We study a simple chemical reaction system and effects of the internal noise. The chemical reaction system causes the same transition phenomenon discussed by Togashi and Kaneko [Phys. Rev. Lett. 86 (2001) 2459; J. Phys. Soc. Jpn. 72 (2003) 62]. By using the simpler model than Togashi-Kaneko's one, we discuss the transition phenomenon by means of a random walk model and an effective model. The discussion makes it clear that quasi-absorbing states, which are produced by the change of the strength of the internal noise, play an important role in the transition phenomenon. Stabilizing the quasi-absorbing states causes bifurcation of the peaks in the stationary probability distribution discontinuously.

Duality relations between continuous-state and discrete-state stochastic processes with continuous time have already been studied and used in various research fields. We propose extended duality relations, which enable us to derive discrete-state stochastic processes from arbitrary diffusion-type partial differential equations. The derivation is based on the Doi-Peliti formalism, and it will be clarified that additional states for the discrete-state stochastic processes must be considered in some cases.

We calculate a pump current in a classical two-state stochastic chemical kinetics by means of the non-adiabatic geometrical phase interpretation. The two-state system is attached to two particle reservoirs, and under a periodic perturbation of the kinetic rates, it gives rise to a pump current between the two-state system and the absorbing states. In order to calculate the pump current, the Floquet theory for the non-adiabatic geometrical phase is extended from a Hermitian case to a non-Hermitian case. The dependence of the pump current on the frequency of the perturbative kinetic rates is explicitly derived, and a stochastic resonance-like behavior is obtained.

We calculate a current and its fluctuation in a two-state stochastic system under a periodic perturbation. The system could be interpreted as a channel on a cell surface or a single Michaelis-Menten catalyzing enzyme. It has been shown that the periodic perturbation induces a so-called pump current, and the pump current and its fluctuation are calculated with the aid of the geometrical phase interpretation. We give a simple calculation recipe for the statistics of the current, especially in a nonadiabatic case. The calculation scheme is based on the nonadiabatic geometrical phase interpretation. Using the Floquet theory, the total current and its fluctuation are calculated, and it is revealed that the average of the current shows a stochastic-resonance-like behavior. In contrast, the fluctuation of the current does not show such behavior.

It is important in computer science, sociology, and so on to investigate complex bipartite graphs from a viewpoint of statistical physics. We propose a model to generate complex bipartite graphs without growing; the bipartite graphs are assumed to have two sets of the fixed numbers of nodes and a fixed number of edges between nodes belonging to different sets of nodes. In this model, essential ingredients are a preferential rewiring process and a fitness distribution function. By using the preferential rewiring process, we confirm that a bipartite graph reaches a stationary state after a sufficiently long time has passed. We find that the obtained bipartite graph has a scale-free-like property when a suitable fitness distribution is used. It turns out that a condensation of edges takes place in the cases of certain fitness distributions.