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On the pair correlations of particle evolution in the direct statistical simulation

On the pair correlations of particle evolution in the direct statistical simulation -- The pair correlation function for the particle evolution system governed by the Kolmogorov equation is evaluated in terms of the eigen-functions of the corresponding integral operator. Introduction It is well known that when solving the nonlinear Boltzmann equation in the spatially homogeneous case by the Monte Carlo method, the original problem is reformulated in terms of the TV-particle Kolmogorov equation (for details see, e.g., [1]). The interrelation of the solutions to these two equations is established under the assumption that there is no pair correlations in the approximating model particle system. This assumption is known as the molecular chaos hypothesis [2]. Therefore, the analysis of the pair correlations is of particular interest. This problem was treated from different points of view in [3], [1], [4]. In the present paper we obtain, under broad assumptions about the particle interaction potentials, the steady state solution to the TV-particle Kolmogorov equation. It is shown that under some restrictions an arbitrary solution (from an appropriate functional space) is expanded into a convergent series in eigen-functions of an integral operator. This enables us not only to represent the general solution of the Cauchy problem for the Kolmogorov equation, but to find also http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Monte Carlo Methods and Applications de Gruyter

On the pair correlations of particle evolution in the direct statistical simulation

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References (9)

Publisher
de Gruyter
Copyright
Copyright © 2009 Walter de Gruyter
ISSN
0929-9629
eISSN
1569-3961
DOI
10.1515/mcma.1996.2.1.25
Publisher site
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Abstract

-- The pair correlation function for the particle evolution system governed by the Kolmogorov equation is evaluated in terms of the eigen-functions of the corresponding integral operator. Introduction It is well known that when solving the nonlinear Boltzmann equation in the spatially homogeneous case by the Monte Carlo method, the original problem is reformulated in terms of the TV-particle Kolmogorov equation (for details see, e.g., [1]). The interrelation of the solutions to these two equations is established under the assumption that there is no pair correlations in the approximating model particle system. This assumption is known as the molecular chaos hypothesis [2]. Therefore, the analysis of the pair correlations is of particular interest. This problem was treated from different points of view in [3], [1], [4]. In the present paper we obtain, under broad assumptions about the particle interaction potentials, the steady state solution to the TV-particle Kolmogorov equation. It is shown that under some restrictions an arbitrary solution (from an appropriate functional space) is expanded into a convergent series in eigen-functions of an integral operator. This enables us not only to represent the general solution of the Cauchy problem for the Kolmogorov equation, but to find also

Journal

Monte Carlo Methods and Applications – de Gruyter

Published: Jan 1, 1996

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