APPLICATION OP THE WIGNER FORMALISM TO A

SLIGHTLY RELATIVISTIC QUANTUM PLASMA

APPROVED:

i£). Major Professor

Minor Professor

y . ' J / .

Director, Departrrjsnt of thy sic a"

f the Graduate School

APPLICATION OP THE WIGNER FORMALISM TO A

SLIGHTLY RELATIVTSTIC QUANTUM PLASMA

THESIS

Presented to the Graduate Council of the

North Texas State University in Partial

Fulfillment of the Requirements

For the Degree of

MASTER OF SCIENCE

By

John H. Harper, B. S,

Denton, Texas

August, 1967

TABLE OP CONTENTS

Page

LIST OP ILLUSTRATIONS iv

Chapter

I. INTRODUCTION 1

II. THE WIGNER FORMALISM 7

The Density Matrix The, Wigner Distribution Function The Kinetic Equation Reduced Distribution Functions Inclusion of Relativistic Effects

III. DEPARTURE FROM EQUILIBRIUM AND DISPERSION • RELATIONS IN THE A3SENCE OP EXTERNAL FIELDS 23

IV. DEPARTURE FROM EQUILIBRIUM AND DISPERSION RELATIONS IN THE PRESENCE OF AN EXTERNAL MAGNETIC FIELD i|3

The Classical Limit The Case of^ug

V. CONCLUSION ^

BIBLIOGRAPHY £6

4 1

LIST OP ILLUSTRATIONS

Figure Page

1. Coordinate System With Along the Z-axis . . 32

2. Coordinate System With K In X-Z Plane . . . . I4.fi>

iv

CHAPTER I

INTRODUCTIOH

The general description of a system of many interacting

particles and determination of its properties Is a very

difficult subject. As expected, only a few idealized sys-

tems have been solved exactly. There are many approaches to

the problem involving approximations or insoluble symbolic

equations with various initial assumptions and a maze of

pathways leading from them. Perhaps the most useful goal

Is the development of a relativistic quantum statistical

mechanics. A brief discussion of these terms, taken in

reverse order, follows.

Statistical mechanics attempts to describe a system of

particles without relying on its detailed microscopic behav-

ior, but rather predicting probable behavior based upon

statistical information or some averaging process. Although

the detailed coupled, equations can be written down, it is

hopeless to grind out solutions involving the initial con-

ditions of 6N +1 variables for an N-particle system where N

may be on the order of 10^, give or take a few orders of

magnitude. In spite of simplifying assumptions, statistical

mechanics can produce quite accurate results. Therefore

some form of statistical mechanics appears to be a logical

foundation for a many-particle system. In tills study the

properties of a system are described by a distribution

function i d e p e n d i n g on the spatial coordinates

••>%«) * the (•?•,-p**—.-M > and

time. The physical meaning of this function is that the

quantity

Jp, c)px-- <Jpw = TT

represents the joint probability that at time t particle one

is within a range e) of ^ and cJ, of , particle two is

within a range c!!. of ^ and Jip of -px , and so forth for

each particle. The integral of this expression is normal-

ized to unity, indicating that each particle is located in a

coordinate and momentum range in phase space. The function

is useful in the sense that the average of a macroscopic

quantity is obtained by integrating the product of £ with

a corresponding weight factor in an appropriate manner

(1, pp. 5-7)• This idea will be developed more fully later.

Since the particles are microscopic entities, quantum

effects may become important. These arise mainly from the

indistinguishability of identical particles and Heisenberg's

uncertainty principle. The former concept leads to the

conclusion that a system of identical particles can be de-

scribed by one of two statistical descriptions, depending on

the spin associated with them (2, pp. 91-137). This study

will be limited to Permi-Dirac statistics, which includes

electron systems.

3

The density matrix (3> pp. 90-109) provides a statis-

tical framework for quantum-mechanical systems. This is

not, however, the quantum analog of classical distribution

functions. In order to develop such an analog so that

classical limits may be easily taken, a transformation is

applied to the density matrix. The resulting quantity is

called the Wigner distribution function. There are many

forms of the Wigner distribution function and methods of its

application. Besides the usual quantum formulation

(4, p. 221; 5, p. 361), many researchers go over to the

formalism of second quantization (1, 6 ), where the number of

particles can vary. The latter approach introduces fields

and their infinite number of degrees of freedom. This study

will pursue the former course, dealing throughout only with

particle coordinates and without using special evaluation

methods such as diagram techniques.

If the particles under consideration have velocities

on the order of magnitude of the speed of light, then rel-

ativistic effects become important. These conditions are

found at high temperatures in the study of the interior of

white dwarf stars (7> p. 20) or at high densities on the or-'

29

der of 6x10 electrons per cubic centimeter (8, p. I4.28).

Relativistic particles may also be emitted in beta decay.

Since the purpose of this study is to investigate the prop-

erties of a randomized gas, such relativistic systems as

particle accelerators where there is a high degree of

k

particle correlation will not be considered. It should be

kept in mind that relativistic effects for a classical

system can be obtained in the limit of

There are various problems involved in relativistic

theories. The concept of action at a distance is damaged

due to retarded potentials arising from the finite times of

propagation. A many-particle theory will not necessarily

have to be cast into covariant form since there is no

Lorentz frame of observation in which relativistic effects

vanish. In addition, only low-order approximations will be

carried out here. There are many starting points for devel-

oping a relativistic statistical theory (7» p. 2-8}, most

beginning with some assumptions about the system^ Hamil-

tonian. It is not obvious, however, that a well-defined

Harailtonian exists in the relativistic case. The method

used in this study involves the two-body Darwin Hamiltonian

(9) correct to order Qf)^ . Leaf (10) shows that while this

provides an adequate description for the case of a particle

pair, a Hamiltonian does not exist when higher-order terms

are included: momentum is no longer conserved. In spite of

the fact that many-body Darwin Hamiltonians have been de-

rived (11), no published calculations have been made using

them. Some work has been done with the two-body form

(12; 13, p» 8J|-90), but not in the form of a kinetic

equation.

5

The following development includes some of the devices

mentioned above. A slightly relativistic fermion gas is

described by the dynamical theory obtained from the Wigner

distribution function. The problem is approached in a

self-consistent manner including the two-body Darwin Harn-

iltonian. The goal is to find the departures from

equilibrium and dispersion relations for wave propagation

in til© *

CHAPTER BIBLIOGRAPHY

1. Balescu, R., Statistical Mechanics of Charged Particles, New York, Interscience Publishers, 1963.

2. ter Haar, D., Elements of Thermostat!sties, New York, Holt, Rinehart and Winston, 19*5^

3. Roman, P., Advanced Quantum Theory, Reading, Mass., Addison-Wesley Publishing Company, Inc., 1965.

I4.. de Boer, J. and G. E. Uhlenbeck, editors, Studies in Statistical Mechanics, Vol. I, Amsterdam, North-Holland Publishing Company, 1962.

