Introduction to

Plasma Theory

Dwight R. Nicholson

University of Iowa

John Wiley & SonsNew York \342\200\242Chichester \342\200\242Brisbane \342\200\242Toronto \342\200\242Singapore

Preface

The purposeof this book is to teach the basic theoretical principles of plasmaphysics. It is not intended to be an encyclopedia of results and techniques. Nor is itintended to be used primarily as a reference book. It is intended to develop thebasic techniques of plasma physics from the beginning, namely, from Maxwell'sequations and Newton's law of motion. Absolutely no previous knowledge ofplasma physics is assumed. Although the book is primarily intended for a one yearcourse at the first or second year graduate level, it can also be used for a one ortwo semester course at the junior or seniorundergraduate level. Such anundergraduate course would make use of that half of the book which assumes aknowledgeonly of undergraduate electricity and magnetism.The other half of the book,suitable for the graduate level, requires familiarity with complex variables, Fouriertransformation, and the Dirac delta function.

The book is organized in a logical fashion. Although this is not the standardorganization of an introductory course in plasma physics, I have found thatstudents at the graduate level respond well to this organization. After theintroductory material of Chapters 1 and 2 (singleparticle motion), the exact theories ofChapters 3 to 5 (Klimontovich and Liouville equations), which are equivalent toMaxwell's equations plus Newton's law of motion, are replaced viaapproximationsby the Vlasov equation of Chapter 6. Further approximations lead to thefluid theory (Chapter 7) and magnetohydrodynamic theory (Chapter 8). The bookconcludes with two chapters on discrete particle effects (Chapter9) and weakturbulence theory (Chapter 10). Chapter 6, and Chapters7 and 8, are meant to beself-contained, so that the book can easily be used by instructors who wish thestandard organization.Thus, the introductory material of Chapters 1 and 2 can beimmediately followed by Chapters 7 and 8. This would be enough material for a

vii

viil Preface

one semester undergraduate course, while the first half of a two semestergraduatecourse could continue with Chapter 6 on Vlasov theory, followed in the secondsemester by Chapters 3 to 5 on kinetic theory and then by Chapters 9 and 10.

It is a pleasureto acknowledge the help of many individuals in writing thisbook. My views on plasma physics have been shaped over the years by dozens ofplasma physicists, especially Allan N. Kaufman and Martin V. Goldman. Thestudentsin graduate plasma physics courses at the University of Colorado and theUniversity of Iowa have contributed many useful suggestions (Sun Guo-Zhengdeserves specialmention).The manuscript was professionally typed and edited byAlice Conwell Shank, Gail Maxwell, Susan D. Imhoff, and Janet R. Kephart. Thefigureswere skillfully drafted by John R. Birkbeck, Jr. and JeanaK.Wonderlich.The preparation of this book was supported by the University of Colorado, theUniversity of Iowa, the United States Department of Energy, the United StatesNational Aeronautics and Space Administration, and the United StatesNationalScience Foundation.

Dwight R. Nicholson

Contents

CHAPTER1. Introduction

1.1

12

1.31.41.51.6

Introduction

Debye ShieldingPlasma ParameterPlasma FrequencyOther ParametersCollisions

References

Problems

1

1

357

91515

2. Single Particle Motion 1717

17

20222425293133

35

35

ix

2.12.22.32.42.52.62.7

2.8

2.9

IntroductionE X B DriftsGrad-B DriftCurvature DriftsPolarization Drift

Magnetic MomentAdiabatic InvariantsPonderomotive ForceDiffusion

References

Problems

X Contents

3. Plasma Kinetic Theory I: KllmontovichEquation 373.1 Introduction 373.2 Klimontovich Equation 393.3 Plasma Kinetic Equation 4J

References 44Problem 44

Plasma KineticTheoryII:Liouville Equationand BBGKY Hierarchy 45

4.1 Introduction 454.2 Liouville Equation 464.3 BBGKYHierarchy 49

References 58Problems 58

5. Plasma KineticTheoryIII:Lenard-BalescuEquation 605.1 Bogoliubov's Hypothesis 605.2 Lenard-BalescuEquation 64

References 68Problems 68

6. Vlasov Equation 706.1 Introduction 706.2 Equilibrium Solutions 716.3 Electrostatic Waves 736.4 Landau Contour 766.5 Landau Damping 806.6 Wave Energy 836.7 Physics of Landau Damping 876.8 Nonlinear Stage of Landau Damping 926.9 Stability: Nyquist Method, Penrose Criterion 976.10 GeneralTheory of Linear Vlasov Waves 1056.11 Linear Vlasov Waves in Unmagnetized Plasma 1086.12 Linear Vlasov Waves in Magnetized Plasma 1106.13 BGKModes 1156.14 Case-Van Kampen Modes 120

References 124

Problems 125

Contents xl

Fluid Equations 127

7.1 Introduction 1277.2 Derivation of the Fluid Equations from the Vlasov Equation 1297.3 Langmuir Waves 1327.4 Dielectric Function 1367.5 Ion Plasma Waves 1387.6 Electromagnetic Waves 1417.7 Upper Hybrid Waves 1447.8 Electrostatic Ion Waves 1467.9 Electromagnetic Waves in Magnetized Plasmas 1507.10 Electromagnetic Waves Along B0 1567.11 Alfven Waves 1617.12 Fast Magnetosonic Wave 1647.13 Two-Stream Instability 1667.14 Drift Waves 1697.15 Nonlinear Ion-Acoustic Waves\342\200\224Korteweg-DeVries Equation 1717.16 Nonlinear Langmuir Waves\342\200\224Zakharov Equations 1777.17 Parametric Instabilities 181

References 184

Problems 185

Magnetohydrodynamics 189

8.1 Introduction 1898.2 MHD Equilibrium 1948.3 MHD Stability 2008.4 Microscopic Picture of MHD Equilibrium 206

References 208Problems 210

Discrete Particle Effects 211

9.1 Introduction 2119.2 Debye Shielding 2119.3 Fluctuations in Equilibrium 219

References 224

Weak Turbulence Theory 226

10.1 Introduction 22610.2 Quasilinear Theory 22610.3 InducedScattering 234

xif Contents

10.4 Wave-Wave Interactions 241References 253

Problem 254

APPENDIX

A. Derivation of the Lenard-Balescu Equation 257

References 266

B. Langevin Equation, Fluctuation-Dissipation Theorem,Markov Processes, and Fokker-PlanckEquation 267B.1 Langevin Equation and Fluctuation-Dissipation Theorem 267B.2 Markov Processes and Fokker-Planck Equation 272

References 278

C. Pedestrian's Guideto ComplexVariables 279References 284

D. Vector and Tensor Identities 285

Reference 285

INDEX 286

CHAPTER

Introduction

1.1 INTRODUCTIONA plasma is a gas of chargedparticles,in which the potential energy of a typicalparticle due to its nearest neighbor is much smaller than its kinetic energy. Theplasmastate is the fourth state of matter: heating a solidmakes a liquid, heating aliquid makes a gas,heating a gas makes a plasma. (Compare the ancient Greeks'earth, water, air, and fire.) The word plasma comes from the Greek plasma,meaning \"something formed or molded.\" It was introduced to describeionizedgases by Tonks and Langmuir [1].Morethan 99% of the known universe is in theplasma state. (Note that our definition excludes certain configurations such as theelectron gas in a metal and so-called\"strongly coupled\" plasmas which are found,for example,nearthe surface of the sun. These need to be treated by techniquesother than those found in this book.)

In this book, weshall always consider plasma having roughly equal numbers ofsingly charged ions (+e) and electrons (\342\200\224e),each with average density n0 (particlesper cubic centimeter). In nature many plasmas have more than two species ofchargedparticles,and many ions have more than one electron missing. It is easy togeneralize the results of this book to such plasmas.

EXERCISE Name a well-known proposed source of energythat involves plasmawith more than one species of ion.

1.2 DEBYE SHIELDINGIna plasma we have many charged particles flying around at high speeds. Considera specialtest particle of charge qT > 0 and infinite mass, located at the origin of a

1

2 Introduction

three-dimensional coordinate system containing an infinite, uniform plasma. Thetest charge repels all other ions, and attracts all electrons. Thus, around our testcharge the electron density ne increases and the ion densitydecreases.The test iongathers a shielding cloud that tends to cancel its own charge (Fig. 1.1).

Consider Poisson's equation relating the electric potential ip to the chargedensityp due to electrons, ions, and test charge,

VV = ~4np = 4ire{ne \342\200\224\302\253,)\342\200\2244nqT

Plasma Parameter 3

If we define the electron and ion Debye lengths

and the total Debye length

Eq. (1.3) then becomes^D \342\200\224^e + ^i

1 d ltd

4 Introduction

Our definition of a plasma requiresthat this potential energy be much less than thetypical particle's kinetic energy

ymsM-|-r^|-mA2 (1.10)where ms is the mass of speciess, ( ) means an average over all particle velocitiesat a given point in space, and we have defined the thermal speed vs of speciess by

n it \\,/2

For electrons, ve *=* 4 X 1Q7,: Te]/2 (eV) in units of cm/s. Our definition of aplasma requires

\302\25301/3e2 \302\253T, (1.12)or

*\"&) \302\273 1 (1.13)Raising each side of (1.13)to the 3/2 power, and recalling the definition (1.4) ofthe Debye length, we have (dropping factors of 4ir, etc.)

A, - ii0A,3 \302\273 1 (1.14)

where Asis called the plasmaparameter ofspeciess.{Note:Someauthors call A/1the plasma parameter.) The plasmaparameter is just the number of particles ofspeciess in a box each side of which has length the Debye length (a Debye cube).Equation (1.14) tells us that, by definition, a plasma is an ionized gas that hasmany particles in a Debye cube. Numerically, A \342\200\236\302\2534X 10* r^/2(eV)/\302\25301/2(cm'3).We will often substitute the total Debye length KDin (1.14), and define the resultA = \302\2530X o to be the plasma parameter.

EXERCISE Evaluate the electron thermal speed, electron Debye length, andelectron plasma parameter for the following plasmas.(a) A tokamak or mirror machine with Te *\302\2531 keV, n0 *=* 1013 cm\"3.(b) The solar wind near the earth with Te \302\25310 eV, \302\2730**> 10 cm\"3.(c) The ionosphere at 300 km above the earth's surface with Te \302\2530.1 eV,

n0 \302\25310* cm3.(d) A laser fusion, electronbeam fusion, or ion beam fusion plasma with

T, \302\253\342\200\2421 keV, n0 - 1020 cm\"3.(e) The sun's center with Tt \302\253*1 keV, n0 \302\2531023 cm\"3.

