  We numerically study the relaxation dynamics of a single, heavy impurity atom
interacting with a finite one- or two-dimensional, ultracold Bose-gas. While
there is a clear separation of time scales between processes resulting from
single- and two-phonon scattering in three spatial dimensions, the
thermalization in lower dimensions is dominated by two-phonon processes. This
is due to infrared divergencies in the corresponding scattering rates in the
thermodynamic limit, which are a manifestation of the Mermin-Wagner-Hohenberg
theorem. It makes it necessary to include second-order phonon scattering in
one-dimensional systems even at $T=0$ and above a crossover temperature
$T_\textrm{2ph}$ in two spatial dimensions. $T_\textrm{2ph}$ scales inversely
with the system size and is much smaller than currently experimentally
accessible.
