Analytical Structure of a Generalized Direct - Interaction Approximation

As a mathematically tractable example, we have investigated the stochastic dynamic problem of an irreversible second-order chemical reaction. A generalized direct-interaction approximation has been devised to close off the hierarchy of moment equations at the arbitrary moment level, and then the results of such a closure technique have been compared term-by-term with the exact moment solutions. This shows qualitatively how the expansion terms summed up in the direct-interaction approximation are different from the classes of expansion terms present in the exact moment solutions. A quantitative comparison of the covariances indicates that the direct-interaction equations which are closed at the triple moment level represent a meaningful statistical approximation of the lowest order for the second-order reactive problem at hand. The problem statement. As a mathematically tractable example, we shall consider in this paper the stochastic dynamical problem governed by #(0M = -KHt)2, (1.1) where K is assumed constant. Physically, (1.1) describes the depletion of the reactant species \p(t) by an isothermal, irreversible second-order chemical reaction with the reaction-rate constant K [1, 2], Considering an ensemble of realizations each governed by (1.1), a stochastic dynamical problem can be formulated by assigning certain statistical properties to the initial ensemble. For the deterministic solution of (1.1) m = i + KHm to)' (1,2) where t0 is the initial time, the stochastic dynamics may be described by the statistical distribution of In particular, the moments of one time argument are completely determined by the initial distribution P[^(<0)]. For t0 = 0, the mean value is *(0 -M0>-/(rTKteW <>-3> * Received October 4, 1971. Dr. Kraichnan's suggestions on the first draft of the manuscript have been extremely helpful.