DMS Applied and Computational Mathematics

Speaker: Evdokiya (Eva) Kostadinova, Auburn University, Department of Physics

Title: Fractional Laplacian Spectral Approach (FLS) to Anomalous Diffusion of Energetic Particles in Magnetized Plasma

Abstract: Fractional-power operators have received much attention due to their wide application in modeling anomalous diffusion. For the fractional Laplacian \((−Δ)^{\it s}\), values \(𝑠∈(0,1)\) correspond to a superdiffusive process, while \(𝑠∈(1,2)\) defines the subdiffusion regime. Physically, subdiffusion can be thought of as superposition of classical diffusion and a small superdiffusive part. Such a process results in overall trapping of the particle ensemble, while also allowing for nonlocal jumps. It can also be shown that for \(𝑠∈(3/2,2)\), the transport process is bounded leading to probability distribution functions with truncated tails. We also argue that in the superdiffusive regime, for \(𝑠∈(2/3,1)\), particles exhibit nonlocal jumps, but transport is described by a true Lévy process only for values \(𝑠∈(0,2/3)\). Thus, we expect that there are at least four regimes of anomalous diffusion: strong trapping, trapping with nonlocal jumps, nonlocal jumps without trapping, and Lévy flights.

To investigate the proposed sub-regimes of anomalous diffusion, we use a Fractional Laplacian Spectral (FLS) method, where the existence of extended states (interpreted as probability for transport) is determined from the existence of a continuous part in the spectrum of a Hamiltonian. Here we are interested in a Hamiltonian with a fractional Laplacian term and a stochastic disorder term. The range of nonlocal interactions and the amount of stochasticity for the Hamiltonian are informed from experiments where energetic electrons (EEs) were detected in the presence of magnetic islands and stochastic magnetic fields. The spectral properties of each Hamiltonian are investigated for fractions representative of each diffusion regime. Comparison of these calculations to experimental data reveals the presence of at least two types of EEs: runaway electrons, best described by a Lévy process, and suprathermal, but non-relativistic, electrons, best described by trapping with nonlocal jumps.

This is joint work with B. Andrew and D. M. Orlov.

Work supported by DE-FC02-04ER54698, DE- FG02-05ER54809, and DE-SC0023061.