Diffusion as a Signature of Chaos

TL;DR

This paper introduces the observable drift as a new measure to identify chaos in classical systems, demonstrating its effectiveness and equivalence to traditional chaos definitions through numerical examples.

Contribution

It extends the sensitivity-based chaos formalism to non-Hamiltonian classical systems using the observable drift, unifying different chaos notions.

Findings

Observable drift correctly identifies chaotic behavior in classical maps

The measure aligns with measure-theoretic chaos via weak mixing

Numerical examples validate the proposed chaos characterization

Abstract

While classical chaos is defined via a system's sensitive dependence on its initial conditions (SDIC), this notion does not directly extend to quantum systems. Instead, recent works have established defining both quantum and classical chaos via the sensitivity to adiabatic deformations and measuring this sensitivity using the adiabatic gauge potential (AGP). Building on this formalism, we introduce the ``observable drift" as a probe of chaos in generic, non-Hamiltonian, classical systems. We show that this probe correctly characterizes classical systems that exhibit SDIC as chaotic. Moreover, this characterization is consistent with the measure-theoretic definition of chaos via weak mixing. Thus, we show that these two notions of sensitivity (to changes in initial conditions and to adiabatic deformations) can be probed using the same quantity, and therefore, are equivalent definitions of chaos. Numerical examples are provided via the tent map, the logistic map, and the Chirikov standard map.