Kinetic Theory of Nonideal Gases and Nonideal Plasmas 0080216714

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Yu.L. Klimontovich

KINETIC THEORY OF NONIDEAL GASES AND NONIDEAL PLASMAS

Other Titles of Interest

AKHIEZER Collective Oscillations in a Plasma CHERRINGTON Gaseous Electronics and Gas Lasers LOMINADZE Cyclotron Waves in a Plasma POZHELA Plasma and Current Instabilities in Semiconductors SITENKO Fluctuations and Nonlinear Wave Interactions in Plasmas

Journals

Plasma Physics Journal of Quantitative Spectroscopy and Radiative Transfer

Full details o f any Pergamon publication and a free specimen copy o f any Pergamon journal available on request from your nearest Pergamon office.

Kinetic Theory o f Nonideal Gases and Nonideal Plasmas by

Yu L KLIMONTOVICH Moscow State University, USSR Translated by

R BALESCU University o f Brussels, Belgium

PERGAMON PRESS OXFORD

NEW YORK

TORONTO

SYDNEY • PARIS

FRANKFURT

U.K. U.S.A. CANADA AUSTRALIA FRANCE FEDERAL REPUBLIC OF GERMANY

Pergamon Press Ltd., Headington Hill Hall, Oxford 0X3 OBW, England Pergamon Press Inc., Maxwell House, Fairview Park, Elmsford, New York 10523, U.S.A. Pergamon Press Canada Ltd., Suite 104, 150 Consumers Rd., Willowdale, Ontario M2J 1P9, Canada Pergamon Press (Aust.) Pty. Ltd., P.O. Box 544, Potts Point, N.S.W. 2011, Australia Pergamon Press SARL, 24 rue des Ecoles, 75240 Paris, Cedex 05, France Pergamon Press GmbH, 6242 Kronberg-Taunus, Hammerweg 6, Federal Republic of Germany Copyright © 1982 Pergamon Press Ltd. All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic tape, mechanical, photocopying, recording or otherwise, without permission in writing from the publishers. First edition 1982 Library of Congress Cataloging in Publication Data Klimontovich, IU. L. (IUrii Lvovich) Kinetic theory of nonideal gases and nonideal plasmas. (International series in natural philosophy; v. 105) Translation of: Kineticheskaia teoriia neideal nogo gaza i neidelrioi plazmy. Includes bibliographical references. 1. Plasma (Ionized gases) 2. Gases, Kinetic theory of. I. Title. II. Series. QC718.K5513 1982 544L7 82-9044 ISBN 0-08-021671-4

In order to make this volume available as economically and as rapidly as possible the author's typescript has been reproduced in its original form. This method un­ fortunately has its typographical limitations but it is hoped that they in no way distract the reader.

Printed in Great Britain by A. Wheaton & Co. Ltd., Exeter

Preface

T here a r e many books on k i n e t i c t h e o r y o f g a s e s and p lasm a s on t h e m a r k e t . What was t h e m o t i v a t i o n o f t h e a u t h o r i n w r i t i n g a n o t h e r book on t h i s s u b j e c t ? The f o u n d a t i o n s o f p r e s e n t - d a y n o n - e q u i l i b r i u m s t a t i s t i c a l t h e o r y o f g a s e s and plasm as i s due t o t h e i m p o r t a n t work o f N.N. Bogolyubov, M. Born, A. A. V l a s o v , H. G reen, J . Kirkwood, J . Yvon, L.D. Landau and I . R . P r i g o g i n e . attem pt

to

present

a p p lie d to plasm as

a

some o f t h e

i d e a s and methods

In

th is

o f t h e i r w ork.

book we

They w i l l be

c o m p le te d e s c r i p t i o n o f th e k i n e t i c p r o c e s s e s i n n o n i d e a l g a s e s and

and t o

the d e r iv a tio n

of the k i n e t i c

theory

of

long-range

fluctuations.

The l a t t e r i s i m p o r t a n t , i n p a r t i c u l a r , f o r t h e d e s c r i p t i o n o f t u r b u l e n t p r o c e s s e s , d e f i n i n g t h e s o - c a l l e d anomalous t r a n s p o r t p r o c e s s e s . C le a rly , the sim p lest case p o ssib le

to

param eter.

in tro d u ce

a

s m a ll

is

t h e one o f weak n o n i d e a l i t y , i n which i t i s

param eter:

the

den sity

param eter

or

the

plasma

For d e n s e r sy ste m s one u s e s model e q u a t i o n s , a s i n e q u i l i b r i u m t h e o r y .