It is fairly easy to see why many ionized gases found in nature are indeed plasmas.If the potential energy of a particle due to its nearestneighborwere greater than itskinetic energy, then there would be a strong tendency for electrons and ions tobind together into atoms, thus destroying the plasma. The needto keep ions andelectrons from forming bound statesmeans that most plasmas have temperaturesin excess of one electron-volt.

Plasma Frequency 5

EXERCISE The temperature of intergalacticplasma is currently unknown, butit could well be much lower than 1 eV. How could the plasma maintain itself atsuch a low temperature? (Hint: n0 *** 10s cm\"3).Of course, it is possible to find situations where a plasma exists jointly with

another state. For example,in the lower ionosphere there are regions where 99%of the atoms are neutral and only 1% are ionized. In this partially ionized plasma,the ionized component can be a legitimate plasma according to (1.14), where Asshould be calculated using only the parameters of the ionized component.Typically,there will be a continuous exchange of particles between the unionized gas andthe ionized plasma, through the processes of atomic recombination andionization.

We can now evaluate the validity of the assumption made before (1.3), thate

6 Introduction

x = 0 x = l

x = S x= 1 + 5

Fig. 1.2 Plasma slab model used to calculatethe plasma frequency.

(force per unit area) = (mass per unit area) X (acceleration), or(-4im02e28L) = (\302\2730wfL)(6) (1.18)

where an overdot is a time derivative.Equation (1.18)is in the standard form of aharmonic oscillator equation,

8 +(*2s\302\243)

d = o

with characteristic frequency

-

= (^-)2 \\l/2

(1.19)

(1.20)

which is called the electron plasma frequency. Numerically, we = 2n X 9000 \302\253/2(cm-3) in units of s\"1.

EXERCISE Calculate the electron plasma frequencywe and oie/2ir (e.g., in MHzand kHz) for the five plasmas in the exercise below (1.14).By analogy with the electron plasma frequency(1.20)we define the ion plasma

frequency wi for a general ion species with density nt and ion charge Ze as= /4ir\302\253iZV\\1/2

The total plasma frequency p\342\200\224

we2 + a>i (1-22)

Other Parameters 7

p1

\302\2730\302\253

0

(X)I

t

L L+fi

M

L +6

(b)

Fig. 1.3 Calculation of the electron plasma frequency, (a) Charge density, (b) Electricfield.

(See Problem 1.3.)For most plasmas in nature we \302\273 /. We willsee in a later chapter that the general response of an unmagnetized plasma to aperturbation in the electron density is a set of oscillationswith frequencies veryclose to the electron plasma frequency (ue.

The relation among the Debye length ks, the plasma frequency cus, and thethermal speed vs, for the species s, is

\\s=

v/ms (1.23)

EXERCISE Demonstrate (1.23).

1.5 OTHERPARAMETERSMany of the plasmas in nature and in the laboratory occur in the presence ofmagnetic fields. Thus, it is important to consider the motion of an individualcharged particle in a magnetic field. The Lorentz force equation for a particle ofcharge q, and massms moving in a constant magnetic field B = B02 is

mjr = \342\200\224(i- X B02) (1.24)For initial conditions r(r = 0) = (x\342\200\236,y0,za)and r(f = 0) = (0, u\302\261,i/-) thesolution of (1.24) is

x(t) = x0 +-j^(l - cosIV)

y{t)= yQ +

-^-sindjz(t) = z0 + v2t (1.25)

8 Introduction

where we have defined the gyrofrequency

a = \342\200\224 (i.26)

EXERCISE Verify that (1.25) is the solutionof (1.24)with the desired initialconditions.

Numerically, ft, =\342\200\2242 X 107 B0 (gauss, abbreviatedG) in units of s\"1, and fl; =

JO4 B0 (gauss) in units of s\"' if the ions are protons.The nature of the motion (1-25) is a constant velocity in the f-direction, and a

circular gyration in the x-y plane with angular frequency [flj and center at theguiding center position tgc given by

*gc~ (*o + vj/ns> y0, z0 + vzt) (1.27)

The radius of the circle in the x-y plane is the gyroradius Wj_/|fl,|. The meangyroradius rs of speciess is defined by setting v\302\261equal to the thermal speed, so

rs=vs/\\Cls\\ (1.28)EXERCISE In the exercise below (1.14), calculate and orderthe frequencies wt,

w\342\200\236jfle|, fit', also calculate the gyroradii rr and r,; take T, = Te and use thefollowing parameters. .(a) Protons,B0 = 10 kG. '(b) Protons, B0 = 10\"5 G.(c) 0+ ions, B0 = 0.5G.(d) Deuterons, B0 = 0 and B0 = 106 G.(e) Protons, B0 = 100G.

At this point, let us briefly mention relativisticand quantum effects. Forsimplicity,we shall always treat nonrelativistic plasmas. In principle,thereisno difficultyin generalizing any of the results of this course to include special relativistic effects;these are discussedat length in the book by Clemmow and Dougherty [2].

EXERCISE To what regime of electron temperature are we limited by thenonrelativistic assumption? Hpw about ion temperature if the ions are protons?Thereare,of course,many plasmas in which special relativistic effectsdo

become important. For example, cosmic rays may be thought of as a component ofthe interstellar and intergalactic plasma with relativistic temperature.

We shall also neglect quantum mechanical effects. For most of the laboratoryand astrophysical plasmas in which we might be interested, this is a goodassumption.There are, of course, plasmas in which quantum effects are very important.An example would be solid state plasmas. As a rough criterion for the neglect ofquantum effects, one might require that the typical de Broglie length k/msvs bemuch less than the average distance between particles n0'vi.

Collisions 9

EXERCISE What is the maximum density allowedby this criterion for electronswith temperature(a) 10 eV?(b) 1 keV?

(c) 100 keV?

In other applications,such as collisions (see next section), one might require thede Broglie length to be much smaller than the distance of closest approach of thecolliding particles.

In addition to these assumptions,we shall also neglect the magnetic field inmany of the sections of this book. This neglect is made for simplicity, in order thatthe basic physical phenomena can be elucidated without the complications of amagnetic field.In practice,the magnetic field can usually be ignored when thetypical frequency (inverse time scale) of a phenomenon is much larger than thegyrofrequencies of both plasma species.

1.6 COLLISIONSA typical charged particle in a plasma is at any instant interacting electrostatically(see Problem 1.5) with many other charged particles. If we did not know aboutDebye shielding, we might think that a typical particle is simultaneouslyhavingCoulomb collisions with all of the other particles in the plasma. However, the fieldof our typical particle is greatly reduced from its vacuum field at distances greaterthan a Debye length, so that the particle is really not colliding with particles atlarge distances. Thus, we may roughly think of each particle as undergoing Asimultaneous Coulomb collisions.

From our definition of a plasma, we know that the potential energy ofinteractionof each particle with its nearest neighbor issmall.Sincethe potential energy isa measure of the effect of a collision, this means that the strongest one of its Asimultaneous collisions (the one with its nearest neighbor) is relatively weak.Thus,a typical charged particle in a plasma is simultaneouslyundergoing A weakcollisions. We shall soon see that even though A is a large number for a plasma, thetotal effect of all the simultaneous collisions is still weak. Of course, a weak effectcan still be a very important effect. In the magnetic bottles like tokamaks andmirror machines currently being used to study controlled thermonuclear fusionplasmas, ion-ion collisions are one of the most important loss mechanisms.

Mathematically, the importance of collisions is contained in an expression calledthe collision frequency, which is the inverse of the time it takes for a particle tosuffer a collision. Exactly what is meant by a collisionof a chargedparticledepends upon the definition, and we will consider two different definitions withdifferent physical content. Our mathematical derivation of the collision frequencyis an approximate one, intended to be simple but yet to yield the correct resultswithin factors of two or so. A more rigorous development can be found in thebook by Spitzer [3]. (See Problem 1.6.)

Considerthe situation shown in Fig. 1.4. A particle of charge q, mass m isincident on another particle of charge q0 and infinite mass with incident speed v0.

10 Introduction

Fig. 1.4 Parameters' used in the discussion of the collision frequency in Section 1.6.

If the incident particle wereundeflected,it would have position x = v0t along theupper dashed line in Fig. 1.4, being at x \342\200\2240 directly above the scattering chargeq0att = 0. The separation p of the two dashed lines is the impact parameter. If thescattering angle is small, the final parallel speed (parallel to the dashed lines) willbe quite close to u0. The perpendicular speed v\302\261can be obtained by calculating thetotal perpendicular impulse

mv\302\261=

/ dtFJt) (1.29)where F\302\261is the perpendicular force that the particle experiences in its orbit. Sincethe scattering angle v\302\261/v0 is small, we can to a good approximation use theunperturbed orbit x = v0t to evaluate the right side of (1.29).This approximationis a very useful one in plasma physics. In Fig. 1.4, Newton's secondlaw with theCoulomb force law is

99a .mi = (1.30)

where r is a unit vector in the r-direction. Then

Fx =qq0

sin 0 = qq0 sin 6 99 o \342\226\240, a\342\200\224t-sin-* 6P

(1.31)(p/sin 0)2wherewe have usedp = r sin 6 since the particle is assumed to be traveling alongthe upper dashed line. Equation (1.29)then reads

v\302\261=

~r\302\243 Asin>6W (1.32)

The relation between 6 and t is obtained from\342\200\224pcos 6

so that

x = \342\200\224r cos 0 =

dt =p

v0

sin 8

ddsin2 6

v (1.33)

(1.34)

EXER CISE Verify (1.34).Using (1.34) in (1.32), we find

99omv0p fJo

dd sin 6 = 299omvap

Defining the quantity

(1.35)

Collisions 11

p\302\260=

iw (I36)we have

v\\ Pa\342\200\224L= \342\200\224 (1.37)

which is strictly valid only when v\302\261\302\253 v0,p \302\273 />\342\200\236.In some books, theparameterp0 is called the Landau length.

EXERCISE Showthat if qqo > 0, then p0 is the distance of closestpossibleapproach for a particle of initial speed v0.

Although (1.37) is not valid for large angle collisions, let us use it to get a roughidea of the impact parameter p which yields a large angle collision;we do this bysetting v\302\261equal to va in (1.37) to obtain p = p0. Thus, any impact parameterp < p0 will yield a large angle collision. Suppose the incident particle is anelectron, and the (almost) stationary scatterer is an ion. (Although Fig. 1.4 shows arepulsive collision, our development is equally valid for attractive collisions.)Thecross section for scattering through a large angle by one ion is 7rp02. Consider anelectron that enters a gas of ions. It will have a large angle collisionafter a timegiven roughly by setting (the total cross section of the ions in a tube of unitcross-sectional area, and length equal to the distance traveled) equal to (the unitarea), or (time) X (velocity) X (number per unit volume) X (cross section) = 1.The inverse of this time gives us the collision frequency vL for /arge anglecollisions; thus

2 4ti7i og2g02 _ 47m0e\"vL

- 7rnnvap0 - 2 3 - , 3 U-38)rrt l/Q rrle V(j

Note that vL is proportional to the inverse third power of the particle speed.Recall that a typical charged particle in a plasma is simultaneously undergoing

A collisions. Only a very lew of these are of the large angle type that lead to (1.38),since a large angle collision involves a potential energy of interaction comparableto the kinetic energy of the incident particle and, by the definition of a plasma, thepotential energy of a particle due to its nearest neighbor is small compared to itskinetic energy. Thus, a particleundergoesmany more small angle collisions thanlarge angle collisions. It turns out that the cumulative effect of these small anglecollisionsis substantially larger than the effect of the large angle collisions, as weshall now show.

