  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 $\sigma$.
