{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $A$ be a $k \\times k$ matrix with entries in $\\set{0, 1}$.\nLet $\\Sigma_A \\subseteq \\set{1, \\dots, k}^{\\N}$ be the one-sided [[subshift of finite type|624XDXX8FN]] of the sequences $x = (x_0, x_1, \\dots)$ with $A_{x_i x_{i+1}} = 1$ for all $i$.\nLet $\\sigma \\colon \\Sigma_A \\to \\Sigma_A$ be the shift, $(\\sigma x)_i = x_{i+1}$.\nLet $\\varphi \\colon \\Sigma_A \\to \\R$ be continuous, and write $S_n \\varphi(x) = \\sum_{j=0}^{n-1} \\varphi(\\sigma^j x)$ for its [[Birkhoff sum|8TD2KCVH5N]].\nFor $x \\in \\Sigma_A$, the cylinder $[x_0 \\dots x_{n-1}]$ is the set of $y \\in \\Sigma_A$ with $y_i = x_i$ for $0 \\le i \u003c n$.\nLet $\\mu$ be a $\\sigma$-[[invariant|B232EZ1FMB]] [[probability measure|059TJN0NZ1]] on $\\Sigma_A$.\nIt is a *Gibbs measure* for $\\varphi$ if there are $C \\ge 1$ and $P \\in \\R$ with\n$$\\frac{1}{C} \\le \\frac{\\mu([x_0 \\dots x_{n-1}])}{\\exp\\left(-nP + S_n \\varphi(x)\\right)} \\le C$$\nfor every $x \\in \\Sigma_A$ and every $n \\ge 1$.\n\n## Remarks\n\nExample: on the full shift $\\set{1, \\dots, k}^{\\N}$ take $\\varphi = 0$, and let $\\mu$ be the uniform [[Bernoulli measure|D33JK3AXRZ]], under which the coordinates are independent and uniform.\nThen $\\mu([x_0 \\dots x_{n-1}]) = k^{-n}$ and $S_n \\varphi = 0$, so the ratio is $1$ with $P = \\log k$, and $C = 1$ works.\n\nNon-example: for the same shift and $\\varphi = 0$, the Dirac measure at the constant sequence $(1, 1, 1, \\dots)$ is invariant.\nIt gives measure $0$ to the cylinder $[2]$, so the lower bound fails.\n\nThe constant $P$ is then the topological pressure of $\\varphi$.\nFor a topologically mixing $\\Sigma_A$ and a [[Hölder continuous|012PBA0FMD]] $\\varphi$, a Gibbs measure exists, is unique, and is the unique equilibrium state of $\\varphi$ (Bowen).\n\nIn statistical mechanics, consider a finite system with states $s$ and energies $E(s)$.\nThe Gibbs measure at inverse temperature $\\beta$ gives $s$ the probability $e^{-\\beta E(s)} / Z$, where $Z = \\sum_s e^{-\\beta E(s)}$ is the [[partition function|0CV69WDRXW]].\nFor infinite lattice systems, Gibbs measures are defined instead by the Dobrushin–Lanford–Ruelle condition on conditional probabilities, and there may be several of them.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "DG9F3W6MZM",
    "kind": "definition",
    "msc": [
      "37D35",
      "82B05",
      "37B10"
    ],
    "name": "Gibbs measure",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Gibbs_measure"
      }
    ],
    "tag": "MC4S4WCGJS",
    "uses": [
      "624XDXX8FN",
      "A1XE59ZHNM",
      "012PBA0FMD",
      "B232EZ1FMB",
      "059TJN0NZ1",
      "8TD2KCVH5N",
      "D33JK3AXRZ",
      "0CV69WDRXW"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/MC4S4WCGJS/",
  "links": [
    {
      "field": "defines",
      "target": "DG9F3W6MZM"
    },
    {
      "field": "uses",
      "target": "624XDXX8FN"
    },
    {
      "field": "uses",
      "target": "A1XE59ZHNM"
    },
    {
      "field": "uses",
      "target": "012PBA0FMD"
    },
    {
      "field": "uses",
      "target": "B232EZ1FMB"
    },
    {
      "field": "uses",
      "target": "059TJN0NZ1"
    },
    {
      "field": "uses",
      "target": "8TD2KCVH5N"
    },
    {
      "field": "uses",
      "target": "D33JK3AXRZ"
    },
    {
      "field": "uses",
      "target": "0CV69WDRXW"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/MC4S4WCGJS.md",
  "msc": [
    "37D35",
    "82B05",
    "37B10"
  ],
  "name": "Gibbs measure",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/MC4/MC4S4WCGJS.md",
  "status": null,
  "tag": "MC4S4WCGJS",
  "uri": "m2://MC4S4WCGJS",
  "url": "https://tamnd.github.io/math2/wiki/gibbs-measure-definition/",
  "writing": {
    "tag": "MC4S4WCGJS",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}