第1章 等离子体动力学方程.doc

第一章 等离子体动力学方程
§1.1 引言
在单粒子理论中，认为等离子体是由一些无相互作用的带电粒子组成的，而且带电粒子仅在外电磁场作用下发生运动。但是，我们知道：等离子体与通常的中性气体的最大差别在于带电粒子的运动能够产生电磁场，反过来这种电磁场又要影响带电粒子的运动，这种电磁场称为自恰电磁场。因此，带电粒子的运动不仅受外电磁场的作用，而且还要受自洽场的影响。由于这种原因，用单粒子理论来描述等离子体的行为有很大的局限性，有必要用一种能够反映出带电粒子相互作用的理论来描述等离子体的状态，这就是等离子体动力学理论。
基本上有两种方法来描述等离子体动力学过程。一种是 BBGKY（Bogoliubov，Born，Green，Kirkwood 及Yvon）的方程链方法。我们已经在《非平衡态统计力学》课程中对该方法进行了较详细地介绍，它是从系统的正则运动方程出发，通过引入系综分布函数及约化分布函数，可以得到一系列关于约化分布函数的方程链，即BBGKY方程链。该方程链是不封闭的，为了得到动力学方程，必须对该方程链进行截断。另一种方法是由前苏联科学家Klimontovich引入的矩方法。在该方法中，同样可以得到一系列关于各阶矩函数的不封闭的方程链。用这种方法描述一些较复杂的等离子体系统，例如有外电磁场存在，是非常有用的。该方法自60年代被提出后，一直在不断的发展。本章将利用后一种方法描述等离子体的动力学过程。可以说，等离子体动力学是把等离子体的微观状态描述引入宏观状态描述的一个桥梁。等离子体的微观状态可用正则运动方程来描述。如果系统有N个粒子组成，则有6N个运动方程。如此多的方程是难以进行求解的，而且包含的微观信息太多。但是我们知道等离子体的宏观状态只需要为数不多的状态参量来描述，如温度、密度、流速及电磁场等。如何把等离子体的微观状态描述向宏观状态描述过渡，这正是等离子体动力学的任务。
§1.2 Klilmontovich 方程
一个带电粒子在t 时刻的微观状态可以用其位矢及速度来描述。在经典力学中，可以引入六维相空间（）来描述粒子的微观状态，每一个粒子在t 时刻的状态对应于六维相空间的一点。对于第个带电粒子，它在相空间的密度为
(1.2.1)
其中 是 Dirac delta 函数。
对于等离子体中第s类粒子有个，则在相空间中其密度为
(1.2.2)
（1.2.2）式表示在相点（）处观察个粒子的运动，若它们均不“占据”该点，则对应的相密度为零；若其中某一个粒子位于该点，则相密度为无穷大。因此，函数具有奇异性，它充分地体现出“点粒子”的性质。函数是由Klimontovich引入的，有时称它为精确分布函数。因为它与用粒子的坐标及速度描述粒子的状态是等价的，没有做任何统计近似。
下面我们建立相函数所满足的方程。第s类粒子中第i个粒子的运动规律服从正则运动方程:
(1.2.3)
(1.2.4)
其中是s类粒子的电荷及质量。分别是总电场和总磁场，既包括外电磁场和带电粒子运动产生的自恰电磁场。电磁场随时空的演化遵从Maxwell方程组：
(1.2.5)
其中是真空磁导率和真空介电常数，分别是微观电荷密度及微观电流密度：
(1.2.6)
In order to obtain an exact equation for the evolution of a plasma one can take the time derivative of the density , from (1.2.2)
(1.2.7a)
where we have used the relations

and where . Using (1.2.4) we can write in terms of , whereupon (1.2.7a) becomes
(1.2.7b)
Using an important property of Dirac delta function

This relation allows us to replace , on the right of (1.2.7b)
(1.2.7c)
But the two summations on the right of (1.2.7c) are just the density (1.2.2); therefore
(1.2.8)
This is the exact Klimontovich equation. 实际上它与运动方程（1.2.3）及（1.2.4）等价，只不过将6个运动方程压缩到一个方程，这样，方程（1.2.8）与Maxwell方程组联立构成了一套封闭的方程组。Klimontovich方程是一个严格的方程，没有做任何统计近似，它与正则运动方程等价。由于该方程包含了太多的微观信息，为了得到有用的动力学方程，必须对它进行统计平均。

