Hanumesha S T
Abstract:
Anomalous and non-diffusive transport in plasmas characterized by intermittent events, long-range correlations, and non-Gaussian statistics often violates the assumptions underlying classical Fickian diffusion. This has motivated a family of nonlocal partial differential equation (PDE) models that replace local gradient-driven fluxes with fractional operators, integral-kernel diffusion, or integro-differential collision dynamics. This paper develops a unified, mathematics-first view of nonlocal plasma transport modeling. We begin with physical mechanisms (turbulent avalanches, coherent structures, long-flight particle trajectories, nonlocal response in gyrokinetics, and collisional scattering) and show how they naturally lead to nonlocal closures in reduced transport equations. We then present core model classes fractional advection–diffusion, finite-horizon kernel diffusion (peridynamic-type), nonlocal gyrokinetic closures, and the Landau/Fokker–Planck collision operator and place each in an analytical framework based on fractional Sobolev spaces, coercivity/dissipation, and well-posedness via variational methods. On the computational side, we compare spectral/FFT discretizations (periodic fractional operators), quadrature-based integral methods (finite-horizon kernels), and structure-preserving solvers for kinetic collision terms. Implementation guidance is provided for stability, boundary treatment, scaling, and verification. Finally, we outline how uncertainty-aware parameterization can be incorporated using intuitionistic fuzzy sets and graph/hypergraph abstractions from the author’s prior work, offering a systematic pathway to robust nonlocal transport simulations without compromising the operator-theoretic foundation.