Klimontovich, Y. L. and V. P. Silin, " K Teopws CneKTpoS Boib^jqemw ManpocnotrwecKiAx CweTeM," Akademiia Nauk SSSR (Doklady), LXXXII (1952), 361-361^

6. Brittin, W. E. and ¥. R. Chappell, "The Wigner Distri-bution Function and Second Quantization in Phase Space," Reviews of Modern Physics, XXXIV (October, 1962), 620-627.

7. Meixner, J., Statistical Mechanics of Equilibrium and Non-equilibrium, Amsterdam, North-Holland Publishing Company, 1965.

8. Jancovici, B., "On the Relativistic Degenerate Electron Gas," Nuovo Cimento, XXV (1962), i|.28-i4.55«

9. Darwin, C., "The Dynamical Motions of Charged Particles," Philosophical Magazine, XXXIX (May, 1920), 537-551.

10. Leaf, B., "On the Relativistic Motion of Two Charged Particles in Classical Electrodynamics," Physlca, XXVIII (February, 1962), 206-216.

11. Kerner, E. H., "Hamiltonian Formulation of Action-at-a-Distance in Electrodynamics," Journal of Mathematical Physics, III (1962), 35>-Ul.

12. Krizan, J. E. and P. Havas, "Relativistic Corrections in the Statistical Mechanics of Interacting Particles," Physical Review. CXXVIII (December, 1962), 2916-2921}..

13. Gartenhaus, S., Elements of Plasma Physics, New York, Holt, Rinehart and Winston, 196ij..

CHAPTER II

THE WIGNER FORMALISM

The Density Matrix

Consider an isolated system on N Identical, interacting,

structureless particles described by the coordinates

and momenta p t l . In general the system can be de-

scribed quantum-mechanically as a mixture of pure states

each of which has a statistical weight \A/C0, by

it I

where Vco W O * O and ^ ^ £ £ ) * 1 is.1

Each of the pure states can be expanded in terms of a

complete set of orthonormal eigenvectors of an operator.

V"' I C„'\

Then the expectation value of some operator 51 in the pure

state V , } is

t-mn < n > , - ( v , s w " ) - i m -• I 0 ' &

The system average of SI is formed by the linear combination

< s i > - - I s u L w o c T ' c ! 0 . i = 7 1

Now define a quantity called the density matrix of the

system for which the matrix elements of this quantity p are

8

/°™»* L w o c ^ c " ' i

Then <£!>-Z = £ (ftpL. -- Tr (H^. «*/» v m

Knowledge of the density matrix therefore provides a

method of calculating the expectation value of a given

operator. The transition to the continuous case is not

difficult.. If the system is in a pure state the density

matrix becomes, in the «p -representation,

Pnmx £ w

Pn*-" C* C„ - \Cnf SO 0 * * 1'.

In the Schrodinger picture operators are time-indepen-

dent and state vectors follow the Schrodinger equation.

- i k f

where is the system Hamilton!an. Then

- I C % . , K c . O - Z c „ ( < L , ^ )

* L n

• a (<*.,$)• at *

Similarly, .

Therefore 3 ~

t L \ y^K C* Ch + (' KicCK^]

"k. P

^ l l . C PoK^Kw\ ~^«K.P K*n)

In terms of commutator brackets this can be written

2"t i - i i M i - (ii-D

The Wigner Distribution Function

The elements of the density matrix in coordinate

representation and Dirac notation, » a r e defined

over a 6N-dimensional configuration space. In order to

develop a theory for quantum-mechanical systems analogous

to the classical method of distribution functions, it is

advantageous to define a quantum distribution function

F t ^ . f ;t) closely related to the density matrix, but

defined over a 6N-dimensional phase space. A function which

provides such a description will not be unique, in the sense

that some other function obtained by another method may

serve just as well. The method below is a straightforward

development of such a function assuming only the definition

of a phase-space distribution function and the correspon-

dence between classical variables and their corresponding

quantum operators. The result is identical to the function

10

proposed in 1932 by Wigner (1), so It will be referred to

as the Wigner distribution function.

Let be a classical function defined over

phase space with the expectation value . According to

Weyl (2), if this function is the Fourier transform of a

function , then the corresponding quantum

operator is defined by the following relations.

G«fUp}> \

The quantities and .p are the operators corresponding to

the classical variables and .p . The first equation is

solved for .. *4

N(W,i?i)>(Siy-\GeiU?i)e IT ,11, dp.

and then substituted into the second equation, giving

C~(k 315 MV —•— I C(i"l s fi +

GiHUpiV{*rtf')Go$ifi)e T e ' IT ji.jpijj.-jsi.

The matrix elements of & now depend on the positive

exponential. This can be simplified somewhat by the use of

McCoy's theorem: if the commutator of two operators SI, and

fl2 is a complex number c. , then a,

e * e e, e . Let and $1%* | . Since

11

eA P ^ e

Ah eB v

The p o s i t i v e exponent ia l then becomes

V ^ O V r ^ - ^ „ ff

- ft V *

The matr ix elements are then

s ince

Now expand the ke t 1%;/ In momentum e i g e n f u n o t i o n a .

where <p;|<E-> • ( ir t .?1 e ^ " * 5 - <tV»fl>* i s t h e l n n 6 r

product of pj wi th ^ •

The remaining matr ix element now becomes

A

Now the matr ix elements of G can be w r i t t e n /

" ( l i S r l

12

( 2 n W Y

Tills expression is inserted into the coordinate represen-

tation of the density matrix for the expectation value of ®

<G>* > ( & p )

* H"k

': (5SS®i IT a?idp:

In order to simplify the argument of G- , use the

substitution ^ % * %

V 2. '

The old variables can b6 written ^ %

Since the Jacobian of the transformation is , then

<G>* ds 5 55GttjW P i ' T i 1 pi%• V ? ) ?

If FCifJ.lp/t) is a distribution function it satisfies

<CG> GCHUf}) F(UUpte) |I. &%i JPi •

Subtracting these last two equations and requiring the

integral to vanish independent of G , the quantum distri-

bution function is found to be

13

- •&] e l* h '% i j u * . ("-2)

This is the Wigner distribution function. Pool (3) shows

that the Weyl correspondence is the only connection between

classical and quantum mechanics that allows the calculation

of expectation values by integrating the corresponding

classical function with the Wigner distribution function

over phase space. Hence, any other quantum distribution

function requires a different correspondence between class-

ical and quantum maehanics. He also claims to have the only

rigorous proof that the Wigner function is square-integrable

over phase space.

