'Steady/equilibrium approximation' in relaxation and fluctuation. II. Mathematical theory of approximations in first-order reaction

A mathematical theory of the steady/equilibrium approximation for first-order reactions is presented. This gives the theoretical basis for the methods of simplifying the complex first-order reactions described in the preceding work The steady/equilibrium relation holds on every fast component after a proper inducation period T degrees T degrees is either of O(1) or less, or nearly of O(1/epsilon) depending on the reaction scheme and on the initial condition but is always less than O(1/epsilon) (as in the preceding paper 1], we use the symbol O(1) to denote a positive number of the order of unity). In the open group, the determinant of the submatrix M(p), representing the interconversion between the fast components in the group and their dissipation, is of O(1). The concentration of the fast components in the open group can thus be expressed as a linear combination of those components neighboring the group after the establishment of a steady/equilibrium relation, and can be eliminated from the reaction scheme leaving the pathway through them. On the other hand, in the closed group the determinant of Mp is of O(epsilon) or less and the components in the group are in quasi equilibrium with each other after T degrees . They are eliminated from the reaction scheme leaving the sum of the components in the closed group as a slow component.