Understanding thermalization of the magnetic gluons in a non-Abelian gauge theory

We explore chaotic dynamics in SU(2) gauge theory by extracting the Lyapunov exponent under both in and out-of-equilibrium conditions to understand how chaos develops and influences the approach to thermalization. In thermal equilibrium at very high temperatures, the hard, electric, and magnetic scales are well separated, and one can write an effective theory for the soft gauge fields (magnetic modes) which are classical due to their large occupation number. In the vicinity of the critical temperature $T_c$, $SU(2)$ gauge theory falls within the same universality class as $Z_2$ scalar field theory. Exploiting this correspondence, we determine the Lyapunov exponent in the scalar theory through out-of-time-ordered correlators (OTOC). To probe non-equilibrium dynamics, we prepare the gauge fields in a highly occupied, non-thermal initial state. Under classical Yang–Mills evolution, the system evolves toward a self-similar scaling regime characterized by a scale separation similar to that of the thermal case. Within this regime, we again compute the Lyapunov exponent and compare it to its equilibrium counterpart at similar energy densities. This comparison allows us to quantify the timescale over which the system approaches thermal equilibrium.