Domain walls out of equilibrium

We study the nonequilibrium dynamics of domain walls in real time for ${\ensuremath{\varphi}}^{4}$ 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 nonequilibrium 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.