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We establish from first principles a perturbative framework that allows us tocompute reaction rates for processes taking place in nonequilibrium $O (N)$linear-sigma systems in broken phase. The system of our concern is quasiuniformsystem near equilibrium or nonequilibrium quasistationary system. We employ theclosed-time-path formalism and use the so-called gradient approximation. Nofurther approximation is introduced. In the course of construction of theframework, we obtain the gap equation that determines the effective masses of$\pi$ and of $\sigma$, and the generalized Boltzmann equation that describesthe evolution of the number-density functions of $\pi$ and of $\sigma$.Comment: 18 page

physical review. d. particles, fields, gravitation, and cosmology/physical review. d. particles and fields

Out of equilibrium<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>O</mml:mi><mml:mo>(</mml:mo><mml:mi>N</mml:mi><mml:mo>)</mml:mo><mml:mn /></mml:math>linear-sigma system: Construction of perturbation theory with gap and Boltzmann equations

Out of equilibriumO(N)linear-sigma system: Construction of perturbation theory with gap and Boltzmann equations