{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $d \\ge 1$, let $m \u003e 0$ be a mass, let $\\gamma \u003e 0$ be a friction coefficient, and let $\\beta \u003e 0$ be an inverse temperature.\nLet $U \\colon \\R^d \\to \\R$ be a continuously differentiable potential, and let $B$ be a $d$-dimensional [[Brownian motion|22735FGJV6]].\nThe *Langevin equation* is the system of stochastic differential equations for a position $X_t$ and a velocity $V_t$ in $\\R^d$:\n$$dX_t = V_t \\, dt, \\qquad m \\, dV_t = -\\nabla U(X_t) \\, dt - \\gamma V_t \\, dt + \\sqrt{2\\gamma / \\beta} \\, dB_t.$$\n\n## Remarks\n\nIn physics notation, $\\beta = 1 / (k_B T)$ for the temperature $T$ and the Boltzmann constant $k_B$.\nThe equation is often written $m \\ddot{X} = -\\nabla U(X) - \\gamma \\dot{X} + \\xi$, where $\\xi$ is white noise.\nExample: for $U = 0$ and $d = 1$, the velocity solves $dV = -(\\gamma/m) V \\, dt + (\\sqrt{2\\gamma/\\beta}/m) \\, dB$.\nThis is an [[Ornstein–Uhlenbeck process|HZM2ASCDY4]], and its stationary law is [[normal|05AQE0QR0D]] with mean $0$ and variance $1/(\\beta m)$.\nNon-example: the equation $m \\, dV = -\\gamma V \\, dt$ lacks the random force, so its solutions decay to $0$ and never reach thermal equilibrium.\nThe overdamped Langevin equation $dX_t = -\\nabla U(X_t) \\, dt + \\sqrt{2/\\beta} \\, dB_t$ is the limit of large friction after a change of time scale.\nWhen $e^{-\\beta U}$ is integrable, the overdamped equation has a stationary law with density proportional to $e^{-\\beta U(x)}$.\nUnder the same condition, the full equation has a stationary law with density proportional to $e^{-\\beta (U(x) + m \\abs{v}^2 / 2)}$ in $(x, v)$.\n",
  "commit": "efe85cee8dbc57d85bedbacfb97b99d1b7bee843",
  "front": {
    "defines": "ZW2RBS9VTR",
    "kind": "definition",
    "msc": [
      "60H10",
      "82C31"
    ],
    "name": "Langevin equation",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Langevin_equation"
      }
    ],
    "tag": "TAQWRWTHX8",
    "uses": [
      "22735FGJV6",
      "HZM2ASCDY4",
      "05AQE0QR0D"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/TAQWRWTHX8/",
  "links": [
    {
      "field": "defines",
      "target": "ZW2RBS9VTR"
    },
    {
      "field": "uses",
      "target": "22735FGJV6"
    },
    {
      "field": "uses",
      "target": "HZM2ASCDY4"
    },
    {
      "field": "uses",
      "target": "05AQE0QR0D"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/TAQWRWTHX8.md",
  "msc": [
    "60H10",
    "82C31"
  ],
  "name": "Langevin equation",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/TAQ/TAQWRWTHX8.md",
  "status": null,
  "tag": "TAQWRWTHX8",
  "uri": "m2://TAQWRWTHX8",
  "url": "https://math2.tamnd.com/wiki/langevin-equation-definition/",
  "writing": {
    "tag": "TAQWRWTHX8",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}