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.
boltzmann-equation
global validity
limit
behavior
mean-field
continuum-limit
gap
3-dimensional rare-gas
binary interactions
Kac's master
stochastic particle approximations
system
Master equation
kinetic equations
spectral
propagation of chaos
vacuum
Author
E. Carlen
Rutgers University
P. Degond
University of Toulouse
Centre national de la recherche scientifique (CNRS)
Bernt Wennberg
Chalmers, Mathematical Sciences, Mathematics
University of Gothenburg
Mathematical Models and Methods in Applied Sciences
0218-2025 (ISSN)
Vol. 23 7 1339-1376Subject Categories
Mathematics
Roots
Basic sciences
DOI
10.1142/S0218202513500115