One of the drawbacks of the Wigner distribution

function is that it is not positive definite, as classical

distribution functions are required to be. In fact,

Sudarshan shows that a positive definite function may be

given a negative expectation value (Ij., pp. IfQ-lQQ). This

is not unexpected since the concept of a quantum-mechanical

phase space is not well defined, due to the uncertainty

principle. But generally these difficulties refer to sit-

uations which are immeasurable in the sense that the

irregularities occur below the uncertainty limit = •

Also, the distribution function can be made positive definite

w _

(5, p. 238). The ultimate evaluation of a theory is based

on its ability to describe and predict, and on this point

Ill

the Wigner distribution function is quite satisfactory.

Some of its properties are listed below. It should be noted

that the first two integrals are positive definite since

the result does not involve the uncertainty principle.

« configuration-space probability density

= momentum-space probability density

ao

00

The last relation shows that FU^Upj.t) is real.

The Kinetic Equation

The time development of is given by the

partial time derivative using equation II-l.

c>F _i

where *

" I tikr\[(vW0 *

TT jfiJpi.

Each of the density matrix elements above can be cast

into the form appearing in the Wigner function by transfor-

mations. These are

3 H

In both cases the Jacobian is 2 . Using these substitutions

the kinetic equation can be written

SF eft i e ^ ' ^ ^ V a ^ f l pia4t»(

Inversion of equation II-2 gives an expression for the matrix

|a-|t) - J e ^ ' T K i & U f W it ap;

Now the kinetic equation becomes

f)] ec^Ti r lTci cjp:ciQ;aft

This equation is simplified somewhat if the ^-dependence

appears only in the exponentials. A convenient transformation

^ > ± ( 3 - ^ IT *«j *

The Jacobian of this transformation is (a) , so

16

=(If 5 w & f e

• ftf 5 H ( 5 ' H p H j ) e 1^ 3''^' v < f'" P i^ 1

Letting p"-P , the kinetic equation finally becomes

(II-3)

Reduced Distribution Functions

Most of the macroscopic quantities of interest whose

averages are calculated with the Wigner function are func-

tions of the positions and momenta of a small number of

particles, usually one or two. Then most of the coordinates

and momenta can be integrated over immediately. This leads

to the definition of distribution functions which depend on

" the positions and momenta of a few particles, called reduced

distribution functions.

If G s G(^pt), then

^G~> • j GC^.pJ FUiUpi.t) | <J|,clp;

• I \F(HUpLt)TT4|;ciji.]

The quantity in brackets is a function of ^ and p,

only, providing the suggested reduction. Since the N par-

ticles are assumed to be indistinguishable, there are N

17

such, integrals which will give the same result since the

subscript one refers to a single particle, and not particle

number one.

where

% ^ r c

In general, the s-body distribution function is defined

r- x M T r- / n

(%.$*.- .p.,--,p*;t) " (IH-S)U

In terms of the density matrix this becomes

S

" Cn-sV. TT df-

" C2rf^ e «Pi'fi<%^Hilpsl^-HS^ciT;

where K=l,2,...,s and

. n\ gfc" TT P :

Since eft. J ;i2 % r , th0 one.feody analogy of

equation II-3 is (Il-lf.)

I Hj)] F„(a,i> 0 v«H>»vci;-r>)

18

i It^j/p.t) * i^i[0<ferV,,3l,.A,^tj.a,-.,f»^V^frH,i!!,,-.#.^ 1,f1,....R,')]-

^{(a/Oe i % T ITjia^p-clfc^: (ii-ij.)

Inclusion of Relativi3tic Effects

Since electromagnetic interactions are transmitted with

velocity o, the speed of light, the force on a particle at

some instant depends on the positions of the particles at

some earlier time. An appropriate mechanical description of

this retardation effect can be formulated by relating the

field and potential of a particle to its position and veloc-

ity at an earlier time.

The equation for the electric potential in terms of

charge density p,

-e-%- -top •

can be solved for <t> as

r per, t - ! » ) , ,

" J i%-tt ^

Expanding the denominator in a binomial fashion for small

velocities with as.© of conservation of charge, it becomes

s'

e £ £ ^ ><KvO *

g- e 3 {

* \%-V \%^\ J

19

where the first term is the expected Coulomb potential and

the second terra is a relativistic correction which can be

neglected for small velocities.

The solution of the equation for the vector magnetic

potential requires only the usual first term

l[j01 A. _ evl_ ~ c ] ^

since it is of the same order as the preceding correction

term. The electric term can be simplified by taking advan-

tage of the gauge invariance of the potentials (6, p. 181)

•fr-* •••cfi A; — Ai- If

by choosing

* • &

The electric and magnetic potentials now have the form

Ai acVivii fe t ^ i 3 /

It is easy to show that the divergence of A is zero and that

its curl is the magnetic field, JL._e. the cross product of v'

and the electric field.

The Lagrangian for a system of charged particles with

relativistic corrections is, approximately,

L- * H-A - e+]

v - £ v

- i y

20

where A w i s an external magnetic vector potential. Retaining

terms through order (cf and omitting the rest-mass energy,

the system Hamiltoniah is found to be

<1

v i i v"4, v- , 1 V e*" , £ v \0ffio ' 2_ 2.<w. • L b » 5 * i L , " h S - L , ^ •

The velocities va can be found in terms of the canonical

momenta by

- /•! v -\ + -A-f , S t p»»svL" *w>U*2iO z ^ u t a J ° «*•

A first approximation is found by neglecting terms with a

coefficient c?.

ptt* m v a - | A w t

W ) -

and substituting this into the previous equation to obtain

^ , y JL. f £kl£je>t . », X L e T

pa.'lfV\VaV ^WlV- J t-* 2.C IV) | |

The velocity is, through order 9

Va_ * *vf ^ i l^-^| """

The Hamiltonian can now be expressed in terms of the

canonical momenta by

x • lAMtff1" - U s *

v- €~ ev v 1 (p&-'£Ac )• Cpt~ c.Ao«)

^ ' M S " * toV«W ^ - V l

+ Cp ~e,Ao,t) • Cpy |

21

The following notation will simplify some of the

equations to be used in later development. Let — - -

i (H-5)

Then the Hamiltonian has the form (II-6)

31* L l 7MI " 8 WC-" L>mlc*" %"• If* % J

The third term was first obtained by Darwin (7)J so this

expression is often referred to as the Darwin Hamiltonian.

Note that for jf£* p; •

CHAPTER BIBLIOGRAPHY

1. Wigner, E. P., "On theQuantum Correction For Thermo-dynamic Equilibrium," Physical Review, XL (June, 1932), 7U-9-759.

2. Weyl, H., The Theory of Groups and Quantum Mechanics, New York, Dover Publications, 1931*

3. Pool, J. C. T., "Mathematical Aspects of the Weyl Correspondence," Journal of Mathematical Physics, VII (January, 1966), 66-7&T

I|.. Rose, M. E. and E. C. G, Susarshan, Lectures in Theoretical Physics, Vol. II, New York, W. A. Ben jamin, Inc., 1962.

