Ultraviolet cascade in the thermalization of the classical phi^4 theory   in 3+1 dimensions
  We investigate the dynamics of thermalization and the approach to equilibrium in the classical phi^4 theory in 3+1 spacetime dimensions. The non-equilibrium dynamics is studied by numerically solving the equations of motion in a light- cone-like discretization of the model for a broad range of initial conditions and energy densities.A smooth cascade of energy towards the ultraviolet is found to be the basic mechanism of thermalization.After an initial transient stage,at a time scale of several hundreds inverse masses,the squared of the field gradient becomes larger than the nonlinear term and a stage of universal cascade emerges. As the cascade progresses, the modes with higher wavenumbers exhibit weaker and weaker nonlinearities well described by the Hartree approximation while the infrared modes retain strong selfinteractions. Two timescales for equilibration appears.For k^2><phi^2>(t) we observe an effective thermalization with a time scale in the thousands of inverse masses and the Hartree approximation holds. By effective thermalization we mean that the observable acquires the equilibrium functional form with an effective time dependent temperature Teff, which slowly decreases with time. Infrared modes with k^2 < <phi^2>(t) equilibrate only by time scales in the millions of inverse masses. Infrared modes with k^2 < <phi^2>(t) equilibrate only by time scales in the millions.Virialization and the equation of state start to set much earlier than effective thermalization.The applicability of these results in quantum field theory for large occupation numbers and small coupling is analyzed. 
C. Destri, H. J. de Vega