  We study thermal behavior of a recently introduced Hartree ensemble
approximation, which allows for non-perturbative inhomogeneous field
configurations as well as for approximate thermalization, in the $\phi^4$ model
in 1+1 dimensions. Using ensembles with a free field thermal distribution as
out-of-equilibrium initial conditions we determine thermalization time scales.
The time scale for which the system stays in approximate quantum thermal
equilibrium is an indication of the time scales for which the approximation
method stays reasonable. This time scale turns out to be two orders of
magnitude larger than the time scale for thermalization, in the range of
couplings and temperatures studied. We also discuss simplifications of our
method which are numerically more efficient and make a comparison with
classical dynamics.
