Logarithmic Density Behavior of a Nonequilibrium Boltzmann Gas

We consider the temporal evolution of the BBGKY hierarchy in the Boltzmann approximation for spatially homogeneous nonequilibrium situations, in the absence of initial correlations. For times of the order of the mean free time or greater, the single particle function f1 is found to be of the form f1=f10+εf11+ε2|lnε|f̃11+O(ε2)with ε=nr03 (n is the density, r0 the range of binary interaction) and f10, f11, and f̃11 of order unity. For times less than the mean free time, with t in units of the duration of a binary interaction, f1 is of the form f1=f10+εf11+ε2(lnt)f̃11+O(ε2). In both cases the same formally higher-order binary correlation functions are neglected.