Kinetic Limits for Pair-Interaction Driven Master Equations and Biological Swarm Models
Journal article, 2013
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.
3-dimensional rare-gas
binary interactions
behavior
global validity
continuum-limit
Master equation
boltzmann-equation
kinetic equations
stochastic particle approximations
Kac's master
gap
spectral
vacuum
system
limit
propagation of chaos
mean-field
Author
E. Carlen
Rutgers, The State University of New Jersey
P. Degond
Universite de Toulouse
CNRS Centre National de la Recherche Scientifique
Bernt Wennberg
University of Gothenburg
Chalmers, Mathematical Sciences, Mathematics
Mathematical Models and Methods in Applied Sciences
0218-2025 (ISSN)
Vol. 23 7 1339-1376Subject Categories (SSIF 2011)
Mathematics
Roots
Basic sciences
DOI
10.1142/S0218202513500115