We derive asymptotic solutions of Kramers-Moyal equations (KMEs) that arise from master equations (MEs) for stochastic processes. We consider both one step processes, in which the system jumps from x to x + x-ε with given probabilities, and general transitions, in which the system moves from x to x + εξ, where ξ is a random variable with a given probability distribution. Our method exploits the smallness of a parameter ε, typically the ratio of the jump size to the system size. We employ the full KME to derive asymptotic expansions for the stationary density of fluctuations, as well as for the mean lifetime of stable equilibria. Thus we treat fluctuations of arbitrary size, including large fluctuations. In addition we present a criterion for the validity of diffusion approximations to master equations. We show that diffusion theory can not always be used to study large deviations. When diffusion theory is valid our results reduce to those of diffusion theory. Examples from macroscopic chemical kinetics and the calculation of chemical reaction rates ("Kramers" models) are discussed.
ASJC Scopus subject areas
- Physics and Astronomy(all)
- Physical and Theoretical Chemistry