5. de Boer, J. and G. E. Uhlenbeck, editors, Studies in Statistical Mechanics, Vol I, Amsterdam, North-Holland Publishing Company, 1962.

6. Pock, V., The Theory of Space, Time and Gravitation, New York, Pergamon Press, 1959.

7. Darwin, C., "The Dynamical Motions of Charged Particles," Philosophical Magazine, XXXIX (May, 1920}, 537-551.

22

CHAPTER III

DEPARTURE FROM EQUILIBRIUM AND DISPERSION RELATIONS

IN THE ABSENCE OP EXTERNAL FIELDS

The Hamiltonian of a plasma consisting of N charged,

weakly interacting particles in the absence of any external

fields is found from a special case of equation II-6.

U - I [ i; - s ||£|" f k^ i 5'' v] * 2 ^ ]

The kinetic equation is then formed by inserting this Ham-

iltonian into equation II-I4. to obtain

i (sM. (Ms? „ OAs?- j£f \

. (iVH)-(QrH-QO P4-(Srtf,-Q lartt.-Sil l^rlx-Qil3

. PrCfr+Vr,-Q.O] y I ex ^ e ij J^frxLtorta-flir

P -) G TT Jfi;

In terms of the two-body distribution, since the equation

involves only sums of pair interactions at the most, it

can be written

23

Zk

3 t ~ ^ - " n i J l Zw(N'i) 5}rWVCN-0 ZKYI L lQ,-%%*Oal tfvVf.'Qtl

, (i^.H§,-Vr~5^ !>.<&,-V?,- Ce-k>(g^r r4) K-(S.+V.-flQ'l \&r\x~8i\3 ~ \&+ \V&P J

\ ev e7- i K _ ^<«^.-(P,-f)} . . _ i W i ^ A t ]] e jKS^eiP.apj^Jx

(in-i)

I f the p a r t i c l e s are not s trongly i n t e r a c t i n g , a f i r s t

approximation of 5a. i s a product of one-body d i s t r i b u t i o n

func t ions kC&.Gx.nA.t) * ^ (Q././O

where ~ ^ , ^ 0 + "HSc&.'O

The term -fo0?0 i s the equil ibrium d i s t r i b u t i o n funct ion and

:K<5,P,-t) i s a small cor rec t ion measuring the departure from

equil ibrium which i s assumed to be small , s o f « f 0 . Retaining

f i r s t - o r d e r cor rec t ions ,

$*(&AI£,Pi;t)»£.Gi,,D£,GP»0 + $0ClP,OH<SiA,t) + kOft.O

The l e f t -hand side of equation I I I - l w i l l then contain

only $ . On the other side both • $ < M | H o 0 f t i ) a n d t e r m s

vanish, as shown l a t e r .

The equation I I I - l can be transformed in to a p a r t i a l

a lgebraic form which i s eas ie r to work with than the integrjo-

d i f f e r e n t i a l equation as i t appears. The new var iab les w i l l

a l so have some use fu l physical s i gn i f i cance . The procedure

i s to ca lcu la te the Fourier transform F(te,p,t} defined in

the following manner: ( r F (Xft) - (210^ e <j|.

25

The transformation Is obtained by multiplying the

equation by and integrating over £ . Each of the terms

is evaluated below.

I. * ^ * St Kfc,p/0

Ix'tiCm)1] *2WUN~0 ii( &.,©*,PMPi,t) z e

^ 5 e d " 3 ^ r p ^ ( P , " ? ) V ^

3 i*C2tTfJ <2 *Jke^ CS,i>)l

-XobSe'H m~& f'*>A5'^ - \ ^ ^ ^

It should be noted that unless specified otherwise,

the term obtained in integrating by parts which is evaluated

at the infinite limits will be zero due to either boundary

conditions on the distribution function or evaluation of a

delta function where its argument does not vanish.

The next term in III-l to consider involves laborious

expansions and combining. In order to save some tedium in

the next chapter* it will be evaluated with an external

magnetic potential which can then be set equal to zero for

this chapter.

26

T I ± i K I R-^rlA(Qr^}H . _ I3" K C ^ J "SmVCN-O JiCcftpi.R.P .t)

It ia sufficient to ohoose A(|)13.fcx% where B is the

external constant magnetic field. It is easy to show that

the condition B'V*^ is then satisfied. The following

symbols will b© usods

<*•- V\ P z

A r k U o - t f ^ A-C. A*c

rp,3t-|A.T-lP-s-SaJ -

- " e C p ^ A - C P ^ 4 o ( P - * ? A z - ( P - 2 ) • % A t ( M 1 "

_ltfj A" A ' ( M • 4 a £

* pW-p-s .H(p.3)^ZPV * { U - f - ^ ^ - 3 - " 6 1 3

-lA-f - c-p-A-a-£•=]*} * SUa'-iXS^-T-Ia'^a-S^c^I

jtP -'ZP-S. . tAP *<L~?-a-2-02)~(A^1A-c*C*) (P-2P-5 +*V)2C1

- ^\U*T.U -»e)U P *c'f-Ai -6J>UMXc.e)(A-p -if -> a-* -c-4 c3 1

e ^CP-3-HP+<^] * [^1"Ca-PX6P)+(X P)C^*^ + A"5)CC*0}

- | ^ (^ C)IAx+C?-]] + i(^c)Lp^4-CA-^lpV]+2.CP-^Cc-2)-2CF4XA P)}

27

+ Z(X-c)CX-P') -2(^-cUc-?)l]

Each of these twenty-four integrals can be evaluated

using appropriate vector identities. One example will be

given and the others will be quoted in the order in whioh

they appear. The Fourier transform is taken later.

Iv6 * ?<. ^ 1 (iTM) ^ ^ ~ olfi.dOxcJilcl Jr

" I W c ? W

-kV" [ 2 iU-(3 ) -(P-pVi _ x |fevnV(an>tj (S»P,t) (B*5'5> 6*^*35 cJadP^^

•)££• Ffc s - n - 5 £lW6T f-CP"f)i *ilfc^(irr)6J5.(Q,m ~ B* 5 J• Bx e dWg«>r

" M-jj - f H C P - j O e ' 1 4 V JSdfjg

* u T ^ S s ] B'l S'5-j - fp" t6*g)l e'514 ? ) J3j§

• 6 - K 6 " P

13j, * 2t«V f~ i3* H

28

JLSL - § dl c , - - r t I v ' 1 p * ^

I v - ^ f e i - C 6 ^ ) - ( 8 x p ) ^ ( ^ , 0 + O i P

I3 j S * iZvcF ~ - f x 5 *^CB-jpx^S.(^pi tO]

1 M ' B- 5p 5.($.?,*>

1*,* * tttfe C&'P * B • (6 * ^ ^ S.<$•?*)

