On the properties of high temperature plasmas : a convergent collision integral and hybrid-kinetic stability theory

On the properties of high temperature plasmas : a convergent

collision integral and hybrid-kinetic stability theory

Citation for published version (APA):

Mondt, J. P. (1977). On the properties of high temperature plasmas : a convergent collision integral and

hybrid-kinetic stability theory. Technische Hogeschool Eindhoven. https://doi.org/10.6100/IR109078

DOI:

10.6100/IR109078

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Published: 01/01/1977

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ON THE PROPERTIES OF HIGH TEMPERATURE PLASMAS:

A CONVERGENT COLLISION INTEGRAL AND HYBRID-KINETIC STABILITY THEORY

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECI-INISCHE WETENSCI-IAPPEN AAN DE TECI-INISCHE HOGESCHOOL EINDHOVEN, OP GEZAG VAN DE RECTOR MAGNIFICUS, PAOF,DR. p, VAN DER LEEDEN, VOOR EEN COMMISSIE AANGEWE2EN DOOR HET COLLEGE VAN DEKANEN, IN HI:T OPENBAAA TE VERDEDIGEN OP

VRIJDAG 3 JUNI 1977 TE 16,00 UUR

DOOR

JOHANNES PETRUS MONDT

GEBOREN TE AMSTERDAM

DIT PROEFSCHRIFT IS GOEOGEKEuRD OOOR DE ?ROMOTORRN

Dr. h-. P. P.J .M. Schram

"Ah God.<Je: h'l.oW that art Is long and Short Ou~ life Often enough my analytical labours Pe~te~ both brain and heart. Ho~ hard it is to attain the means By wnich one climbs to the fOunta~n head; Before: 3 poor devil can reach the half~ay house. Like as not he is dead. I'

(I<"'gnel; in Goethe I s"Faust")

Dedicated to my deal; parents and to my late grand~father.

Opged,agen aan mijn tieve ouoers en aan mijn overleden grootvader.

TABLE OF CONTENTS

Samenvatting .•••...•....•...• , ...•...••• , " , .,'" ••••• , . 1-3 Summary ...•..•....•..•. ,., .•• ,., •••••.•.•...•.... 4-7 FAR! 1:8-98

Ch , 1. In t roduc t ion •...•....• , .•.. , .•.•• , . , _ , •..•.•...•. 8-1 3 Ch.II.Som~ nonunitormly valid app~ox~mations •.••...•....•... 14-25 Ch.lll.A spa~ially uniforro app~oximation for the binary correlation

. . . 26-40 Ch. IV. Thermodynamic Equilihrium ...•... , ..•.. 41-42 Ch_V.F~operties of th~ lin~ariz~d colli~ion integral. " •••... , •• ,43-47 Ch.VI.On th~ high-treG~en~y electrical conductivity in a

spatially homogeneous plasma ... _ ...•... 48-56 Appendix A •••••••••••••••••••••••••••••• , •• , ••• , ••• _, • ,.' ••• , ••• 57-65 Appendix B . . • . . . • . • . . . • • . " .• , _ •.. , . . . • . . . . • . . . 66-69 Appendix C ••• , •• _ ••• " •••••••••••••••••••••••••••••••••••.•••.•• 70~86

Appendix D •••••••••••••••••••••••••••••••••••••••••••••••••••••• 87-95 Appendix E ••••••••••••••••••••••••••••••••••• , . , . _ •••• ' . " •••••• 96-97 List of R~ferenc<i's of Fa-,;t I. . . 98 PART II: 99~ 189

Ahstract ... 99-100

I. Introduction ...•..•...• , ...••...• , •.. _ .... 101-103 2.Assumptions and Ba.ic Eq~ations •• , •...•.•...•...•..••....••• 104-107 3.Equilibrium Properties, ... 108-113 4.General Stahility Analysis:

4.A:Linea-,;ized Equations ... I 14-118

4.B:Perturb~d Electron Prop~rties ••... , ...•.. , ...•. 119-126 4. C: Per turbed IOn P);"operti<i's .. , •.• , ••. , .•••. , •. , •. , ••• , ••••...• , 127-1

n

4.0:General Eigenvalue Equation ...•... · ..•... 132-137

5.Effect of finite ~l~ctron temperature on low frequency

long wavel~ngth modes ~n a near-th~ta~pinch with sharp boundary and r·

.!R

<;

1.

L:l P

5. Fl.: Bas i C orde-,;ing ••..•....••..••• , . " , •...•• ,.·· •• ····•·•·· l 38~ I 39 5.B:Simplified Eigenvalue Equation ... ,., •.... 139-143

J.C: Solution to Eigenvalue Equation .••••••••...•.•...•• 14")-147 S.D:Discussion of Dispersion Relation ...•••..••...• 147-148 6. Influence of ion energy ani.socropy on theta-pinch st3bi 1 i.ty

properties . . . 149-157 7.Effec:ts of finiCe electron ternperatur" and ion anisotro!'y on

low fJ:eq ... ",ncy long wav .. length modes in a ne.~J:-theta-!,inch with shu!, hOUlldary and rL;fRp~l ... 158-166

8.A critic;ll comment Oll Ohm's law:finire resistivity effects on the stability prop~rt~es of long w3velength electromaglletic

peTturoations in a near-theCa-pinch with 5h3rp boundary ... 167-176 9.Conclusions . . . • . . . • . . • . • • • . . . 177-179 Appendix A . . . • • . . . . • . . . • . • • • 180-181 Appendi)(' lL . . • . . . • . . . • . • . • . . • . . • . . . 182-185 List of References (If P~:rt U . . . 186-188 figure Captions ... _ •.••...•... 189.

~/-SAMENVA'I'tING

Voor een ge~atigd verdund,moleculair gS$ kunnen de

~scroacopLsche trsnspm;-tcoetticiencen Horden herekend uitgaande van de Eoltzmann

~ergelijking.I-5De ~otsings­

integrsal in deze v~rgelijking repreBenteert slechta de invloed van binaire hotsingen.

In een vol~edig geioniseerd gas of plas~ is de ~rscht

van de wi~selwerldngsl'otentiaal dermate groot dat veel deel tj ,",s geli jkdjdig interagerel;l: Rie:rdoor i" een vanuit Sen hoger standpunt ontwikkelde theotie van hotsingen in een plasma noodzskelijk.Een uitgangspunt voor een derg~liike theorie is de zogenaamde Born-BogolVuhov-r.reen-Kirkwood-Yvon ~ierarchie,een oneindig stelael van vergeliikingen welke de een-deeltje~ verdelingsfunctie in verband hrengt

. . _ . 5·'f .

met aIle wult11'1e correlatles ln de faSeruLmte. Verscheldene hotsingsintegralen voar een plasma ziin a.m de hand vsn deze

,,, -111

hierarchi .. verkregen. In ait l'roehcl<jrift ,,,ordt ech~er I.dt de hierarehi~ aen ruimtelijk uniforme benadering voor de b~naire correlatie funetie sfgelejd.met een axuliciete evaluat ie van de

nauwkeurighe~<;\

van het re"ul taat

.'9

Hi~rbi

i

is geen gebruik gemaakt van ad hoc a£kappingen van de hii!rarchie vergelijkingen,ol' van in<::onsistente machtreeks-ontwikkelingen in de plasma parsrneter.Evenmin ziin vorm~n van functies gebruikt welke ontleend ziin aan argumenten welke slechts geldiS zijn in het geval van thermodynami5ch evenwicht.Da re~ulterende uitdrukking voor de vriie energie in ~hermodynamisch evenwicht blijkt eindig te ziin en in overeens·telllnlirtg met het resul tas~ verkregen op gron!! van evenwichtstheod.e.De verkregen hotsingsintegrasl convergeert volledig en is tOegepast ter berekening van het elektrische geleid~ngsvcrtllogen V"r> ruimteliik hornogene plasmas onder invloed van hoog£requentc velden.

