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  "body": "\n## Definition\n\nLet $S$ be a countable set of sites, such as $\\Z^d$, and let $W$ be a finite set of local states.\nA configuration is an element $\\eta \\in X = W^S$, and $X$ carries the product topology, in which it is compact.\nAn *interacting particle system* on $X$ is a [[Feller process|4HS3C12472]] $(\\eta_t)_{t \\ge 0}$ on $X$ whose transitions change finitely many coordinates at a time.\nThe rates of these transitions depend on the configuration.\nThe two main classes, with $W = \\set{0, 1}$, are given by their [[generators|VPA41PWRR1]] on functions $f$ that depend on finitely many coordinates:\n\n1. a *spin system* with flip rates $c(x, \\eta) \\ge 0$ has\n$$Lf(\\eta) = \\sum_{x \\in S} c(x, \\eta) \\big( f(\\eta^x) - f(\\eta) \\big),$$\nwhere $\\eta^x$ is $\\eta$ with the coordinate at $x$ changed;\n2. an *exclusion process* with jump rates $p(x, y) \\ge 0$ has\n$$Lf(\\eta) = \\sum_{x, y \\in S} p(x, y)\\, \\eta(x) \\big(1 - \\eta(y)\\big) \\big( f(\\eta^{x,y}) - f(\\eta) \\big),$$\nwhere $\\eta^{x,y}$ is $\\eta$ with the coordinates at $x$ and $y$ exchanged.\n\n## Remarks\n\nThe rates must be such that this formula defines the generator of a Feller process.\nFor a spin system it suffices that $\\sup_x c(x, \\cdot)$ is bounded, each $c(x, \\cdot)$ is continuous, and $\\sup_x \\sum_{u \\in S} \\sup_\\eta \\abs{c(x, \\eta^u) - c(x, \\eta)} \u003c \\infty$.\nExample: on $S = \\Z^d$, the rates $c(x, \\eta) = 1$ if $\\eta(x) = 1$ and $c(x, \\eta) = \\lambda \\#\\set{y : \\abs{y - x} = 1, \\eta(y) = 1}$ if $\\eta(x) = 0$ define the [[contact process|9RVKQJQ5JJ]]; they are bounded by $\\max(1, 2d\\lambda)$ and depend only on the $2d + 1$ nearest sites.\nNon-example: a probabilistic cellular automaton, in which all sites update together at each discrete time step, is not of this form, since each of its transitions changes infinitely many coordinates.\n",
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