1 ^ * 4-w"'c' ^""f

V • i ^ ? f " 6 ' 4 ^• s ' tS'f ' t ' )

I y , • "iifc* f l 4 5 , t ^ f ' d '

-•fte s _ 3 s S r . A

I%1»%' f ' '%%H^*•

*~€- 9 i W H * f *

i v s • Q$A F • U , i & f ^ 4 ( H ) * + t H J 4 * < « H

I , . , . * ^ f e , C B " ^ t f ; % < . t f f ' - t >

I^.n * •xrn'C 6* (8 "?v l - f Sf * 2j> £<vf'Tl3

I , , * * S S 5 ? S"gVB>(6' ' t ,V^i5,Cvf '^

29 0* X c r

lip* 6 4 ^ f"&*l^'^'P'r)

x^a, * \Gm\i5" b * V c&*$ia]

iw- * ^ 4 ' W ^ ' t&^*!,v These integrals are then added together and combined

to form the expression

laa 2 f e { " C f + IcC^-I^yCS**> Jp + |(S*A>pS,

* \ A ' C 4 + l c B A | p t ^ * | •&* 5p

~^CfxfeV Jp(l| +fcB*^^,

+ ! i B - ^ C & * § j ^ ( *lcp4tt*f-a^ + tf-CA*5H,

- , e. ^ n \ 1 (f c.p " 2>| v C^ c otpJ^tj'

The change of variables given in equation II-5 will

further simplify this equation by eliminating -p . The

chain rule of differentiation gives

c) _ e. < A ^ — St. at cat"^>jT "" 2>t,

jr- Ij "* Jt" 4 " l U * " h) ' Sr

30 3 ^ 3

2>jr

The integral then reduces to the compact form

Is® zjnJ%- £<($.p;t)

In the case of no external field, %-*0 and the integrals

I^and I3;i are retrieved. Since this equation contains

only in the function f, and in derivatives, the Fourier

transform effectively replaces derivatives with respect

to ^ by -if£. The final transformed form is then

The terms in brackets in equation III-l can be divided

into two groups, those which are multiplied by ^ and allow

immediate Integration over ij and 5, , and those that are

multiplied by ?) . Replacing g by a derivative with respect

to the exponential and integrating by parts, these terms are

te. c? KCxnJ6Jl* z \S,+ f-0zl J

*t. \ £ 5 Jl } t -* V»VOn)3)3$ U^Vr-cM i

In each of the integrals arising from the terms in

brackets of III-l, can be replaced by since

one group of terms cancels in pairs when integrated over (3

with its infinite limits because the form of the terms is*

the same, and the others vanish because the terms can be

combined into the form of the divergence of A» which is zero.

31 L r f>.pv * «* —

The 5 and Q, integrals have already been performed. Using

the substitutions

?-Qx dx-cl|

which is common in-the following integrals, it becomes

It 2mVk(2nf'lifi^fexM U^Ja)

The X -integration is performed as follows:

U i e i f c x J x - i n x £ l W V * > X-o 10

«o

f*\9 J<f49olx

' . ^ l i e ^ s r n k x ^ f 1

The term e"* is a convergence factor. It can be shorn by

more rigorous means that this gives the correct answer.

- 2iuev f - _

I,* mv-iicun)5 T,% diiL JWPi

* w v i V F(K,i,rtdf,

0^ f j F

V w 5 q w ' ) i | - UP,)W5,,it)eiT'<f'-FVK? dMP,di<Kd5

* W c ' d m ' j J* • d e ; k" 4.(PJeiT'tK ?t*6WA,t)eifc®,d%4cipltRdj

The X-integration is performed as follows:

32

V ^ -4r I - r4 — - »-< LjT \ 2>x 1*1 £ dx • -1 l€ iXi e 4x » -i-fe

I r v ^ e ^ n ? *11 $0CfO SCP.-f A%) S i k A t ) e^"° l J^olP.JP,

-Cne?Ji „ r -3 l i f + J ? x

I a * A J i ! 4 r t ^

^ ^VnVU^TiJ4^ €^'% ^ot?0^T ^ € ^ S l c S Q ^ I ^ X

The i n t e g r a l involving 5? can be w r i t t e n i n dyadic form.

I n order t o eva lua te t h i s i n t e g r a l , the matrix form of a

dyad i s used. Choose a coordina te system as shown i n Pig . 1

where £ p o i n t s along the z - a x i s . Then in terms of c a r t e s i a n

coo rd ina t e s , «~a A X-Xe

€ a ( s i n$ cos1?, sin® sin<i>, cos® ) = un i t

vec tor

&

Pig . 1—Coordinate system wi th Je along the z - a x i s

33

-te-X * -k* 6i*£ x -k*C«o©

(sin2<0 cosS sin2<0 oos<e sin<?> sin$ cos^ cos^

sin2<& cos<e sin<f sin2® sin2<p slnft 6os&- sin<f

sin©- cosS cos<p sin® cosfc sint cos2<&

srn<0d<Bcl<pcix

The integrations involving off-diagonal elements are zero

due to the ^-integration. „i<r X clX |j Co <f d<f

* 1 r i X t l X U l " / ) e L f e ^ /

4rr * \e

The other dyadic integrals are evaluated similarly.

1 . )£*•

T _tU ^s,>3* k.1"

I a5 w f r o i oCp.) e^'a^(p,-f4te)4(5JLJp,!t)

-2mV*- r _ n O o\ _ ^ vnV C-p^ I^O-] "(o o-i) PC^Pa.^^^-

3k ex f 3 t |Y-3x) _ A i-CP.-p51 iH-« - ,£ .x

V W ( W j 3 | ' e o^4 i , I

= W c f a n ] t j 4 ^ r ^ ' ! k < M e ! f ' t P , " f * H c < w ^ ^

• -i t- ( £ s) i("f.) S (Vf * 1 W f&*#i eit5- M M ,

- ^ W f ^ j O 4 .

I 71 i t ^ n !^|f-^i elT"(p'? ellc'^ c i & d ^ ^ f

• * •ktof \ Si etfcs fetf.) eite'®2 e^'(* P 1 ^ d M ^ i *

• e'14'5* Stf.-f 1 VvO

* W f 3 ^ ) \

These integrals can now be inserted into equation III-l

to obtain the kinetic equation. It can be simplified some-

what by recalling that a vector may be written as the product

of that vector with the unit dyad. Then the z-z elements

cancel out. The net effect is the removal of the

z-component of P; i.e., P will have no component in the

direction of propagation, which is the direction of the wave-

vector- -j£. The equation is now /

~ 5^" U ^ p - W - h + d e f • ?-(stl)] FAS, P.tf J P

3$ This equation can be made more algebraic by removing

the partial time derivative through a Laplace transform.

