1、1第一章第一章第一章 等离子体等离子体等离子体 动动动 力学方程力学方程力学方程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
2、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
3、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2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)(tX)(tV经典力学中，可以引入六维相空间
4、（ ）来描述粒子的微观状态，每一个粒子在x,vt 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i度为(1.2.1)(,)()()i iiNxvtXtvVt其中 是 Dirac delta 函数。()i iiixXtyYzZ对于等离子体中第 s 类粒子有 个，则在相空间中其密度为0(1.2.2)1(,)()()Ns iiixvtxXtvVt（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，,0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sN(x,vt) sN
5、(x,vt)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sN(x,vt)规律服从正则运动方程:(1.2.3)iiX(t)V(1.2.4)mmsisiiitqE,(t)BX,t)3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m,qmE,B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)mmmmE(x,t)(,t)/B,t,t/()(J)E(x,t)/000其中 是真空磁导率和真空介电常
6、数， 分别是微观电荷密度及微观电,0 ,J流密度：(1.2.6)mssss(x,t)qdvN(x,t)JIn order to obtain an exact equation for the evolution of a plasma one can take the time derivative of the density , sNfrom (1.2.2)(1.2.7a)00sNixiii1ivii(,vt)X(t)vV(t)Vttwhere we have used the relationsfg(t)ff(ab)f(ab), and where . Using (1.2.4) we
7、 xyzxxyzvv,), can write in terms of , whereupon (1.2.7a) iX,V miVEB4becomes(1.2.7b)00sNixii1mmsiiiviii(,vt)VX(t)vV(t)qE,tB,txX(t)vV(t) Using an important property of Dirac delta functiona(b)(a)This relation allows us to replace , on i itwh v,and (t)hxthe right of (1.2.7b)(1.2.7c)Nsxiiimms viiiN(x,vt
8、)X(t)V(t)q E(,t)B(,t)xtvt 0011But the two summations on the right of (1.2.7c) are just the density (1.2.2); therefore(1.2.8)sxsvsmmsN(,vt)aNqaE(,tB(,t)0This 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N0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
9、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。51.3 Plasma kinetic equation The Klimontovich equation tells us that the density of a particle at a given point ( ) in the phase space is infinite x,vor vanish whether or not a particle can be found there. What we really want to kn
10、ow is how many particles are likely to be found in a small volume of phase space whose xvcenter is at ( ). Thus, we really are not interested in the x,vspiky function , but rather in the smooth functionsNt)(1.3.1) sf(x,vt)where denotes ensemble average. （系综平均） The interpretation of the distribution
11、function is sf(x,vt)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 sf(x,vt)equation (1.2.8) by ensemble averaging. We define fluctuation function bysN,EB(1
12、.3.2)s sm(xvt)f(,t)(x,vt),t,t,t6where . Inserting these msB,E andNEB0definitions into (1.2.8) and ensemble averaging, we obtain(1.3.3)sxss vss sf(,vt)fqE,tB(,t)fmNor (1.3.4)xsvss(/ta)f(,tI(x,vt)(1.3.5)sI,t(1.3.6)sqaE(,t)B(,t)m(1.3.7)svEquation (1.3.4) is the exact form of the plasma kinetic equation
13、. 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 Maxwells equations (1.2.5) and (1.2.6) ensemble averaging, we obtain(1.3.8)msss(x,t)(,t)qdvf(x,t)JJ7(1.3.9)E(x,t)(,t)/B,t,t/()J()E(x,t)/000(1.3.10)ssmssEx,t,t/(),tB(,t)/JE(x,t)/(x,t)qdvNJ(,t)00the fluctuation function satisfy follow
14、ing equations(1.3.11)xvssvss(/ta)(x,t)afNa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。f(x,vt)8附 A Liouville equationIntroductionKlimontovich equation describes the behavior of individual particles. By contrast,
15、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 N0(A1)Niiiii(x,v,.t)xX(t)vV(t)0121Liouville equati
16、onTaking the time derivative of (A1) and using the relation(A2)iii ixixX(t)/t(t)The time derivative of (A1) is(A3)iiNNixiiiiiviiii/tV(t)X(t)vV(t)ttt00001111 0Using a(b)(a)(A4)i iNNxivimmsii i ii/tvV(t)qV()E(,tBx,tc00110which is the Liouville equation. When combined with Maxwells equations and the Lo
