  The features for the unsteady process of thermal equilibration ("the fast
motions") in a one-dimensional harmonic crystal lying in a viscous environment
(e.g., a gas) are under investigation. It is assumed that initially the
displacements of all the particles are zero and the particle velocities are
random quantities with zero mean and a constant variance, thus, the system is
far away from the thermal equilibrium. It is known that in the framework of the
corresponding conservative problem the kinetic and potential energies oscillate
and approach the equilibrium value that equals a half of the initial value of
the kinetic energy. We show that the presence of the external damping
qualitatively changes the features of this process. The unsteady process
generally has two stages. At the first stage oscillations of kinetic and
potential energies with decreasing amplitude, subjected to exponential decay,
can be observed (this stage exists only in the underdamped case). At the second
stage (which always exists), the oscillations vanish, and the energies are
subjected to a power decay. The large-time asymptotics for the energy is
proportional to $t^{-3/2}$ in the case of the potential energy and to
$t^{-5/2}$ in the case the kinetic energy. Hence, at large values of time the
total energy of the crystal is mostly the potential energy. The obtained
analytic results are verified by independent numerical calculations.
