For homogeneous initial conditions, Hartree (gaussian) dynamical approximations are known to have problems with thermalization, because of insufficient scattering. We attempt to improve on this by writing an arbitrary density matrix as a superposition of gaussian pure states and applying the Hartree approximation to each member of such an ensemble. Particles can then scatter via their back-reaction on the typically inhomogeneous mean fields. Starting from initial states which are far from equilibrium we numerically compute the time evolution of particle distribution functions and observe that they indeed display approximate thermalization on intermediate time scales by approaching a Bose-Einstein form. However, for very large times the distributions drift towards classical-like equipartition.11.15.Ha, 11.10.Wx

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 ϕ 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. *

The on-shell imaginary part of the retarded selfenergy of massive ϕ 4 theory in 1+1 dimensions is logarithmically infrared divergent. This leads to a zero in the spectral function, separating its usual bump into two. The twin peaks interfere in time-dependent correlation functions, which causes oscillating modulations on top of exponential-like decay, while the usual formulas for the decay rate fail. We see similar modulations in our numerical results for a mean field correlator, using a Hartree ensemble approximation.In our numerical simulations of 1+1 dimensional ϕ 4 theory using the Hartree ensemble approximation 1 we found funny modulations in a time-dependent correlation function. Fig. 1 shows such modulations on top of a roughly exponential decay. The correlation function is the time average of the zero momentum mode of the mean field, F mf (t) = ϕ(t)ϕ(0) − ϕ(t) ϕ(0), where the over-bar denotes a time average, X(t) = t2 t1 dt ′ X(t + t ′ )/(t 2 − t 1 ), taken after waiting a long time t 1 for the system to be in approximate equilibrium. This equilibrium is approximately thermal and F mf (t) is analogous to the symmetric correlation function of the quantum field theory at finite temperature, F (t) = 1 2 {φ(t),φ(0)} conn . A natural question is now, does F (t) also have such modulations?The function F (t) can be expressed in terms of the zero momentum spectral function ρ(p 0 ),and the latter in turn in terms of the retarded selfenergy Σ(p 0 ),The selfenergy can be calculated in perturbation theory. The one and two loop diagrams in the imaginary time formalism which have nontrivial energymomentum dependence are shown in Fig. 2. Diagrams not shown give only rise to an effective temperature dependent mass, which we assume to be the a Presented by J. Smit.