A Stochastic Model of Inward Diffusion in Magnetospheric Plasmas

TL;DR

This paper develops a theoretical stochastic model explaining inward particle diffusion in magnetospheric plasmas, accounting for magnetic field inhomogeneity and invariants, and demonstrates its validity through numerical simulations showing realistic density profiles.

Contribution

The paper introduces a novel diffusion model based on proper phase space coordinates and invariants, explaining inward diffusion phenomena in magnetospheric plasmas.

Findings

The model reproduces peaked density profiles similar to radiation belts.

Numerical simulations confirm the coupling between perpendicular and parallel diffusion.

The approach aligns with the entropy principle despite inward particle flux.

Abstract

The inward diffusion of particles, often observed in magnetospheric plasmas (either naturally created stellar ones or laboratory devices) creates a spontaneous density gradient, which seemingly contradicts the entropy principle. We construct a theoretical model of diffusion that can explain the inward diffusion in a dipole magnetic field. The key is the identification of the proper coordinates on which an appropriate diffusion operator can be formulated. The effective phase space is foliated by the adiabatic invariants; on the symplectic leaf, the invariant measure (by which the entropy must be calculated) is distorted, by the inhomogeneous magnetic field, with respect to the conventional Lebesgue measure of the natural phase space. The collision operator is formulated to be consistent to the ergodic hypothesis on the symplectic leaf, i.e., the resultant diffusion must diminish gradients on the proper coordinates. The non-orthogonality of the cotangent vectors of the configuration space causes a coupling between the perpendicular and parallel diffusions, which is derived by applying Ito's formula of changing variables. The model has been examined by numerical simulations. We observe the creation of a peaked density profile that mimics radiation belts in planetary magnetospheres as well as laboratory experiments.