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For a Maxwellian,f falls faster than any power of v as v ! 1, and the integral therefore vanishes. Thesecond integral vanishes because v B is perpendicular to ∂/∂v. Finally, the fourthterm in Eq. (7.26) vanishes because collisions cannot change the total number ofparticles (recombination is not considered here). Equations (7.27)–(7.30) then yieldthe equation of continuity:∂nþ ∇ ðnuÞ ¼ 0∂tð7:31Þ7.3 Derivation of the Fluid Equations223The next moment of the Boltzmann equation is obtained by multiplying Eq.
(7.19)by mv and integrating over dv. We haveððð∂f∂fdv þ m vðv ∇ Þ f dv þ q vðE þ v BÞ dvm v∂t∂vð ∂f¼ mvdv∂t cð7:32ÞThe right-hand side is the change of momentum due to collisions and will give theterm Pij in Eq. (5.58). The first term in Eq. (7.32) givesð∂f∂∂v f dv m ðnuÞdv ¼ mm v∂t∂t∂tðð7:33ÞThe third integral in Eq. (7.32) can be writtenðð∂f∂dv ¼ ½ f vðE þ v BÞdv∂v∂vðð∂∂ ðE þ v BÞdv f ðE þ v BÞ v dv fv∂v∂vvð E þ v B Þ ð7:34ÞThe first two integrals on the right-hand side vanish for the same reasons as before,and ∂v/∂v is just the identity tensor I. We therefore haveð∂fq vðE þ v BÞ dv ¼ q ðE þ v BÞ f dv ¼ qnðE þ u BÞ∂vðð7:35ÞFinally, to evaluate the second integral in Eq.
(7.32), we first make use of the factthat v is an independent variable not related to ∇ and writeðððvðv ∇ Þ f dv ¼ ∇ ð f vvÞdv ¼ ∇ f vv dvð7:36ÞSince the average of a quantity is 1/n times its weighted integral over v, we haveð∇ f vv dv ¼ ∇ nvvð7:37ÞNow we may separate v into the average (fluid) velocity u and a thermal velocity w:v¼uþwð7:38ÞSince u is already an average, we have∇ ðnvvÞ ¼ ∇ ðnuuÞ þ ∇ ðnww Þ þ 2∇ ðnuwÞð7:39Þ2247 Kinetic TheoryThe average w is obviously zero. The quantity mnww is precisely what is meant bythe stress tensor P:P mnwwð7:40ÞThe remaining term in Eq. (7.39) can be written∇ ðnuuÞ ¼ u∇ ðnuÞ þ nðu ∇ Þuð7:41ÞCollecting our results from Eqs.
(7.33), (7.35), (7.40), and (7.41), we can writeEq. (7.32) asm∂ðnuÞ þ mu∇ ðnuÞ þ mnðu ∇ Þu þ ∇ P qnðE þ u BÞ ¼ Pi j∂tð7:42ÞThe first term in Eq. (7.42) represents ionization drag, since u is small when an ionis first created. Combining the first two terms in Eq. (7.42) with the help ofEq. (7.31), we finally obtain the fluid equation of motion:∂uþ ðu ∇ Þu ¼ qnðEþu BÞ ∇ PþPi jmn∂tð7:43ÞThis equation, an extension of Eq. (3.44), describes the flow of momentum. To treatthe flow of energy, we may take the next moment of Boltzmann equation bymultiplying by 12mvv and integrating. We would then obtain the heat flow equation,in which the coefficient of thermal conductivity κ would arise in the same manneras did the stress tensor P.
