Energetic Particle Superdiffusion in Solar System Plasmas: Which Fractional Transport Equation?

Superdiffusive transport of energetic particles in the solar system and in other plasma environments is often inferred; while this can be described in terms of Lévy walks, a corresponding transport differential equation still calls for investigation. Here, we propose that superdiffusive transport can be described by means of a transport equation for pitch-angle scattering where the time derivative is fractional rather than integer. We show that this simply leads to superdiffusion in the direction parallel to the magnetic field, and we discuss some advantages with respect to approaches based on transport equations with symmetric spatial fractional derivates.

Intracellular transport of electrotransferred DNA cargo is governed by coexisting ergodic and non ergodic anomalous diffusion

The ability of exogenous DNA cargo to overcome the active and viscoelastic eukaryotic cytoplasm is a principal determinant for the gene delivery efficacy. During DNA electrotransfer, DNA forms complexes with the membrane (DNA cargo) which is transported through the cytoplasm through a combination of passive diffusion and active transport. However, this process is poorly understood limiting rational optimization of DNA cargo to be delivered to different cell types. We have investigated the intracellular transport of DNA cargo (of sizes 100 bp, 250 bp and 500 bp) delivered by electrotransfer to non-cancerous and cancerous mammalian cells. We demonstrate that intracellular DNA cargo transport is governed by coexisting ergodic and non ergodic anomalous diffusion for all the tested DNA sizes and cell types. The apparent diffusion coefficient of the electrotransferred DNA cargo in the cytoplasm decreases when the DNA size is increased from 100 bp to 500 bp. Interestingly, the electrotransferred DNA cargo (500 bp) transport is strongly dependent on the cell’s cancer state. Intracellular electrotransferred DNA cargo transport has a higher probability of superdiffusive transport and lower probability of caging in metastatic cells compared to malignant cells followed by benign cells.

Inverting the variable fractional order in a variable-order space-fractional diffusion equation with variable diffusivity: analysis and simulation

Variable-order space-fractional diffusion equations provide very competitive modeling capabilities of challenging phenomena, including anomalously superdiffusive transport of solutes in heterogeneous porous media, long-range spatial interactions and other applications, as well as eliminating the nonphysical boundary layers of the solutions to their constant-order analogues. In this paper, we prove the uniqueness of determining the variable fractional order of the homogeneous Dirichlet boundary-value problem of the one-sided linear variable-order space-fractional diffusion equation with some observed values of the unknown solutions near the boundary of the spatial domain. We base on the analysis to develop a spectral-Galerkin Levenberg–Marquardt method and a finite difference Levenberg–Marquardt method to numerically invert the variable order. We carry out numerical experiments to investigate the numerical performance of these methods.

Superdiffusive transport of energy in one-dimensional metals

Metals in one spatial dimension are described at the lowest energy scales by the Luttinger liquid theory. It is well understood that this free theory, and even interacting integrable models, can support ballistic transport of conserved quantities including energy. In contrast, realistic one-dimensional metals, even without disorder, contain integrability-breaking interactions that are expected to lead to thermalization and conventional diffusive linear response. We argue that the expansion of energy when such a nonintegrable Luttinger liquid is locally heated above its ground state shows superdiffusive behavior (i.e., spreading of energy that is intermediate between diffusion and ballistic propagation), by combining an analytical anomalous diffusion model with numerical matrix-product–state calculations on a specific perturbed spinless fermion chain. Different metals will have different scaling exponents and shapes in their energy spreading, but the superdiffusive behavior is stable and should be visible in time-resolved experiments.

On the Fractional Diffusion-Advection Equation for Fluids and Plasmas

The problem of studying anomalous superdiffusive transport by means of fractional transport equations is considered. We concentrate on the case when an advection flow is present (since this corresponds to many actual plasma configurations), as well as on the case when a boundary is also present. We propose that the presence of a boundary can be taken into account by adopting the Caputo fractional derivatives for the side of the boundary (here, the left side), while the Riemann-Liouville derivative is used for the unbounded side (here, the right side). These derivatives are used to write the fractional diffusion–advection equation. We look for solutions in the steady-state case, as such solutions are of practical interest for comparison with observations both in laboratory and astrophysical plasmas. It is shown that the solutions in the completely asymmetric cases have the form of Mittag-Leffler functions in the case of the left fractional contribution, and the form of an exponential decay in the case of the right fractional contribution. Possible applications to space plasmas are discussed.