Relationship of transport coefficients with statistical quantities of charged particles

TL;DR

This paper derives formulas for transport coefficients in charged particle transport equations with various spatial derivatives, linking them to statistical quantities like skewness and kurtosis, aiding the study of particle transport processes.

Contribution

It introduces new transport coefficient formulas based on statistical quantities and explores their relation to higher-order spatial derivatives in transport equations.

Findings

Transport coefficients are determined by statistical quantities.

Skewness and kurtosis relate to specific transport coefficients.

Formulas for coefficients up to fifth-order derivatives are provided.

Abstract

In the previous studies, from the Fokker-Planck equation the general spatial transport equation, which contains an infinite number of spatial derivative terms $T_n=\kappa_{nz}\partial^n{F}/ \partial{z^n}$ with $n=1, 2, 3, \cdots$, was derived. Due to the complexity of the general equation, some simplified equations with finite spatial derivative terms have been used in astrophysical researches, e.g., the diffusion equation, the hyperdiffusion one, subdiffusion transport one, etc. In this paper, the simplified equations with the highest order spatial derivative terms up to the first-, second-, third-, fourth-, and fifth-order are listed, and their transport coefficient formulas are derived, respectively. We find that most of the transport coefficients are determined by the corresponding statistical quantities. In addition, we find that the well-known statistical quantities, skewness $\mathcal{S}$ and kurtosis $\mathcal{K}$, are determined by some transport coefficients. The results can help one to use different transport coefficients determined by the statistical quantities, including many that are relatively new found in this paper, to study charged particle parallel transport processes.