Kinetics of the chiral phase transition in a linear $\sigma$ model

We study the dynamics of the chiral phase transition in a linear quark-meson $\sigma$ model using a novel approach based on semiclassical wave-particle duality. The quarks are treated as test particles in a Monte-Carlo simulation of elastic collisions and the coupling to the $\sigma$ meson, which is treated as a classical field. The exchange of energy and momentum between particles and fields is described in terms of appropriate Gaussian wave packets. It has been demonstrated that energy-momentum conservation and the principle of detailed balance are fulfilled, and that the dynamics leads to the correct equilibrium limit. First schematic studies of the dynamics of matter produced in heavy-ion collisions are presented.