Arnold diffusion of charged particles in ABC magnetic fields

TL;DR

This paper demonstrates the existence of Arnold diffusion for charged particles in ABC magnetic fields, combining geometric methods and computer-assisted proofs to analyze global instability in a Hamiltonian system relevant to plasma physics.

Contribution

It provides the first rigorous proof of Arnold diffusion in a physically relevant magnetic field configuration using geometric and computational techniques.

Findings

Existence of diffusing solutions in ABC magnetic fields

Explicit conditions for transition chains of invariant tori

Computer-assisted verification of instability conditions

Abstract

We prove the existence of diffusing solutions in the motion of a charged particle in the presence of an ABC magnetic field. The equations of motion are modeled by a 3DOF Hamiltonian system depending on two parameters. For small values of these parameters, we obtain a normally hyperbolic invariant manifold and we apply the so-called geometric methods for a priori unstable systems developed by A. Delshams, R. de la Llave, and T.M. Seara. We characterize explicitly sufficient conditions for the existence of a transition chain of invariant tori having heteroclinic connections, thus obtaining global instability (Arnold diffusion). We also check the obtained conditions in a computer assisted proof. ABC magnetic fields are the simplest force-free type solutions of the magnetohydrodynamics equations with periodic boundary conditions, so our results are of potential interest in the study of the motion of plasma charged particles in a tokamak.