{
  "id": "2208.08818",
  "version": "v1",
  "published": "2022-08-18T13:20:32.000Z",
  "updated": "2022-08-18T13:20:32.000Z",
  "title": "Burgers turbulence in the Fermi-Pasta-Ulam-Tsingou chain",
  "authors": [
    "Matteo Gallone",
    "Matteo Marian",
    "Antonio Ponno",
    "Stefano Ruffo"
  ],
  "comment": "accepted for publication on Physical Review Letters; 6 pages + Supplemental Material. 4 Figures",
  "categories": [
    "cond-mat.stat-mech",
    "math-ph",
    "math.MP",
    "nlin.SI"
  ],
  "abstract": "We prove analytically and show numerically that the dynamics of the Fermi-Pasta-Ulam-Tsingou chain is characterised by a transient Burgers turbulence regime on a wide range of time and energy scales. This regime is present at long wavelengths and energy per particle small enough that equipartition is not reached on a fast time scale. In this range, we prove that the driving mechanism to thermalisation is the formation of a shock that can be predicted using a pair of generalised Burgers equations. We perform a perturbative calculation at small energy per particle, proving that the energy spectrum of the chain $E_k$ decays as a power law, $E_k\\sim k^{-\\zeta(t)}$, on an extensive range of wavenumbers $k$. We predict that $\\zeta(t)$ takes first the value $8/3$ at the Burgers shock time, and then reaches a value close to $2$ within two shock times. The value of the exponent $\\zeta=2$ persists for several shock times before the system eventually relaxes to equipartition. During this wide time-window, an exponential cut-off in the spectrum is observed at large $k$, in agreement with previous results. Such a scenario turns out to be universal, i.e. independent of the parameters characterising the system and of the initial condition, once time is measured in units of the shock time.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-08-18T13:20:32.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fermi-pasta-ulam-tsingou chain",
      "transient burgers turbulence regime",
      "burgers shock time",
      "fast time scale",
      "long wavelengths"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}