Unlikethe large angle collisions, the many small anglecollisionscan produce alarge effect only after many of them occur. But these small angle collisionsproduce velocity changes in random directions, somt up, somedown, some left, someright. We need to know how to measure the cumulative effect of many smallrandom events.

Consider a variable Ax that is the sum of many small random variables Ax,-,i = I, 2, . . . , N,

Ax = Ax, + Ax2 + . . . + Ax* (1.39)

12 Introduction

Suppose (Ax,) = 0 for each /' and ((Ax^)\") is the same for each i, where < )indicates ensemble average [4]. Furthermore, suppose(Ax, Ax;) = 0 if i \302\245\"_j, sothat Ax, is uncorrected with Axy, / ^ j. Then by (1.39) we have (Ax) = 0, and

((Ax)2)

X \302\253^2>

= N ((Ax,)2) (1.40)Consider a typical particle moving in the ^-direction through a gas of scatteringcenters. As it moves, it suffers many small anglecollisionsgiven by i/j_ which can bedecomposed into random variablesAvx and An,,. These latter have just thepropertiesof our random'variable Ax, above. For one collision,with a given impactparameter/? (Fig. 1.5),we have from (1.37)

(v\302\261>)= P^1) + ((A,,)2) = ^

Since Avx must have the same statistical propertiesas Avy, we must have1 Va2Po

(1.41)

(1.42)tfAiO2) - ({Avyf) = 2 p2Then by (1.40) we have, for the total x velocity Avxtot,

((Avx\302\253*y) = A = \302\243^d. pSince we are considering a particle moving through a gas of scattering centers, it ismore useful for our purposes to have the time derivative of (1.43), where on the

(1.43)

Fig. l.S The incident particle is located at the origin and is traveling into the paper. Itmakes simultaneous small angle collisions with all of the scattering centers randomlydistributed with impact parameters between p and p + dp.

Collisions 13

right we shall have dN/dt = 2wp dp n0v0 as the number of scattering centers, withimpact parameter between p and p + dp, which our incident particle encountersper unit time. The time derivative of (1.43) is then

-^-{{Av^)2) = rm0v0W y (1-44)

We have calculated (1.44) for only one set of impact parameters between p andp + dp.Thesamelogicthat led to (1.40) also allows us to sum (integrate) the rightside of (1.44) over all impact parameters to obtain a total change in mean squarevelocity in the i-direction. Likewise, we can add the total x-direction and the totalJ/-dtrection mean square velocities to obtain a total mean square perpendicularvelocity {(Av\302\261tot)2). With this final factor of two we have

d rP\342\204\242*dp\342\200\224

((Av\302\261=^r^lnA (1.48)A reasonable definition for the scattering time due to small angle collisions is thetime it takes ((Ai;\302\261,ol)J) to equal v02 according to (1.48); the inverseof this time isthe collision frequency vc due to small-angle collisions:

_

8irn0e\302\260 In AVc

~~

\342\200\242\302\2732,, .1mt v0

(1.49)

Note again the inversecubedependenceon the velocity v0. One important aspectof ve is that it is a factor 2 In A larger than the collision frequency vL for large

14 Introduction

angle collisionsgiven by (1.38). This is a substantial factor in a plasma (In A = 14if A = 106). Thus, the deflection of a chargedparticle in a plasma ispredominantlydue to the many random small angle collisionsthat it suffers, rather than therare large angle collisions.

Throughout one's study of plasma physics,it is useful to identify eachphenomenonas a collective effect or as a singleparticle effect. The oscillation of the plasmaslab in Section 1.4, characterized by the plasma frequency a}e, is a collective effectinvolving many particles acting simultaneously to produce a large electric field.The collisional deflection of a particle, represented by the collision frequency vc'm(1.49), is a singleparticle effect caused by many collisions with individual particlesthat do not act cooperatively.

EXERCISE Is the Debye shielding described in Section 1.2 a collective effect ora single particle effect?It is instructive to calculate the ratio of vc to a)c, which is, taking a typical speed

v0=

v, in (1.49),ve

___

8nn0e4 In A_

In A_

In A

ws me2ve2aje 2irn0\\e} 2irAe

By crudely dropping the factor In A/27T and replacing Ae by A, we have the easilyremembered but very approximate expression

me A(1.51)

Thus, the collision frequency in a plasma is very much smaller than the plasmafrequency. In this respect, single particle effects are less important than collectiveeffects. A wave with frequency near eoe will oscillate many times before beingsubstantially damped because of collisions.

EXERCISE What is the ratio of the collisional mean free path, for a typicalelectron, to the electron Debye length?The collision frequency i^that we calculated in (1.49) is the one appropriate to

trie collisions of electrons with ions, vei. The collision frequency vee of electronswith electrons could be calculated in the same way, by moving to the center-of-mass frame rather than taking the scattering center to have infinite mass. Thisprocedure would only introduce factors of two or so, so that within such factorswe have vee ^ vei. Next, consider ion-ioncollisionsbetween ions having the sametemperature as the electrons that have collision frequency vee. Equation (1.49)yields, with mereplaced by m, and u, - (w/w!)1/2 uf instead of v\342\200\236,vii='{m/m^w2vee. Finally, consider ions scattered by electrons (or Mack trucks scattered bypedestrians). This calculation in the center-of-mass frame would introduce anotherfactor of (w/w,)1/2, so that vie \302\273{m/m^vfe.

Suppose an electron-proton plasma is prepared in such a way that the electronsand protonshave arbitrary velocity distributions, and comparable but not equaltemperatures. On the time scale ve^ *** vei~l \302\253\302\253A

Problems 15

lize via electron-electron and electron-ioncollisionsand obtain a Maxwelliandistribution. On a time scale 43 times longer,the ions will thermalize and obtain aMaxwellian at the ion temperature via ion-ion collisions.Finally, on a time scale43 times longer still, the electrons and ions will come to the same temperature viaion-electron collisions.

This completes our brief introduction to the important basic concepts of plasmaphysics.In the next chapter, we shall consider the motion of singlechargedparticles in electric and magnetic fields.

REFERENCES

[1] L. Tonks and I. Langmuir, Phys. Rev., 33, 195(1929).[2] P.C.Clemmow and J. P. Dougherty, Electrodynamics ofParticles and

Plasmas, Addison-Wesley, Reading, Mass., 1969.

[3] L. Spitzer,Jr., Physics of Fully Ionized Gases, 2nd ed., Wiley-Interscience,New York, 1962.

[4] F. Reif, Fundamentals of Statistical and Thermal Physics, McGraw-Hill,NewYork, 1965.

PROBLEMS

1.1 Debye ShieldingIn the discussion of Debye shielding in Section 1.2, suppose that the ions areinfinitely massive and thus cannot respond to the introduction of the test charge.How does the answer change?

1.2 Potential Energy (Birdsall's Problem)A sphere of plasma has equal uniform densities n0 of electrons and infinitelymassive ions. The electrons are moved to the surface of the sphere, which theycover uniformly. What is the potential energy in the system? Sketch the electricfield and electricpotential as a function of radius. If the electrons initially hadtemperature Te, and it is found that the potential energyis equal to the total initialelectron kinetic energy, what is the radius of the sphere in terms of the electronDebye length?

1.3 Total Plasma FrequencyIn the discussion of the plasma frequency in Section 1.4, suppose the ions are notinfinitely massive but have mass m,. Modify the discussion to show that the slabsoscillate with the total plasma frequency defined in (1.22).

1.4 Plasma in a Gravitational Field

Consider an electron-proton plasma with equal temperatures T = Te = Tif nomagnetic field, and a gravitational accelerationg in the \342\200\224f-direction. We desire the

16 Introduction

densities ne(z) and n,{z), where 2 = 0 can be thought of as the surface of a planet.If the electrons and ions were neutral, their densities would be given by theBoltzmann law

ntti\302\253

exp (-mei gz/T). Then the scale height T/mci g would bequite different for electrons and ions. However, this would give rise to hugeelectric fields that would tend to move ions up and electronsdown. Taking intoaccount the electric field,usethe Boltzmann law and the initial guess that ne(z)

CHAPTER

Single Particle Motion

2.1 INTRODUCTIONA plasma consists of many charged particlesmoving in self-consistent electric andmagnetic fields.The fields affect the particle orbits, and the particle orbits affectthe fields. The general solution of any problem in plasma physics can be quitecomplicated. In this chapter, we consider the motion of a singlecharged particlemoving in prescribed fields. After studying this part of the problem in isolation,we can proceed in following chapters to include these particle orbits in the self-consistent determination of the fields.

2.2 E x B DRIFTSConsider a particle with vz = 0 gyrating in a magnetic field B\342\200\236in the 1-direction,

with an electric field E() in the \342\200\224y-direction perpendicular to the magnetic field asin Fig, 2.1. (The symbol & always means a unit vector in the a-direction.) Theelectric field. E0 cannot acceleratethe particle indefinitely, because the magneticfield will turn the particle. (The component of electric field E2, which we ignorehere, can accelerateparticles indefinitely. In a plasma, the resulting current usuallyacts to cancel the charge that caused the electric field in the first place. There are,however, important cases where this cancellation is hindered: for example,theearth's aurora, and tokamak runaway electrons.) What does happen? When thecharge qsis positive, the ion is accelerated on the way down. This gives it a largerlocalgyroradius at the bottom of its orbit than at the top; recall that the gyroradiusis rs = Vj/n,5. Thus, the motion will be a spiral in the x-y plane as shown in Fig.2.2, where we have used the symmetry of the situation to draw the upward part ofeach orbit. We see that the orbit does not connect to itself, but has jumped a

2

18 Single Particle Motion

i

Fig. 2.1 Configuration that leads to an E * B drift.

certain distance to the left during one orbit. The net result is that the particle has adrift velocity \\d to the left. Let us guesshow big the drift speed is. If we averageover many gyroperiods, we see that the average accleration is zero. Thus, the netforce must be zero. The force downward is qsE0> while the force upward is(qs/c)yd X Bu. We must have

\302\253>= 0 = 9jEo +

-^- vrf X B0 (2.1)

where ( ) indicates an average over one gyroperiod. Taking the cross productof (2.1) with B0) and assuming vrf \342\200\242B0 = 0, we find

vrf= c

E0 X B0Bo1

(2.2)

Note that the drift velocity does not depend on the particle's charge or mass.

