Approach to thermalization in the classical ${\phi }^4$ theory in $1+1$ dimensions: Energy cascades and universal scaling

We study the dynamics of thermalization and the approach to equilibrium in the classical ${\phi }^4$ theory in $1+1$ spacetime dimensions. At thermal equilibrium we exploit the equivalence between the classical canonical averages and transfer matrix quantum traces of the anharmonic oscillator to obtain exact results for the temperature dependence of several observables, which provide a set of criteria for thermalization. In this context, comparing to the exact results we find that the Hartree approximation is remarkably accurate in equilibrium. The nonequilibrium dynamics is studied by numerically solving the equations of motion in light-cone coordinates for a broad range of initial conditions and energy densities. The long time evolution is described by several distinct stages, all characterized by a cascade of energy towards the ultraviolet. After an initial transient stage, the spatiotemporal gradient terms become larger than the nonlinear term, and there emerges a stage of universal cascade. This cascade starts at a time scale $t_0$ independent of the initial conditions (except for very low energy density). During this stage the power spectra feature universal scaling behavior and the front of the cascade $\overline{k}\mathrm{}\left(t\right)\mathrm{}$ moves to the ultraviolet as a power law $\overline{k}\mathrm{}\left(t\right)\mathrm{}\sim t^{\alpha }$ with $\alpha \lesssim 0.25$ an exponent weakly dependent on the energy density alone. The wake behind the cascade is described as a state of Local Thermodynamic Equilibrium (LTE) with all correlations being determined by the equilibrium functional form with an effective time dependent temperature $T_{\mathrm{eff}}\mathrm{}\left(t\right),$ which slowly decreases with time as $\sim t^{-\alpha }.$ Two well separated time scales emerge: while $T_{\mathrm{eff}}\mathrm{}\left(t\right)\mathrm{}$ varies slowly, the wave vectors in the wake with $k<\mathrm{}\overline{k}\mathrm{}\left(t\right)\mathrm{}$ attain LTE on much shorter time scales. This universal scaling stage ends when the front of the ultraviolet cascade reaches the cutoff at a time scale $t_1\sim a^{-1/\alpha }.$ Virialization starts to set much earlier than LTE. We find that strict thermalization is achieved only for an infinite time scale.