Lyapunov spectrum scaling for classical many-body dynamics close to integrability

TL;DR

This paper introduces a new framework using Lyapunov spectrum scaling to analyze how classical many-body systems thermalize near integrability, revealing distinct behaviors for long-range and short-range perturbations.

Contribution

It presents a novel approach to characterize thermalization near integrability using Lyapunov spectrum scaling, including analytical and non-analytic behaviors for different perturbation types.

Findings

Long-range limits show a single-parameter scaling with an analytical spectrum.

Short-range limits exhibit a slowdown in thermalization with a non-analytic spectrum.

An additional diverging length scale suppresses Lyapunov exponents exponentially.

Abstract

We propose a novel framework to characterize the thermalization of many-body dynamical systems close to integrable limits using the scaling properties of the full Lyapunov spectrum. We use a classical unitary map model to investigate macroscopic weakly nonintegrable dynamics beyond the limits set by the KAM regime. We perform our analysis in two fundamentally distinct long-range and short-range integrable limits which stem from the type of nonintegrable perturbations. Long-range limits result in a single parameter scaling of the Lyapunov spectrum, with the inverse largest Lyapunov exponent being the only diverging control parameter and the rescaled spectrum approaching an analytical function. Short-range limits result in a dramatic slowing down of thermalization which manifests through the rescaled Lyapunov spectrum approaching a non-analytic function. An additional diverging length scale controls the exponential suppression of all Lyapunov exponents relative to the largest one.