In t h i s f i e l d , t h e de v e lo p m e n ts a r e a t p r e s e n t s t i l l v e r y p r e l i m i n a r y . The

theory

is

constructed

i d e a l g a s e s and p l a s m a s .

as

a

g e n era liz atio n

of

the

k in etic

th eo ry of

I t is t h e r e f o r e im p o rtan t to analyze the l i m i t a t i o n s of

the u su a l k i n e t i c e q u a tio n s .

The book i s w r i t t e n w i t h g r e a t d e t a i l ;

therefore i t

sh ould be o f u s e n o t o n l y t o r e s e a r c h p h y s i c i s t s , b u t a l s o t o p r o f e s s o r s , and t o graduate stu d e n ts of v a rio u s s p e c i a l i z a t i o n s . The book c o n s i s t s o f t h r e e p a r t s .

The f i r s t p a r t i s d e v o te d t o th e c l a s s i ­

c a l k i n e t i c t h e o r y o f n o n i d e a l g a s e s , t h e second t o t h e c l a s s i c a l k i n e t i c t h e o r y o f f u l l y i o n i z e d p l a s m a s , and t h e t h i r d t o t h e quantum k i n e t i c t h e o r y o f n o n i d e a l g a s e s and p l a s m a s .

The c o n c l u d i n g c h a p t e r p r e s e n t s a s h o r t a c c o u n t o f t h e k i n e t i c t h e o r y

o f c h e m i c a l l y r e a c t i n g sy s te m s and o f p a r t i a l l y i o n i z e d p l a s m a s . inclu d ed in o rd e r present problem .

re su lts,

to and

in d icate to

some

a ttra c t

d irec tio n s

a tte n tio n

of

upon

fu rth er th is

T h is c h a p t e r was

g e n era liz atio n s

im portant

and

of th e

in te re stin g

PREFACE

X

The main s t r e s s R e la tiv e ly

little

sp a c e

is

l a i d h e r e on

is

given

to

th e

the

fu n d a m e n ta l

a p p lic atio n s .

aspects

of

Whenever

the

th eo ry .

p o ssib le ,

the

r e a d e r i s d i r e c t e d to w a rd s a d d i t i o n a l l i t e r a t u r e . In

several

w i t h my s t u d e n t s

p laces and

of

t h i s book

c o llab o rato rs:

W. K r a e f t , V.A. Puchkov, E .F . S l i n ' k o .

I

use d r e s u l t s o b t a i n e d i n c o l l a b o r a t i o n

V.V. B e l y i ,

YU. A. Kukharenko,

W. E b e l i n g ,

The c o l l a b o r a t i o n w i t h them was f o r me n o t

only u s e f u l , but a ls o p l e a s a n t. K.P. Gurov was th e f i r s t p e r s o n who r e a d my work on th e k i n e t i c t h e o r y of n o n i d e a l g a s e s and p l a s m a s . g r a t e f u l t o him f o r h i s h e l p .

He a l s o r e a d th e m a n u s c r i p t of t h i s book.

I am v e r y

I a l s o g r a t e f u l l y acknowledge t h e r e m a rk s and d i s ­

c u s s i o n s a b o u t t h e m a n u s c r i p t w i t h V.V. B e l y i , L.M. Gorbunov, M.E. M arinchuk and A.A. Rukhadze.

The Author

PART I

Kinetic Theory of Nonideal Gases

INTRODUCTION The b a s i s

of th e

kin etic

p a r ti c l e d i s t r ib u t i o n fu n ctio n :

theory c o n s is ts

of

the

the k i n e t i c eq u atio n s.

equations

for

the

one-

T y p ical k i n e t i c e q u atio n s

a r e t h o s e o f Boltzm ann, o f V l a s o v , o f L andau, and o f B a le s c u and L e n a r d . A ll sim p lified are

the

phenomena,

In o rd e r

k in etic

d escrip tio n

to

which

include

equations of

the

cannot

are

approxim ate;

sta tistic a l

processes

therefore

they

provide

i n g a s e s and p l a s m a s .

a There

be d e s c r i b e d i n te r m s o f t h e known k i n e t i c e q u a t i o n s .

such phenomena

in

the

d escrip tio n ,

the

a s s u m p t io n s

made

i n th e d e r i v a t i o n s o f t h e s e e q u a t i o n s must be weakened and t h e s e e q u a t i o n s m ust be g e n era liz ed . Thus, i n d e r i v i n g t h e Boltzmann e q u a t i o n from t h e L i o u v i l l e e q u a t i o n ( o r from t h e c o r r e s p o n d i n g s i t y param eter

e .

