Thermalization of classical fields is investigated in a \\phi^4 scalar field theory in 1+1 dimensions, discretized on a lattice. We numerically integrate the classical equations of motion using initial conditions sampled from various nonequilibrium probability distributions. Time-dependent expectation values of observables constructed from the canonical momentum are compared with thermal ones. It is found that a closed system, evolving from one initial condition, thermalizes to high precision in the thermodynamic limit, in a time-averaged sense. For ensembles consisting of many members with the same energy, we find that expectation values become stationary - and equal to the thermal values - in the limit of infinitely many members. Initial ensembles with a nonzero (noncanonical) spread in the energy density or other conserved quantities evolve to noncanonical stationary ensembles. In the case of a narrow spread, asymptotic values of primary observables are only mildly affected. In contrast, fluctuations and connected correlation functions will differ substantially from the canonical values. This raises doubts on the use of a straightforward expansion in terms of 1PI-vertex functions to study thermalization.
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
Tc,
SU(2) gauge theory falls within the same universality class as
Z2 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.
A new approach to the dynamics of relaxation and kinetics of thermalization in a scalar field theory is presented that incorporates the relevant time scales through the resummation of hard thermal loops. An alternative derivation of the kinetic equations for the "quasiparticle" distribution functions is obtained that allows a clear understanding of the different "coarse-graining" approximations usually involved in a kinetic description. This method leads to a systematic perturbative expansion to obtain the kinetic equations including hard thermal loop resummation and to an improvement including renormalization, off-shell effects, and contributions that change chemical equilibrium on short time scales. As a by-product of these methods we establish the equivalence between the relaxation time scale in the linearized equation of motion of the quasiparticles and the thermalization time scale of the quasiparticle distribution function in the "relaxation time approximation" including hard thermal loop effects. Hard thermal loop resummation dramatically modifies the scattering rate for long wavelength modes as compared to the usual (semi)classical estimate. Relaxation and kinetics are studied both in the unbroken and broken symmetry phases of the theory. The broken symmetry phase also provides the setting to obtain the contribution to the kinetic equations from processes that involve decay of a heavy scalar into light scalar particles in the medium.