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Kinetic Limits for Pair-Interaction Driven Master Equations and Biological Swarm Models

Journal article
Authors E. Carlen
P. Degond
Bernt Wennberg
Published in Mathematical Models and Methods in Applied Sciences
Volume 23
Issue 7
Pages 1339-1376
ISSN 0218-2025
Publication year 2013
Published at Department of Mathematical Sciences, Mathematics
Pages 1339-1376
Language en
Links dx.doi.org/10.1142/S021820251350011...
Keywords Master equation, kinetic equations, binary interactions, propagation of chaos, Kac's master, stochastic particle approximations, 3-dimensional rare-gas, mean-field, limit, boltzmann-equation, global validity, continuum-limit, spectral, gap, behavior, vacuum, system
Subject categories Mathematics

Abstract

We consider a class of stochastic processes modeling binary interactions in an N-particle system. Examples of such systems can be found in the modeling of biological swarms. They lead to the definition of a class of master equations that we call pair-interaction driven master equations. In the spatially homogeneous case, we prove a propagation of chaos result for this class of master equations which generalizes Mark Kac's well-known result for the Kac model in kinetic theory. We use this result to study kinetic limits for two biological swarm models. We show that propagation of chaos may be lost at large times and we exhibit an example where the invariant density is not chaotic.

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