The equation of state p / ργ is a simple form of the heatflow equation for κ ¼ 0.7.4Plasma Oscillations and Landau DampingAs an elementary illustration of the use of the Vlasov equation, we shall derive thedispersion relation for electron plasma oscillations, which we treated from the fluidpoint of view in Sect. 4.3. This derivation will require a knowledge of contourintegration. Those not familiar with this may skip to Sect. 7.5. A simpler but longerderivation not using the theory of complex variables appears in Sect. 7.6.In zeroth order, we assume a uniform plasma with a distribution f0(v), and we letB0 ¼ E0 ¼ 0. In first order, we denote the perturbation in f(r, v, t) by f1(r, v, t):f ðr; v; tÞ ¼ f 0 ðvÞ þ f 1 ðr; v; tÞð7:44ÞSince v is now an independent variable and is not to be linearized, the first-orderVlasov equation for electrons is7.4 Plasma Oscillations and Landau Damping225∂f1e∂fþ v ∇ f 1 E1 0 ¼ 0m∂t∂vð7:45ÞAs before, we assume the ions are massive and fixed and that the waves are planewaves in the x directionf 1 / eiðkxωtÞð7:46Þe ∂fiω f 1 þ ikvx f 1 ¼ Ex 0m ∂vxð7:47ÞThen Eq.
(7.45) becomesf1 ¼ieEx ∂ f 0 =∂vxm ω kvxð7:48ÞPoisson’s equation givesðððε0 ∇ E1 ¼ ikε0 Ex ¼ en1 ¼ ef 1 d3 vð7:49ÞSubstituting for f1 and dividing by ikε0Ex, we have1¼e2kmε0ððð∂ f 0 =∂vx 3d vω kvxð7:50ÞA factor n0 can be factored out if we replace f0 by a normalized function ^f 0 :ω2p1¼kð11dvzð1dv y1ð11∂ ^f 0 vx ; v y ; vz =∂vx∂vxω kvxð7:51ÞIf f0 is a Maxwellian or some other factorable distribution, the integrations over vyand vz can be carried out easily. What remains is the one-dimensional distribution^f 0 ðvx Þ. For instance, a one-dimensional Maxwellian distribution is^f m ðvx Þ ¼ ðm=2πKT Þ1=2 exp mv2 =2KTxð7:52ÞThe dispersion relation is, therefore,1¼ω2pk2ð11∂ ^f 0 ðvx Þ=∂vxvx ðω=kÞð7:53Þ2267 Kinetic TheorySince we are dealing with a one-dimensional problem we may drop the subscript x,being careful not to confuse v (which is really vx) with the total velocity v usedearlier:1¼ω2pk2ð11∂ ^f 0 =∂vdvv ðω=kÞð7:54ÞHere, ^f 0 is understood to be a one-dimensional distribution function, the integrations over vy and vz having been made.
Equation (7.54) holds for any equilibriumdistribution ^f 0 ðvÞ; in particular, if ^f 0 is Maxwellian, Eq. (7.52) is to be used for it.The integral in Eq. (7.54) is not straightforward to evaluate because of thesingularity at v ¼ ω/k. One might think that the singularity would be of no concern,because in practice ω is almost never real; waves are usually slightly damped bycollisions or are amplified by some instability mechanism.
Since the velocity v is areal quantity, the denominator in Eq. (7.54) never vanishes. Landau was the first totreat this equation properly. He found that even though the singularity lies off thepath of integration, its presence introduces an important modification to the plasmawave dispersion relation—an effect not predicted by the fluid theory.Consider an initial value problem in which the plasma is given a sinusoidalperturbation, and therefore k is real.
If the perturbation grows or decays, ω will becomplex. The integral in Eq. (7.54) must be treated as a contour integral in thecomplex v plane. Possible contours are shown in Fig. 7.14 for (a) an unstable wave,with Im(ω) > 0, and (b) a damped wave, with Im(ω) < 0. Normally, one wouldevaluate the line integral along the real v axis by the residue theorem:ððG dv þG dv ¼ 2πiRðω=kÞð7:55ÞC1C2Fig.
7.14 Integration contours for the Landau problem for (a) Im(ω) > 0 and (b) Im(ω) < 07.4 Plasma Oscillations and Landau Damping227Fig. 7.15 Integrationcontour in the complex vplane for the case of smallIm(ω)where G is the integrand, C1 is the path along the real axis, C2 is the semicircle atinfinity, and R(ω/k) is the residue at ω/k. This works if the integral over C2 vanishes.Unfortunately, this does not happen for a Maxwellian distribution, which containsthe factorexp v2 =v2thThis factor becomes large for v ! i1, and the contribution from C2 cannot beneglected.
