Spectral Approach to Anomalous Diffusion in Plasma

Anomalous diffusion is a microscopic process that leads to non-linear mean squared displacements and non-Maxwellian velocity distributions caused by non-local interactions, correlations, memory effects, and/or stochasticity in complex systems. The presence of energetic or trapped particles in plasmas is a signature of anomalous diffusion. Modeling these particles is challenging as it requires working with nonlinear differential equations and with distribution functions that may not converge when integrated.



This tutorial explores the use of a spectral approach to analyze anomalous diffusion in different plasma systems, including dusty plasma, magnetized (fusion and low temperature) plasma, and space plasma. In this method, processes driving anomalous diffusion are represented by a Hamiltonian operator with a fractional Laplacian kinetic term and a random or correlated disorder potential term. The spectrum of possible energy states for the system operator is studied in an infinitely-dimensional Hilbert space. Specifically, the identification of a continuous spectrum is interpreted as transport, energy mixing, and onset of turbulence in the system at hand. The main advantage of the spectral method is the ability to map a non-linear problem in phase space to a linear problem in Hilbert space, thus utilizing linear algebra and avoiding the need to assume boundary conditions.



In this tutorial, we will introduce the main ideas of spectral theory and fractional calculus. Then, we will discuss how to adapt the spectral approach to several processes leading to anomalous diffusion in plasmas. Those include anisotropic interaction and turbulence in low temperature and dusty plasma, magnetic island bifurcations in fusion plasmas, and Fermi acceleration in space plasma. Finally, we discuss how such reduced-models in Hilbert space (or the space of all possible measurements) are appropriate for use with system identification ML techniques.