Distributions with power-law decrease and the approach to equilibrium

Abstract

The main purpose of our work concerns the long-time behavior of initial distributions with diverging second moments and to show that in the mean the behavior is still exponential. We consider solutions $F(x,t)$ of the Boltzmann equation for a Maxwell-like model for which $\ensuremath{\int}{0}^{\ensuremath{\infty}}\mathrm{dx}{x}^{2}F(x,0)=\ensuremath{\infty}$ and $\ensuremath{\int}{0}^{\ensuremath{\infty}}\mathrm{dx}F(x,0)=\ensuremath{\int}{0}^{\ensuremath{\infty}}\mathrm{dx}\mathrm{xF}(x,0)=1$. It is shown that $F(x,t)$ can be approximated arbitrarily well in the mean by $\overline{F}(x,t)$ for $0\ensuremath{\le}tl\ensuremath{\infty}$ and $\overline{F}(x,t)\ensuremath{\simeq}{e}^{\ensuremath{-}x}[1+K{e}^{\ensuremath{-}\frac{t}{3}}{L}_{2}(x)]$ as $t\ensuremath{\rightarrow}\ensuremath{\infty}$. Consequently, the Hilbert space of square-integrable functions, viz., $\ensuremath{\int}{0}^{\ensuremath{\infty}}\mathrm{dx}{F}^{2}(x,0)l\ensuremath{\infty}$, is complete in the sense of convergence in the mean for $0\ensuremath{\le}tl\ensuremath{\infty}$, and thus the time evolution of initial singularities---for instance, an $F(x,0)$ with $\ensuremath{\delta}$-function peaks, jump discontinuities, etc.---is not physically meaningful.