The Diffusion Coefficient with Displacement Variance of Energetic Particles Caused by Adiabatic Focusing

Abstract

The equation κ zz = dσ 2 /(2dt) (hereafter DCDV) is a well-known formula of energetic particles describing the relation of parallel diffusion coefficient κ zz with the parallel displacement variance σ 2 . In this study, we find that DCDV is only applicable to two kinds of transport equations of isotropic distribution function, one is without cross terms, the other is without convection term. Here, by employing the more general transport equation, i.e., the variable coefficient differential equation derived from the Fokker-Planck equation, a new equation of κ zz as a function of σ 2 is obtained. We find that DCDV is the special case of the new equation. In addition, another equation of κ zz as a function of σ 2 corresponding to the telegraph equation is also investigated preliminarily.

Diffusion of cosmic-ray electrons in M 51 observed with LOFAR at 54 MHz

“…We note that there is some freedom of choosing the shape of the diffusion kernel. Numerical studies on CR transport fit diffusion coefficients using the second moment of their spatial distribution (Qin & Shalchi 2009;Wang & Qin 2019;Xu & Yan 2013;Snodin et al 2016;Seta et al 2018). Based on magnetohydrodynamical turbulence simulations, Sampson et al (2023) find that super-diffusion is ubiquitous in the ISM: super-diffusion would result in kernel shapes that have a narrower core and more extended wings than a Gaussian.…”

Relationship of Transport Coefficients with Statistical Quantities of Charged Particles

“…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 . Although the general spatial transport equation is a generalized form of the transport equation with finite spatial derivative terms and can describe more propagation processes, this equation is too complex to be used in relevant studies.…”

“…The topic of this paper is to explore the parallel transport coefficients of various STGEs that describe the parallel propagation of charged particles. For the parallel diffusion coefficient, generally speaking, there are three different definitions (Wang & Qin 2019), i.e., the displacement variance definition d dt lim 2…”

“…However, it has been demonstrated that for some scenarios different definitions of parallel diffusion are not equivalent. Wang & Qin (2019) has proved that zz DV k , rather than zz FL k and zz TGK k , is the most appropriate definition. In this paper, the transport coefficients expressed by the moments of distribution function are derived, and their relations with statistical quantities are explored.…”

“…In this paper, we list all the simplified equations belonging to the first-, second-, third-, fourth-, and fifth-order spatial transport equations. The transport coefficients expressed by the moments of distribution function can be obtained using the method proposed by Wang & Qin (2019). This paper is organized as follows.…”