Model for the Derivation of Kinetic Theory

Abstract

A soluble model for the derivation of kinetic theory is discussed. It is shown that Bogoliubov's adiabatic assumption is not valid even to leading order in the kinetic regime. Furthermore, the adiabatic assumption gives rise to a singular behavior in the correlation function which is sufficiently strong to violate the ordering assumed in the expansion. The exact result for the model correlation is neither adiabatic nor singular throughout the kinetic regime. We first show that the singularity is removed by properly reordering the correlation in the relevant domains; then a method for obtaining uniformly valid asymptotic solutions for a wide class of integrodifferential problems (that includes our model) is presented. We apply our technique to the BBGKY hierarchy and show that for a weakly coupled gas (i) the familiar Landau equation is satisfied in lowest order by the velocity distribution function and (ii) the two-particle correlation function can be calculated consistently to leading order, yielding a nonadiabatic and nonsingular behavior.