Quantum kinetics and thermalization in an exactly solvable model

We study the dynamics of relaxation and thermalization in an exactly solvable model with the goal of understanding the effects of off-shell processes. The focus is to compare the exact evolution of the distribution function with different approximations to the relaxational dynamics: Boltzmann, non-Markovian and Markovian quantum kinetics. The time evolution of the occupation number or distribution function is evaluated exactly using two methods: time evolution of an initially prepared density matrix and by solving the Heisenberg equations of motion. The former allows to establish a connection with the stochastic nature of thermalization and the fluctuation dissipation theorem, whereas the latter leads to the interpretation of an interpolating number operator to ‘count’ quasiparticles. There are two different cases that are studied in detail: i) no stable particle states below threshold of the bath and a quasiparticle resonance above it and ii) a stable discrete exact ‘particle’ state below threshold. Email: [email protected] Email: [email protected] Email:[email protected] Email: [email protected] Laboratoire Associé aui CNRS, UMR 7589. The exact solution for the evolution allows us to investigate the concept of the formation time of a quasiparticle and to study the difference between the relaxation of the distribution of particles and quasiparticles. In particular we compare the quasiparticle distribution for asymptotic times with the equilibrium canonical distribution. For the case of quasiparticles in the continuum (resonances) the exact quasiparticle distribution asymptotically tends to a statistical equilibrium distribution that differs from a simple Bose-Einstein form as a result of off-shell processes such as the strength of the quasiparticle poles, the width of the unstable particle and proximity to thresholds. In the case ii), the distribution of particles does not thermalize with the bath. We study the kinetics of thermalization and relaxation by deriving a non-Markovian quantum kinetic equation which resums the perturbative series and includes off-shell effects. A Markovian approximation that includes off-shell contributions and the usual Boltzmann equation are obtained from the quantum kinetic equation in the limit of wide separation of time scales upon different coarse-graining assumptions. The relaxational dynamics predicted by the non-Markovian, Markovian and Boltzmann approximations are compared to the exact result of the model. The Boltzmann approach is seen to fail in the case of wide resonances and when threshold and renormalization effects are important. Implications for thermalization in field theory models are discussed.