BBGKY h i e r a r c h y ) one makes u s e o f t h e s m a l l n e s s o f t h e d e n ­

S i m i l a r l y , f o r a plasm a one assum es t h a t t h e plasm a p a r a m e t e r

y

i s sm all. The p a r a m e t e r s

e and

y c h a r a c t e r i z e th e r o l e of th e i n t e r a c t i o n s in the

k i n e t i c eq u atio n s fo r gases or plasm as.

T h is r o l e i s t w o f o l d .

On t h e one h a n d , i t

d e f in e s th e r e l a x a t i o n p r o c e s s e s r e s p o n s i b l e , f o r i n s t a n c e , f o r th e approach to eq u ilib riu m .

In o th e r words, th e i n t e r a c t i o n determ in e th e d i s s i p a t i v e p ro c e ss e s

i n g a s e s and p l a s m a s . On t h e o t h e r hand, t h e i n t e r a c t i o n s c o n t r i b u t e t o t h e n o n - d i s s i p a t i v e p r o p e r ­ t i e s , e . g . , t h e thermodynamic f u n c t i o n s ( i n t e r n a l e n e r g y , p r e s s u r e , e n t r o p y , e t c . ) . These c o n t r i b u t i o n s o f t h e i n t e r a c t i o n s a r e r e s p o n s i b l e f o r t h e d e v i a t i o n s o f t h e s e q u a n t i t i e s from t h e i r i d e a l v a l u e . I n t h e Boltzm ann, Landau o r B a le s c u - L e n a r d e q u a t i o n s , t h e i n t e r a c t i o n s d e t e r ­ mine o n l y t h e d i s s i p a t i v e c h a r a c t e r i s t i c s .

1

I n t h i s s e n s e , t h e s e e q u a t i o n s c an be

KINETIC THEORY OF NONIDEAL GASES AND NONIDEAL PLASMAS

2

c a l l e d k i n e t i c e q u a t i o n s f o r th e i d e a l gas o r p la s m a . present

book

plasm as.

is

th e

construction

of

k in etic

One o f t h e problem s o f th e

equations

for

n cn id eal

gases

and

W it h i n t h e i r c o r r e s p o n d i n g models — t h e b i n a r y c o l l i s i o n a p p r o x i m a t i o n o r

th e p o l a r i z a t i o n a p p r o x i m a t i o n — t h e s e e q u a t i o n s t a k e a c c o u n t o f t h e c o n t r i b u t i o n s o f t h e i n t e r a c t i o n s t o b o t h th e d i s s i p a t i v e and t h e n o n - d i s s i p a t i v e p r o p e r t i e s [1 7 , 18].

(see a ls o r e fs

[ 6 , 67, 6 8 , 71, 7 3 ] ) .

The se cond p roblem o f t h e book i s th e c o n s t r u c t i o n o f k i n e t i c e q u a t i o n s f o r dense g a s e s .

The f i r s t d i f f i c u l t y i n t h i s d i r e c t i o n i s t h e d e r i v a t i o n o f e q u a t i o n s

t a k i n g i n t o a c c o u n t b o t h b i n a r y and t r i p l e c o l l i s i o n s .

Such an e q u a t i o n was d e r i v e d

by Choh and U hlenbeck [ 5 ] , by u s i n g B ogolyubov?s e x p r e s s i o n f o r th e t w o - p a r t i c l e d i s t r i b u t i o n fu n c tio n to the f i r s t o rd e r in the d e n s i ty p a ra m e te r. equation i s not q u ite c o n s i s t e n t .

In

th e

d issip a tiv e

T his k i n e t i c

c h ara cte ristics

it

takes

a c c o u n t of b o t h b i n a r y and t r i p l e c o l l i s i o n s , b u t i n th e n o n - d i s s i p a t i v e p r o p e r t i e s i t r e t a i n s only b in a r y c o l l i s i o n s .

I n t h i s book we d e r i v e an e q u a t i o n i n w hich t h e

t r i p l e c o l l i s i o n s a r e t r e a t e d more c o m p l e t e l y . In r e f .

[ 4 ] , Bogolyubov d e v e lo p e d

a method

by w h ic h ,

assum ing

th e

com­

p l e t e w eakening o f th e i n i t i a l c o r r e l a t i o n s , he expands t h e two-body c o r r e l a t i o n s s y s t e m a t i c a l l y i n powers o f t h e d e n s i t y p a r a m e t e r .

