Random initial data and average shock time in the Fermi-Pasta-Ulam-Tsingou chain

TL;DR

The paper studies the Fermi-Pasta-Ulam-Tsingou chain with long-wavelength random initial data to understand prethermalization and Burgers-type shock formation in the low-energy regime. By reducing the dynamics to a pair of near-Burgers equations and applying advanced probabilistic methods from Dudley and Talagrand, the authors derive sharp asymptotics for the average shock time in the thermodynamic limit. They show that for a large number of excited modes $p$, the average inverse shock time scales as $\langle t_s^{-1}\rangle \sim p\sqrt{\log p}$, implying $t_s \sim (p\sqrt{\log p})^{-1}$ up to constants, with a logarithmic correction. The results demonstrate the robustness of the Burgers turbulence scenario to random initial phases and provide a rigorous statistical framework that clarifies the differences between coherent and incoherent initial data and informs the interpretation of tygers and prethermalization in nonlinear lattices.

Abstract

We investigate the dynamics of the Fermi-Pasta-Ulam-Tsingou chain with long-wavelength random initial data. When the energy per particle is small, thermal equilibrium is not reached on a fast timescale and the system enters prethermalization. The formation of the prethermal state is characterized by the development of a Burgers-type shock and the onset of a turbulent-like spectrum with a time dependent exponent $ζ(t)$ in the inertial range. We perform a significant step forward by demonstrating that these features are robust under generic long-wavelength random initial conditions. By employing advanced probabilistic techniques inspired by the works of Dudley and Talagrand, we derive a sharp asymptotic expression for the average shock time in the thermodynamic limit. For large $p$, this time scales as $(p \sqrt{\log p})^{-1}$, where $p$ is the number of excited modes proving that it is an intensive quantity up to a logarithmic correction in the size of the system.

Figures (6)

• Figure 1: Fourier Energy Spectrum (full orange line) at the shock time with $\epsilon=10^{-3}$ for $\alpha=1$, $\beta=0.1$ and $N=16\,384$. $20$ modes are initially excited. The black dashed line is the theoretical prediction $\log E_k =-8/3 \log k + 0.86$.
• Figure 2: Numerical estimate of the inverse shock time as a function of the number of excited modes $p$ with all phases equal to zero (points), eq. \ref{['eq:InitialCondi']}, \ref{['eq:InitialCondi2']}. The solid blue line is the interpolation curve $4.774 \times 10^{-5} p^{\frac{3}{2}}$. $\alpha=1$, $\beta=0.1$, $\epsilon=10^{-3}$, $N=2048$.
• Figure 3: Average of the inverse shock time $\langle t_s^{-1} \rangle$ vs. the number of initially excited modes $p$ (points) eq. \ref{['eq:InitialCondi']}, \ref{['eq:InitialCondi2']}. Each point is the average over $1\,000$ realizations of the phases. The solid orange line is the theoretical interpolation curve $p \sqrt{\log p}$. $\alpha=1$, $\beta=0.1$, $\epsilon=10^{-3}$, $N=16\,384$.
• Figure 4: Space profile of the solution of the FPUT system at the shock time and after the shock time. $N=4096$, initial solution with only one mode initially excited with $R=0$ at $t=0$.
• Figure 5: Fourier Energy spectrum of the FPUT chain with random initial data in a short interval around the shock time. In the first panel, the blue line represents the FES at $\sim 97\%$ shock time; in the second panel it is represented the FES at the shock time and in the third panel the FES at $\sim 103\%$ shock time. In the last panel we superimpose the three FES.