Landau showed that when the problem is properly treated as an initialvalue problem the correct contour to use is the curve C1 passing below thesingularity. This integral must in general be evaluated numerically, and Fried andConte have provided tables for the case when ^f 0 is a Maxwellian.Although an exact analysis of this problem is complicated, we can obtain anapproximate dispersion relation for the case of large phase velocity and weakdamping.
In this case, the pole at ω/k lies near the real v axis (Fig. 7.15). Thecontour prescribed by Landau is then a straight line along the Re(v) axis with asmall semicircle around the pole. In going around the pole, one obtains 2πi timeshalf the residue there. Then Eq. (7.54) becomes1¼ω2pk224Pð11∂ ^f 0 =∂v∂ ^f 0 dv þ iπv ðω=kÞ∂v v¼ω=k35ð7:56Þwhere P stands for the Cauchy principal value.
To evaluate this, we integrate alongthe real v axis but stop just before encountering the pole. If the phase velocityvϕ ¼ ω/k is sufficiently large, as we assume, there will not be much contributionfrom the neglected part of the contour, since both ^f 0 and ∂ ^f 0 =∂v are very smallthere (Fig. 7.16). The integral in Eq. (7.56) can be evaluated by integration by parts:ð11"#1ð1ð1^f 0^f 0 dv∂ ^f 0 dv ^f 0 dv¼¼22∂v v vϕv vϕ1 v vϕ1 v vϕ1ð7:57ÞSince this is just an average of (v vϕ)2 over the distribution, the real part of thedispersion relation can be written2287 Kinetic TheoryFig. 7.16 NormalizedMaxwellian distribution forthe case vϕ vth1¼ω2p k2v vϕ2ð7:58ÞSince vϕ v has been assumed, we can expand (v vϕ)2:v vϕ2¼ v2ϕv1vϕ2¼ v2ϕ2v 3v2 4v31 þ þ 2 þ 3 þ vϕ vϕvϕ!ð7:59ÞThe odd terms vanish upon taking the average, and we havev vϕ2v2ϕ3v21þ 2vϕ!ð7:60ÞWe now let ^f 0 be Maxwellian and evaluate v2 .
Remembering that v here is anabbreviation for vx, we can write1mv2x212¼ KT eð7:61Þthere being only one degree of freedom. The dispersion relation (7.58) thenbecomesω2p k2k2 KT e1¼ 2 2 1þ3 2ω mk ωω2 ¼ ω2p þω2p 3KT e 2kω2 mð7:62Þð7:63ÞIf the thermal correction is small, we may replace ω2 by ω2p in the second term.We then haveω2 ¼ ω2p þ3KT e 2kmð7:64Þwhich is the same as Eq. (4.30), obtained from the fluid equations with γ ¼ 3.7.4 Plasma Oscillations and Landau Damping229We now return to the imaginary term in Eq. (7.56). In evaluating this smallterm, it will be sufficiently accurate to neglect the thermal correction to the realpart of ω and let ω2 ω2p .
From Eqs. (7.57) and (7.60), we see that the principalvalue of the integral in Eq. (7.56) is approximately k2/ω2. Equation (7.56) nowbecomesω2pω2p ∂ ^f 0 ð7:65Þ1 ¼ 2 þ iπ 2ωk ∂v v¼vϕ01"#ω2p ∂ ^f 0A ¼ ω2ω2 @1 iπ 2ð7:66Þp∂vkv¼vϕTreating the imaginary term as small, we can bring it to the right-hand side and takethe square root by Taylor series expansion. We then obtain0"#ω2p ∂ ^f 0πω ¼ ω p @1 þ i2 k2 ∂vv¼vϕ1Að7:67ÞIf ^f 0 is a one-dimensional Maxwellian, we have 2 2∂ ^f 0 2 1=2 2vv2vv¼ πvthpffiffiffiexp¼exp∂vv2thv2thv2thπ v3thð7:68ÞWe may approximate vϕ by ωp/k in the coefficient, but in the exponent we mustkeep the thermal correction in Eq.
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