Entropy Analysis of a Nearest-Neighbor Attractive/Repulsive Exclusion Process on One-Dimensional Lattices

Stationary measures for an interactive exclusion process on $\mathbb{Z}$ are considered. The process is such that the jump rate of each particle to the empty neighboring site is $\alpha > 0$ (resp., $\beta > 0$) when another neighboring site is occupied (resp., unoccupied) by a particle, and that $\alpha \neq \beta$. According as $\alpha < \beta$ or $\alpha > \beta$ the process becomes nearest-neighbor attractive or repulsive, respectively. The method of relative entropy is used to determine the family $\mathscr{M}_{\beta/\alpha}$ of stationary measures. The member of $\mathscr{M}_\gamma$ is simply described as the probability measure having the regular clustering property which is a generalization of the exchangeable property of measures. It is shown that extremal points of $\mathscr{M}_\gamma$ are renewal measures. Thus the structure of stationary measures for the process is completely determined.