The Boltzmann Equation for Bose–Einstein Particles: Velocity Concentration and Convergence to Equilibrium

Abstract

Long time behavior of solutions of the spatially homogeneous Boltzmann equation for Bose–Einstein particles is studied for hard potentials with certain cutoffs and for the hard sphere model. It is proved that in the cutoff case solutions as time $t\rightarrow\infty$ converge to the Bose–Einstein distribution in L¹ topology with the weighted measure $(\varrho +|v|^2)dv$, where $\varrho=1$ for temperature $T\geq T_c$ and $\varrho=0$ for T<T _c . In particular this implies that if T<T _c then the solutions in the velocity regions $\{v\in{\bf R}^3|\,\,|v|\leq \delta (t)\}$ (with $\delta(t)\rightarrow 0$) converge to a unique Dirac delta function (velocity concentration). All these convergence are uniform with respect to the cutoff constants. For the hard sphere model, these results hold also for weak or distributional solutions. Our methods are based on entropy inequalities and an observation that the convergence to Bose–Einstein distributions can be reduced to the convergence to Maxwell distributions.