17、rentz force equation, the liouville equation is an exact description of a plasma.9附 B: BBGKY Hierarchy We define(B1)NNNf(x,v,.x,vt)dxv,.d0 0 012 12to be the probability that a particular system is at the point (in the 6N0-dimensional phase space, that is Nx,v,.,)012the probability of particle 1 havi
18、ng coordinates between and particle 2 having coordinates (,)and(x,vd)111between , and etc. an(x,vd)222We may also define reduced probability distributions(B2)00k12kk12Nf(,.,t)Vdxvd.xvfwhich give the joint probability of particles 1 through k having the coordinates and and (,)to(,d)111, irrespective(
19、不考虑) of the coordinates of kkk(x,v)to(dx,v)particles k+1, k+2,N0. The factor on the right of (B2) is kVa normalization factor, where V is finite spatial volume in which is nonzero for all . At the end of our Nf0 Nx,.012theoretical development, we will take the limit , in N,V0such a way that (1) is a
20、 constant giving the average number of n/V0particles per unit real space. We assume that (2) as or for any i. Nf0ixiv(3) Since we do not care which one of the particles is 0N10called particle 1, etc. Thus, we always choose probability densities that are completely symmetric with respect 0Nfto the pa
21、rticle labels, that is NijNjif(.x.)f(.x.)0 0(4) is the number of particles per unit real space per f(x,vt)1unit velocity space, it has the same meaning as the function introduced in the previous section.sf(x,t)(5) To keep the theory as simple as possible, we shall ignore any external electric and ma
22、gnetic fields, and we shall deal with only one species of particlesN0(6) We adapt the Coulomb model, which ignore magnetic fields produced by the charged particle motion. In this model, the acceleration(B3)NiijV(t)a01where(B4)sij ijijqa(x)m23is the acceleration of particle due to the Coulomb electri
23、c field of particle . jia0The liouville equation i iNNixivNfv(t)fV(t)ft0 00 011becomes 11(B5)i iNNixijvNifv(t)faft0 00 011Integrating the Eq.(B5) over all , we will get k,.x0the equation for the reduced distribution , for example, to kfobtain the equation for we integrate (B5) over all , 01Nf Nxv0ob
24、taing 000 0i000i0NNixN1ijvi1(1)2(3)fdxvdx(t)ftafThe term (1) is easy, since we can move the time derivative outside the integral to obtain(B6)00000N1N1f()dxvttVfIn term (2), in the first terms in the sum, the integration variables are independent of the operator ,this operator can then be moved outs
25、ide the iixvintegration and we again obtain a term proportional to . Nf01The last term in the sum, with iN0Nxdxvf000since vanishes at the boundaries of the system that Nf012have been placed at , thus,Nx0(B7)iixN()Vv(t)f00112Term (3) is not much harder. Splitting the double sumNNijijNjiNi jggg0000001
26、111we get(B8)0i000N000i0N1ijv1ij1jvj1Niv(3)VafdxafWhere the term has discarded because .In ij0 Na0(B8), the second term on the right vanishes after direct integration with respect to and evaluation at . The 0Nvd v0remaining terms in (1),(2) and(3), after multiplication by , areNV01(B9)00i0i0000i0N1x
27、1i1ijvijN1Nivi(a)b(c)dfftaVdxafThis is the desired equation for . Notice that it does not f0113depend only on ; the last term depends on . We have Nf01 Nf0made no approximation in deriving (B9); Within Coulomb model, it is exact. Let us proceed to derive the equation for . To do this, we integrate (
28、B9) over all Nf02 Nx,v001 The term (a), yields as in (B6) tVf02000000tN1N1t Nt2fdxvdxvfVf The term (b),as in (B7), yields one term that vanishes upon integration, leaving a sum from 1 to 02 In term (c) we do as in (B8); we split the double ( ) N01sum into a double ( ) sum plus two single ( ) N02 2su
