The Invariance of the Diffusion Coefficient with Iterative Operations of the Charged Particle Transport Equation

Abstract

The spatial parallel diffusion coefficient (SPDC) is one of the important quantities describing energetic charged particle transport. There are three different definitions for the SPDC: the displacement variance definition , the Fick’s law definition with , and the Taylor–Green–Kubo (TGK) formula definition . For a constant mean magnetic field, the three different definitions of the SPDC give the same result. However, for a focusing field, it is demonstrated that the results of the different definitions are not the same. In this paper, from the Fokker–Planck equation, we find that different methods, e.g., the general Fourier expansion and iteration method, can give different equations of the isotropic distribution function (EIDFs). But it is shown that one EIDF can be transformed into another by some derivative iterative operations (DIOs). If one definition of the SPDC is invariant for the DIOs, it is clear that the definition is also invariant for different EIDFs; therefore, it is an invariant quantity for the different derivation methods of the EIDF. For the focusing field, we suggest that the TGK definition is only an approximate formula, and the Fick’s law definition is not invariant to some DIOs. However, at least for the special condition, in this paper we show that the definition is an invariant quantity to the DIOs. Therefore, for a spatially varying field, the displacement variance definition , rather than the Fick’s law definition and TGK formula definition , is the most appropriate definition of the SPDCs.

Transport of energetic particles in turbulent space plasmas: pitch-angle scattering, telegraph, and diffusion equations

“…Analytical and numerical investigations of pitch-angledependent transport equations have been the subject of several papers published during recent years. In addition to studies of the basic pitch-angle scattering equation (Shalchi et al, 2011;Tautz and Lerche, 2016;Lasuik and Shalchi, 2019), authors have explored the impact of so-called focusing, an effect which is related to a non-constant mean magnetic field (Danos et al, 2013;Litvinenko and Schlickeiser, 2013;Effenberger and Litvinenko, 2014;Wang and Qin, 2020;Wang and Qin, 2021;Wang and Qin, 2023). Even more complicated cases, including perpendicular particle transport, have been investigated by Wang and Qin (2024).…”