This is used instead of a Fourier transform because the

lower limit will provide an Initial-value form for the

equation. The Laplace transform G is defined by

When the equation just derived is multiplied by e. and

integrated over "t, the first term is the only one which

requires any manipulation.

\a * F(£ip^ eT,wt [ - £ F(k,p,t)

Then the kinetic equation becomes

il05" T?{l" S w W I G(K,p,u)

" f'?•CkU)]«K.R<*)Jf

This can be solved by dividing both sides by the coefficient

of G , but the same function still appears on the right-hand

side. The following symbols will be used:

A 5 oc=- (mo1 1 = F(te,jpjO) M = \oo%)

H(f)s )]

36

GCp' A acpj G(p)dp + ckj\ P*J G-(p f>

^GC-p^-p * * A^&(^)d£^G(p)dP + d A d|5 5 P-/ GrfWdP

Except for the dummy variable of integration, one of the

factors on the right-hand side is the same as the left-hand

side. Using the symbols

J, * Jf J 3(f) dip

this can be written

\r/aw£ • rri * JWf)d|. 1-AJ,

This is substituted into the equation at the top of this

page to eliminate one of the troublesome integrals. A

similar procedure is used to eliminate the other integral

problem.

GCj5> B(f)+ o,Aaip[T^3lSdp/A(|y) jo' + f]*$p./£GCp)dp

S f ' U U p J f • Sf^3f f)d P -

+ *a) f.% P-ai

Sip-M G(^)c^2 \ f>- 9cp)aj5 *• bipdf]*

GC^> IK-p) + ^ - *A Atf)l_ KJU, <Af' Mp) p • jf\*

U - ^ A ^ /itp-oL^Aap' Uf ^ - f ' f ' { I f

37

This is the desired equation. It describes the

behavior of G» which is the Fourier-Laplace transform of

the function S, which described the departure from equili-

brium of the system. If the initial departure F(fe>p.o} is

known, the integral can be evaluated explicitly.

The equation describing departures from equilibrium

may have singularities arising from two sources, the

numerator or denominator. The former case will arise from

particular initial conditions which give rise to transient

oscillations damped by free motion of the particles. There

Is nothing fundamental about these oscillations. The latter

case is more interesting since it depends on the system's

parameters, being independent of the initial disturbance

except in the way in which may be restricted.

These singularities may give rise to steady, growing,

or damped oscillations at natural frequencies which are

roots of the dispersion relation obtained by setting the

denominator equal to zero. The resulting dispersion relation

gives the relation between frequency and the wave vector

in terms of the plasma frequency

where n is the particle density. The plasma frequency is

a fundamental quantity found in all plasma descriptions,

corresponding to the collective oscillation of a homogeneous

plasma set up by motion to screen out a charge distribution.

38

The dispersion relation arising from the denominator

WUi is

UrteM 1- w e ) co- *

The plasma under study is assumed to be composed of

fermions. For small temperatures the Fermi-Dirac equilibrium

distribution function: can be approximated by

fop-where H(js-f,vf 1 l f ' P 1 4 1 ^

I 0 1 ? Ipl^lpM

and pP is the Fermi velocity.

The function is normalized such that

"f

Yi- fMf *

The integral will be evaluated in the long-wavelength

approximation, i.e. for small The following relations

will then hold:

m w ^

2rvi\o *•

The denominator can then be approximated by a binomial

expansion. Since £ is small, terms will be neglected which

involve products such as

M \ *(!?.

i * 11- i P — i - ^ V 1 Ti, / I-1 W'cx 2i»fc?i [' IvftT I1 2H.V-J W i 5wWJ

39

where cos®.

x Hp r h V | * + s « w *** Wtf-

-- ( s f [ 1 r l ( l ? i ] • 4 ^ & » . l " , ^ ^ "S<l) - lXwl>Tl

Since the terms in brackets which involve £ are assumed

to be small, co can be approximated by on the right-hand

side. The same result can be obtained in the same approx-

imation by use of the quadratic formula.

cos 4 1 w * -7 w - - u&] The non-relativistic limit is obtained by letting o».

This agrees with the dispersion relations obtained by

other methods (1, p. 332; 2, p. 361).

Another dispersion relation can be obtained from the

kinetic equation. This occurs when the inverse matrix

becomes infinite, or when the determinant vanishes due to

the matrix relations .m

^ 4 ^ " <^^*1 > £ * where 4 is the transpose of the cofactor of ^ . The

required condition is then

C/ioo\ MTTe1 V , AOO'VTs. A \ . .. v ,1 -^P4fe)~^P+\-V) 1 ^ det \ fe) • W P - f e W W f *lPwF> J u>- l£f " 0

The second term in brackets makes no contribution

since it has only a z-component due to integration over <p

and is multiplied by a vector which has no z-component.

ko . iTe1- ^ ~noo\- "ScCP-^V ^ P'UloS)P CO-^P

yv>

The small terms In the denominator have been neglected

since there is already a small coefficient in front of the

integral. This result will then apply to first-order

corrections only. The dyad can be written

~ f\° °\

{?% fxR, P*P*\ / W $ go<p r«\ h9ces><f %\nW£$<os>

P'Cooo^ P * I ^ R,P;t j " P I Cc6<> sirt'© Siril(f> cc6© sin<f

\ 0 o o J \ o O o

The off-diagonal components vanish due to the ^-integration.

2 jf Co >x /»p. j"ir J'l*

XX~ RwVnVuf) ]04t^Lt} Cttft + £>55l?>9Co# Co<^

+ 2>Cfet>T sivftcoS* cosl(f]

'Ifesfel C Polp i i l + W f c + ^ <**«

The other integral is performed in a similar manner.

4 B-sfe ( S r i K K ^ * C E W

Solution of the determinant equation yields a degenerate

pair of solutions which correspond to

kl

Drummond indicates a procedure by which damping terms

can be obtained by assuming the frequency to be complex

(3, pp. 19-21). This Landau damping is expressed by a term

€** where co RT -A U-Ti1"

and is the Debye screening length. In the approximation

of small •€ the damping term becomes insignificant. Since

the limit of small M is taken above, damping effects are

not expected to contribute to the waves. For larger values

of , damping becomes predominant.

CHAPTER BIBLIOGRAPHY

1. Balescu, R», Statistical Mechanics of Charged Particles, New York, Interscience Publishers, 1963#

2. Klimontovich, Y. L. and V. P. Silin, "K Teopwn CneKTpos ManpocKonwMtCKWX C cTem Akademiia Wauk SSSR

(Dokladv). LXXXII (1952), 361-361}..

3. Druramond, J. E., Plasma Physics, New York, McGraw-Hill Book Co., Inc., 19&1.

i. o

CHAPTER IV

DEPARTURE FROM EQUILIBRIUM AND DISPERSION RELATIONS

IN THE PRESENCE OP AN EXTERNAL MAGNETIC FIELD

The inclusion of an external magnetic field complicates

the situation somewhat since the Hamiltonian now includes

not only the canonical momenta p but also the magnetic

vector potential in the form of equation II-£.

