{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Problem\n\nLet $N \\ge 2$ and $\\alpha \\in \\R$.\nConsider $N$ particles on a line with displacements $q_1, \\ldots, q_N$ and momenta $p_1, \\ldots, p_N$, with fixed ends $q_0 = q_{N+1} = 0$, and the [[Hamiltonian system|0EDKGA2FAK]] with Hamiltonian\n$$H = \\sum_{k=1}^{N} \\frac{p_k^2}{2} + \\sum_{k=0}^{N} \\left( \\frac{1}{2} (q_{k+1} - q_k)^2 + \\frac{\\alpha}{3} (q_{k+1} - q_k)^3 \\right).$$\nFor $\\alpha = 0$ the chain is a set of $N$ independent normal modes, each with a conserved energy.\nStart with all energy in the lowest normal mode and $\\alpha \\ne 0$ small.\nExplain the long-time behaviour of the energies of the normal modes.\nIn particular, explain why the energy does not spread evenly over all modes.\nInstead it stays in a few low modes and returns almost completely to the initial mode after some time.\n\n## Remarks\n\nFermi, Pasta, Ulam and Tsingou expected the nonlinear term to share the energy evenly between the modes, as statistical mechanics predicts for an ergodic system.\nTheir computer simulation on the MANIAC computer at Los Alamos showed the opposite: near recurrences to the initial state.\nIn 1965 Zabusky and Kruskal studied a continuum limit of the chain, which is the [[Korteweg–De Vries equation|KTT75QC72Z]], and discovered [[solitons|63D4AXYCT4]] in it.\nOther explanations use the closeness of the chain to an [[integrable system|N0N7CMRCMH]], namely the [[Toda lattice|ADBEEPPKRR]], and KAM-type and Nekhoroshev-type stability.\nIt is now understood that for small energies the approach to equipartition, if any, happens only on very long time scales.\nA full quantitative description is still a subject of research.\n\n## Explanations\n\n### L1 human\n\nTake a chain of masses joined by springs that are slightly nonlinear, and pluck it in its simplest way of vibrating.\nPhysicists expected the energy to spread to all the ways of vibrating, like heat spreading out.\nOne of the first computer experiments showed that it does not: the energy moves to a few nearby ways of vibrating and then comes back almost entirely.\nExplaining this surprise led to the discovery of solitons and to a large part of modern nonlinear science.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "N0N7CMRCMH"
    ],
    "history": {
      "first_stated": "1955",
      "posed_by": "Enrico Fermi, John Pasta, Stanislaw Ulam and Mary Tsingou"
    },
    "kind": "problem",
    "msc": [
      "37K60",
      "37N20",
      "70K75"
    ],
    "name": "Fermi–Pasta–Ulam–Tsingou problem",
    "same_as": {
      "wikidata": "Q462254",
      "wikipedia": "Fermi–Pasta–Ulam–Tsingou problem"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Fermi–Pasta–Ulam–Tsingou_problem"
      }
    ],
    "tag": "EBXRM4PFRC",
    "uses": [
      "0EDKGA2FAK",
      "N0N7CMRCMH"
    ]
  },
  "kind": "problem",
  "link": "https://tamnd.github.io/math2/node/EBXRM4PFRC/",
  "links": [
    {
      "field": "about",
      "target": "N0N7CMRCMH"
    },
    {
      "field": "uses",
      "target": "0EDKGA2FAK"
    },
    {
      "field": "uses",
      "target": "N0N7CMRCMH"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/EBXRM4PFRC.md",
  "msc": [
    "37K60",
    "37N20",
    "70K75"
  ],
  "name": "Fermi–Pasta–Ulam–Tsingou problem",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/EBX/EBXRM4PFRC.md",
  "status": null,
  "tag": "EBXRM4PFRC",
  "uri": "m2://EBXRM4PFRC",
  "url": "https://tamnd.github.io/math2/wiki/fermi-pasta-ulam-tsingou-problem/",
  "writing": {
    "tag": "EBXRM4PFRC",
    "kind": "problem",
    "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
      }
    ]
  }
}