Nonequilibrium process in the $\sigma$ model and chemical relaxation time in a homogeneous pionic gas

In a homogeneous pionic gas system, chemical nonequilibrium process is considered to understand its effect in the expansion processes that are realized immediately after heavy ion collisions. The chemical relaxation time is calculated by incorporating the $\pi+\pi \leftrightarrow \pi+\pi+\pi+\pi$ reaction, which is given in the second order of perturbation in the $\sigma$ model. The $\pi+\pi \leftrightarrow \pi+\pi+\pi+\pi$ reaction is assumed to be less frequent than the $\pi+\pi \leftrightarrow \pi+\pi$ scattering that is expected to establish the local equilibrium, and hydrodynamical equation is solved for various initial conditions. It is shown that the relaxation time is of the order of 100fm and does not have a significant effect on the expansion process, which implies that the pion number freezeout takes place at an early stage of the expansion.