Burgers turbulence in the Fermi-Pasta-Ulam-Tsingou chain

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.

accepted for publication on Physical Review Letters; 6 pages + Supplemental Material. 4 Figures; corrected a typo in the second equation of Supplemental Materials

References in corpus (3)

• Tail resonances of FPU q-breathers and their impact on the pathway to equipartition
• Korteweg-de Vries and Fermi-Pasta-Ulam-Tsingou: asymptotic integrability of quasi unidirectional waves
• (Non equilibrium) Thermodynamics of Integrable models: The Generalized Gibbs Ensemble description of the classical Neumann Model

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• Energy cascade and Burgers turbulence in the Fermi-Pasta-Ulam-Tsingou chain
• Hamiltonian field theory close to the wave equation: from Fermi-Pasta-Ulam to water waves
• One dimensional energy cascades in a fractional quasilinear NLS
• Random initial data and average shock time in the Fermi-Pasta-Ulam-Tsingou chain