§1.3 Plasma kinetic equation
The Klimontovich equation tells us that the density of a particle at a given point () in the phase space is infinite or vanish whether or not a particle can be found there. What we really want to know is how many particles are likely to be found in a small volume of phase space whose center is at (). Thus, we really are not interested in the spiky function , but rather in the smooth function
(1.3.1)
where denotes ensemble average. （系综平均）
The interpretation of the distribution function is the number of particles of species s per unit configuration space per unit velocity space.
An equation for the time evolution of the distribution function can be obtained from the Klimontovich equation (1.2.8) by ensemble averaging. We define fluctuation function by
(1.3.2)
where . Inserting these definitions into (1.2.8) and ensemble averaging, we obtain

(1.3.3)
or
(1.3.4)
(1.3.5)
(1.3.6)
(1.3.7)
Equation (1.3.4) is the exact form of the plasma kinetic equation. 可见，方程（1.3.4）的左边是与光滑分布函数，与平均电磁场有关，而右边则是与涨落矩的统计平均有关。
Likewise taking Maxwell’s equations (1.2.5) and (1.2.6) ensemble averaging, we obtain
(1.3.8)
(1.3.9)
(1.3.10)
the fluctuation function satisfy following equation
(1.3.11)
可见：一阶涨落函数所满足的方程与二阶矩函数有关，同样，可以证明二阶涨落矩所满足的方程与三阶涨落矩有关，…。这样方程（1.3.3）与各阶矩方程组成了一个不封闭的方程链，类似于BBGKY方程链。为了得到一个光滑分布函数所满足的封闭方程，必须对上述方程链进行截断。

附A Liouville equation
Introduction
Klimontovich equation describes the behavior of individual particles. By contrast, the Liouville equation describes the behavior of systems. Klimontovich equation is in the six-dimensional phase space, while Liouville equation is in the 6N0-dimensional phase space. Suppose that we have a system of particles. The density of systems
(A1)
Liouville equation
Taking the time derivative of (A1) and using the relation
(A2)
The time derivative of (A1) is
(A3)
Using
(A4)
which is the Liouville equation. When combined with Maxwell’s equations and the Lorentz force equation, the liouville equation is an exact description of a plasma.

附B: BBGKY Hierarchy
We define
(B1)
to be the probability that a particular system is at the point ( in the 6N0-dimensional phase space, that is the probability of particle 1 having coordinates between and particle 2 having coordinates between , and etc.
We may also define reduced probability distributions
(B2)
which give the joint probability of particles 1 through k having the coordinates and …and , irrespective(不考虑) of the coordinates of particles k+1, k+2,…,N0. The factor on the right of (B2) is a normalization factor, where V is finite spatial volume in which is nonzero for all . At the end of our theoretical development, we will take the limit , in such a way that
(1) is a constant giving the average number of particles per unit real space. We assume that
(2) as or for any i.
(3) Since we do not care which one of the particles is called particle 1, etc. Thus, we always choose probability densities that are completely symmetric with respect to the particle labels, that is

(4) is the number of particles per unit real space per unit velocity space, it has the same meaning as the function introduced in the previous section.
(5) To keep the theory as simple as possible, we shall ignore any external electric and magnetic fields, and we shall deal with only one species of particles
(6) We adapt the Coulomb model, which ignore magnetic fields produced by the charged particle motion. In this model, the acceleration
(B3)
where
(B4)
is the acceleration of particle due to the Coulomb electric field of particle .
The liouville equation

becomes
(B5)
Integrating the Eq.(B5) over all , we will get the equation for the reduced distribution , for example, to obtain the equation for we integrate (B5) over all , obtaing

