Slow dissipation and spreading in disordered classical systems: A direct comparison between numerics and mathematical bounds

TL;DR

This paper compares numerical simulations and mathematical bounds to understand wave spreading in disordered classical systems, revealing that current numerics may not capture the true long-time behavior due to slow dissipation.

Contribution

It provides a mathematical theorem on decorrelation time bounds and demonstrates numerical evidence of power-law behavior in a disordered classical system.

Findings

Mathematical theorem shows decorrelation time exceeds any inverse power law in anharmonicity.

Numerical simulations indicate decorrelation time follows a power law across various parameters.

Numerics may not fully capture the slow long-time spreading behavior in disordered classical systems.

Abstract

We study the breakdown of Anderson localization in the one-dimensional nonlinear Klein-Gordon chain, a prototypical example of a disordered classical many-body system. A series of numerical works indicate that an initially localized wave packet spreads polynomially in time, while analytical studies rather suggest a much slower spreading. Here, we focus on the decorrelation time in equilibrium. On the one hand, we provide a mathematical theorem establishing that this time is larger than any inverse power law in the effective anharmonicity parameter $\lambda$, and on the other hand our numerics show that it follows a power law for a broad range of values of $\lambda$. This numerical behavior is fully consistent with the power law observed numerically in spreading experiments, and we conclude that the state-of-the-art numerics may well be unable to capture the long-time behavior of such classical disordered systems.