The terms which eventually are placed on the left-

hand side of the kinetic equation are not too difficult to

determine. One method is to expand JT in terms of p and A

and substitute this into the original kinetic equation.

£-h. f ' H + " E

E,= 2m \ carrf tS| 5>*T Jig JfJa i

« - £• • i- x

= -Ztoc\

ex ~ * W o*ap»(g^)

ij-3

1

14J-

These terms can then be regrouped In to the form

E -

The r e l a t i v i s t i o c o r r e c t i o n to the k i n e t i c energv was

worked out i n genera l i n Chapter I I I . Combining t he se ,

the l e f t - h a n d s ide of the k i n e t i c equat ion becomes

( I t + m l > 3 ^ * P * &" i i ' 2 j f e J t * \ (|g'* ] ) * i p *

Appl ica t ion of the Fourier and Laplace t ransforms

e l imina tes d e r i v a t i v e s with respec t t o ^ and "t . The use

of these t ransforms i s s t r a igh t fo rward and leads t o

j h i ] ] )

The r igh t -hand side involves i n t e g r a l s of the Darwin

i n t e r a c t i o n terms. Some s i m p l i f i c a t i o n w i l l be gained by

the t r ans fo rmat ion i®,,©»,?,,?*} """* . Since the

Jacobian of t h i s t r ans fo rmat ion i s one, jp can be replaced

by in the Wigner equat ion:

^ 3 £ ciirj1 A A ,JTx+^R>xS0

- p _ V, - o > itvfS,- )*T-(SrX-+ ,S*L®ril)3 - Q . , A* >U > , iOe A

I n each case the only term of the two-body approx-

imation which does not produce a vanishing i n t e g r a l i s

4o^M(1S$). The i n t e g r a l s t h e r e f o r e have the form

& . r * e \ [ J T V C S , ( J t e cI..3 St. IWc1] t dQido^Jr^^^tol^

A similar expression applies to the Coulomb terms. This

is just the previously derived equation in the last chapter

with p replaced by jf . By taking advantage of this cor-

respondence the kinetic equation for a plasma subject to

an external magnetic field B can be written as (IV-1)

(,iu> •*" KV!C JT*'ciji ~ + ~ ^ ) ]^ =

Since this equation is rather difficult to solve in

general, .several special cases will be considered.

The Classical Limit

The first special case is the classical limit of 1r*0

Equation IV-1 can be made into a first-degree equation by

assuming the magnetic field to be small so that terms

involving B1oan be neglected. The coordinate system is

shown below.

Pig. 2—Coordinate system with in x-z plane

U6

The magnetic field will be directed along the z-axis.

Without loss of generality $ can be chosen to lie in the

x-z-plane.

The equilibrium distribution function can be expanded

in a Taylor's series. Recalling that V o ,

-> ---fc-lit)

In this case equation IV-1 can be written

( l*4* ** vvic" * ~ *!) G C **0 ~

K l ^ o ) 4 S * ^ f J l l " w & x C o o SW-T)]GC TTJW )<JTT

The matrix term appears different in this case since it

was defined to have the effect of removing the component

of TP parallel to •£. Now introduce the cyclotron frequency

. e6 w«' wic

The kinetio equation can be written in the form

t W j . J V t t . i t 1 (l-•SS&) tCfefl,") =

F l M , o > T O h s h~ k '•* ->jl a t ) at

This can be solved by introducing the integrating factor

Now multiply the equation by the integrating factor, divide

by -vc and integrate with respect to <? from -*> to <p . This

kl

will make the lower limit on the left-hand side vanish

upon evaluation. Define

When both sides are multiplied by the inverse of the

integrating factor after integration, the equation for G is

~ . f FtM.o) -MUfr'-'f) - mi a U j1 e J \

<f

\ 1 b" fc § A - * ? j r J " V -A?

The solution of this equation proceeds in the same

manner as in Chapter III. With the definition of

I;--]i =

r'* ri

v **» -« Ki

the solution for G- can be written

I,*B^ C t e . ^ K o o V ^ - S t c a ) J U 3W-- - f ^ V 1

f* **

- vJ^ ^U)JjV - t^^3(ir0c)j/jjT-H 0 ? ^ ^ ' J ^ H ^ ) €^ "Vc^]

*il + ] ? + 4i,S^r°(k £r-+ taa? irr)$oLx")e^'^Jjr''] tr-M *

•^(jrOe (>„ +\?xcgv -if*)e

The dispersion relations are found by setting the

denominator equal to zero and the inverse matrix infinity.

ij.8

In the classical case "$o0t) is the Maxwell-Boltzmann

distribution. Since the interaction term is considered

small, use the non-relativistic function

where n is the particle density and p>*

kt is the reciprocal

of the product of Boltzmann's constant and the absolute

temperature of the system.

With this relation the dispersion relation from the denom-

inator is

1 * CJC& J OP.V^^JCL) 6. —do

Let - (p'~<p -> dip * cl

The exponentials can eventually be simplified somewhat

by expanding them in terms of Bessel functions.

.U*ih»V * • I e'Tj.rt

3 ^ L nuc-u>* ig&aCa} Jiidjr jL

49

The sum Is not easy to carry out, but in the long-

wavelength approximation of small the Bessel function

can be expanded in second order in M..

J*(x) 3 (x') m [l" + •"}

WVU)c. S ' C n t 2 > w t J \ •* 5

The only contributing terms to the order of approximation

assumed are the values for n= -1, 0, 1. Using this with

the relation j.a>o- c-o"j„(x)

the integral becomes

O

( W w V p ) ,44?

where the symbol £l- 3 ^ was used. The dispersion

relation can now be written as

10L (1 * * " 0 ( 1- 2 * V p

The wave dispersion for waves parallel to & is found by

setting \-o.

X r . CO - tOp (1-

so

The waves perpendicular to B have the relation

(jS ~ ( u > p ( 1 " 2'wiCp)

There will also be dispersion relations obtained by

setting the determinant of the inverse matrix in the

kinetic equation equal to zero.

e L JV^"JV * 1 ^ r ° v TfN SiftaC /.

J

Nothing new is involved in evaluating the integrals

outside of the fact that dyads are formed and care must

be taken to keep track of which variables go with which

integrations when the integrals are compounded. The

results of the integrations are

c . /[ £&]cos<( o o \

drtUSo?)t|lwp(''S^) o &..%» o \*

This corresponds to three roots obtained by multiplying

diagonal elements when the three dyads are added together.

In this case the degeneracy of a double root is removed.

