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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.

Analytical structure of a generalized direct-interaction approximation