EXERCISE What is the drift speed of an electron in the earth's magnetosphercif |B0| = 0.1G and |E0| = 10^ Kcirf1? (Remember 1 sV = 300 V.) What ifthe particle were a uranium atom with charge qs = +57 efLet us now make sure that our guess is correct, and that the kind of motion we

have in mind, namely a gyration about the magnetic field lines,accompanied by adrift, is an exact solution to the equation of motion. This equation is

9sm,\\ qsE0 +

-f- v * B\342\200\236 (2.3)

A

Fig. 2.2 Motion of a particle of positive charge leading to the E x B drift.

E x B Drifts 19

We define a new variable v by splitting v into a piece v (which will turn outoscillatory) and a piecevd as given by (2.2). Taking

(2.3) becomesw,v

v + vd

vrf.

Note that this entire discussion would apply if an arbitrary (temporally andspatially constant) force Fx such that \302\245\302\261\342\200\242B0 = 0 were to replace

20 Single Particle Motion

2.3 GRAD-B DRIFT

First, let us calculate the so-calledgrad-Bdrift. To do this we need to have a feelingfor expansion techniques. Suppose we have an equation for a variable x, with onesmall term of size t, expressedas

f(x) - eg(x) = 0 (2.9)Here, e is a small constant, / and g can represent a general integro-differentialoperator,and the solution of (2.9) when e \342\200\2240 is x0, so that/(x0) = 0. We canlook for a solution x to (2.9)of the form

x = x0 + ex, + (2x2 + . . . (2.10)Inserting (2.10) in (2.9) yields

/(x0 + ex, + e2x2 + ...) = \302\253\302\243(*o+ \302\253*.+ \302\2532*2+ \342\200\242\342\200\242\342\200\242)(2-11)After Taylor expanding/and g, one obtains

/(*\342\200\236)+^

(\302\253i+ *2*2 + \342\200\242\342\200\242\342\200\242)+ \\ -\302\2434

Grad-BDrift 21

Fig. 2.4 Ion VB drift.

ions and in Fig, 2.5 for electrons. Thus, electrons and ions drift in oppositedirections and, in a plasma, a net current results.

The force on a charged particle is

m,y =-y- (v X B) (2.15)

Taylor expanding B about the guiding center of the particle,B = Bu + (r \342\200\242V)B\342\200\236 (2.16)

where B;, is measured at the guiding center, and where r is measured from theguiding center (see Section 1.5), and inserting in (2.15), one obtains

m,v =~y- (v X B()) + -2f- [v X (r

\342\200\242V)B\342\200\236] (2.17)

EXERCISE What assumption is being made in (2,16)?Expanding

v = v0 + v, (2.18)we have

\342\204\242,v=

-y- (ToX B.\302\273) (2-19)

which yields gyromotion, and

\302\253.*.=

\"T\" (v, X B0) + -^ [v(1x (r \342\200\242V)B0] (2.20)

where we are treating rv\" as a small quantity, and to be consistentr must becalculated using only v\342\200\236.

.MSLFig. 2.5 Electron VB drift.

22 Single Particle Motion

Now we are only interested in that part of v, that represents steady drift motion;therefore, after averaging both sides of (2.20) over a gyroperiod, upon which theleft side vanishes, we have

tfS ?.

Curvature Drifts 23

force Fc outward (Fig.2.6),equal to

Our general drift equation (2.8) then predictsFc -

-jJ- R,Kb

Vw =_c_ F\302\261

X Bu

24 Single Particle Motion

2.5 POLARIZATION DRIFT

We discuss next a drift that is the result of an electric field which varies with time.Since the drift is opposite for oppositely charged particles, it leads to a currentcalled the polarization current.

Considera constant magnetic field Bnz, and an electric field E(t) = \342\200\224Ely,where E is a constant (Fig.2.7).The force equation is

mj = qsE +-^

v X B() (2.33)

We expect an E X B0 drift in the (\342\200\224)x-direction, which will be increasing withtime. Thus, the particle is being accelerated in the (\342\200\224^-direction and, therefore,feels an effective force in the \302\243-direction. (The effective force is in the directionopposite to the acceleration; when one steps on the gas pedal of a car, one is forcedbackwardinto the seat.) This effective force should give rise to an F X B drift in the^-direction. Using this intuition, we consider a solution of (2.33)of the form

v = v\342\200\236+ vEic + vpp (2.34)where v0 will contain all gyromotion, v\302\243isthe E X B drift, and vp is thepolarizationdrift, which is assumed to be constant. Substituting (2.34) into (2.33), weobtain

wA + V,) - -q>Ety +-^

vuX Bu

- -^- vEBJ +-y- vpB0t (2.35)

The assumed nature of the solution indicates the separation of this equation intopieces, with

msvB = ~ v0 x B0 (2.36)

representing gyromotion,

w,v\302\243=

~^-vpB0x (2.37)

giving the polarization drift, and

0 = - qfiiy ~ \342\200\224TVeBJ (2.38)

y

tEft)

2/

Fig. 2.7 Configuration that leads to a polarization drift:

Magnetic Moment 25

giving the E X B drift, namely,

vEX -

Then (2.37)yields the polarization driftcEt_ , E X B0Ba

X 'B0>

which in vector form is

cms cE_

c

qf0 ( } B0 n

v = E

4 -5(1

(2.39)

(2.40)

(2.41)

We see that (2.34) is an exact solution to (2.33).

EXERCISE If E(t) \342\200\224E0 cos atl, can you invent a criterion for the validity of(2.41) at each instant of time?

The polarization drift leads to a polarization current J of electrons andprotons, given by

n0c2 dE~bT ~dT

or

Jp = n0e(vp, - v) = -j^-j- \342\200\224(me + m,) (2.42)

' V dt (2.43)

where pm is the mass density.

2.6 MAGNETIC MOMENTThe preceding sections have discusseddrifts due to \"forces\" perpendicular to themagnetic field. There are also forces parallel to the magnetic field that are veryimportant, leading to the concepts of magnetic moment and adiabatic invariants.

Recall that the magnetic moment of a current loop with current /, area A, inc.g.s. units, is

JAr=Jf- (2-44)

A charged particle gyrating in a magnetic field is such a current loop, with currentqsfls/2TT, area np/ \342\200\224irv\302\261/ns2 (ps is the gyroradius), and magneticmoment

H \342\200\2242ttC

or

n.

v--

left,, B

B

(2.45)

(2.46)

26 Single Particle Motion

y

Fig. 2.8 Magnetic field with nonzero grad-B. The orbit shown is a circle in a planeperpendicular to the paper. For the discussion following (2.51), the orbit is actually a spiralwith a velocity component uy in the ^-direction.

where W\302\261= Vimsv\302\261 is that portion of a particle's kinetic energy which isperpendicular to the magnetic field.

We know that magnetic moments feel a force \342\200\224pVBin an inhomogeneousmagnetic field. How doesthis work out for a charged particle?Considera particlegyrating about the axis of a cylindrically symmetric magnetic field, whosemagnitude is changing along the axis, as shown in Fig. 2.8.

EXERCISE Does the field in Fig. 2.8 satisfy Maxwell's equations?In the figure, the vertical line is a side view of the gyrating particle. Notice that atthe top of the orbit (Fig. 2.9), for a positively chargedparticle, the v X B force hasone component giving gyromotion about the field, and another componentpointingin the (\342\200\224)\302\243-direction.This latter component is constant around the gyro-orbit,and the particle is steadily accelerated away from regions of strong field.

EXERCISE Showthat this works the same for either sign of the charge.The force in the (\342\200\224)\302\243-direction,evaluated anywhere on the orbit, is

/\342\226\240=-*T(v X B)x=^f-v\302\261Br (2.47)

where r is the distance from the x-axis, and Br is the component of the magneticfield in the y-z plane in Fig. 2.8. In cylindrical coordinates,

dx r drSolving this equation with Br = 0 at r \342\200\2240, and Br \302\253 Bx everywhere, one ob-

\302\256\302\253

vXB

Fig. 2.9Fig. 2l8.

The vectors v, B, and v x B for a particle with qs > 0 at the top of the orbit in

Magnetic Moment 27

tains

*' =~ f IT = \" T |Vi?' (148)

Inserting (2.48)in (2.47), with /- equal to the gyroradius p.,, we obtain\\9,\\ P* \342\200\236\342\200\236_ \302\253^F - - -^-

-^- WjTO ^ V5 (2.49)2\302\243or

F =-/iVB (2.50)

as expected.Knowing the force on the particle allows the calculation of its orbit. First one

needsto know how the magnetic moment ^ changesalong the orbit. Theremarkablefact is that the magnetic moment is constant along the orbit, provided the fielddoes not change much in one gyroperiod. Let us prove this.

The Lorentz force on a charged particle (with E \342\200\2240) is F = (q/c) v X B. Asmall component \302\245\302\261of this force acts to accelerate the particle in the directionperpendicular to the local magnetic field and parallel to the component of theparticle velocity vj_ used in the definition of p. (In Fig.2.8,let the particle shownhave a positive velocity component v., along J?.Then at the top of the orbit, v X Bhas a component of magnitude | v,, Br\\ pointing into the paper.) This force is intothe paper at the top of the orbit of the particle in Fig. 2.8, and is given by

F\302\261= ~

\"7\" w\342\200\236a, (2.51)

where v.. is the component of velocity in the \302\243-direction and Br is negative. Theperpendicular energy of the particle then changes with time according to

~af (y m'vl)=

v\302\261F\302\261=

-f\" v\302\261v\"Br (2'52)

When we use (2.48), this becomes (with r = ps at the location of the particle)d

W7 _ \302\260s Ps dBx I 2 1 dB

where Bx >=* B, or

4rw\302\261= \">\" i \302\247 (2-54)

EXERCISE Combine (2.50) and (2.53) to show that total energy is conserved.The time rate of change of the magnetic moment is

d^_ __d IWL\\

_

1 dW\302\261 1 dB

It~

~dx \\B~)~

~B ~dt IF W\302\261~dx

I dB I dB

= 0 (2.55)

26 Single Particle Motion

where the rate of change of B along the particle orbit dB/dt \342\200\224v., dB/dx hasbeen used. Thus,

H\342\200\224constant (2.56)

along the particle orbit, to within the accuracy of this calculation. This is anexample of an adiabatic invariant, a quantity that is constant under slowchangesin an external parameter. Although our derivation treated spatially varyingmagnetic fields, the same result holds for magnetic fields with slow time variation.

