A Stochastic process behind Boltzmann's kinetic equation and issues of coarse graining

Abstract

We consider a stochastic process behind Boltzmann's kinetic equation that is obtained by identifying the infinitesimal generator of a Markov process. Within an intrinsically probabilistic interpretation, the nonlinear nature of Boltzmann's equation, which can be regarded as an expression of self-consistency, can be achieved via weakly interacting Markov processes. Whereas the Markov process associated with the Boltzmann equation is known as a powerful tool both to study fundamental mathematical issues of existence and to solve practical engineering problems, we here consider fundamental physical issues of coarse graining and, in particular, the role of diffusion in hydrodynamic equations. As a provocative conclusion, it may be less important to solve the Boltzmann equation than to coarse grain it. © 2006 Springer-Verlag Berlin Heidelberg.