Adiabatic Gauge Potential as a Tool for Detecting Chaos in Classical Systems

TL;DR

This paper introduces a classical analog of the quantum adiabatic gauge potential (AGP) to detect chaos, revealing how its variance over time distinguishes between different chaotic regimes and correlates with diffusion and thermalization.

Contribution

It extends the concept of the quantum AGP to classical systems, providing a novel tool to identify chaos and analyze the transition from integrability to chaos in classical models.

Findings

AGP variance distinguishes nearly integrable, weakly chaotic, and strongly chaotic regimes.

Strongly chaotic regimes exhibit normal diffusion, while weakly chaotic regimes show anomalous diffusion.

Hierarchy of timescales: Lyapunov time < chaos onset time < thermalization time.

Abstract

The interplay between chaos and thermalization in weakly non-integrable systems is a rich and complex subject. Interest in this area is further motivated by a desire to develop a unified picture of chaos for both quantum and classical systems. In this work, we study the adiabatic gauge potential (AGP), an object typically studied in quantum mechanics that describes deformations of a quantum state under adiabatic variation of the Hamiltonian, in classical Fermi-Pasta-Ulam-Tsingou (FPUT) and Toda models. We show how the time variance of the AGP over a trajectory probes the long-time correlations of a generic observable and can be used to distinguish among nearly integrable, weakly chaotic, and strongly chaotic regimes. We draw connections between the evolution of the AGP and diffusion and derive a fluctuation-dissipation relation that connects its variance to long-time correlations of the observable. Within this framework, we demonstrate that strongly and weakly chaotic regimes correspond to normal and anomalous diffusion, respectively. The latter gives rise to a marked increase in the variance as the time interval is increased, and this behavior serves as the basis for our probe of the onset times of chaos, which is interpreted as a ``mixing" time. Numerical results are presented for FPUT and Toda systems that highlight integrable, weakly chaotic, and strongly chaotic regimes. Further, a hierarchy of $t_{\text{Lyapunov}} < t_{\text{chaos}} < t_{\text{thermalization}}$ is found in these models. We conclude by commenting on the wide applicability of our method to a broader class of systems.