Logarithmic Density Behavior of a Nonequilibrium Boltzmann Gas

Abstract

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 $$f_1{\mathrm{=f}}_1^0{\mathrm{+\epsilon f}}_1^1{\mathrm{+\epsilon }}^2|\ln {\mathrm{\epsilon |f̃}}_1^1{\mathrm{+O(\epsilon }}^2)$$ with ${\mathrm{\epsilon =nr}}_0^3$ (n is the density, r₀ the range of binary interaction) and $f_1^0$ , $f_1^1$ , and ${\mathrm{f̃}}_1^1$ 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 $$f_1{\mathrm{=f}}_1^0{\mathrm{+\epsilon f}}_1^1{\mathrm{+\epsilon }}^2(\ln {\mathrm{t)f̃}}_1^1{\mathrm{+O(\epsilon }}^2)$$ . In both cases the same formally higher‐order binary correlation functions are neglected.