Paper Info
Title
Ordinary differential equation system for population of individuals and the corresponding probabilistic model
Abstract
The key model for particle populations in statistical mechanics is the Bogolyubov-Born-Green-Kirkwood-Yvon (BBGKY) equation chain. It is derived mainly from the Hamilton ordinary differential equation (ODE) system for the particle states in the position-momentum phase space. Many problems beyond physics or chemistry, for instance, in the living-matter sciences (biology, medicine, ecology, and sociology) make it necessary to extend the notion of a particle to an individual, or active particle. This challenge is met by the generalized kinetic theory. The corresponding dynamics of the state vector can also be regarded to be described by an ODE system. The latter, however, need not be the Hamilton one. The question is how one can derive the analogue of the BBGKY paradigm for the new settings. The present work proposes an answer to this question. It applies a very limited number of carefully selected tools of probability theory and common statistical mechanics. It also uses the well-known feature that the maximum number of the individuals which can mutually interact directly is bounded by a fixed value of a few units. The proposed approach results in the finite system of equations for the reduced many-individual distribution functions thereby eliminating the so-called closure problem inevitable in the BBGKY theory. The thermodynamic-limit assumption is not needed either. The system includes consistently derived terms of all of the basic types known in kinetic theory, in particular, both the ''mean-field'' and scattering-integral terms, and admits the kinetic equation of the form allowing a direct chemical-reaction reading. The approach can deal with Hamilton's model which is nonmonogenic. The results may serve as the basis of the generalized kinetic theory and contribute to stochastic mechanics of populations of individuals.
Year
DOI
Venue
2009
10.1016/j.mcm.2008.09.010
Mathematical and Computer Modelling
Keywords
DocType
Volume
active particle,ordinary differential equation system,ordinary differential equation,generalized kinetic theory,particle population,ode system,finite system,kinetic equation,probabilistic population dynamics,bbgky theory,corresponding probabilistic model,kinetic theory,statistical approximation,probability theory,particle state,population of individuals,biophysics,signal processing,natural sciences,biological sciences,applied mathematics,computational mathematics,computer and information science,mathematical physics,mechanical engineering,statistical mechanics,information systems,physical sciences,bioinformatics,structural biology,fluid mechanics,sociology,mathematics,condensed matter physics,statistical physics,biostatistics,social sciences,theoretical chemistry,chemical sciences
Journal
49
Issue
ISSN
Citations 
7-8
Mathematical and Computer Modelling
0
PageRank 
References 
Authors
0.34
5
1
Name
Order
Citations
PageRank
E. Mamontov1114.24