Kinetic limits for pair-interaction driven master equations and biological swarm models

Eric Carlen (1) , Pierre Degond (2) , Bernt Wennberg (3)

1 Department of Mathematics - Rutgers School of Arts and Sciences
2 IMT - Institut de Mathématiques de Toulouse UMR5219
3 Department of Mathematical Sciences

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. We prove a propagation of chaos result for this class of master equations which generalizes Mark Kac's well know 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.

Domaines: Physique mathématique [math-ph]

Eric Carlen, Pierre Degond, Bernt Wennberg. Kinetic limits for pair-interaction driven master equations and biological swarm models. Mathematical Models and Methods in Applied Sciences, 2013, 23, pp.1339-1376. ⟨hal-00625542⟩