The first two frequencies will differ by a factor of cos<* •

If ot*o they will not coincide. These frequencies are

• 1 ^ ' " ^ ,

51

These frequencies differ little from the correction

terras of the original dispersion frequency for the system.

If cl*o , these are the only frequencies obtained from

the inverse dyad. If ot*o , there is a non-trivial root

from the z-z element of the dyad. It can be written approx-

imately as

o 1 " ,. [S + k -£^ »«]

« « u£smV - sm a .

Care must be exercised with the apparent discontinuity

with which these frequencies appear as <*-»o .

The Case of -fe" B

Another special case which can be handled in explicit

form is where oC- o and quantum effects are retained. For

a good degree of accuracy compared with other terms which

will arise, the first two non-vanishing terms of the Taylor

expansion of the equilibrium distribution function will be

retained.

If the magnetic field is again weak, then waves which

travel parallel to the field satisfy the kinetic equation

(toj + |c5f *8 - +

- Hkj.o)

52

G can be solved for in the same systematic manner as

in the previous problems. The only difference will be

that the components will be a little different and that

J \ /

If &$0 is used as a symbol for the two expanded terms of

the equilibrium distribution function, the solution is

( H l M . u O * h £ x ( 5 M , <2, ^ " W

3(3rtiif *

U"* Jtufff"

' —GO

The main dispersion relation is found by setting the

recurring denominator equal to zero. The Fermi-Dirae

function is again approximated by a step function for the

integration.

. r)_ ^nieM C - £Ju-^S» 9 Ujk)] (*'-*)

1 c 6l, = UJk_\z% J ^ °l<f

, ^Jiex\ 1 f s> W 3^ ~\e v-

The Integration variables may be changed from cylindrical

to spherical coordinates to make some of the integrations

easier. The first Integral becomes

T - TfeM'' c^r 5 ^ _ -1

•*•1 * W " Co- 3Cjv>) = & L3 4 *""J

U3* -

53

The second Integral can be written

- WeMt ^-ICjtV i, I •U" 24- J) C0-1 >

This forms the second derivative of a delta-function in

the numerator. It is evaluated by two successive integrations

by parts. The result is

T . wpM 3 H 5 yy i lj.~ Zo1 L 2-° JTf1 1 4 ^ It \auk3-J

These two solutions are then added together to obtain the

dispersion relation

11 1 f - * L - 3 [MR f

This' result is approximately the same as was obtained

in Chapter III for the case of M.* £. The slight discrep-

ancy arises somewhere in the second term of the expansion

of $o(3ri when used to evaluate the integral.

Again, there is another dispersion relation obtained

from the inverse dyad. It will have components only along

the diagonal with the value one in the z-z position as was

found in the corresponding case in Chapter III. The

Integration is straightforward If the techniques of

evaluating the previous integrals have been mastered. The

result is

t-i, idL 1 5*1,1. + i g a M

There are no magnetic effects on the frequencies in this

case, which is not surprising.

CHAPTER V

CONCLUSION

There appear to be no essential difficulties in applying

the Wigner distribution function to a plasma in the first-

order self-consistent approximation of two-body correlations.

Equations of departure from equilibrium both with and

without an external magnetic field were derived in the form

of an initial-value problem. If the form of an initial

disturbance F(ft,p,o) is specified, these equations can

be solved to obtain- an explicit expression for the non-

equilibrium behavior of the plasma. The theory is not

complete since no mechanism is employed to cause return to

equilibrium, i_.£. the equations are time reversible and no

H-theorem applies.

The dispersion relations agree with accepted results

in the appropriate limits. No published dispersion relations

were found which included relativistic effects in' an explicit

form. This is apparently the first attempt to derive such

relations from the Darwin viewpoint. Besides the expected

dispersion relations, other resonant conditions arose from

the Inverse tensor obtained from the equation for equi-

librium departure. They appear to be low-frequency terms

of relativistic origin. In the limit c+oo the terms vanish.

55

Dispersion relations of this form have not been previously

reported.

A magnetic field does not alter the frequencies of

waves parallel to it, but increases the frequencies of

perpendicular waves. Relativistic corrections lower the

frequencies. Such an effect is due in part to the more

sluggish motion of the particles as their mass increases

with velocity.

BIBLIOGRAPHY

Books

Balescu, R., Statistical Mechanics of Charged Particles, New York, Interscience Publishers, 1963.

de Boer, J. and G. E. Uhlenbeck, editors, Studies in Statistical Mechanics, Vol. I, Amsterdam, North-Holland Publishing Company, 1962.

Drummond, J. E., Plasma Physics, New York, McGraw-Hill Book Co,, Inc., 1961.

Pock, V., The Theory of Space, Time and Gravitation. New York, Pergamon Press, 1959.

Gartenhaus, S., Elements of Plasma Physics, New York, Holt, Rinehart and Winston, 1964.

Meixner, J., Statistical Mechanics of Equilibrium and Non-equilibrium, Amsterdam, North-Holland Publishing C ompany, 1965.

Roman, P., Advanced Quantum Theory. Reading, Mass., Addison-Wesley Publishing Company, Inc., 1965.

Rose, M. E. and E. C. G. Sudarshan, Lectures in Theoretical Physics,, Vol. II, New York, W. A. Benjamin, Inc., 1962.

ter Haar, D., Elements of Thermostat!sties. New York,, Holt, Rinehart and Winston, 1966.

Articles

Brittin, W. E. and W. R. Chappell, "The Wigner Distribution Function and Second Quantization in Phase Space," Reviews of Modern Physics, XXXIV (October. 1962). 620-627.

Darwin, C., "The Dynamical Motions of Charged Particles," Philosophical Magazine, XXXIX (May, 1920), 537-551.

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Jancovici, B., "On the Relativistic Degenerate Electron Gas," Nuovo Cimento, XXV (1962), I}.28-l|.55»

Kerner, E. H., "Hamiltonian Formulation of Action-at-a-Diatance i n E l ec t rodynamics , " Jou rna l of Mathematical P h y s i c s , III (1962), 35-fcL.

Kliffiontovich, Y. L. and V. P . S i l i n , "KTeopvw CneHTpoB MdKpocKpnusecKwx Cwae " Akademiia Nauk SSSR

(Doklady), LXXXII (1952), 351-361;.

Krizan, J. E. and P. Havas, "Relativistic Corrections in the Statistical Mechanics of Interacting Particles," Physical Review, CXXVIII (December, 1962), 2916-2921+.

Leaf, B., "On the Relativistic Motion ot Two Charged Particles in Classical Electrodynamics," Journal of Mathematical Physics, III (1962), 35- 11*

Pool, J. C. T., "Mathematical Aspects of the Weyl Corre-spondence," Journal of Mathematical Physics, VII (January, 1966), 66-T&•

Wigner, E. P., "On the Quantum Correction For Thermodynamic Equilibrium," Physical Review, XL (June, 1932), 714.9-759.