; / ,

-Behalve het hierboven be.chreven prOCes. van verstrooilng door COUlomb interactie zi;n er andere proceSsen in plasmas welke aanlciding gcven tot tI"anBPortver~ch

i:i

nse 1 en: gO If-c\eeltj es en golf-golf interact.lcs.A'ls het plasma slechts zwa1< turbulent is. hoeft in "'HSt. .. instcmtie slec',ts de invloed van lineahe golf-dee1tjes interactie~ OJ'> de ont"ikkeling

20 . ,,2 van de e;;.n-deeltje" v"rdeling.functi e te wOl'den beschouw<l, Dit r"sult.~ert in eel' quasi-J,i.neaire heschrijving van

(anomaa!) transpo1:t,welke heschrijvi.ng h"staat uit ",en ldnetische vergeIijking waarin de diffusietensor afhangt Van de

g:n)eisndh"i. d verkregcn ~li t lineaire stahO i t"i tsanal),se. Derhalve , . lineaire st~bl1iteitsan81y5e niet aIleen nodig om de ti jdsc"a .• l te berden imilrin het pl,'a"ma

uiteenv~lt ten gevolge van macro-instabiliteiten:het is ,",venzecr een noodza\<"liik" stap in het onderzoek na"1: anorn .. le t~"nsportvergchi.inselen in z",ak t.urhul"nte

pla~maR,verc,orza8kt door micro~insLabiJ,iteit~n.

II) het hiizonder kan genoemde anal-yg" dicnen t€r "erk I aring van anomadl transport ge~urend~ de im'Plos i.e-fase en de fase welke da('lr onmiddellijk o'P vol-gt in £len 'Pinch

eJ{periment.~5 In dit g"V('l,l kan de aand"cht ni.et h"uerkt blijven tot el-ect1:oStatische verstoringen en is iuist ess€ntj eel ~en volle,jig elecrromagnetische hehatldBling nODdzak"ll 11<,.1l'a.,. de vBrllOuding h~ta van kineti.che tot magnetisehe druk il'\ een t~oisch pinch experiment vri.; hoog is"

In deel Tl van dit pcoefschdft ""dt electro-m"3gnetische stilbiJiteit van hot~ingnou hoog-beta p1asmas

o~der.ocht op basis van laa~ste ord~ hybried-kinetische the-ori.e (D' 11'1'oli to en Davidson);:-" De frequentie van de ver-storitlgetl 1<an hierbij v"n de zelrM orde van grootee zijn

-

5-als de ionencyclotronfrequentie of lager.Dit hybried-kinetische model best~at uit een vol1edige Vl~50V beschrijving van de ionen en een dri£t-kinetig~he ver-gelijking voor de electronen. rwee eigen"'auclevel;geli,ikingen worden afgeleid;een voor verstoringen ten opzichte van een lineaire sehroefpineh evenwichts~on£iguratie gekarakteriseerd door een isotrope verdeling van de ionenenergie,Z}en een voor verstoringen ten opzichte van een mogelijkerwijs door eell anisotropie in de iOne" energie verde1ing gekarakteriseerd' theta-pinch

evenwic::.ht~Deze

\/ergeliikingen tonen dui(leiijk (Ie invloed van eindige electronentemperatuur,elec::.tronkinetische effecten langs het magnetische veld en anisotropie in de ionenenergieverdeling aan op electroroagnetische stabi~itei •. In de limiet waarin de electronentemperatuur en de ionenergie-anisotropie naar nul ga.ln.wol;(lt de v1asQv-fluidum vergelijking van Fre idberg ,;tIs speciaal genl teruggevon(ien

~1' ~t

»1ii kt numeriek tot de mogeliikhp.den te he~oren de stabil\teit~­ criteria en groeisnelheden uit deze eigenwaardevergelijkingen

. 28·,Ll

t

,

te berekenen,zonder verdere beperk~ngen.Met hehulp van analyt\Qch~

methoden zijn e>:pliciete uitdrukJdngen VOo~ de groeisnelheden verkrege.n voor het "eval van langgolvige verstoringell van e.en scherprandige evenwichtsconfiguratie.De frequenties liggen hierbij in het magnetohydrodynamische gebied en er is gebruik gemaakt van een additionele Eindige

l,.armo~

Straal

ordening,~C

en,voor een ander gehie.d van parameterS,van Freidberg's

probHrf~nctieroethode.2.

7

Tenslotte wordt aangetoond dat voor zeer langgolvige ver~toringen in een scherprandige oineh met kleine he1ische magneetcomponent het hybried-kinetische model moet en kan worden veralgemeend ten einde de invloed op stabiliteit van effe~ten verband houdend met de eindige resistiviteit in re.kening te. hrengen,Een eigen-waardevergelijking voor dit gev.a1 is afge1e.id.

SUMMARY

Jo a moderately dilute,molecular gas the macro~ccpic transport coefficients can be c .. lc.ulat",d from the BoltZlnann equation.' -

~

The collision integral j.n this ",quation

represent~

binary collisions only.

In .. fully ionized gas or plasma the ~ange of the inter-partiele potential is so lHge that many particles simult .. n.,ously interact." Therefore, a mor., sophis tiGated the"ry of colI is ions in a pla~ma is needed.A point of departure for such a theory c~ th~ so-called Born-Bogolyubov-Green-Kir~wood-Yvon hierarchy, .. n infinite set of equations,relating the one-body di5tribution

.

. .

;-'1 .

fLLnction to all nluJ.t1ple c.orrela.tLon5 in phase spa~.e. ThH hierarc.hy is formally equi.val~.))t; with the Liouville equation,

de~<:ribing the exact dynamical evolution of a many-particle system. Several collision integrals fOr a plasma have been obtained

~o _i~

[rom thi£ hierarchy- In the present thesis,however,fram this hierarchy a spatially uniform ~pp~oximation for the hinary corl"elati()n fl)nction b derived,with an explicit evaluadon of the accuracy of the result:

r

l'!o use i5 made of ad hoc truncations of

the open set of hierarchy ~qu~tio~s~or of incon~i8tent power series

expansions in the plasma p .. rameter.No functional formE are used that arc only valid in therl1lOdynamic equilibrium. The ,esuHing expression for the free energy in thermodynamic equilibrium is shown to be Hnl. te and i.n "greement wi th equ i Ii brium theory. The resulting collision integral converges completelY and is used lo calculate the high-frequency electrical conductivcty in a spatially homogeneous plasma.

In addition to the abovadescribed dissipatiQnal process of Coulomb-collisional scattering,there are other processes in pl"5mao that give rise to transport phenomena:wave-particle and wave-Wave interactiorls.If the plasma is only weakly turbulent.,then in leading order only the influence of ~inear ~ave-particle interactions on the

-

5-development of the one body distribution has to be

consider~d.~o-~~ Thi~ ~e~ults in a quasi-linea~ description of (,'anomalous") tJ:ansport, consistin~ of a kinetic equation in which the diffusion tensor depends on the growth ~ate obtained from linear stability analysis. TheJ:efore, linear stability analysis is not only

neceSSa~y to p~edict the time scale at which the plasma undergoes complete disJ:uption due to ~ross instabilities: it i5 also a necessary step in the investigation of anomalous transport processes in weakly turbulent plasmas due to micro-instabilities.

In particular, this analysis may 5erVe to explain th~ anomalous transport phenomena that are observed during the implosiOn and

immediate post-implosion phases of a pinCh experiment.

~~

In that case tbS attention cannot be restricted to electrostatic ~des, hut it is essential to give a fully ele~tromagnetic treatment, since the ratiO

S

of kinetic versus magnetic pressure in a typical theta-pinch experiment is rather high.

