Nonextensive Statistics Approach to Anomalous Diffusion in Plasmas: Applications and Scaling to Other Models

Abstract

Plasmas, known for their multiscale physics phenomena and rich dynamics, often exhibit anomalous diffusion that deviate from classical models, challenging the assumptions of locality, linearity, and Gaussianity. This dissertation investigates anomalous diffusion in two distinct plasma regimes—microgravity dusty plasmas and magnetically confined fusion plasmas—through the unified lens of nonextensive statistics and fractional spectral models. We first establish formal connections between nonextensive statistical mechanics and fractional derivative operators, showing how $q$-Gaussian distribution functions relate to Lévy flights and how the nonextensive index \( q \) maps to the fractional Laplacian exponent \( s \). These scaling relations are used to connect a spectral transport model based on an Anderson-type Hamiltonian with a fractional Laplacian to the nonextensive statistical behavior often observed in complex systems, such as plasmas.

The nonextensive framework is first applied to video data from the PK-4 dusty plasma experiment aboard the International Space Station. The PK-4 experiment uses video cameras to track individual dust particles suspended in low temperature plasma, which allows the collection of large amounts of statistical information on the dust particle positions and velocities. These statistics are used to study anomalous dust diffusion caused by anisotropies in the plasma-mediated dust-dust interactions in PK-4. Using $q$-Gaussian fits to histograms of particle displacements and velocities, we define an anisotropic kinetic temperature and identify inverse correlations between the nonextensive parameter \( q \) and local diffusivity, thus quantifying deviations from thermal equilibrium. A spatial disorder metric and analysis of particle jumps are further used to identify microscopic processes contributing to the observed anisotropic anomalous diffusion of the dust particles.

To further understand how the interplay between nonlocality and stochasticity leads to different regimes of anomalous diffusion, we introduce a Fractional Laplacian Spectral Method (FLSM) that calculates probability for diffusive transport at different scales in Hilbert space from the spectrum of the discrete random fractional Schr{\"o}dinger operator. We perform a large-scale parameter sweep across 55,000 realizations of the operator that represent different combinations of nonlocality, stochastic disorder and Hilbert space vector scales, including all combinations of parameters extracted from PK-4 data using scaling relations. The spectral simulations reveal "islands of enhanced transport" in Hilbert space—regions where transport is amplified due to constructive interplay between nonlocality and stochasticity. We compare the predictions from the spectral model to the dust dynamics observed in the PK-4 experiments.

Finally, nonextensive statistics the framework is applied to simulations of magnetic island topology in the NSTX-U tokamak. Of specific interest are cases where the island structure undergoes successive bifurcations under the action of coil perturbations. The reconstruction of magnetic field line diffusion in NSTX-U is used to understand how changes in magnetic topology will alter electron diffusion in magnetized plasmas. By treating the normalized poloidal magnetic field flux $\Psi_N$ as a statistical distribution, we extract non-Gaussian signatures via $q$-Gaussian fits to histograms of magnetic field line displacements and apply a spatial KD-tree disorder metric to quantify field-line divergence. Both metrics increase monotonically with applied perturbation coil current, tracing the emergence of bifurcations, stochasticity, and topological complexity in the magnetic geometry. We find that increasing the perturbation leads to a crossover from subdiffusion, to classical diffusion, followed by superdiffusion, and eventually, L{\'e}vy flights. These measures provide potential tools for diagnosing magnetic field instability and precursor activity for plasma instabilities and disruptions in fusion devices. Together, the investigations presented here offer a unified statistical-spectral approach to modeling anomalous diffusion in plasmas.