The term (1) is easy, since we can move the time derivative outside the integral to obtain
(B6)
In term (2), in the first terms in the sum, the integration variables are independent of the operator ,this operator can then be moved outside the integration and we again obtain a term proportional to . The last term in the sum, with

since vanishes at the boundaries of the system that have been placed at , thus,
(B7)
Term (3) is not much harder. Splitting the double sum

we get
(B8)
Where the term has discarded because .In (B8), the second term on the right vanishes after direct integration with respect to and evaluation at . The remaining terms in (1),(2) and(3), after multiplication by , are
(B9)
This is the desired equation for . Notice that it does not depend only on ; the last term depends on . We have made no approximation in deriving (B9); Within Coulomb model, it is exact. Let us proceed to derive the equation for . To do this, we integrate (B9) over all
l The term (a), yields [as in (B6)]

l The term (b),[as in (B7)], yields one term that vanishes upon integration, leaving a sum from 1 to
l In term (c) we do as in (B8); we split the double () sum into a double ( ) sum plus two single () sums. The term vanishes since . The term (c) becomes
(B10)
the second term on the right vanishes upon direct integration with respect to .
l From (d) we have
(B11)
Where the term in the sum vanishes upon doing the integration. The variables and are simply dummy variables of integration on the far right of (B11). Therefore, we can switch the labels and , so that becomes . The density can stay the same, because it has the symmetry property (3). Equation (B11) becomes
(B12)
which is identical with the last term on the right of term (c ) in (B10). Collecting all of the remaining terms in (a),(b),(c),and (d) and dividing by V, we obtain
(B13)
This equation for is quite similar in structure to (B9) and (B13), for the arbitrary k, satisfies:
(B14)
This is the BBGKY hierarchy (Bogoliubov, Born, Green, Kirkwood and Yvon). Each equation for is coupled to the next higher equation through the term. It consists of coupled integro-differential equations. When we take the first few equations, for k=1,2…, then use an approximation to close the set and cutoff the dependence on higher equations.
From (B14) the k=1 equation is
(B15)
(1) is the probability that a given particle finds itself in the region of phase space between ) and
(2) is proportional to the joint probability that particle 1 finds itself at and particle 2 finds itself at . It turns out that has an intimate relation of .
(3) For a plasma, we define the correlation function by
(B16)
This is the first step in the Mayer cluster expansion ( 梅耶簇展开) Insert the form (B16) into the (B15)
(B17)
where . Assume that the correlation function vanishes, that is no correlation
(B18)
where is the ensemble averaged acceleration experienced by particle 1 due to all other particles.
(B19)
(B18) is the Vlasov equation.
The Vlasov equation is probably the most useful equation in plasma physics, and a large portion of this book is devoted to its study. For our present purposes, however, it is not enough. It does not include the collisional effects that are represented by the two-particle correlation function g. We would like to have at least an approximate equation that does include collisional effects and that, therefore, predicts the temporal evolution of due to collisions. We must therefore return to the exact k = 1 equation (B18) and find some method to evaluate g. Since g is defined through (B17) as , we must go back to the k= 2 equation in the BBGKY hierarchy in order to obtain an equation for and, hence, for g. Setting k = 2 in (B14) and using one has
(B20)
Mayer cluster expansion is
(B21)
where we have introduced a simplified notation: .
Our procedure is to insert (B21) into (B20) and neglect h(123). This means that we neglect three-particle correlations, or three-body collisions. Thus, we have truncated the BBGKY hierarchy while retaining the effects of collisions to a good approximation.
Inserting (B21) for into the k = 2 BBGKY equation (B20), we find for the numbered terms:
(B22)
where and means that all of the preceding terms on the right side are repeated with the symbols 1 and 2 interchanged. Recall that by the symmetry of . Many of the terms in (B22) can be eliminated using the k=1 BBGKY equation (218). For example, (a)+(b)+(c)+(d)=0. Term (e) likewise combines with three of the terms to vanish, leaving
(B23)
together with
(B24)
We have two equations in the two unknowns . We have truncated the BBGKY hierarchy by ignoring three-particle correlations.
In practice, (B23) and (B24) are impossibly difficult to solve, either analytically or numerically. They are two coupled nonlinear integro-differential equations in a twelve-dimensional phase space. The present thrust of plasma kinetic theory consists in finding certain approximations to g(12) that are then inserted in (B24). Using the definition of the acceleration a in (B19), we rewrite (B24) as
(B25)
which is in exactly the same form as the plasma kinetic equation (1.13).
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