In part II of the present thesis electromagnetic 5ta~ility of collisionlesg high 6 plasmas is investigated On basis of the lowest-order hybrid-kinetic model of D'Ippolito and David50n.L~ The frequency of the perturbations is ~llowed to be of the order of the ion cyclotron frequency or Lower. This hybrid~kinetic model consists of a complete Vlasov descript~on o~ the ions and a drift-kinetic equation for the electrOns. Two eigenvalue equations are derived: one fo~ perturbations around a linear ~crew-pinch equilibrium ~onfiguration characterized by an

isot~opic

10n energy

di5tribution,~5

and another for the case of a linear theta-pinch equilibrium configur~tion allowing for an anisotropic, ion

ene~gy

dis tribution. to In both derivations the den"i ty profile i~ arbitrary. These ei~envalue equations clearly exhibit the influence of finite electron temperature, electron kinetic effects parallel to the magnetic field and ion energy anisotropy On electro-magnetic stability. In the limit of zeto ele~tron temperature and isot~opic ion energy, the Vlasov-fluid eigenvalue equation of Freidberg is recovered:7 It seems numerically tractable to obtain the instability thre$ho~ds and growth rates from these eigenvalue equatiOnS, witho~~ further

restrictions~~-t1

Analytically,

expli~it

expressions for the gJ:owth tate$ ate obtain~d for the case of long-wavelength peJ:turhations

-6-around a sh~~p-bounda~y equilibrium con£igura~ion. The frequencie~ are in the magnetohydrodyn~m~c range and use i~ ro~de of au additionnal

~~

Finite-Larmor-Radius ordering and, for another parameter region, of Preidberg's trial function method:?

Finally. it is shown th~t for extremely ~ong wavelength rerturbatiofis in a near-theta-pinch plasma with sharp bounda~ies, the hybrid-kinetic model should and can be generalized to include finite resistivity effects on stability. An eigenvalue equation for this case is derived.

-7-List of References of Summ!EY

1.5.C\1apman and T .G.Cowl ing. "The Mathematicl>~ T\1eory of Non-uniform GI>ses".C"ml;>rid~e University Press (London. 1960) .

2 .D. ter Raar, "Elements of Statistical Mecnani<;a", Holt, Rineh'll,t and l.inston,Inc. (New York, 1954)

3. H. G,ad, "Handbuch de:r Phlsik". vol. 12, "Principle$ of the Kinetic l'heo-ry of Gases" ,Springer-VeJ;'l'llg OHG(Berlin,1957) 4.Cf.Ref.20 o~ Pa.t J.

~-9.Cf.Ref31-5 of Part I.

10~18.Cf.Refs.9,1 1-17 "nd 19 of Part I. 19.Cf.Part 1.

20. A. A. Vedenov, E. P. Velikhov "nd R. Z. Sagdeev,Nucl. Fusion

l.82(1961);Nucl.Fusion 1,465(1962).

21.W.E,D.ummond and D.Pines,Nu<;L.Fu$ion SuPpl'l.1049(1962).

22. R. C. Davidson, "Method5 in Nonlinear Plasma theory" ,AcI>d, Press(New York, 1972).

23.Cf.Ref.8 of Part II. 24.Cf.Ref,35 of Part II.

25.J.P.MDndt and R.C.David"on,Bull.Amer.Phys. Soc.ll,648(1976). 26.R.C.Davidson and J.P,Mondt,Bull.Amer.Phys.Soc.~,II08(1976).

27.Cf.Ref. I of Part II.

2S.C~.Ref.4~ of Part II,

29tJ.P.Freidber~ and D.W,Rewett,submitted to Phys.Fluids(1977). 30.Cf.Ref.33 of Part II.

PART 1

DERIVATiON FROM THE B.B.G.K.Y.-HIE~RCHY

OF A COMPLETELY CONVERGENT COLLISION INTEGRAL

FOR A FULLY IONIZED PLASMA

(through an eval~ation of the binary correlation function,with application to the free energy and to the high-frequency e12ctrical conductivity)

-8-CHAPTER I: INTRODUCTION

Consi.;jer3tions lI);e restric.te(i to a homogeneoll~ stabl" classi.:al electron. plaslll3 in th" el"ttrostati.: approximation, moving in a con ti nuous homogeneo1).s 03ckp:ound oJ; ililIIlObi 1" posi ti vely Charged i"ns. ,]le system 3~ a whole i~ aSSutned to b", electrically n"utrll! and infini tely large. The);e fore its Hamd toni an is

N

1'~/2m

N

HI<

,:

+ J; ~<j> - • (1)

;:1 1 _{i< .} .i cJ ]

N bei.ng the tot.:.l number of partid,," and ~ is the Coo-lomb ene!:!.':y. Severa L me tllO(!~ 1-7 for lnv"stigation of transport properties of this system have lJeen constr ... cted ~tart;il:\g ir()m the Liouville ",quatiotl for the distribution flltlctiotl DN of the dynamical states of the whole system:

(2)

wh"re ( ; ) detlot;e5 the familiar Poisson bracket. 1-5

Perhaps the ,ncSt sl'stemat).c met;hod is based upon a set of equa~Ions for the 50-called r~du~~d distribution functions

(3)

v

denoting the system '" volume in configuration "pac,," These equations d;i.rec~~y

re~ult hom j.jouvi ile' s theorem [Eq. (2)

J

under th", ... "u"l symmetry condition for DN" In the thermodynamic limit N " "', V .,. =, N!V : n "onstan this set of eq'lations, knOWlJ as the 1l.1l.G. 1(:, Y .-hierarchy, reads

(5)

is the time-derivative taken along the free s-~article ~rajectories;

-I 5 :l~ ..

I m

r

_!-.J.

5

i.ej °;r;ij 1

is the operator rep~esenting the direct inte~actions between s particles;

L "n/m

s i=1

~

fox

~s+1 ~s+1 a~i

dv

~:l

__

~.

1,s+1

a

avo

'Vi

(6)

(7)

is the "phase-mi:dng" operator representing the deviation of the evolution of Fa from that of D_s' caused by the interactions with the remainder of ~he 6y~ tern.

Seve~al authorsl.8.9·~~lved

this hierarchy by expansion of Fs into a power Ee~ies in the plasmaparameter E.

Depending on which terms in the hiera~"-hy e<p,lati.ons were considered Bm(>ll perturbations, the results were inadequate for small 1,8,9,15 or Large interparticle distances.lgiving cause to short- and long-range

divergen~es

in the cDllisiDn integral,

resp~ctively.

10

TI,e aim of the p~esent investigation is to develop a perturbation approach to the B.B.G.K.Y. hierarchy that takes into account the spatial ~nhomogeneity of the ratio of magnitudes of the terms ISF? and LsFs+I in Eq. (4). Besides, a ~epa~ate assumption about the order of magnitude

. 7,9,15

of the i~reducible correlations, as made ~n one of the approa~he8

should be avoidable in any systematic perturbation theory.

1 0

-form of the two-body distribution function, - as made ,n Refs. (",12)

is avoided in the present wo~k.

Finally, no match\ng procedure, as put forward either phenomenologically 14 _or

based on the f.orm of the hierarchy equations i.n thermodynamic _ . . 13171827

equ,llbrlUm,' 'lS 'needed to obtain a ~ompletel1 convergent collision integr.3L In contrast with, e.g., the result of Ref. (16), the occurrence

of a logari thmic de:-pendeI'lc:e oIl thE:'! plasma paratnete.r doe~ not r~veal an

inconsistency in th" method, and unlike the result of Ref.(19), the obt~ined collisioll int~Kral is c"plicit.

As function~ of ~(s) e (~l' .... , ~) the operators Is and La have the scaling properties [Eqs. (6), (7)J

(8)

These properties give rise to a seal"- dependent

orderin~O,2~n

tne nondimensionBlized hiera~chy, depending on the lmj t of length chosen, the unit o£ v"-loclty being invariantly (2E/mN)!, where E is the total kinetic energy of the system,

Tne smallest length Ulli.t i" the Landau-length r_L = e2fT.

~.e.

the characteristic length of the "colU.,ion region" in spa£:e of interpa..-ticl, coordinate>; where di ree t i nte..-ac tions dominate.

Th" second unit of length conddered is the Debye-length

i.e., th~ characteristic l~~gth for processes related to the i~te~aetion with th~ ;!;"mainde~ of th~ syHe11l.

In the C~~e of a plasma the Debye-length is much gr~ater than the Landau-length, i.e.

( 10)

whe;!;e £ is called the ?la6IDa parameter.