29、ms. The term vanishes since . The ij1N,a01term (c) becomes(B10)0i0000N10000i0N2ijv2i1jN1jvj21iv1(c)Vafdxfvathe second term on the right vanishes upon direct integration with respect to .N01 From (d) we have14(B11)iiNNiNvi ivi(d)Vdxdafx 000000000001121 1Where the term in the sum vanishes upon doing t
30、he 0integration. The variables and are Ndv01 N(,)0 N(,v)001simply dummy variables of integration on the far right of (B11). Therefore, we can switch the labels and , so 00that becomes . The density can stay the same, iNa0 iNa01 0Nfbecause it has the symmetry property (3). Equation (B11) becomes(B12)
31、00000000021111() iiNNivNiNividVdxadxfvfwhich 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)0000 000022211211i iiNNt xjvNi ijNiNvifvfafdfV This equation for is quite similar in structur
32、e to (B9) 02fand (B13), for the arbitrary k, satisfies:kf15(B14)110 1() 0i iikktxjvkiikkivkifvfafNdfVThis is the BBGKY hierarchy (Bogoliubov, Born, Green, Kirkwood and Yvon). Each equation for is coupled to the kfnext higher equation through the term. It consists of 1kfcoupled integro-differential e
33、quations. When we take the 0Nfirst 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)1 101 22()0tx vNfvfdxafV (1) is the probability that a given particle finds 1(,)fxditself in the region of phase sp
34、ace between ) and 1(,xv11(,)v(2) is proportional to the joint probability that 22fxtparticle 1 finds itself at and particle 2 finds itself 1(,)xvat . It turns out that has an intimate 2(,)v 212,)ftrelation of . 1f(3) For a plasma, we define the correlation function by12(,)gxvt16(B16)2121212(,)(,)fxv
35、tfgxvtThis is the first step in the Mayer cluster expansion ( 梅耶簇展开) Insert the form (B16) into the (B15)(B17)11021212(,)(,)(,)0txvffndatfxvtgxvtwhere . Assume that the correlation function /NVvanishes, that is no correlation(B18)1 1102121(,)(,)(,)0txvtfvfndafxtfvtt where is the ensemble averaged ac
36、celeration 1(,)aexperienced by particle 1 due to all other particles.(B19)0212(,)(,)axtndvafxt(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 n
37、ot 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 1fth
38、erefore return to the exact k = 1 equation (B18) and find 17some method to evaluate g. Since g is defined through (B17) as , we must go back to the k= 2 equation in the 21gfBBGKY hierarchy in order to obtain an equation for and, 2fhence, for g. Setting k = 2 in (B14) and using one has00(2)/NVn(B20)1
39、21212() (3)(1)2 2(4)0333()0txxvvvvffafndafMayer cluster expansion is(B21)21311(,)()(,23 ,),(3),2()ffgff fghwhere we have introduced a simplified notation: .(),ixvOur procedure is to insert (B21) into (B20) and neglect h(123). This means that we neglect three-particle correlations, or three-body coll
40、isions. 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 32,f(B20), we find for the numbered terms:18(B22)1 11()()()22()0311 ()() 2(4)32,aebxxvvcvffgafandffgf(),(),2dgwhere and means that a
41、ll of the preceding 3dxvterms on the right side are repeated with the symbols 1 and 2 interchanged. Recall that by the symmetry of . (12)g2fMany 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 te
42、rms to 12vanish, leaving(B23)121122103 ()(,2)(),3(1xxvvggafndtogether with (B24)1 1102()2()0txvfvfafgWe have two equations in the two unknowns . We have 1,fgtruncated the BBGKY hierarchy by ignoring three-particle correlations.In practice, (B23) and (B24) are impossibly difficult to solve, either an
43、alytically or numerically. They are two coupled 19nonlinear 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)11 11 02()txvvfvffndag which is in exactly the same form as the plasma kinetic equation (1.13).
