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  "body": "\n## Example\n\nLet $p \\in [0, 1]$ and $q = 1 - p$.\nThe *asymmetric simple exclusion process* (ASEP) on $\\Z$ is the [[interacting particle system|GR49AM3PSZ]] on $\\set{0, 1}^{\\Z}$ in which $\\eta(x) = 1$ means that site $x$ holds a particle.\nEach particle tries to jump one step to the right at rate $p$ and one step to the left at rate $q$.\nA jump is suppressed if the target site is occupied.\nIts generator, on functions $f$ that depend on finitely many coordinates, is\n$$Lf(\\eta) = \\sum_{x \\in \\Z} \\Big( p\\, \\eta(x) \\big(1 - \\eta(x+1)\\big) + q\\, \\eta(x+1) \\big(1 - \\eta(x)\\big) \\Big) \\big( f(\\eta^{x,x+1}) - f(\\eta) \\big),$$\nwhere $\\eta^{x,x+1}$ exchanges the values at $x$ and $x + 1$.\nThis 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.\n\n## Remarks\n\nParticles are never created or destroyed, and two particles never share a site.\nFor each $\\rho \\in [0, 1]$, the product measure under which the $\\eta(x)$ are independent Bernoulli variables with mean $\\rho$ is invariant.\nThe case $p = 1$ is the totally asymmetric simple exclusion process (TASEP), and the case $p = q = 1/2$ is the symmetric exclusion process.\nIn the asymmetric case the current of particles in the Bernoulli state of density $\\rho$ is $(p - q)\\rho(1 - \\rho)$.\n\n## Explanations\n\n### L1 human\n\nPicture cars on a one-lane road divided into cells, at most one car per cell.\nEach car tries to move forward at random times, but it cannot move into a cell that is already taken.\nThis simple rule produces traffic jams and shock waves, and the ASEP is a basic model of such traffic.\n",
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