-I

At the scal~ of the m£a~ free path for 5trong cDllisio~s, rmfp = E r_D,

th~ ordering of term~ as suggested by the corresponding nondimcnsional B,B.G,K.Y. hierarchy leaves some doubt for the possibility of truncation at such large i~terparticle distances. The probl~m of how to give a good description of this region is beyond the scope of the present work which only aims to obtain a spatially uniform solution of the hierarchy

21 equations for interpartid~ dis t.lnces ml,l<::h sm'l.l1er than rm. f. p, . The dimensio~les~ hierarchy corresponding with r

L as unit of length

reads:

(II)

and the one <::orr~sponding with rn as u~it of length reads:

D(S)P- IDT =cl

~

+ L

F

I s s S S 5+

~ 'Z

-Eqs. (11) and (12) then serve as a starting point for the perturbation analys~s, They are supplemented by the well-known initia1 condition of Sandri 8, 22:

8

n ~. 1 ()1;) , i=1

( 13)

related to the point of view that th~ relevant part of the correlations is caused by interactions. The well-known adiabatic assumption 1,20 ~ill be adopted, i.e., the correlations reach their time-asymptotic

fOl"nI in a ti.me very shor t compared wi eh the rel",xatioTI ti_ of Fl. (0) (0) PI/ttj,ng ( =

a

in llq. (11) yields approl<im",tions 82 _{and 3 3 for} the binary aud ternary correlations for interparticle distances not exc.eeding r by order of ma~nitude.

L

The remainder of intet;"p"'t;"ti"le space can b~ divided into three regions which partly overlap:

Region R1(I,Z): A cylinder parallel to :¥.12 with the origin on its axis and with radius A equal to

(14)

Region R2(1,2): The complement of the union of RI and the sphere around the origin wi.th radius ~ equal to:

( 15)

-

'3-The radii \ and ~ shall be determined in the sequel such as to ~ke che approximation for the binary correlation optiroal, i.e. the accuracy of the resulting collision integral as high as possible.

-

''-1-CHAPTER II: SONt NON~lfORMLY VALID APPROXIMATIONS

1n deriving a sp<ltiaUy uniform approximatl,;m for tbe binary correlation function "''' part of a solution C)f the B.B.G.K.Y. hierar<;hy equations. \l~e shall be made of expre~sions for and properti"s of some nOrluniforrnly

val~d approximations to be discussed in ~hi~ chap~er.

2

On toh" ~cille of the Landa1rJengrh, r

L

=

~

•

and in lo",,,-st order in £, the asymptotic two-body dis tribution function F (0) is the 801ution for

2A

t -> '" of

I

Eq" (I I)

I

d

a

[at

+

~12

.

(l~12

m with [Eq. CI3)] F,(O)Ct = Q) =

F(~I)

2 I '

Th" charact<!ristics

or

Eq. (16) are

)J

FCO )

~

0, 2 ( 16) f(V Z) as initial condition. I~'

the traj,,-ctories in a two-partIcle system. The initial velocities of the trajectories are det",=ined by

the conserva t.ion laws II - i .. e. tho8e of energy t momen turn, a.ngular

momentum and an additional vector quantity, being conse,ved only in the ca~~ of Coulomb interaction,

(a) CDnservation of energy yields:

say E, (17)

2

with \.I = m/2 the reduced mass, e /x12 ~ <!> is the Coulomb energy, ~12

=

~I - ~2' ~12

=

~1 - Z2'

-15'-(b) Conservation of momentum imp~ies that the mean velocity

!

(~I + ~Z) is conserved.

(c) Conservation oi ang~lar momentum implies that

k "

~ x_l2 x ~12 is conserved. (18)

23

(d) The ~ditional conserved quantity is the vector ~ defined by

2 /;';12

~":j:;12

><k+

e -_x (19)

l2

In order to make use of these con$~rvation laws we put Eq. (l6) in the following form:

(20)

with:

(21 )

Let .\512 (1), ~l (1), )(,2 (1) form the solution of Eq. (21) subje~ted

o

0 0

to the boundary conditions:

Then Eq. (20) implies that:

(0) _ (0)

FZ (/;';12' ~l' ~2' t) - F2 [.~12 (0),11:1 (0), II:z (0),0]. (23)

0 0 0

(2.4)

In genO l"al:

(25)

since v. (0) is de.termined by tracing ba(Jk the trajectories along

"~o

the <.:haracteristic5 given by llq. (21).

In p .. rticl)J,.~r it. follows from Eq. (24) and Eq. (5) th"t

(0)

FZA '" lim F2 (O) ( -I; 12' t; :t I' ~2 )

t .... .,

(26)

or:t using momentum conRervation:

r

){.1 + .l{,z

FI ~-2--

-(27)

Renlains the problem of finding vIZ(oo) '" :t12 (A5I2; 'iI' )(2; .. ),

o

0

i.e. the relative veloci ty of the. partic:l~~ ill the e.xtreme. paS t.

Energy conservation [Eq. (17)] yields:

so that

~17-I

v

(~)I ~

(v 1Z 2 ₊

4~2 )~

"'12

o

mx_l2

By means of (28) we can write (24) as!

(28)

~l

(29) where ~(~12; ~l' ~2) is the vnit vector in the direction of ~12 (~).

o

mis vector is det~rmined in Appendix A,by ~ing u~e of con~ervation of angular motJlentum a ... d of

J2

[Eq". (IS) and (19)]. Thereby t.he asymptotic two-body distribution functiOn is completely d~termined. Making u~~ of Eq, (A,I I) for ~ and Eq.

the

a~ymptotic

binary correlation

g~~)

F [{I + :1::2 +

!/V72

(0)

_n

glA ~ _I

2

- +,-where 2

J2

/2

4e2 _L2 ~

"

-

e C2 + "' vl2 +~-mX l2 (0)

(29) for FZA we obtain for

F;~)

- F_I(:;:I)F₁(:;:2)

th~

following 2 I

+~)

_mX

_tJ

- F I (:{ I ) F I (~2) l2

,

(30) ~ x C 'v

Ik

x

~

I

2 (31 )

~ ~

I

~12

x

~12

•

and where the product +~_ is over the plus and minus ~i~ in front of the _quare root.

-

18-Th~ approximation (30) for the binary correlation function breaks down on the scale of the Debye-length and can be ~hown to l~ad

to ll"ltzmann'8 collision integral I, \>Ihen substituted into the first B.fI.G.K.Y. hierarchy equation. A.::cordingly, We refer to the

(0) (0) solutions g2A and 8

3A of Eq. (11) to low~st order in E as the "Boltzmaml .::orrelation?".

{OJ (0)

In the ne~t se~tion, series representations for g2~ and g3A are derived in order to determine how these function~ behave for large irnpll.ct parameterS.

Cons idBr Eq. (I I) in lowest order in e, for s ~ 2 and for " = 3. The "s traigh t path apprOX:Llrration" consists of consi tiering the force

5

convecti.ve part Is g" to be l;<m:;ller than the source terro1 s il:11 Fl(~~)'

yielding in leadillg order and for R = 2:

:'(;1:1, • d{IL lIiD) (J.;U' Y.I' :(2) .. 1_{2 (;:;12;} t l '

Z)

F I(:(I)F I (:(2) ,

(32) (0)

where 6A (~12; :(1' :(2) is the lead~ng order Hrai.gh~ path approximation of

>:1°);

6iO) yields, whell sllbstituted into the first hierarchy

equation, the "Fokker-Planck" or "Landau" colliaion integral and will therefore henceforth be referred to as the Landau ~orrelation

(cf. lI.2bfor an explicit expression).

In nth ordel;, by iteration, we get corrections !lin) defjned by:

such that

(34)

As will be shown in appendix A, the function

biO)

can be obtained by a Taylor expansion of g(O) with

respe~t

to the projection

~12

of

~12

1

perpendicular to :t12. Eq. (34)can therefore he e:qoe~t;:ed to be valid outside a cylinder witb :::12 as axis, the axis going: tb:r:ough the origin.

Since

g;~)

is completely independent of " the radius of thi"

~ylinder

must be of o~der ~ity.

By means of scaling property (8) for the operator I2 it is readily ve [Hied that

(35)

for any )..

f.

O.

