Understanding thermalization in a non-Abelian gauge theory in terms of its soft modes

We measure the maximal Lyapunov exponent ${\lambda }_L$ of physical states in a SU(2) gauge theory consisting of soft momentum modes both in and out-of-thermal equilibrium conditions using ab-initio lattice techniques. We have implemented different algorithms to appropriately describe the dynamics of soft-modes for a wide range of temperatures and under non-equilibrium conditions. The non-equilibrium state has been realized starting from an over-occupied initial condition for low momentum soft gluons whereas the thermal state comprises of strongly interacting soft gluons at temperatures where these are well separated from the hard momentum modes. Spectra of positive Lyapunov exponents is observed in both these states, similar to a chaotic dynamical system. From the Kolmogorov-Sinai entropy rate measured in terms of this spectrum, we estimate a typical time-scale of $\sim 0.50(3)$ fm/c to achieve thermalization at $T\sim 600$ MeV starting from the non-thermal state. We also measure, for the first time, the ${\lambda }_L$ for long wavelength critical modes of SU(2) using the out-of-time-ordered correlator of a classical $Z_2$ scalar field theory, which shares the same universal behavior with SU(2), near the deconfinement phase transition. The ${\lambda }_L$ is observed to maximize at the transition temperature.