  The real time evolution of field condensates with soft length scales
k^{-1}>(eT)^{-1} is solved in hot scalar electrodynamics, with a view towards
understanding relaxational phenomena in the QGP and the electroweak plasma. We
find that transverse gauge invariant non-equilibrium expectation values of
fields relax via {\em power laws} to asymptotic amplitudes that are determined
by the quasiparticle poles. The long time relaxational dynamics and relevant
time scales are determined by the behaviour of the retarded self-energy not at
the small frequencies, but at the Landau damping thresholds. This explains the
presence of power laws and not of exponential decay. Furthermore, we derive the
influence functional, the Langevin equation and the fluctuation-dissipation
theorem for the soft modes, identifying the correlation functions that emerge
in the classical limit. We show that a Markovian approximation fails to
describe the dynamics {\em both} at short and long times. We also introduce a
novel kinetic approach that goes beyond the standard Boltzmann equation and
incorporates off-shell processes and find that the distribution function for
soft quasiparticles relaxes with a power law through Landau damping. We also
find an unusual dressing dynamics of bare particles and anomalous (logarithmic)
relaxation of hard quasiparticles.