In an analogous way. starting from the equation

fo~ gi~)

[Eq. (11) for s - J, 8 = 0], 1. e.. ,

- 2.0

-- the 5unh'Ilcltion [ being over all permutations of {I,2,:)}, iterative p

straight path approximations a);"e now made by means of t.he function

sequenc~ {D(n)} 00 defined by:

A n:O and, for n. yielding: (0) [;3A !.. 1, (37) (39)

I f x_l2 >, I, t.hen at least two particles are far apart so that l)g3

in Eq. (36) can be considered smaller than

If, mor~over, 1~I3

I

. .L I,

1~12.

1.

I

s~raight

path approximation tor

g~~)(i,j)

holds for every i and j.

Series (39) can therefore be expected to hold, provided x 12 » 1

n

I

~

12

I

~

I

n

1.:151 3

I

> 1 <

I

i'S23

I·

~ l l

Making U$e of s~aling property (8) and of Eq. (38) it is readily vedHed ~hat

(40) for any A / 0, and for n ~ O.

From series expansions (34), (39) and scaling laws (35), (40) it follows that the binary Bolt~mann correlation falls off inversely proportional with the impact parameter ~12 whereas the te~na~y

~

Boltzmann correlation falls off quadratically when all impact parameters increase indefifii~y, a differenc~ important fo~ the justification of truncating the B.!.C.K.Y. hiera~chy equations, a~ will be discuss~d later.

The asymptotic Landau con-dation

LI~O)"

lim

6(O)(~12.t;~I'~2)

t .,. ..,

can be determilled from Equation (32), i.e.

<1

a

("'IT

+ :'1:12'

~;l;)12

- Zl

-su~ple.mented by the init,-"l condition (llq. (13)]

[,(O)(t = 0) = 0 (42)

By inc"-sr"tion along th" (straight) <;.hara;:t"rlsr.ics we find:

and in parti~ular:

I

m f d5V~(~12

o

the last equality d~fining ~.

(43)

(44)

In ol;"d.:or to evaluate \It, we <;.hoo"~ a Cartesian frame "i,th "12-"xi.

J dS(s x l2 2 xl2 _{)2 +} 2 -3/2- _f

It,

= v l2 0 vl2 )lv I2(s '1 12 P IZ ) I\; x -Y12 0 d~ 2 xlZ ) 2 2) ~ :< )(12 Z + 2 )-3/2~ {v 12 (.

-

+ Pl2 ~Y - zl2

r

{v 12 (s - - ) _{Pl2 I\;z·} VIZ 0 vl2

where ~~. ~y and ~2 ar-e unit vecto~g along the C~rt~"ian axes. Tile integration is readily performed to yield

XLI

{I + - - ) .

1~121

(45)

(46)

wh~re ~12 is tile proj,,-ction of ~12 in a plane perpendicular to ~12

l-and x j2 ~ the component of .\\12 pa~allel to ~12'

l.andau-correlation is singular around zero impact parameter (i.c .• for 1'<:;₁₂.L_{1 ...} 0), el<cept if <LI -

!v/FI(~I)FI(l:')

only has a

component alon$ the ~IZ-direction,as is the case in thermodynamic equilibrium.

In order to construct a collision integral describing particle interactions at the ecale of the Debye-length,Balcscu, 7 Guern6eyl5 and Lenard 9 formulated approaches equivalent with adopting ordering (12) with the supplementary ass~tion that the irreducible s-particle correlation is of order £s-I

ti~s

the uncorrelated

s

part IT FI(~) of the s-body distr~bution F

s' for all 6 ~ I.

i"l 6

This leads to the following equation for the bina~y cor~elation function,first put

fo~ard

by Bogolyubov: 1

+

(47)

(I) (2)

where the operator. LI and LI are defined by

(48)

and where we wrote g2(i,j)

=

gZ(~ij' ~i' ~j,t) as a shorthand. Although the collision integral can be obtained without explicitly solving for

-

c..t;-g2' the asymp tocic binary correlation fuI'l<'.tiol1 ~ determined by Eq. (47) and iI'll tiaL "oudi tion (13) has been evaluated in fourie.

apace ~

10

\-Ie OnAY give lI"re the final result,

JF I (vI)

""---

<luI

~~I(U2)

+ -il li2 {(

}

~ I (u 2) d I :il,l~ (u2) :;:-(\12) I a~ 1 (u_I) -~-u-I~ )+ + ( 'V

~

1 (\12)

}J

-ilJ'1 (u₂) + ~Z(u2) J]

-where • 1,2 , 2 2 -2 2 2 1.1 _=' Ul p k • Ul =~ P P m ~ 1 (u) - !dvFl~) 3 is

(I

~

·t -

u)

_•

"";I: Z±(u)

-

• 2 of 1 (u) and

₌

I + ₂1l 1U p ~

The reader is reminded o~ the general definitions

1 001/1(1.1') du'

-

~

2TIl

-oo~ u' - u

For small impact parameter (~12 + 0) the relative difference 1(0)

between the Landau correlation ~A and gB tends to zero, as shown in Appendix C ,i.e.,

(51)

(52)

CHAPTER III. A SPATIALLY UNHORM APPROXlHATIOll FOR THE BINARy CORRELATlON.

,,1

this c:hilpter "e shall derive an e"Pli<;i t expression for the asympcotic: binary correlation,as part of a ~patially uniformly valid solution of th., B. B.C.K.Y. hierarchy equations [Eq. (i.)

J.

To this end we distinguish three region,; which partly overlap «Ii discussed in eh. L

Due to "crc~ning the «symptotic binary and ternary correlations g2A and "3A depart from their zero order O1pproxim«tions

gi~)

O1[ld

g~~)

as soon as the relevant impact parameters become comparabl..: with the Del>y,,-length.

In order to <.:alculate this departure we d",fin., correction factors

g;~)

(1)

«nd g3A by

(55)

and notice chilt the series expansions [Eqs. (34) and (J9)) for

g~~)

d (0) , . d b - - \ " 1- d

<In gSA o"t"~ne y 3\lcce3 J;nye 6trallpt path appro,amat10nS are va l within R

2(I,2) and the complement of the union of R1(I,3) and R1(2,3), >;e'pec ti ve 1y. In view of this limi ted validi ty of the 6 trai ght path

. . (I) (2)

apprOX1Il\atlOn, thf! operators LI ,L₁ and L2 [~;qs. (8) and (48») <Ire d~composed into two terms

-2.7-L (j) 1 = L{j)* 1 + L (j)

t

1 • (56) LZ = L2 + L t 2

such that the *-operato.s involve integ.ation ave. the regions where the straight path approximation yields a convergent series repre8ent~tion. so that substitution of series (34) and (39) is valid in the integral terms wi th aste.isk. but not in tbose wi th dagger index,

Since the Debye length is a typical unit of length within R Z(I,2) the ordering of terms in the second hierarchy equation is given by

]l:q. (12).

Making use of Eq. (12), series expansions (34) and (39), scaling laws (35) and (40) and of decomposition (S6),we obtain the following

- f h t" (1) h ' b' l '

equat~on or t e correc ~on g2A to t e asymptot1c lnary carre at,on function:

- 2.8 ~

(57) where the symbol (I ++ Z) d'i'f1otes the IHeced~ng term with pa~l;ticle

i ndi eM and 2 inte~changed.

N~xt we investigate to which extend truncation of th~ B.B.G.K.Y. hierarchy is poaaibl8 by neglect of the terms in Eq, (57) involving thl! ternary ~orr"latioll functicm.

We first consider the expression

(58)

(59)

where the prim" in f ' d \ indicates that integration is restricted to

I~R l-R

{k:~.L'o lolJi,L+k12 1>£ I} .

.L

Making USe or a simple transfOrmll1;:ion of integration variable~,'i/e get

whe~e now the ~rime denotes restriction of the integration to the interval

{~:'~>I

n

I~~

+

~~

I>I}, i.e. to an interval where

D~O)

is at most of order unity. As can be trivially checked, the orders of magnitude of the? - and? -terms in Eq. (60) are identical.

