Charged particle isotropization in collisionless environments: turbulent magnetic fields as a conduit for random walks

Abstract

We develop a simple model for the kinematics of charged particles in regions of magnetic turbulence. We approximate the local magnetic field as smoothly varying in strength and direction, where adiabatic invariance prevails, or as presenting rapid changes in direction or ‘kinks’. Particles execute guiding centre gyromotion around a field line. However, in analogy to kinetic theory for collisional environments, when the particle undergoes a rapid change in direction by some angle $\unicode[STIX]{x1D703}$ , it would instantaneously transition to Larmor motion around the new field line. This mimics Brownian motion wherein we replace collisions with other particles by rapid transitions or ‘collisions’ with other field lines. Using standard methods drawn from Brownian motion, we follow the evolution of the parallel and perpendicular components of the velocity, namely $v_{\Vert }$ and $v_{\bot }$ , and rigorously show that kinetic energy isotropization necessarily emerges.