Asymmetric simple exclusion process

Example

Let $p \in [0, 1]$ and $q = 1 - p$. The asymmetric simple exclusion process (ASEP) on $\Z$ is the interacting particle system on $\set{0, 1}^{\Z}$ in which $\eta(x) = 1$ means that site $x$ holds a particle. Each particle tries to jump one step to the right at rate $p$ and one step to the left at rate $q$. A jump is suppressed if the target site is occupied. Its generator, on functions $f$ that depend on finitely many coordinates, is $$Lf(\eta) = \sum_{x \in \Z} \Big( p\, \eta(x) \big(1 - \eta(x+1)\big) + q\, \eta(x+1) \big(1 - \eta(x)\big) \Big) \big( f(\eta^{x,x+1}) - f(\eta) \big),$$ where $\eta^{x,x+1}$ exchanges the values at $x$ and $x + 1$. This is the exclusion process with jump rates $p(x, x+1) = p$ and $p(x+1, x) = q$, so it is an interacting particle system.