  Acoustic-phonon-assisted thermalization kinetics of excitons in quantum wells
(QWs) is developed for small concentrations of particles, $\rho_{\rm 2D}
\lesssim 10^9$ cm$^{-1}$, when particle-particle interaction can be neglected
while Bose-Einstein statistics already strongly influences the relaxation
processes at low temperatures. In this case thermalization of QW excitons
occurs through nonequilibrium states and is given by the following scenario.
During the first transient stage, which lasts a few characteristic scattering
times, the correlations with an initial distribution of QW excitons disappear.
The next, adiabatic stage of thermalization usually takes many characteristic
scattering times, depends only upon two control parameters, the lattice
temperature $T_b$ and the degeneracy temperature $T_0 \propto \rho_{\rm 2D}$,
and is characterized by a quasi-equilibrated distribution of high-energy QW
excitons with effective temperature $T(t)$. We show that the thermalization law
of high-energy particles is given by $\delta T(t) = T(t) - T_b \propto
e^{-\lambda_0 t}/t$, where $\lambda_0$ is a marginal value of the continuous
eigenvalue spectrum of the linearized kinetics. By analyzing the linearized
phonon-assisted kinetics of statistically-degenerate QW excitons, we study the
dependence $\lambda_0 =\lambda_0(T_b,T_0)$. Our numerical estimates refer to
high quality GaAs and ZnSe QWs. Finally, we propose a special design of
GaAs-based microcavities, which considerably weakens the bottleneck effect in
relaxation of excitons (polaritons) and allows us to optimize the
acoustic-phonon-assisted thermalization processes.
