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.