Ultraviolet cascade in the thermalization of the classical ϕ4 theory in 3+1 dimensions

Abstract

We investigate the dynamics of thermalization and the approach to equilibrium in the classical ${\ensuremath{\phi}}^{4}$ theory in $3+1$ spacetime dimensions. The nonequilibrium 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 hundred inverse masses, the squared magnitude of the field spatial gradient becomes larger than the nonlinear term and there emerges a stage of universal cascade, independent of the details of the initial conditions. As the cascade progresses, the modes with higher wave numbers, but well behind the forefront of the cascade, exhibit weaker and weaker nonlinearities well described by the Hartree approximation, while the infrared modes retain strong self-interactions. As a consequence, two time scales for equilibration appear as characteristic of two time-dependent wave number regions. For ${k}^{2}\ensuremath{\gtrsim}\overline{{\ensuremath{\phi}}^{2}}(t)$, we observe an effective equilibration to a time-dependent powerlike spectrum with a time scale in the hundreds of inverse masses; cutoff effects are absent and the Hartree approximation holds for ${k}^{2}\ensuremath{\gg}\overline{{\ensuremath{\phi}}^{2}}(t)$. On the other hand, infrared modes with ${k}^{2}\ensuremath{\lesssim}\overline{{\ensuremath{\phi}}^{2}}(t)$ equilibrate only by time scales in the millions of inverse masses when the cutoff effects become dominant and complete thermalization is setting in. Accordingly, we observe in the field correlator a relatively large and long-lived deviation from the Hartree behavior of a nonperturbative character. There corresponds an effective mass governing the long distance behavior of the correlator which turns out to be significantly smaller than the Hartree mass which is exhibited by the modes with ${k}^{2}\ensuremath{\gtrsim}\overline{{\ensuremath{\phi}}^{2}}(t)$. Virialization and the equation of state start to set in much earlier than thermalization. The applicability of these results in quantum field theory for large occupation numbers and small coupling is analyzed.