Discreteness-Induced Transitions in Autocatalytic Systems

To study the dynamics of chemical processes, we often adopt rate equations to observe the change in chemical concentrations. However, when the number of the molecules is small, the fluctuations cannot be neglected. We often study the effects of fluctuations with the help of stochastic differential equations. Chemicals are composed of molecules on a microscopic level. In principle, the number of molecules must be an integer, which must only change discretely. However, in analysis using stochastic differential equations, the fluctuations are regarded as continuous changes. This approximation can only be valid if applied to fluctuations that involve a sufficiently large number of molecules. In the case of extremely rare chemical species, the actual discreteness of the molecules may critically affect the dynamics of the system. To elucidate the effects of the discreteness, we study an autocatalytic system consisting of several interacting chemical species with a small number of molecules through stochastic particle simulations. We found novel states, which were characterized as an extinction of molecule species, due to the discrete nature of the molecules. We also observed a strong dependence of the chemical concentrations on the size of the system, which was caused by transitions to the novel states.