带电粒子输运方程迭代运算中扩散系数的不变性
The invariance of the diffusion coefficient with the iterative operations of charged particles' transport equation
摘要：
空间平行扩散系数（SPDC）是描述高能带电粒子输运的重要物理量之一。对国家和平与发展委员会有三种不同的定义，即，位移方差定义$κ_{zz}^{DV}=\lim_{t\rightarrow t_{\infty}}dσ^2/(2dt)$、具有$X=\partial{F}/\partial{z}$的菲克定律定义$κ_{zz}^{FL}=J/X$和TGK公式定义$κ_{zz}^{TGK}=\int_0^{\infty}dt \langle v_z(t)v_z(0) \rangle$。对于恒定平均磁场，三种不同的SPDC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相同的。本文从福克-普朗克方程出发，发现不同的方法，例如，利用广义傅立叶展开和微扰理论，可以给出各向同性分布函数的不同表达式。但它表明，一个EIDF可以转化为另一个由一些衍生迭代运算（DIO）。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也是不变的，因此它对于EIDF的不同推导方法（DME）是不变量。对于聚焦场，我们认为TGK定义$κ_{zz}^{TGK}$只是近似公式，Fick定律定义$κ_{zz}^{FL}$对某些DIO不是不变的。然而，至少在特殊情况下，本文证明了定义$κ_{zz}^{DV}$是DIO类的不变量。因此，对于空间变化场，位移方差定义$κ_{zz}^{DV}$，而不是Fick定律定义$κ_{zz}^{FL}$和TGK公式定义$κ_{zz}^{TGK}$，是最合适的SPDCs定义。
英文摘要：
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 $κ_{zz}^{DV}=\lim_{t\rightarrow t_{\infty}}dσ^2/(2dt)$, the Fick's Law definition $κ_{zz}^{FL}=J/X$ with $X=\partial{F}/\partial{z}$, and the TGK formula definition $κ_{zz}^{TGK}=\int_0^{\infty}dt \langle v_z(t)v_z(0) \rangle$. 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 $κ_{zz}^{TGK}$ is only the approximate formula, and the Fick's Law definition $κ_{zz}^{FL}$ is not invariant to some DIOs. However, at least for the special condition, in this paper we show that the definition $κ_{zz}^{DV}$ is the invariant quantity to the kinds of the DIOs. Therefore, for spatially varying field the displacement variance definition $κ_{zz}^{DV}$, rather than the Fick's law definition $κ_{zz}^{FL}$ and TGK formula definition $κ_{zz}^{TGK}$, is the most appropriate definition of the SPDCs.
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