Some remarks on the stochastic aspects of a simple autocatalytic reaction

The stochastic behavior of one of the simplest autocatalytic reactions, A+X -> X+X has been analyzed. Assuming the perfect mixing of substrate and autocatalytic particles in a closed volume to be uniform, an exactly solvable system of differential equations for the probability of finding n autocatalytic particles at a given time moment was derived. It was found that the time dependence of the mean number of the autocatalytic particles is clearly different from that obtained from the corresponding rate equation. The hierarchial system of the moment equations derived from the generating function of the number of the autocatalytic particles has been also analyzed. It was shown that neglecting the semi-invariants of order larger than two brings about the most suitable approximation for the first moment. In the case of closed reaction constituents, the probability distribution of the system lifetime has been determined, and the dependence of the expectation and variance of the lifetime on the number of substrate particles has been calculated.