Interacting particle system

Definition

Let $S$ be a countable set of sites, such as $\Z^d$, and let $W$ be a finite set of local states. A configuration is an element $\eta \in X = W^S$, and $X$ carries the product topology, in which it is compact. An interacting particle system on $X$ is a Feller process $(\eta_t)_{t \ge 0}$ on $X$ whose transitions change finitely many coordinates at a time. The rates of these transitions depend on the configuration. The two main classes, with $W = \set{0, 1}$, are given by their generators on functions $f$ that depend on finitely many coordinates:

1. a spin system with flip rates $c(x, \eta) \ge 0$ has $$Lf(\eta) = \sum_{x \in S} c(x, \eta) \big( f(\eta^x) - f(\eta) \big),$$ where $\eta^x$ is $\eta$ with the coordinate at $x$ changed;
2. an exclusion process with jump rates $p(x, y) \ge 0$ has $$Lf(\eta) = \sum_{x, y \in S} p(x, y)\, \eta(x) \big(1 - \eta(y)\big) \big( f(\eta^{x,y}) - f(\eta) \big),$$ where $\eta^{x,y}$ is $\eta$ with the coordinates at $x$ and $y$ exchanged.