Distribution Functions of Energetic Particles Experiencing Compound Subdiffusion

Abstract

There are indications that the perpendicular transport of energetic particles is sometimes subdiffusive for intermediate timescales. This corresponds to a scenario where particles follow diffusive magnetic field lines while they also move diffusively in the parallel direction. This type of transport should occur at times after the ballistic regime but before the particles experience the transverse complexity of the turbulence. In this article we present a detailed analytical investigation of distribution functions of particles experiencing compound subdiffusion. Simple approximations of particle distributions are derived which can easily be used in applications. We also compare our findings with test-particle simulations performed for slab turbulence corresponding to the case of vanishing transverse turbulence structure.

Relationship of Transport Coefficients with Statistical Quantities of Charged Particles

“…4. The spatial 1D subdiffusion equation deduced by Shalchi & Arendt (2020) for large enough time t and z…”

“…As shown in Section 2.4, the fourth-order transport equation has eight different STGEs, which include the hyperdiffusion equation derived by Malkov & Sagdeev (2015) and the subdiffusion transport equation deduced by Shalchi & Arendt (2020). The relationships of transport coefficients with statistical quantities are listed in Table 2.…”

“…The relations of transport coefficients and statistical quantities can be obtained as For the perpendicular subdiffusion process x t 2 á ñ ~, through lengthy derivation Shalchi & Arendt (2020) found its governing equation. In fact, the deduction is also applicable to the parallel transport.…”

“…In addition, in order to explore particle transport in turbulent plasmas, perturbation theory was employed by Malkov & Sagdeev (2015) to obtain the hyperdiffusion equation from the Fokker-Planck description. Meanwhile, Shalchi & Arendt (2020) obtained a transport equation with the fourth-order spatial derivative term for the subdiffusion process. By integrating the Fokker-Planck equation over pitch angle, Wang & Qin (2019) derived the general spatial transport equation, which contains an infinite number of spatial derivative terms T n = κ nz ∂ n F/∂z n with n = 1, 2, 3, L .…”

Generalized Karagiannidis–Lioumpas Approximations and Bounds to the Gaussian Q-Function With Optimized Coefficients

“…In particular, the GKL approximation/bound can be used to calculate the sampling bit error probability of binary phase shift keying [16], to approximate the phase noise probability density function in the system considered in [17], and to derive the coherent LoRa R symbol error rate under additive white Gaussian noise [18]. Beyond communications, it allows to approximate the distribution functions of particles experiencing compound subdiffusion [19] and to derive the predictive error of the probability of failure [20], for instance.…”