans. Image Process..

[20]  R. Tibshirani,et al.  An Introduction to the Bootstrap , 1995 .

[21]  Lebowitz,et al.  Scaling functions, self-similarity, and the morphology of phase-separating systems. , 1991, Physical review. B, Condensed matter.

[22]  G. V. Chester,et al.  Solid State Physics , 2000 .