Out of equilibrium O (N) linear-sigma system - Construction of   perturbation theory with gap- and Boltzmann-equations
  We establish from first principles a perturbative framework that allows us to compute reaction rates for processes taking place in nonequilibrium $O (N)$ linear-sigma systems in broken phase. The system of our concern is quasiuniform system near equilibrium or nonequilibrium quasistationary system. We employ the closed-time-path formalism and use the so-called gradient approximation. No further approximation is introduced. In the course of construction of the framework, we obtain the gap equation that determines the effective masses of $\pi$ and of $\sigma$, and the generalized Boltzmann equation that describes the evolution of the number-density functions of $\pi$ and of $\sigma$. 
A. Ni\'egawa