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, The State University of New Jersey

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 1339-1376

Subject Categories

Mathematics

Roots

Basic sciences

DOI

10.1142/S0218202513500115

More information

Latest update

9/7/2018 1