We are now in the position to understand the principle of mirror confinement,which is the basis for one of the two major approaches to magneticfusion, and isalso the reason for the existence of the earth's magnetosphere. Consider amagneticfield created by two coils, as shown in Fig. 2.10. A particle that starts at x \342\200\2240with energy Wn and magnetic moment n conserves both of these quantities.Suppose the particle initially has d, > 0; it moves to the right and feels a force to theleft, F \342\200\224\342\200\224/iVB.Does the particle get reflected by the force, or does it go pastx = x0 to be lost from the machine? This will depend on its initial Uy\" and v\302\261\302\260.Wehave

Vimvj_ '/iwufM

-

B B\342\200\236\342\200\224const

and

W - Vim(vl + v\\\\)~

wa- const

(2.57)

(2.58)As the particle moves to the right, B increases and IV\302\261\342\200\224Vimv^ must increase tosatisfy (2.57). If WL ever reaches W^, then all the energy will be in perpendicularmotion, v\302\253will vanish, and the particle will be reflected back toward the mirrormachine. This happens if

M-

Vimvj\342\226\240Omin

> 3l-Ornaa

Vimjvf + v[)Bmix

or

>5,

(2.59)

(2.60)

Defining the pitch angle 6 \342\200\224tan ](v\"/vu), we have from (2.60)

Fig, 2.10 Simple mirror machine configuration.

Adiabatlc Invariants 29

sin0 >J/2

R J/2 (2.61)

where we have introduced the mirror ratio R = Bm3*/Bmm.Particles whose pitchangles satisfy (2.61) at the center of the machine are confined. Those that do notare lost out from the ends. While our derivation appliesonly to particles circlingthe central magnetic field line, a similar statement is true for off-axis particles.

2.7 ADIABATIC INVARIANTS

The magnetic moment, constant under slow spatial or temporal changes in themagnetic field, is one example of an adiabatic invariant. It often turns out that in asystem with a coordinate q, and its conjugate momentum/), the action, defined by

,/ = $pdq (2.62)is a constant under a slow change in an external parameter. Here, we haveassumed that when there is no change in the external parameter, the motion isperiodic, and^ represents an integral over one period of the motion. In the case ofa charged particle in a magnetic field, we could take, for example, measuring xfrom the guiding center: p \342\200\224mvx, q \342\200\224x, and

/- msvx dx~i> msvl sin2 (ft,f) dt

2m\342\200\236irclW,\\ 2m,ircnm,v ,

h \\B } ?, (2.63)which is the magnetic moment to within a constant.

One famous example of the constancy of action was derived at the 1911Solvayconference, where Lorentz asked: \"What happens if we slowly shorten the stringof a swinging pendulum?\" The next morning, Einstein answered:\"Action f\342\200\224energy/frequency] is conserved.\"

Let us now demonstrate the invariance of the action in the general case. Wehave in mind the picture of a particle bouncing in a potential well, with the shapeof the well changing slowly with time, as shown in Fig. 2.11. In Hamiltonianmechanics,we have a Hamiltonian H(p,q,\\) where/) is the momentum (mx in the

*1 *2Fig. 2.11 Periodicmotion in a slowly changing potential well.

30 Single Partlcl* Motion

figure), q is the coordinate (x in the figure), and k is the parameter that determinesthe shape of the well. When k is constant, the Hamiltonian is constant; when kchanges,

dH_

dH dkdt

~

dk dtThe Hamiltonian equations of motion are

(2.64)

9 = 1F- p = ~^ (165)The time derivative of the action

J =6pdq = 2 f9i p dq (2.66)i\\

,sr 92

J=2} pdq + 2p(q2)qi~ 2p(qi)ql (2.67)

The last two terms vanish since the momentum is zero at the turning points; then

r n dHJ = -2

J^--qj*1

=-2[H(q2) - H(q,)]= 0 (2.68)

since dH/dq is to be taken at fixed \\, implyingH(q2)and H{qt) are to be taken atfixed k, and H is a constant except for its variation with k. We have thus shownthe approximate constancy of any action variable in the presence of periodicmotion and slow changes of external parameters. Note that the present derivationis heuristic, since it has been assumed that the variation is so slow that the turningpoints q2(t) and q\\(t) can be treated as continuous functions of time and can bedifferentiated as in (2.67) [4]. A rigorous treatment of this problem can be found inthe fundamental paper of Kruskal [5]; seealso Goldstein [6],

The knowledge of an adiabatic invariant can be very useful in predicting particlebehavior. Let us return to the mirror machine, where the constancyof themagneticmoment adiabatic invariant has already enabledus to determine which particleswill be confined and which will be lost. Consider a confinedparticle.The confinedparticle executes periodic motion between jfj and x2. Thus,

J2 =6 pdq =Smvxdx (2.69)must be a constant of the motion, even when the entire mirror field undergoesslow temporal changes, or when the mirror field is not axisymmetric.

There is yet a third adiabatic invariant. As the chargedparticle bouncesfrom Xito Xi, it drifts perpendicular to the field lines, with speed vd. Eventually, it comesall the way around the mirror machine. Because this is like a huge gyro-orbitabout the axis of the magnetic field, we define a new adiabatic invariant

Ti=(f)vdd[ (2.70)

where / is the distance around the mirror machine measured at some fixedx, forexample x{. It turns out that J3 is proportional to the total magneticflux enclosedby the drifting motion. This invariant is useful when the mirror fieldis notaxisymmetric, or when it undergoes slow temporal changes.

Ponderomotive Force 31

EXERCISE Sketch the earth's magnetosphere, and discussthe motion leadingto /j, J2, and J3.

2.6 PONDEROMOTIVEFORCEAll of the examples of single particle motion considered in previous sectionsinvolve motion in a magnetic field, with or without an electric field. There is onevery important single particle effect, the ponderomotive force or Miller force, thatoccurs in spatially varying high frequency electric fields, with or without anaccompanying magnetic field. Consider a charged particle oscillating in a highfrequency electric field E(l) \342\200\224if,, cos (to/). The motion is then a sinusoidal variationof distance with time, as shown in Fig. 2.12. Now supposethe electric field has anamplitude that varies smoothly in space, E{x,t) \342\200\224EQ(x) cos (wt), being strongerto the right and weaker to the left. Then the first oscillation brings the particle intoregions of strong field, where it can be given a strong push to the left (see Fig.2.13). When the field turns around, the particle is in a region of weaker field, andthe push to the right is not as strong. The net result is a displacementto the left,which continues in succeeding cycles as an acceleration away from the region ofstrong field [7].

Mathematically, the force equation ismsx \342\200\224q\302\243

\342\200\224q\342\200\236E0(x)cos wt (2.71)

It is convenient to decompose x into a slowly varying component x\342\200\236,called theoscillation center (compare this concept to the guiding center in a magnetic field)and a rapidly varying component x,, x ~ x0 + x,. Here,x0 \342\200\224T where ( )indicates a time average over the short time 27r/

32 Single Particle Motion

Weak Strong

Fig. 2.13 Motion of a charged particle in a high-frequency electric field that is weaker tothe left and stronger to the right.

E0(x) about the oscillation center x0, (2.71)becomesmAXo + x,) = q, \\Ea + xx

-^-Jcos wt

where dE0/dx is to be evaluated at x0. Averaging (2.72) over time, we get

msx0= qs

dEAdx

x, cos wl

(2.72)

(2.73)

To obtain an equation for xu we note that x, \302\273 x0 since jc, is high frequency;moreover, in the spirit of the Taylor expansion we havefo \302\273 x^dE^/dx);therefore (2.72) is approximately

msX! = q3E0 cos wt (2.74)with solution x, \342\200\224\342\200\224(^jEo/wijft^Xcos cot)'-, inserting this in (2.73) and performingthe time average one obtains

q[Eu dE0X\302\260

2m*a2 dx

so that the ponderomotive force Fp \342\200\224msx0 is

F = - Si d\342\200\224(E 2)*\302\273

4msw2 dx( a)

(2.75)

(2.76)

This formula will be easier to remember if we introduce the jitter speed v =(iiW = qsE0/msw; then

m, d(2.77)

This force is very important in such applications as laser fusion', electron beamfusion, radio frequency heating of tokamaks,radio frequency plugging of mirrors,radio frequency modification of the ionosphere, and solar radio bursts. The study

Diffusion 33

of the effects of ponderomotive force is one of the areas of current basic plasmaphysicsresearch.Notice that the overall mass dependence is as given in (2.76), sothat the ponderomotive force acts much more strongly on electrons than on ions.A more complete derivation of the ponderomotive force, including the magneticfield in an electromagnetic wave, can be found in Schmidt [4].

2.9 DIFFUSION

We conclude this chapter with a brief discussion of the effects of collisions on thelocation of the guiding center of a particle in a magnetic field. The discussion ofChapter1 shows that the effects of many small angle collisionsin a plasma aremore important than the effects of rare large angle collisions.However, it issimplest to consider here a single large angle collision between two chargedparticles; the results can then be qualitatively applied to determine the effects of manysmall angle collisions.

Consider first the head-on collision between two electrons at x = 0, as shownin Fig. 2.14. The last gyro-orbit,and the guiding center, of each particle before thecollision are indicated in the upper half of the figure. After the collision, electron

Before

x, B

x,B

After

Fig. 2.14 Head-on collision between two electrons in a magnetized plasma. Numbersindicate the location of the guiding centers of the two electrons.

34 Single Particle Motion

number 2 has the same orbit that electron number 1 had before the collision, andvice versa. Thus, the locations of the two guiding centers have been interchanged,and there isno net motion of the electrons. We conclude that collisions betweenlike particles do not lead to diffusion of those particles across magnetic field lines.

Next, consider the (almost) head-on collision betweenan electron and a slightlymore energeticpositronat x = 0, as shown in Fig. 2.15. The last gyro-orbit andthe guiding center of each particle before the collision are indicated in the upperhalf of the figure. After the collision, the electron has slightly more energy than thepositron, and both guiding centers have moved by two gyro-radii in the (\342\200\224)z~direction. Thus, the center-of-mass of the system has moved a substantial distancein the (\342\200\224)f-dire'ction. We conclude that collisions between unlike particles cancause significant diffusion of particles across magnetic field lines. Furtherdiscussionof diffusion can be found in Ref. [I].

This completes our discussion of single-particlemotion in prescribed electricand magnetic fields. In the next chapter, we begin a systematic treatment ofplasma physics in which the electromagnetic fields and the particle orbits aredetermined self-consistently*

BEFORE

x, B

x, B

AFTER

Fig. 2.15 Head-on collisionbetween an electron and a slightly more energetic positron ina magnetized plasma.

Problems 35

REFERENCES

[1] F. F. Chen, Introduction to Plasma Physics, Plenum, New York, 1974.[2] A. Baflos,Jr.,J. Plasma Phys., 1, 305 (1967).[3] T. G. Northrop, The Adiabatic Motion of Charged Particles, Wiley, New

York, 1963.

