Nonequilibrium Quantum Fields and the Classical Field Theory Limit
  We calculate the far-from-equilibrium dynamics and thermalization both for the quantum and the classical O(N)--model. The early and late-time behavior can be described from the 2PI--loop expansion for weak couplings or the nonperturbative 2PI--1/N expansion of the effective action beyond leading order. A comparison with exact simulations in 1+1 dimensions in the classical limit shows that the 2PI--1/N expansion at next-to-leading order gives quantitatively precise results already for moderate values of N. We derive a criterion for the validity of the classical approximation and verify it by comparing far-from-equilibrium quantum and classical dynamics. At late times one observes the expected deviations due to the difference between classical and quantum thermal equilibrium. 
J. Berges