Predicting properties of the stationary probability currents for two-species reaction systems without solving the Fokker-Planck equation

Abstract: We derive methods for estimating the topology of the stationary probability current $\vec{j}_s$ of the two-species Fokker-Planck equation (FPE) without the need to solve the FPE. These methods are chosen such that they become exact in certain limits, such as infinite system size or vanishing coupling between species in the diffusion matrix. The methods make predictions about the fixed points of $\vec{j}_s$ and their relation to extrema of the stationary probability distribution and to fixed points of the convective field, which is related to the deterministic drift of the system. Furthermore, they predict the rotation sense of $\vec{j}_s$ around extrema of the stationary probability distribution. Even though these methods cannot be proven to be valid away from extrema, the boundary lines between regions with different rotation sense are obtained with surprising accuracy. We illustrate and test these method using simple reaction systems with only one coupling term between the two species as well as a few generic reaction networks taken from literature. We use it also to investigate the shape of non-physical probability currents occurring in reaction systems with detailed balance due to the approximations involved in deriving the Fokker-Planck equation.