Singular Perturbation Expansion for Markovian Master Equations

Abstract

Asymptotic properties of the master equations for chemical reactive systems whose macroscopic rate equations have more than one stationary state are discussed using generating function method. The systematic singular perturbation expansion method for equation of generating function is generalized to include the case of multi-stationary system beyond the bifurcation point and the following conclusions are proved: For such systems, there is a Gaussian fluctuation before next genuine bifurcation point; a critical fluctuation at a genuine bifurcation point; and a macroscopic fluctuation when the system is on the coexistence line which is determined by the master equation. Furthermore, the relation between the cumulant of generating function and the stochastic potential is also established. Our discussion, however, is limited to homogeneous system of one variable only.