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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 $π$ and of $σ$, and the generalized Boltzmann equation that describes the evolution of the number-density functions of $π$ and of $σ$.