Kinetics of the chiral phase transition in a quark-meson $\sigma$ model

Using the two-particle irreducible (2PI) $\mathrm{\Phi }$-functional formalism for self-consistent approximations of a linear-$\sigma$ model for quarks and mesons in and out of equilibrium, the build-up of fluctuations of net-baryon number during the time evolution of an expanding fireball is studied within a kinetic theory for the order parameter ($\sigma$ field) and quark distribution functions. Initializing the system with purely Gaussian fluctuations a fourth-order cumulant is temporarily built up due to the evolution of the $\sigma$-field. This is counterbalanced, however, by the dissipative evolution due to collisions between quarks, anti-quarks, mesons, and the mean field, depending on the speed of the fireball expansion.