Collective Behavior: Macroscopic versus Kinetic Descriptions

Invariant manifolds and hydrodynamic limit in kinetics

Alexander Gorban

University of Leicester


The problem of the derivation of macroscopic equations from kinetic equations is formulated as the problem of slow invariant manifold in the space of distributions. We review a few instances where such hydrodynamic manifolds were found analytically by Gorban and Karlin. The exact solutions of the reduction problem are represented both as the result of summation of the Chapman-Enskog asymptotic expansion and by the direct solution of the invariance equation. We present also several methods for approximate solution of the invariance equation and solution of the problem of projection of the kinetic equation onto the approximate hydrodynamic invariant manifold using entropy concepts. This is a joint work with Ilia Karlin (ETH Z├╝rich).