Mattias Sundén

Brownian approximation in the Kac model

Abstract:

The Kac model is a Markov process which describes random collisions of
gas atoms under the assumption of constant energy. Thus, the process
can be considered to describe the movement of particle living on a
sphere of radius \sqrt{energy} and whose dimension is the number of
gas atoms. The generator of this process is the right hand side of a
much simplified Boltzmann equation, namely the so called Kac master
equation. Though simplified, the Kac model captures many interesting
properties, such as relaxation (to equilibrium) times and chaos
propagation or Boltzmann property. In the original model by Kac (1956)
collision angles are assumed to be uniformly distributed. In this
joint work with Bernt Wennberg we consider other collision kernels to
model more realistic collisions of gas atoms. Our main focus will be
on approximation of small jumps (angles) by a diffusion process for
the Kac model with unbounded collision kernel.