VI v₁ Let us consider the \I -term. Since

(0) vZ _₁

DA (~+~,

kl

~l'

ZZ' Z3)

O(~ ) as ~~ ~ 00 (cf. Appendix A

>,

we arrive at the estimate:

2+R 00 = 2 2 -I -I I {I)I

g I~ ~ dz

f

d~ ~ (p +:t) p g3A '

2+R '" -II (I) I

::. ~ 1

f

do" _{8 M} . (61)

The factor

g~) represent~

the decrease of g3A due to screening. (I)

Th~refore it is reasonab Ie to lI~ ~ume t:hat S3A .. 0 as p .... , 50 that

00 -I (I)

11 dp P g3A con~crges and is at: most of order ln~. Hence:

ILI~~2,j-Rll,,~-I, (62)

uniformly over R 2{I,2).

Since the source

term/I2(~12;

:;:1'

~2)FI(:;:I)FI(Z2)

inEq. (57) is at least of order ,2 for interparticle distances not

e~ceedins

the Oebye length in order of ~gnitude (all con5iderations in this report Olive been resticted to such lengths anyway), i~ can therefore be conCAuded that the term EL; in Eq. (57) ClIn be neglected, the relative error being of the order (RI In(

Next we consider the terro

-

30-(1)

in liq. (57) and put il3A c I as a,~ overestimate. ,'irst Wi!, notice

1 f h . 1 ~ (0). 1

that at eaRt one 0 t e spat~a arguments 0,. 83A ~s very arge:

(64)

iR, therefore. much larger than 13(~13/£.

{n/

q

:Yl'

:YZ' J:3)

g~~)(~13!~' k23!~;

Yl' ?t2' .1::3)' whereas for the part

{t~3:

t_l3 <E l-Rl} the

t~rm

_{1 2}

<-bjf.;

JG2'

Y3)Fl(:Y2)gi~)(~13/q

!(l' :Y3) dominates

(0) (0)

1

3g3A . We may conclude that in ~q. (36) the cerm 13g3A Can he

(0) .

negl.:cted wh':n evR.luatine 3 3A (~13/c, ~231"; ''(;1' :t2' 1:3) within

t

the (loml),i,n of integration contained in L 2: (65) -R -R , -It I . . provided x 12 >8 2, xl3 <: ~ I J x23 <: < (w~th rl,. as un~t) . il') J .L L

In Ilie", of the hct that

gi~)

iR l)niformly bounded [Ilq. (30)] it fol.101'8 (0)

from Eq. (65) chac g3A behaves approximately like a homogeneous function ::>f degree - I , i.e.:

l-Rj I-R within R₂(l,2) and provided ~13 <: < Ul;23 < ~ I.

J, .L

(66)

"faki!\g uSe of l::q. (66) we get the following estimate for T:

3 the prime denoting restriction of the integration over d ;3 to

{~3:~13

< .I-Rl U

~23

< .I-R I}. Clearly, to the right hand side

L !

of Eq. (67) the regions

contribute te~ of the same Order of magnitude.

Let us consider the contribution TI to the right hand side of

I-R I} . . . . , b '

Eq. (67) coming from {ii;3:~13L ( g • rutt~ng I;l !!' {,IS "e 0 tatn:

, 'I I

_i

2] 3 f t 3

I {"'"

(n ) II 'J~ (>" ~) II } (0 ) (f ,n • )

T

I'"

T I;f.E d v 3 d " ' ... ¥U • vI + T -'G 12 +;c • v Z g 3A '" 12"'"

it

I '~2 >4! 3 (68)

where the prime denotes that integration over d3n

i~ ~estricted

to

< I~l-RI.

(0)

:t'r(lv~ded the velocity gradi~nt~ of the ternary cOrJ;elatiofi gJA are bounded in I;onfiguration space ,there e~ists sOlIle function 11 'l,lch

that:

1,2.. (69 )

~ven tor situations outside equilibrll,lm it is rea~onable to assume a~L

velocity integrations to be convergent; in particular:

(70)

We now have the following estimate for the term T:

(7i)

where the prime denotes in~egration to be restricted to ~! < gl-R1• In order to d~termine the order of magnitude of

-

~z-we perform the intesrat~on~ over d~~ and dnll' 1.lsing cYlindric.al coordinates:

(73)

tiThere If> b the angle between

4.L

ffild lJ~' Int"gration oVer do yields

(74)

whe~e the. integration interval is the interior of " circle, arouIld the od gin, wi th radius" l-R l

(75)

since eStj,ro"te (75) applies uniformly over ](2 (j ,2) we conClude that for interparticle distances not exc.eeding the Debye length in order of magni tude:

over R 2(1,2).

Combining this result with (62), we conclude that in Eq. (57) tr.uncation can be made at the expense of a relative error. of order maximum

{ £ Rj 1n£, £ l-R I} uniformly over R2 (1,2). The resulting dosed equation for

g~!)

can be simp1ified to yield

-H-~2~12'

a/0.t12 (!I·iO)

(k

12; :t I' .1:':2)

si2

('%12/"" C; :tl' :tZ)}

~

, (l)t (0) , ) (l)e / ' ) 1:' ( )} + (1+->2) + HI {g2A ('23/~, /(;2' -1::3 •

g2A ~23 t ,

E;

~2' ~3 I ~l ~

without further loss of accu~acy.

The right hand si~e C~ be easily rearranged to yield:

(77)

If the

la~t

cwo terma, i.e. those involving the ti-operator;. are neglected then we obcain for

gil)

the following equation:

(78)

-H-j

As $h~ in appendix B,th~s negle~t is consistent in the sense that substitution of.

th~

solution of Eq. OS} into the terms F;L;I,3)t

d L(2,3)t. E (77) d

an f- I ,n q. an compar~son with the source term

in Eq. (77) yield8:

where again considerations are ,estricted to interparticle di8tancas not exceeding the Debye l~ngth in order of magnitude.

The ~olution uf Eq. (78), which is exactly Bugolyubov's equation [Eq.

h b . (0)« ) (Il(

I

(47)] for t e ,nary correlat10n tA ~12; ~I' ~2 g2A kl2 £, 8; ~I' ~2 yidds:

(30)

with (Y:: maximum{<.Rllln c

I '

.I-Itl}.

In the next 8e<::tions it will be shown that the solvtio.., (80). obtained in ~egion 1t2(I,2).i~ al~o a good approximation for the asymptotic binary correlation in the rest of ~12-5pace, i.e. in regions R

Next we try to obtain a good approximatio~ for S2A in the region

-R2 ~

R₃(I,2) ~ {~12: x_lZ < ~ r_L} with 0 < RI R_Z <I. To this end we define a correction factor

gil)

for the deviation from the

"'( 1) .

Bogolyuhov correlation gB auch that S2A ~e of order unity for \ distances of the order of the Debye length, i.e.,

Scaling the second hierarchy equat~Qn hy means of the ~its (£-R2 r L, Vth

!

we obtain where we defined (81) (82) (83) r L

where t is measured in units The orders of magnitude of the correlations have been estimated to equal those of the ~ero order approximations

g~~)

and

g~~)

for impact parameters of order

~-R2rL'

Since the importance of the integral Cel:jllB d"-,,reases wi th decreasing interparticle dia~ance, we can neglect these Lntegral tenna £or

-il.2

distances smaller than 0 r

L, up to a relative accuracy of order ~R2. But since the interaction operator 12 is invel;sely propol:tional to the square of the interparticle distance WI< can

!!2!

neglect the

3!l.2

term E 1

2(1,2) G2A(I,Z) in Eq. (82) uniformly OVer R/I,2).

,,( I)

Thereby we obtain th" following equation for g2A '

(84)

Since over R}(1,2):kljI21 < E

1

-R2, Eq. (53) imp1i.e~ that up to an accuracy of ord"r f I-R2

(85)

can be allowed as a \miforrn approximation within R

3(I,Z), so that

Eq. (86) is identical with the equat£on for the binary ~orrelation a~~ord~ng to the Boltz~nn theory [Eq. (16)]. Hence,

Combination of result~ (80) and (87) leads to the con~lU6ion that:

gi~\~lz;

-1:1' -1:Z)·gll(£.lS12>

\':1'

ltZ)

to)

{:,A (~.\'j12; ~I' .'1:2)

+ O[max {E<\(lnE-I)~Rl,gl-R}] , (88)

uniformly over the union of R₂(1,2) and R 3(I,2).

