Burgers Turbulence in the Fermi-Pasta-Ulam-Tsingou Chain

We prove analytically and show numerically that the dynamics of the Fermi-Pasta-Ulam-Tsingou chain is characterized 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 timescale. In this range, we prove that the driving mechanism to thermalization is the formation of a shock that can be predicted using a pair of generalized 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 wave numbers $k$. 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 cutoff 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 characterizing the system and of the initial condition, once time is measured in units of the shock time.

Figure 1

Numerical simulation of the FPUT model Eq. (1) for $ϵ=0.07$, $N=4096$, $\alpha =1$, and $\beta =1/2$. The colored solid lines are the profiles corresponding to a left traveling wave excitation (TWE) plotted at the shock time $t_s$, Eq. (7), and at later times. Note the evolution toward a sawtooth profile (black solid line) followed by fast oscillations (discussed in the text).

Figure 2

Normalized FES of the FPUT model Eq. (1) for $\alpha =1$, different values of $\beta$, and $N=4096$ at the shock time $t_s$, Eq. (7). The initial condition is given by Eq. (4) with different values of $\theta =\phi -\pi /4$ and $ϵ$. The dashed line is the theoretical prediction given in Eq. (12) $E_k/E\simeq 0.8k^{-8/3}$. Note the exponential cutoff at large $k$. Inset: FES at $4t_s$ for the same initial conditions. The dashed line is the theoretical prediction $E_k\sim k^{-2}$.

Figure 3

$\mathrm{Slope}-\zeta (t)$ of the power law that interpolates the FES at small $k$ and for $N=4096$; see Eq. (2). One should remark that the data collapse follows from measuring the time in units of $t_s$, Eq. (7), which incorporates all the different values of the initial conditions and the parameters of the Hamiltonian.

Figure 4

FES versus $k/N$, left TWE at $4t_s$, $\alpha =1$, $\beta =1/2$, $ϵ=0.05$, and different values of $N$. Inset: the same at $t_s$.