[4] G. Schmidt, Physics of High Temperature Plasmas, Academic, New York,1966.

[5] M. Kruskal, J. Math. Phys., 3, 806 (1962).[6] H. Goldstein, Classical Mechanics, 2nd ed., Addison-Wesley, Reading,Mass.,

1980,Sect.11-7.[7] L. D. Landau and E. M. Lifshitz, Mechanics, Addison-Wesley, Reading,

Mass., 1960, Sect. 30.

PROBLEMS2.1 Example of a Drift(a) Consider a particle of charge q and massm, initially at rest at (x,y,z) \342\200\224

(0,0,0), in the presence of a static magnetic field B = B02 and E = EQp.Taking En, Bn > 0, sketch the orbit of the particle when q > 0.

(b) Derive an exact expressionfor the orbit [x(t), y(t), z(t)] of the particle inpart (a). Does this result agree with the sketch of part (a)?

(c) Show that the orbit in (b) can be separatedinto an oscillatory term and aconstant drift term. After averaging in time over the oscillatory motion, isthere any net acceleration? If not, how are the forces in the problembalanced?

(d) In what direction is the drift for q > 0? For q < 0? If there were manyparticles of various charges and massespresent,would there be any netcurrent?

(e) Supposethe electric field were replaced by a force F0 in the ^'direction.What would be the drift velocity? [Hint: Guess the answer using the last partof(c).]

2.2 Grad-B DriftLet us derive the grad-B drift in a different way. With the force equation F ={q/c)y X B, insert the zero order orbit and the Taylor expanded magnetic field.Average F over one gyroperiodto obtain an average force. Insert this averageforce into the general drift equation (2.8), and compare the resulting drift to thegrad-B drift (2.28).

2.3 Polarization DriftLet us get the polarization drift (2.41) in a faster but less rigorous manner. Withthe given electric field E = \342\200\224Etp,calculate an E x B drift. Relate the resulting

36 Single Particle Motion

accelerated drift to a force (being careful with signs), plug in the F * B formula(2.8), and compare the result to (2.41).

2.4 Mirror Machines

(a) A mirror machine has mirror ratio 2. A Maxwellian group of electrons isreleased at the center of the machine. In the absence of collisions,whatfraction of these electrons is confined?

(b) Supposethe mirror machine has initially equal densities n \"=> 1013 cm\"3 ofelectrons and protons, each Maxwellian with a temperature I keV \302\253107oC.The machine is roughly one meter in size in both directions. Recalling ourdiscussion of collisionsfrom Chapter 1, estimate very roughly the time for(1) loss of the unconiined electrons;(2) loss of the unconffhed ions;(3) loss of many of the initially confined electrons (due primarily to which

kind of collision?); why do not all of the electrons leave?;(4) loss of the initially confined ions (due primarily to which kind of

collision?).For fusion purposes (supposing the protons were replacedby deuterium ortritium) which of these numbers is the most relevant?

2.5 Drift EnergyA particle of mass m and charge q in a uniform magnetic field B = B0z is set intomotion in the ^-direction by an electric fieldE(t)j>that varies slowly from zero to afinal value \302\2430.Thus, at the final time the particle has an E X B drift v0.

(a) Use energy arguments to show that the particle's guiding center musthave been displaced a distancevu/D. in the direction of the electricfield.

(b) Integrate the polarization drift velocity from time zero to time infinityto obtain a displacement. Does the answer agree with (a)?

CHAPTER

Plasma Kinetic Theory I:

Klimontbvich Equation

3.1 INTRODUCTION

In this chapter, we begin a study of the basic equations of plasma physics.Theword \"kinetic\" means \"pertaining to motion,\" so that plasma kinetic theory is thetheory of plasma taking into account the motions of all of the particles. This canbe donein an exact way, using the Klimontovich equation of the present chapter orthe Liouville equationof the next chapter. However, we are usually not interestedin the exact motion of all of the particles in a plasma, but rather in certain averageor approximate characteristics.Thus, the greatest usefulness of the exactKlimontovich and Liouville equations is as starting points for the derivation ofapproximateequations that describe the average properties of a plasma.

In classical plasma physics,we think of the particles as point particles,each witha given charge and mass. Supposewe have a gas consisting of only one particle.This particie has an orbit Xj(r) in three-dimensional configuration space x. Theorbit Xj(/) is the set of positions x occupied by the particle at successive times /.Likewise,the particle has an orbit V\\{t) in three-dimensional velocity space v. Wecombine three-dimensional configuration space x and three-dimensionalvelocityspace v into six-dimensional phase space (x,v).The density of one particle in thisphase space is

Ar(x,v,0 =

38 Plasma Kinetic Theory I

EXERCISE At any time /, the density of particles integrated over all phasespace must yield the total number of particles in the system. Verify this for thedensity (3.1).Next, supposewe have a system with two point particles, with respective orbits

[X,(/); V,(0] and [X2(0>V2(0] in phase space (x,v). By analogy to (3.1),the particledensity is

N(x,r,t) = 2\302\253[x - XXOMv - V,(/)] (3.2)

EXERCISE Repeat the previous exercise for (3.2).Now supposethat a system contains two species of particles,electronsand ions,

and each species has N0 particles. Then the density Ns of speciess is

NA*,v,t) - %B[* ~ X;(f)]S[v- V,(/)] (3.3)and the total density N is

N(x,\\,t) = 2X(X>V) (3.4)

EXERCISE Repeat the previous exercise for (3.4).If we know the exact positions and velocitiesof the particles at one time, then

we know them at all later times. This can be seen as follows.The positionX,(/)ofparticle / satisfies the equation

UO = TO (3.5)where an overdot means a time derivative.Likewise,the velocity V,(0 of particle isatisfies the Lorentz force equation

mjtt) = q,E\"[XM,t] +-^- VXO

* B^X/*),'] (3-6)

where the superscript m indicates that the electric and magnetic fields are themicroscopic fields self-consistently produced by the point particles themselves,together with externally applied fields. [On the right of (3.6), the portion of Em andBm produced by particle i itself is deleted.]The microscopicfields satisfy Maxwell'sequations

V \342\200\242Em(x,t) = 4npm(x,t) (3.7)

V \342\200\242BM

Klimontovich Equation 39

The microscopicchargedensity is

Pm(*J) = X q,JdvNjLx,v,t) (3.11)while the microscopic current is

J\"(x,/) - X ?,/rfvvJVJ(x,v,0 (3.12)

EXERCISE Convince yourself that (3.11) and (3.12) yield the correct chargedensity and current.

Equations 3.7 to 3.12 determine the exact fields in terms of the exact particleorbits,while (3.5) and (3.6) determine the exact particleorbits in terms of the exactfields. The entire set of equations is closed, so that if the positions and velocities ofall particles, and the fields, are known exactly at one time, then they are knownexactly at all later times.

3.2 KLIMONTOVICHEQUATIONAn exact equation for the evolution of a plasma is obtained by taking the timederivative of the density Ns. From (3.3),this is

^X;V-= - t x,-vx5[x - x,

40 Plasma Kinetic Theory I

This relation allows us to replace V,{/) with v, and X;(0 with x, on the right of(3.14) (but not in the arguments of the delta functions) so that (3.14) becomes

^^=

-v.Vxis[x-Xi]5[v-V,.]_\342\226\240\\l\302\261-

E\"(x,t) + \342\200\224v X Bm(x,/) 1 \342\226\240V\302\2452 fi[x - X,]5[v - VJ(3.15)

But the two summations on the right of (3.15) are just the density (3.3); therefore

dNjjx,\\,t)dt + T'VXN, + ^\"(E\" + 7>

Plasma Kinetic Equation 41

in phase space we know that

dt= v (3.18)

orbit

and

Thus,

d\\dt

orbit

=

-\302\243-IE-CM) + T x B\"(x'^

157= ~t + v'v* + ~t~[\302\243m(x,/) + \"7 x Bm(x,r)]'v* (320)and the Klimontovich equation (3-16) simply says

-^-iV\342\200\236(x,v,/)~= 0 (3.21)

The density of particles of species s is a constant in time, as measured along theorbit of a hypothetical particle of species s. This is true whether we are movingalong the orbit of an actual particle, in which case the density is infinite, orwhether we are moving along a hypothetical orbit that is not occupied by an actualparticle, in which case the density is zero. Note that the density is only constant asmeasuredalongorbits of hypothetical particles; in (x,v) space at a given time it isnot constant but is zero or infinite.

There is yet a third way to think of the Klimontovich equation. Any fluid inwhich the fluid density/(r,f) is neither created nor destroyedsatisfiesa continuityequation

d,/(r,0 + V,- (/V) =0 (3.22)where Vr is the divergence vector in the phase space under consideration, and V isa vector that gives the time rate of change of a fluid element at a point in phasespace. (See, for example, Symon [5],p. 317.)In the present case, Vr = (VX,VV)and V = (rfx/\302\253//|orbit, dv/dt\\0Mi)- Since the particle density is neither created nordestroyed,it must satisfy a continuity equation of the form

B,NAx,v,t) + Vx'(vNs) + Vv-[-Me\342\204\242

+ yX B\"]n,) = 0 (3.23)

It is left as a problem to demonstrate that the continuity equation (3.23) isequivalent to the Klimontovich equation (3.16).

3.3 PLASMA KINETICEQUATIONAlthough the Klimontovich equation is exact, weare really not interested in exactsolutions of it. Thesewould contain all of the particle orbits, and would thus be fartoo detailed for any practical purpose. What we really would like to know are theaverage properties of a plasma.TheKlimontovich equation tells us whether or nota particle with infinite density is to be found at a given point (x,v) in phase space.

42 PIbsma Kinetic Theory I

What we really want to know is how many particles are likely to be found in asmall volume Ax Av of phase space whose center is at (x,v). Thus, we really are notinterested in the spikey function N,(x,\\,t), but rather in the smooth function

/s(x,v,/) = (3.24)The most rigorous way to interpret ( ) is as an ensemble average [6] over aninfinite number of realizations of the plasma, prepared accordingto someprescription. For example, we could prepare an ensembleof equal temperatureplasmas, each in thermal equilibrium, and each with a test charge

Pies ma Kinetic Equation 43

the left side of (3.26) contains smoothly varying functions representing collectiveeffects, while the right side represents the colhsional effects. We have seen inSection 1.6 that the ratio of the importance of collisionaL effects to the importanceof collectiveeffects is sometimes given by 1/A, which is a very small number. Wemight guess that for many phenomena in a plasma, the right side of (3.26)has asize 1/A comparedjp each of the terms on the left side; thus the right side can beneglectedfor the study of such phenomena. This indeedis the case, as shown in thenext two chapters.

