  We study the non-equilibrium dynamics of domain walls in real time for
$\phi^4$ and Sine Gordon models in 1+1 dimensions in the dilute regime. The
equation of motion for the collective coordinate is obtained by integrating out
the meson excitations around the domain wall to one-loop order. The real-time
non-equilibrium relaxation is studied analytically and numerically to this
order. The constant friction coefficient vanishes but there is dynamical
friction and relaxation caused by off-shell non-Markovian effects. The validity
of a Markovian description is studied in detail. The proper Langevin equation
is obtained to this order, the noise is Gaussian and additive but colored. We
analyze the classical and hard thermal loop contributions to the self-energy
and noise kernels and show that at temperatures larger than the meson mass the
hard contributions are negligible and the finite temperature contribution to
the dynamics is governed by the classical soft modes of the meson bath. The
long time relaxational dynamics is completely dominated by classical Landau
damping resulting in that the corresponding time scales are not set by the
temperature but by the meson mass. The noise correlation function and the
dissipative kernel obey a generalized form of the Fluctuation-Dissipation
relation.
