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

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 ${\kappa }_{zz}^{DV}={lim}_{t\to t_{\infty }}d{\sigma }^2/\left(2dt\right)$, the Fick's law definition ${\kappa }_{zz}^{FL}=J/X$ with $X=\partial F/\partial z$, and the Taylor–Green–Kubo (TGK) formula definition ${\kappa }_{zz}^{TGK}={\int }_0^{\infty }dt⟨v_z\left(t\right)v_z\left(0\right)⟩$. 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 ${\kappa }_{zz}^{TGK}$ is only an approximate formula, and the Fick's law definition ${\kappa }_{zz}^{FL}$ is not invariant to some DIOs. However, at least for the special condition, in this paper we show that the definition ${\kappa }_{zz}^{DV}$ is an invariant quantity to the DIOs. Therefore, for a spatially varying field, the displacement variance definition ${\kappa }_{zz}^{DV}$, rather than the Fick's law definition ${\kappa }_{zz}^{FL}$ and TGK formula definition ${\kappa }_{zz}^{TGK}$, is the most appropriate definition of the SPDCs.