  We study how oscillations of a scalar field condensate are damped due to
dissipative effects in a thermal medium. Our starting point is a non-linear and
non-local condensate equation of motion descending from a 2PI-resummed
effective action derived in the Schwinger-Keldysh formalism appropriate for
non-equilibrium quantum field theory. We solve this non-local equation by means
of multiple-scale perturbation theory appropriate for time-dependent systems,
obtaining approximate analytic solutions valid for very long times. The
non-linear effects lead to power-law damping of oscillations, that at late
times transition to exponentially damped ones characteristic for linear
systems. These solutions describe the evolution very well, as we demonstrate
numerically in a number of examples. We then approximate the non-local equation
of motion by a Markovianised one, resolving the ambiguities appearing in the
process, and solve it utilizing the same methods to find the very same leading
approximate solution. This comparison justifies the use of Markovian equations
at leading order. The standard time-dependent perturbation theory in comparison
is not capable of describing the non-linear condensate evolution beyond the
early time regime of negligible damping. The macroscopic evolution of the
condensate is interpreted in terms of microphysical particle processes. Our
results have implications for the quantitative description of the decay of
cosmological scalar fields in the early Universe, and may also be applied to
other physical systems.