!!!~~~_~~_~~gi£~_~I!~~1_2f_~~~l!_i~~~~-E~E~~!~E~~

Finally

~onsider

region R₁(1,2) M

{.lS12:I~I2

I < £-RI r

L} where

~12'~12ltI2 ~

~12 a ~12 - .2 and 0 < RI ~ I, as in~roduced in Chapter I.

1 v

l2

For a typical point in R₁(I,2)

(89)

sinee RI > O,expansion (34) for the Boltzmann correlation ~onverges for a typi~al point in Rl (I ,2). and we find [Eq. (35)

1

-

38-for the order of magnitude of the Boltzmann co:n:elation in a ( 1)

typical point in R

1(I,2). The correction facto~ gZA due to

~creening tends to decrea~e the correlation,but can be taken of order unicy in the region where cor~elations are nonvanishing.

(91 )

wne~ea~ in the rest of space, where typicallYI~121~ Y_D isotropically,

(92)

as follow. frOm Eqs. (4) and (35). ],'urthermore, the ce~nary correlation S3A can be ass\l1l\ed not to exceed F I (;{;l)F 1 CY;Z)F I

(;r)

ill order of

magnitude, i.~.,

(93)

wit~in R

1(I,2), R)(1,3) and R)(2,3), whereas in the rest of space (Eqs. (39), (40)), where typically ~ij ~ r_D,

(94)

From (89) it follows that

(95)

since the Coulonb potelltial is all isotropic function. 3

of the ~e~ond hier~r~hy equation [Eq. (4)1 must be ordered according to the scale dependence of the integr~d, yielding

where d323 is the integration element measured in units r L, if integration is to be perforwed over a region R_I, and

(96)

(97)

-RI

I

Adopting t r_L as unit for ~12' rO as unit for 1~12 ,and making 1

USe of estimates (90) - (97) ,we obtain the foLlowing form for the nondimensional hierarchy equation relevant to region R

I(I,2);

(96)

from which it follows that

(99)

Hence, within a good ~~CuraCy the binary correlation function does not vary along the ~12-axis within region R₁(I,2). and can eherefore be equated to its form within region R](1,2). i~p11ing [Eq$. (80),

-

~o­ (0) g2A (;;;;12; ,';;1' ';52) ,(0) (e:x . "P, 'V12' fJ -1 Rj I-R + O[max {~ , (lnE: ) " , c I,

~niformly over R_l(I,2) U R

_z

(I,2) U R₃(1,2).

+

( 100)

An accuracl' of order:

8\

In

0

is now achieve.d by putting RI=RZ=J,

~. + e. ~

uniformly in ~12.

InEq. (101),

g~~)

is e.xplicitly given by Eq. (~O). ~A (Q) by (10 I)

Eqs. (44) and (46), Yhil~ th~ Fouri~r-transform of gBis given by

Eq. (50). ThTough Eq. (101) an(\ the. first hierarchy eq~ation [E<j. (4), with s =

IJ

the collision integral is determined, i.e.,

( 102)

Note that the dependence. on the plasma parameter " only occurs through the space coordinates in the Eo~olyubov and Landau correlations, gB

g}O)

and

<0).

respectiv·ely. Since

~

.. > I if 1S12 +

~

[l!q. (A.21)] and

I> .L

I i f x

l2

-~

0 [App.C]

ou~

collision illtegn.l evidently

converge~.

~

'(I

-CHAPTER ~V: U!IlRMODYNAMIC EQ!JlLIBRIUM

In this chapter the free energy density in thermodynamic eq~il~brium is de(ived from our formula [Eq, (101)] for the binaty correlation function.

As is well known the free energy density ~n equilibrium is given by24

2 I n f " ' f +-2 f o 0 3 ,,2 2 d~fd x

-X

G(Ae ; x) • ( 103)

where f _o is a

~onstant

and where G(e2; x) is the space-dependent £a~tor in the expression for the equilibrium binary correlation fun~tion., j,.e ... ~

-J(

~exp(-)

r_D

in our case [Eq. (10 J)

J.

Defining

r L

[I - exp(-

;-»)

and making \;lse of Ilq. (104) we obtain fl"om Eq. (103)

In view of !

o

I f

o

(104) (105) ( 106) ( 107)

and

-'1Z-I 1 x l+n

2,(-'2-

r

n=O .... ~ (2+n)! - !ljJ(n+3)] , [In (I08)

we r(ladily obtain the following approximation for the free en"rgy

den~i ty [Eq, (106)

I

F 2 F _{o - nT}

_{[3 ...}

~

-rz--

E; lne 0(,;:2)] + ( 109) where c " rL/r

D, Thj.s expression i~ ~n agreement with Guernsey's result 16 and differs from the divergent expr"5sion Obtained by Frieman and !look,I;!, 17 It should b", noted that the t"'rms 0(02) cannot be

tru~t~d

in vie", of the relative error OrE!) made in d"riving our formul~ for th~ binary correlation function [Eq, (101»),

CHAPTER V pROpERTIES OF THE LINEARIZED COLLISlON INtEGRAL

If the deviation from the:rmodynamic:: equilibrium is small, then the b~naty correlation only slightly d~f£ere from its equilibrium form and the collision integral can be written, - in dimensional

fOl:Ill, as

of

-m;=A+B+C, where with ,,(x) with ~(x) (eq. ) (eq. )

1

( 110) (112) ( 113)

-

~y-with

-y(x) .. - ( 116)

In Eqs. (112), (114) and (116) thO! symbol [ j(eq.) denote" that the equilibrium iorm of the quantity between the square br3ckets should be taken; hence, ", Sand y only depend on x. In .l;qs. (III),

(0) (0) .

(113) an~ (115) the correlations gl)' g2A and "'A ,respe.ctively, can be further li.neari~ed around thermodynamic eqoilibrhlm. Conservation of numher density and o~ particle momentum density

are obviously checked from Eqs. (110), (111), (113), (liS). Conserv3tion o~ kinetic energy density is checked for terms A, Band C in

Eq. (110) separately, as follows:

Ad. A:

Introduce the potential ~(x) defined by

JjJ{x) ( 117)

in terms of which [Eq. (111)]

( I 18)

Making use of Farsefal's theorem we then obtain

A =11_(211)3 _~_ • !d3k l'k m(k)l [!d 3 ~ (lc )]

'" 3~1 ",'" m v2gB,";:X:I'~2' (119)

-

H-and where l~[

1

denote~ the ~maginary part of the quantity between brackets. !t is straightfo~ard to show9,20that the right hand ~~de can be ~itten 35

where

where the dielectric function is defined by af 2 . ~ (lZO) (oj e 1~ 3

a;;:

~+(~,p)

_ -

~

./d v p +

1~;;:

(lZZ) Therefore (123) since ~12.~ = 0 • (124)

Introduce a potential ;(x) defined by

in terTIlS of which

(126) Integrating by part~ with respect to both ~ and ~l and using ~ symmetry argument we reao;lily ohtain

(17. 7)

Since [Eq. (16)1

the right hand side of Eq. (127) is seen to be equal to zero using

Gauss I theorem.

Ad C!

Quite similarly we find

(128) where the potential n is defined by

'i?-I) (x) '" - ! (129)

x

and where the right hand side of Eq.(128) vanishes in view of Gauss' theorem.

-

'18-cHAPTER VI ON THE HIGH-FREQUENCY ELECT~ICAL CONDUCTIVlTY IN A SPATIALLY HOMOGENEOUS ELECTRON-ION PLASMA

In this ~hapt~r we consider an unmagnetized and horoogeneouR plasma under the influence of a spatially independent b~t

~me-varying electric fi~ld. ~ ~ exp(-iwt), and we will restrict

considerations to the case when

Iwl

« ~l

pe ( 1.30)

where v . is the relaxation e1

with ions, 20 ,2~nd where w pe

frequency of ele(!trons due to collisj.on,

2

m

(4

1Tne

)1

is the electron plasma

m

e

fr£qu"ncy. Ihc unperturbed ele<;:tron and ion distributions are assumeo Co l)e Maxwellian" with constant and equal temperatures,

f~ 1 2 m v e exp(- ~) o (131 )

As a result of the strong inequality in (130) the plasma can be assumed qLlasi-neutral. T1\e other two inequalities imply a possibility to(analytically)solve the kinet~c equations for hoth £lec.trons

27

ano ions by expanding in the small parameter

Since the relaxation freqtiency of ions is much lower than ve~' (132)

the ~ons can be descr~bed by the (collisionless) Vlasov-equation,

at.