This important point can be illustrated by a hypothetical exercise. Imagine thatwe break each electron .into an infinite number of pieces, so that.w0 \342\200\224\302\253,m( \342\200\224\342\226\2400,and e \342\200\2240, while hae \342\200\224constant, e/m. = constant, and; v., = constant.

EXERCISE Show that in this hypothetical exercise, we \342\200\224constant, \\e \342\200\224constant, but T, \342\200\2240 and Ae \342\200\224\302\273.

Then any volume, no matter how small,would contain an infinite number of pointparticles, each representedby a delta function with infinitesimal charge. Statisticalmechanicstells us that the relative fluctuations in such a plasma would vanish,since the fluctuations in the number of particlesN0 in a certain volume isproportional to the square root of that number: Thus, on the right side of (3.26) we have6NX

~

A7,)1-'2'~ A,,1,2, and 6E and 5B, which are produced by SNS behaving like

(from Poisson's equation) ~ eSNs -~ NU~'N0]/2 ~ Nu~h'2 ~ A/1'2, so that the rightside becomes constant. On the left, however, each term becomes infinite as/j \342\200\224\302\260\302\260.Thus, the relative importance of the right side vanishes -~ AV1 ~ Ae~\\ and wehave

d/*(x,v,0 , _ ,. ,

44 Plasma Kinetic Theory I

In the next two chapters we shall approachthe plasma kinetic equation (3.26)from another direction,and shall use approximate methods to evaluate the cotli-sional right side. In Chapter 6 we shall take up the study of the Vlasov equation(3.27).

REFERENCES

[1] J. D.Jackson, Classical Electrodynamics, 2nd ed., Wiley, New York, 1975.[2] K. Gottfried, Quantum Mechanics, Benjamin, New York, 1966.[3] Yu. L. Klimontovich, The Statistical Theory of Non-equilibrium Processes in a

Plasma, M.I.T. Press, Cambridge,Mass.,1967.[4] T. H. Dupree, Phys. Fluids, 6, 1714(1963).[5] K. R. Symon, Mechanics, 3rd ed., Addison-Wesley, Reading, Mass.,1971.[6] F. Reif, Fundamentals of Statistical and Thermal Physics,McGraw-Hill,New

York, 1965.

[7] A. A. Vlasov,/. Phys. (U.S.S.R.), 9, 25 (1945).

PROBLEM

3.1 Klimontovich as ContinuityProve that the continuity equation (3.23) is equivalent to the Klimontovichequation (3.16).

CHAPTER

Plasma Kinetic Theory II:

Liouville Equation

4.1 INTRODUCTION

In addition to the Klimontovich equation, there is another equation, the Liouvilleequation,which provides an exact description of a plasma.Like the Klimontovichequation, the Liouville equation is of no direct use,but provides a starting pointfor the construction of approximate statistical theories. One of the most usefulpractical resultsof this approach is to provide us with an approximate form for theright side of the plasma kinetic equation (3.26),which tells us how the distributionfunction changes in time due to collisions.

The Klimontovich equation describes the behavior of individual particles. Bycontrast, the Liouville equation describesthe behavior of systems. Consider first a\"system\" consisting of one charged particje.Supposewe rrjeasure this particle'sposition in a coordinate systemiX^; then the orbit of the particle Xt(/) is the set ofpositionsxf occupied by the particle at consecutive times t. Likewise, in velocityspace we denote the orbit of the particle V^/); this is the set of velocities taken bythe particle at consecutive times t\\ these velocities are measured in a coordinatesystem v,. We thus have a phase space (x^y^ = (x1,yl,z1,vxi,vyi,v:i).In this six-dimensional phase space there is one \"system\" consisting of one particle. Thedensity of systems in this phase space is

tfOc\342\200\236v,.r)= 3[x, - X,(fMvi \" V,(0] (4.1)

Next, consider a system of two particles. We introduce a set of coordinateaxesforeach particle. Particle 1 has (x^v,) coordinate axes as before. Particle 2 has (x2,v2)coordinate axes that lay right on top of the (x^v,) coordinateaxes.The orbit X,(0iV^O of particle 1 is measured with respect to the (x^Vi) coordinateaxes,while theorbit X2(/), V2(t) of particle 2 is measured with respect to the (x2,v2) coordinate

4

46 Plasma Kinetic Theory IIaxes. We now introduce an entirely new phase space, having twelve dimensions.The phase space is

(\\1,\\l,\\2,\\2)= (xuyuZy,v]li,vyi,v2t,x1,y1,z.1,vXi,vv_,vZ2) (4.2)

In this twelve-dimensional phase space,there is one system that is occupying thepoint [x, = X((0, V! = V,(/), x2 = X2(f).v2 = V2(r)] at time /. The density ofsystems in this phase space isN(x^XJx2^2,t)= fi[x, - X.COMvi - V,(f)]\302\253[x2 \" X2(016[v2 - V2(r)] (4.3)

EXERCISE Show that there is indeed one system in the phase space byintegratingthe density (4.3) over all phase space.

Note that the density i^in (4.3) is completely different from the densityNsusedinthe previous chapter in the discussion of the Klimontovich equation. The densityNs in Ch. 3 is the density of particles in six-dimensional phase space. The density Nin (4.3) is the density of systems(each having two particles) in twelve-dimensionalphase space.

Finally, suppose that we have a system of N0 particles. With each particle /,i = 1,2, .. . JV0, we associate a six-dimensional coordinate system (x;,v;).Usingthese 6Af0 coordinate axes, we construct a 6JV0-dimensional phase space, analogousto the twelve-dimensional phase spacein (4.2). There is one system in 6JV0-dimen-sional phase space; therefore the density of systems,by analogy with the density ofsystems(4.3),is

\342\226\240v\302\273

iV(x1,v1,x2,v2 . . . x^.v^.O= JJ

Liouville Equation 47

Using aS(a \342\200\224b) \342\200\224bd(a \342\200\224b) to replace V, by v\342\200\236and similarly for V, so that forthe remainder of this chapter

V,

48 Plasma Kinetic Theory II

Then the Liouville equation (4.8)becomesdN \"\302\273 ,v\"

-fr+ 2 Vx, ' (v,JV) + Y, V\302\245;

\342\200\242(V,JV) -0 (4.14)i-1

BBGKY Hierarchy 49

continuity equation in 67vydimensional phase space of the form (4.14).Thus,/V\342\200\236must satisfy

dt+ t ^x,-(^A) + X V(V,/;0 = 0 (4.15)

where \\,{t)is,as usual, calculated from the Lorentz force equation (4.7)and thefields Em and Bm are the exact fields appropriateto the system that occupies thisparticular point in 67V(1-dimensional phase space.

We shall only be concerned with smooth functions/*-,,. Thus, we might think ofa drop of ink placed in a glass of water. The initial drop contains all those systemsthat have a finite probability of being representedin the ensemble of systems attime ta. Ignoring diffusion, the drop may lengthen, contract, distort, squeeze,break into pieces, deform, etc., as time progresses.However, the total volume ofink is always constant; the total probability is always unity. The convection of theprobability ink is expressed mathematically hy reversing the steps that led fromthe Liouville equation (4.8) to the continuity equation (4.14). (See Problem 4.1.)Equation (4.15)becomes

3/m,

dt+ X v V*. + X Vvv/v, = 0 (4.16)

which by (4.10) is

Wn.Dt

= \302\260 (417)

Equation (4.16) is the Liouville equation for the probability density/v\342\200\236. Thus, thedensity of the probability ink is a constant provided that we move with the ink.The probability density fNn is incompressible.

4.3 BBGKY HIERARCHY

As discussed above, the density/^ represents the joint probability density thatparticle 1 has coordinates between (x|,Vj) and (x| + dx[,vl + d\\{)andparticle 2has coordinatesbetween (x2,v2) and (x2 + dx2,\\2 + dvj), and etc. We may alsoconsider reduced probability distributions

/fe(x,,v,,x2,vj, . . . ,Xa,v^0 ^ Vk)Jdxk+ld\\k., . . . rfxM,rfvM/w,-: (4.18)which give the joint probability of particles 1 through k having the coordinates(x^Vj) to (\\i + d\\i,\\, + d\\j) and. . . and(xk,\\k) to (xk + dxk,\\k + d\\k),irrespective of the coordinates of particles k + ],k + 2, . . . , N0. The factor Vk onthe right of (4.18)isa normalization factor, whereul^is the finite spatial volume inwhich/*,, is nonzero for all x,,x2, . . . , xA.0(Fig. 4.2). At the end of our theoreticaldevelopment, we will take the limit N0 \342\200\224\302\260\302\260,V -* \302\260\302\260,in such a way that n0 \342\200\224N0/V is a constant giving the average number of particles per unit real space. Forthe present, we assume that fNl, \342\200\224Oasx, \342\200\224\302\261\302\260\302\260ory,\342\200\224\342\200\242\302\261\302\273orZj\342\200\224\302\261\302\260\302\260for any

50 Plasma Kins tic Theory II

Fig; 4.2 Finite spatial volum^f in which/V|. is nonzero for any x,, / = I A',,.

/. Likewise, because there are no particleswith infinite speed, fNil \342\200\2240 as vXl \342\200\224+\302\260\302\260or vyi

\342\200\224i\302\260\302\260or v7.\342\200\224\302\261\302\260\302\260for any /.

In this development, we do not care which one of the N0 particles is calledparticle number 1, etc. Thus, we always choose probability densitiesfNa that arecompletely symmetric with respect to the particle labels. For example,

/Vc (. . . z7 = 2 cm . . . zn = 5 cm . . . 0=

/a'\342\200\236(\342\200\242- \342\200\242*7 = 5 cm . . . z13 = 2 cm ... 0 (4.19)provided all of the other independent variables are the same. Here, we mustinterchange all of the / = 7 variables with all of the i \342\200\22413 variables. This meansthat when we set k = 1 in (4.18), the function fl(xl,\\],t) is (to within anormalizationconstant) the numbeT of particles per unit real space per unit velocity space.Thus, this function/|(x,,V|,7) has the same meaning (to within a normalizationconstant) as the function/J(x,v,7) introduced in the previous chapter in connectionwith the plasma kinetic equation.

