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Monte-Carlo approach to particle-field interactions and the kinetics of the chiral phase transition

The kinetics of the chiral phase transition is studied within a linear quark-meson-$σ$ model, using a Monte-Carlo approach to semiclassical particle-field dynamics. The meson fields are described on the mean-field level and quarks and antiquarks as ensembles of test particles. Collisions between quarks and antiquarks as well as the $q\overline{q}$ annihilation to $σ$ mesons and the decay of $σ$ mesons is treated, using the corresponding transition-matrix elements from the underlying quantum field theory, obeying strictly the rule of detailed balance and energy-momentum conservation. The approach allows to study fluctuations without making ad hoc assumptions concerning the statistical nature of the random process as necessary in Langevin-Fokker-Planck frameworks.