Transition phenomena induced by internal noise and the discreteness

We study a simple chemical reaction system and effects of the internal noise. Although stochastic differential equations are often used in order to study the fluctuations, they can not treat the discreteness in the number of components (molecules, individuals, and so on) because they are based on the continuum approximation which is justified by the large system size. The discreteness of components would not be negligible when we treat the system with small size, such as biochemical reactions in a cell. To elucidate the effects of such discreteness, we propose a simple chemical reaction system, and show that the internal noise and the molecular discreteness cause an anticipated transition phenomenon. The transition phenomenon is characterized by the change of the number of peaks in the stationary probability distribution. The transition point is analytically determined by using a random walk model. PACS. 05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion – 02.50.Ey Stochastic processes – 87.16.-b Subcellular structure and processes