~+'="'E

_at

m. '" _1. ( 133)

On the other hand, the collision terms in the ele~tron kinetic equation are finite, though small [Eq. (132)J,

H

_{e e} Of _{e n}

F-m~·~a m

e '" e

( 134)

whe re gee .. g as gillen by Eq. (10 I). and 8ei denotes the "nalog of g corresponding to electron-ion correlation.

since me <~ wi the current is predominantly carried by the electrons,

(135)

The kinetic equations (133) and (134) ean be used to determine the perturbed electron distribution 6f_e, explicitly in terms of the applied

field ~. Substitut~on into the eXpress~on for the current density [Eq. (135)] then yields an expression for the conductivity, in each

ox-del'" in 1;:,

In first order in ~, Eqs, (133) "nd (134) yield

(136)

for the first order perturb"tions 1n fe and fi' The resulting conductivity

50

-In second order in E ,

( 137)

where G

ei is the (lineari zed) e lec tron~ion collis ion teut, for which, in analosy with the linearized electx-on-electx-on collision term, can be written rEq~. (110)-(116)]

( 138) where [cf. App.

EJ

with .p(x)

~

To f di [I

~

exp(- :L)] x ( 140) G(~) n

a

fd3)( Vn()() 3 (0) _;;;2) ( 11.2) - +~~ _{fd II;.:} lIeiA(~; ~1 ' el m _e

_ai':

1

.

with '1(x) T

r

d{: [I _ J: L -

,

( 143)

"

0 _x E; e:x:p(-T)] e:X:P(-E;J:"o )

-;;1

~

(0) (0)

In ~q5. (139-1_43), ~eiB' _{geiA and} ~eiA ar~ _{the electron-ion}

analogi~s

of gR'

glO)

and

~~O)

[Eqs. (50), (30), (45)] ana T

2 -I (~)!. . (

r_L ~ e To ' r_D ~ • l In d~riv1ng Eq. 137), note

s'rne

that th~ lineariz~d el~ctron-electron ~olliaion te~ can be disca~ded, since, due to conservation o£ linear moment~,

its contribution proportional to H( I) vanishes. Also it e

is

noti~~d

that a term

.!....

R.t- 6

f(l) on

th~

left hand side of

Ille 'I; o~ e

Eq. (134) has been dis~arded, aince, as this term is symmetric in

X,

it does not contribut~ to the current density [Eq. (135»). Ne~t the three collision te~ms in Eq. (137) a~e evaluated. First consider

c(~)

_e. as defined by Eqs. (139) and (140). Making use of Parseval's theorem, we obtain

where the Fourier-transfo~ ~ - (2~) -3 3 fd x ~(x) exp(-i~.~) of W equals

T

i)J(k) = _ 0 _ i dp p sin(kp)

2~2k 0 p i

and

whe~~

the expression for !d3_v2

geiB(~; ~l' ~2)

c

(145)

-3 3 3

= (2~) fd v₂/d x geiR(~; ~I' ~2)e~(-i~.~) as given by ~.g. fm(7.45) of Reference (20)can be substituted wi ehout any further, yielding

( 146)

- 'i

2. -where where w 2

i.ll;

.!d3v 1 - ~+(~,p) ;;~ k2

In

lin~arizing C(~)nQtice

that e1 ( 147) ( 148) df H.

"

2 .~ l.

~

UJ , J. .Jd3v

~

+~ +

i.ll;.y

P + i~.y k2 P ( Wi) ~ =

o •

( 150)

There£Qre, the equilibriwn distributions fQ and f~ may be substituted

e 1 into ,6+ t i.~. ~ where Z is defined by 1 2(x) i i

71T

.. 2 f dE; ellpC-E; ) ( - x (15 I) ( 152)

yielding for ~ [Eq. (148)]

and hereby Eq. (154) ~educes to

Making use of vi « v_e' and 5ubstituting Eq~, (136) and (156) into Eq. (146), we obt~in

;

( 1.53)

(154)

( 155)

. (157)

tn order to evaluate

C(~)

[Eq, (141»). notice that the expression e1

for the electron-electron Boltzmann correlation function [Eq.(30)] was derived by 50lving the characterist~c differential equations

-

5''1-deB~ribing the p~~ticle trajectories, after red~ct~on to those

~orrasponding with a center of fo~ce problem. Therefore, the corresponding electron~ion formul~ ~8 obtained by s~bstitutins: appropriate values for the reduced mass, interaction strength and center of mass velo~ity. yielding

m -~ _{m +m.} e L rn"ZI+mi;r2 "he r"

X,

i i m +rn-e L m. ~ + -ro +m. e ~

Also not~,ce that the influeI1ce of electron captu~e is excluded ( 153)

in a cla~sic"l, i.e., nonquantal. theory of a fully ionized pla~ma.

_ • . 3 2 2e2 •

We shall account for th,s by wntlng fd V

2U(v1Z--;;}'X) lnstead of

3 ( ) "

fd V

z

in th", e~ression for C ~ [Eq. (140») ,Here U is the unit ~tep el

function. Substitution of the e"l'ansion around equilibri1)rn [Eq. (136)] of the distribution functions fe and

£i

into Eq. (158) yields

( 159)

Thereby we obtain [Eq, (141)J

(6) en

~

3 x 3 2 2e2

C . . fd x 17Hx) ""11(- rn)fd v_{1 U(v 1o - meX})

e1 me(-iw+O) ~ ~

Finally, we evaluate C(?) [Eq. (142)]. this is nOW mo$t easily .done

e:t-by noting that Jd3v

₂

~~~~

can be obtained

f~om J~3V2 &B(~; ~I' ~2)

by taking the Limit r_D + ~. The result is

where ff(k)

=

---1--- Jd

3

x

~(x) eXp(-i~.~)

is equal to [Eq.

(143)]

(211")3 T ff(k) ~ ___ 0_ J dp p sin(kp) 2,,2k 0 d< _{r L} c J ~ [I ~ exp(- ~»)exp(-~) • ~ ( ~D ( 162)

Equation$ (157), (160) and (161) together determine the total electron-ion collision operator C

ei On the ~ight hand side of Eq. (137), thereby

de~e~i~ing of~Z)

in terms of the

electri~

£ield ~. Substitution of this expression into Eq. (135) for the current density then yi~ld5 the contribution to second order in to the electri~al ~onductivity. Thereby we find the folloWing formula for the electrical conductivity;

2

~

w _

4:r:

~

+

~(A)

+

~(B)

+

~(C)

,

( 163)

where

(164)

(8) Z 2 3 3 x 3 2 2 2

~ - - ~Jd vI J d x "<I>(x) exp(- - ) [d

Vz

U(v₁₂ -...!...)

m w I rD mex

(C) ~ = -16 :3 2 4

""3

~ n " '" 2 2 L m W o e !

o

dk k l1(k) ~ , ( 165) (166)

wh~r~ Wand 11 are def~ned by Eqs. (145) and (162), respectively, (Uld { is the unit dyadic. In Eq. (163), the first term On the dght i6 the first orde.r (in f.) contribution and all other t~rn'lS are collisional contributions. found in second ordeJ:: in it.

'{he i.ntegrals in EqB.(164). (165) and (166) were performed by R.L. Guernsey [cf. Appendi~ DJ and the result is

2 3 3

W 2

w

/(4TI)T

- ~ ~e _{411.lw +}

_'3

₂

8 [-In e- 20y -

!

+ 2 In 2J (oj

rr

wheJ::e f "- ~ is tbe plasma param~ter and y is EuleJ::'s constant,

Xu

y = 0 • .577 ••.

I 16,26 . f .

The above expression agrees with R.L. Guernsey s result, 1 hlS re.sult is corre.cte.d for the fact that R.L. Guexnsey originally considered only two out of three contributions to the conductivity,

. . 17,26 whereas all three turn out to be of the same order a posten.O);1.