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, i.e., the Displacement Variance definition $\kappa_{zz}^{DV}=\lim_{t\rightarrow t_{\infty}}d\sigma^2/(2dt)$, the Fick's Law definition $\kappa_{zz}^{FL}=J/X$ with $X=\partial{F}/\partial{z}$, and the TGK formula definition $\kappa_{zz}^{TGK}=\int_0^{\infty}dt \langle v_z(t)v_z(0) \rangle$. For constant mean magnetic field, the three different definitions of the SPDC give the same result. However, for 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 perturbation theory, can give the 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 an invariance for different EIDFs, therewith it is an invariant quantity for the different Derivation Methods of EIDF (DMEs). For the focusing field we suggest that the TGK definition $\kappa_{zz}^{TGK}$ is only the 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 the invariant quantity to the kinds of the DIOs. Therefore, for 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.

中文翻译：

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带电粒子输运方程迭代运算的扩散系数不变性

空间平行扩散系数 (SPDC) 是描述高能带电粒子传输的重要量之一。SPDC 有三种不同的定义，即位移方差定义 $\kappa_{zz}^{DV}=\lim_{t\rightarrow t_{\infty}}d\sigma^2/(2dt)$，菲克定律定义 $\kappa_{zz}^{FL}=J/X$ 与 $X=\partial{F}/\partial{z}$，以及 TGK 公式定义 $\kappa_{zz}^{TGK} =\int_0^{\infty}dt \langle v_z(t)v_z(0) \rangle$. 对于恒定平均磁场，SPDC 的三种不同定义给出了相同的结果。然而，对于聚焦场，证明了不同定义的结果是不一样的。在本文中，从 Fokker-Planck 方程我们发现不同的方法，例如一般的傅里叶展开和微扰理论，可以给出各向同性分布函数 (EIDF) 的不同方程。但事实证明，一个 EIDF 可以通过一些衍生迭代操作 (DIO) 转换为另一个。如果 SPDC 的一个定义对于 DIO 是不变的，那么很明显，该定义对于不同的 EIDF 也是一个不变的，因此对于 EIDF (DME) 的不同推导方法，它是一个不变的量。对于聚焦场，我们建议 TGK 定义 $\kappa_{zz}^{TGK}$ 只是近似公式，而 Fick 定律定义 $\kappa_{zz}^{FL}$ 对某些 DIO 不是不变的。然而，至少对于特殊条件，本文我们证明定义$\kappa_{zz}^{DV}$ 是DIO 种类的不变量。所以，