Thermalization of Classical Weakly Nonintegrable Many-Body Systems

TL;DR

This paper investigates the relationship between thermalization and chaos in weakly nonintegrable many-body systems, introducing a novel Lyapunov spectrum-based framework to diagnose dynamics and analyze timescales near integrability.

Contribution

It develops a new method using Lyapunov spectra to characterize weak nonintegrability and applies it to large systems close to integrable limits, advancing understanding of thermalization timescales.

Findings

Thermalization timescales grow rapidly near integrable limits.

Lyapunov spectrum scaling reveals the structure of observable networks.

Unitary maps enable large-scale, error-free numerical simulations.

Abstract

We devote our studies to the subject of weakly nonintegrable dynamics of systems with a macroscopic number of degrees of freedom. Our main points of interest are the relations between the timescales of thermalization and the timescales of chaotization; the choice of appropriate observables and the structure of equations coupling them; identifying the classes of weakly nonintegrable dynamics and developing tools to diagnose the properties specific to such classes. We discuss the traditional in the field methods for thermalization timescale computation and employ them to study the scaling the timescale with the proximity to the integrable limit. We then elaborate on a novel framework based on the full Lyapunov spectra computation for large systems as a powerful tool for the characterization of weak nonintegrability. In particular, the Lyapunov spectrum scaling offers a quantitative description allowing us to infer the structure of the underlying network of observables. Proximity to integrable limit is associated with the rapid growth of thermalization timescales and, thus, potential numerical challenges. We solve these challenges by performing numerical tests using computationally efficient model - unitary maps. The great advantage of unitary maps for numerical applications is time-discrete error-free evolution. We use these advantages to perform large timescale and system size computations in extreme proximity to the integrable limit. To demonstrate the scope of obtained results we report on the application of the developed framework to Hamiltonian systems.