Range of validity of transport equations
  Transport equations can be derived from quantum field theory assuming a loss of information about the details of the initial state and a gradient expansion. While the latter can be systematically improved, the assumption about a memory loss is in general not controlled by a small expansion parameter. We determine the range of validity of transport equations for the example of a scalar $g^2 \Phi^4$ theory. We solve the nonequilibrium time evolution using the three-loop 2PI effective action. The approximation includes off-shell and memory effects and assumes no gradient expansion. This is compared to transport equations to lowest order (LO) and beyond (NLO). We find that the earliest time for the validity of transport equations is set by the characteristic relaxation time scale $t_{\rm damp} = - 2\omega/\Sigma^{\rm (eq)}_\varrho$, where $-\Sigma^{\rm (eq)}_\varrho/2$ denotes the on-shell imaginary-part of the self-energy. This time scale agrees with the characteristic time for partial memory loss, but is much shorter than thermal equilibration times. For times larger than about $t_{\rm damp}$ the gradient expansion to NLO is found to describe the ``full'' results rather well for $g^2 \lesssim 1$. 
Jurgen Berges, Szabolcs Borsanyi (Heidelberg)