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

TL;DR

This paper studies the Fermi-Pasta-Ulam-Tsingou chain with random initial data, revealing robust prethermalization features and deriving an asymptotic expression for average shock time that scales with system size and excited modes.

Contribution

It demonstrates the robustness of prethermalization features under generic random initial conditions and provides a sharp asymptotic formula for average shock time in the thermodynamic limit.

Findings

Prethermalization occurs with long-wavelength random initial data.

Shock formation time scales as (p√log p)^{-1} for large p.

Features are robust under generic initial conditions.

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 $\zeta(t)$ 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.