To keep the theory as simple as possible, we shall ignore any external electricand magnetic, fields. We shall deal with only one speciesof Af0 particles; it is easyenough to generalizethe results to a plasma with two species ofNn/2 particles eachat the end of the development. For some purposes, such as calculating electron-electron collisional effects, the second speciescan be introduced as a smeared-oution background of density n0, which simply neutralizes the total electron charge.Finally, we adopt the Coulomb model, which ignores the magnetic fields producedby the charged particle motion. In this model, the acceleration

V,(0 - X a'> (4'20)where

BBGKY Hierarchy 51

is the acceleration of particlei due to the Coulomb electric fieldof particle/.Sincea particleexertsno force on itself, we use (4.21) only if / \302\245^j; if / = j, we useaf,

= 0. Equation (4.21) replacesMaxwell'sequations and the Lorentz force law.The Liouville equation (4.16) becomes

N N N

-% + 2 v, \342\200\242vXi/\342\200\2360+ \302\243X \302\273\302\253\342\200\242v,,/*. = o

52 Plasma Kinetic Theory II

\302\251= v-n\" t i vv,,/vr

*i fr\\

r %l+ J d\\Nvd\\Na 2, aM.>\" vv\342\200\236)n

+ jdxNodvm % aWo-VVj/W(, (4.27)

where the / \342\200\224N0,j \342\200\224N0 term has been discarded becauseajVllJV(l = 0. The secondterm on the right vanishes after direct integration with respect to dvXN andevaluationat vxs

= \302\261\302\260\302\260,etc. The remaining terms in \302\256,\302\251,and \302\251,after multiplicationby VN

BBGKY Hierarchy 53

For term \302\251we have

\302\251= yf/,,-1 \302\243

/ dxNrl dyj,^ dxNo dyNo aWo \342\200\242VV/fNo\342\200\242Fl

J

= yN

54 Plaama Kinetic Theory II

As it stands, the BBGKY hierarchy (4.33)is still exact (within the Coulombmodel) and is just as hard to solve as the original Liouville equation (4.22). Itconsists of N0 coupled integro-differential equations. Progress will come onlywhen we take just the first few equations, for A: ~ l,k = 2, etc., and then use anapproximation to close the set and cut off the dependence on higher equations.

From (4.33) the k \342\200\2241 equation is3,/i(xi.Vi.O + *i \342\200\242VXJ,

+\302\260

~

/rfx2

BBGKY Hierarchy 55

But by (4.35) this is justP*(x,y) = P,{x)P>(y) (4.37)

This separationalways occurs when two quantities are statistically independent;that is, the value of one quantity does not depend on the value of the otherquantity. Thus, it is always useful in considering joint probability distributions tofactor out the_pjiece that would be there if the two quantities were uncorrelated.Thus, for the dice we have

Pi(x,y) = PMPdy) + dP(x,y) (4.38)where dP(x,y) = 0 by (4.37). For a plasma,we define the correlation functiong(x,,vux2,vj:0by

/2(xi,Vi,x2,Vj,0 = /^i.VnO/^j.^r)+

^(xlVlXi.Vj.O (4.39)

This is the first step in the Mayer [6] cluster expansion.

EXERCISE From the definitions oifNls and/t, convince yourself that/2 has thesame units asfif,.

We are ready to insert the\342\200\236form,(4^39) into the equation (4.34) for /,, whichbecomes

3,/i(Xi.v..O + \342\226\274\342\226\240\342\200\242VX|/(+

\302\253oJdx2 dv2 aI2 \342\200\242V\302\245[r/i(x,,v,,0/i(x2,v2,0

+ g(x\342\200\236Vi,x2,\302\2452lr)]- 0 (4.40)where we have replaced (N0 \342\200\224\\)/V by n0 because we are interested only insystems with N0 \302\273 1.

Suppose one assumes that the correlation function vanishes. That is, weassumethat the particles in the plasma behave as if they were completely independent ofthe particular positionsand velocities of the other particles. This assumptionwould be exactly valid if we performed the pulverization procedure discussed inthe previous chapter, in which n0 \342\200\224\342\226\240\302\253>\342\226\240,e \342\200\2240, me \342\200\224\342\200\2420, A \342\200\224\302\260\302\260,n0e = constant,e/mc = constant, ve = constant, to, = constant, and kt = constant.Then eachparticle would have zero charge, and its presencewould not affect any otherparticle. Collectiveeffects could of course still happen, as these involve only/i andnot g. When we set g equal to zero, (4.40) becomes

3, A + v.-V,,/,+ [n0fdx2 dv2 \342\226\240,i/,(x\342\200\236v\342\200\236/)]

\342\200\242VTl/,(x\342\200\236v,,0

= 0 (4.41)But the quantity in brackets is just the acceleration aI2 produced on particle 1 byparticle 2, integrated over the probabilitydistribution/1(x2,v2,0of particle 2. Thisis the ensemble averaged acceleration experiencedby particle 1 due to all otherparticles,

a(x,,0 s\342\200\2360J

d\\2 d\\2 a12/,(x2,v2,0 (4.42)

EXERCISE Convince yourself that a is normalized correctly.

56 Plasma Kinetic Theory il

Then (4.41) becomes

B, /,+*\302\273\342\200\242VXi /, + a\342\200\242

Vy, / = 0 (4.43)which we recognize as our old friend 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 collisionat effects that arerepresented by the two-particle correlation function g. We would like to have atleast an approximate equation that does include collisionaleffects and that,therefore, predicts the temporal evolution of/[ due to collisions. We must thereforereturn to the exact k = 1 equation (4.40)and find some method to evaluate g.

Sinceg is defined through (4.39) as g \342\200\224f2 \342\200\224/( /(, we must go back to thek = 2 equation in the BBGKY hierarchy in order to obtain an equation for/2and, hence, for g. Setting k = 2 in (4.33) and using (A^ \342\200\2242)/V \302\273n0, one has

\302\251 \302\251 \302\251

3,/i + (v, \342\200\242V,, + v2 \342\200\242VXj)/2 + (\342\226\240\342\200\236\342\200\242Vv, + a21 \342\200\242Vv,)/2

C\302\256

+n0j rfx,rfv,(\302\253\342\200\236

\342\200\242V\302\245[+ a23

\342\200\242VYj)/3

- 0 (4.44)We have seen that it is useful to factor out the part/i/j of/2 \342\200\224ftf} + g, which

exists when the particles are uncorrelated. Likewise, it is useful to factor from/3the part that would exist when the particles are uncorrelated,plus those parts thatresult from tWo-particlecorrelations.This leads to the next step in the Mayercluster expansion, which is

/,(I23) = /,(I)/i(2)/,(3) + /10M23)+ /,(2)\302\243(13) + /,(3)\302\243(12) + ft(123) (4.45)

where we have introduceda simplified notation: (1) = (xl,v1),(2) = (x2,v2),and(3) = (x3,v,)- Equation (4.45) will be explored further in Problems 4.4 and 4.5.

Our procedure is to insert (4.45) into (4.44)and neglecth{123).This means thatwe neglect three-particle correlations, or three-body collisions. It turns out thatthese correlations are of higher order in the plasma parameter A; therefore theirneglect is quite well justified for many purposes. The resulting set of equationsconstitute two equations in two unknowns/! and g. Thus, we have truncated theBBGKY hierarchy while retaining the effects of collisions to a goodapproximation.

Inserting (4.45) for/, and f7 = / /, + g into the k = 2 BBGKY equation(4.44),we find for the numbered terms:

\302\251 \302\251

\302\256=/i(0/.(2) +/,{2)/,(I) +\302\243(12)

\302\251

\302\251= Ti \342\200\242Vx,/1(l)/,(2) + v, \342\200\242Vx.\302\243(12) + {1 - 2}

BBGKY Hierarchy 57

\302\251=

\302\253i2'

VV[/1(1)/1(2) + a,2 \342\200\242VV[^(12) + {I - 2}\302\251

\302\251=n0J

dl a\342\200\236\342\200\242VV| [/,(l)/,(2)/,(3) + /,(l)g(23)\302\251

+ /,(2k(13) +/,(3)*(I2)] + (1 - 2} (4.46)where d3 = d\\} d\\} and {1 ** 2} means that all of the preceding terms on the rightside are repeated with the symbols 1 and 2 interchanged. Recallthat #(12) = #(21)by the symmetry of /2. Many of the terms in (4,46) can be eliminated using thek = 1 BBGKY equation (4.40).Forexample,

\302\251+ \302\251+ \302\251+ \302\251= {/,(!) + v, \342\200\242Vx,/,(1)

+n0jd3 a,3

\342\200\242Vf, [/,(!)/, (3) + *:vVv/,= -

n0fd2 a,2 Vv. *(I2) (4.50)which is in exactly the same form as the plasma kinetic equation (3.26).

Most of the discussion in this chapter has been exact, in particular, thederivationof the Liouville equation and the BBGKY hierarchy. Even theapproximationsthat lead to (4.48) and (4.49)are extremely good ones, for example, 1 \302\253 iV0and the neglect of three-particlecollisions.By contrast, the approximations neededto convert (4.48)and (4.49)into manageable form are sometimes quite drastic and

58 Plasma Kinetic Theory II

less justifiable, as will be seen in the next chapter. Further discussion of theLiouville equation and the BBGKY hierarchy can be found in the books ofMontgomery and Tidman [7], Montgomery [8], Clemmowand Dougherty [9], Kralland Trivelpiece [10], and Klimontovich [11].

REFERENCES

[1] N. N. Bogoliubov,Problemsof a Dynamical Theory in Statistical Physics,State TechnicalPress,Moscow, 1946.

[2] M. Born and H. S.Green,A Genera/ Kinetic Theory of Liquids, CambridgeUniversity Press, Cambridge, England, 1949.

[3] J. G. Kirk wood, /. Chem. Phys., 14, 180(1946).[4] J. G. Kirkwood, / Chem. Phys., 15, 72 (1947).[5] J. Yvon, La TheoriedesFluides et /'Equation d'Etal, Hermann et Cie, Paris,

1935.[6] J. E.Mayer and M. G. Mayer, Statistical Mechanics,Wiley, New York,

1940.

[7] D. C. Montgomery and D. A. Tidman, Plasma Kinetic Theory, McGraw-Hill, New York, 1964.

[8] D. C. Montgomery, Theory of the Unmagnetized Plasma, Gordon andBreach,New York, 1971.

[9] P. C. Clemmowand J. P.Dougherty, Electrodynamics of Particles andPlasmas, Addison-Wesley, Reading, Mass., 1969.

[10] N. A. Krall and A. W. Trivelpiece, Principles ofPlasma Physics,McGraw-Hill, New York, 1973. %

[11] Yu. L. Klimontovich, The Statistical Theory of Non-equilibrium Processes ina Plasma, M.I.T. Press, Cambridge, Mass., 1967.

PROBLEMS4.1 Continuity vs. ConvectiveDemonstrate the equivalence between the convective derivative form of theLiouville equation (4.16) and the continuity equation (4.15).

4.2 BBGKY Hierarchy

Integrate (4.32)over all xNr2 and v^,^ to obtain the k = N0 \342\200\2243 equation of theBBGKY hierarchy,and compare your result to (4.33).

4.3 Normalization

Explain in detail the normalization of (4.42).

Problems 59

4.4 Three-Point Correlations (Coins)In (4.45) we define a three-point joint probability function /3 in terms of theone-point probability/,, the two-point correlation function g, and the three-pointcorrelation function h