C l e a r l y , t h i s method a l s o l e a d s

to an e x p a n s io n o f t h e c o l l i s i o n i n t e g r a l o f t h e k i n e t i c e q u a t i o n i n powers o f t h e de n sity .

However, th e r e a l i z a t i o n o f B ogolyubov’ s programme f a c e s some d i f f i c u l ­

t i e s of p r i n c i p l e .

The i n v e s t i g a t i o n s o f W einstock [ 1 9 ] , Goldman and Freeman [ 2 0 ] ,

Dorfman and Cohen [ 2 1 ] , showed t h a t t h e c o l l i s i o n i n t e g r a l , i n c l u d i n g f o u r - b o d y and h ig h er o rd e r c o l l i s i o n s , d iv e rg e s [ 22]. The s o l u t i o n o f t h e s e d i f f i c u l t i e s l e a d s t o t h e m o d i f i c a t i o n o f t h e b a s i c a s s u m p tio n s u n d e r l y i n g t h e k i n e t i c e q u a t i o n s .

It

was

shown t h a t

the

c om p le te

w eakening o f th e i n i t i a l c o r r e l a t i o n s must be r e p l a c e d by t h e more f l e x i b l e assum p­ tio n

of

the

a s s u m p t io n ,

p a rtial

w eakening o f

we d e r i v e

from

the

these

co rrelatio n s

L io u v ille

d i s t r i b u t i o n f u n c t i o n i n phase s p a c e . t h e smoothed d i s t r i b u t i o n f u n c t i o n s .

equation

[ 2 3 , 24] .

By u s i n g

th is

an e q u a t i o n f o r t h e smoothed

From t h e l a t t e r we d e r i v e a h i e r a r c h y f o r I t d i f f e r s from th e

BBGKY h i e r a r c h y i n r e ­

t a i n i n g e x p l i c i t l y th e d i s s i p a t i o n due t o b i n a r y c o l l i s i o n s .

If th is h ierarchy is

so l v e d by assum ing t h e c om p le te w eakening o f t h e i n i t i a l c o r r e l a t i o n s i n a tim e s h o r t e r than the b in a ry

c o llisio n

re la x atio n

t im e ,

the

Boltzm ann

equation

is

recovered. For d e n s e r g a s e s , we o b t a i n from t h e smoothed h i e r a r c h y whose c o l l i s i o n i n t e g r a l i s c o n v e r g e n t . equations

tak in g

in to

account

In

fo u r-b o d y ,

e q u a t i o n s become more and more c o m p l i c a t e d .

th is

way,

five-body

one

a k i n e tic equation

may c o n s t r u c t

c o llisio n s, e tc .

But

k in etic these

One t h e r e f o r e u s e s a more c o n v e n i e n t

method, a n a lo g o u s t o t h e one use d i n e q u i l i b r i u m s t a t i s t i c a l m ec hanic s o f de n se

INTRODUCTION g a s e s and f l u i d s .

3 In ste a d of a k i n e t i c e q u a tio n f o r th e o n e - p a r t i c l e d i s t r i b u t i o n ,

one r a t h e r u s e s a s e t of e q u a t i o n s f o r t h e o n e - p a r t i c l e d i s t r i b u t i o n and f o r t h e binary c o r re la tio n s . From t h e h i e r a r c h y f o r t h e sm oothed d i s t r i b u t i o n f u n c t i o n s we may d e r i v e k i n e t i c eq u atio n s ta k in g in to account the long-range f l u c t u a t i o n s .

From t h e l a t t e r

we may d e r i v e hydrodynam ic e q u a t i o n s i n which t h e v i s c o s i t y and t h e t h e r m a l c onduc­ tiv ity

are

d e te r m i n e d

not

o n l y by

the

c o llisio n s,

but

a lso

by

th e

lo n g -ran g e

flu ctu atio n s. I n t h e d e r i v a t i o n o f t h e Boltzm ann e q u a t i o n , one assumes i m p l i c i t l y t h e c o n t i n u i t y of t h e c o l l i s i o n p r o c e s s d e f i n i n g t h e c o l l i s i o n i n t e g r a l .

T h is amounts

to d e s c r i b i n g t h e d i s t r i b u t i o n f u n c t i o n a s a d e t e r m i n i s t i c ( n o n - f l u c t u a t i n g ) quan­ tity .

T aking i n t o a c c o u n t t h e d i s c r e t e n e s s o f t h e c o l l i s i o n p r o c e s s e s l e a d s t o

f l u c t u a t i o n s of t h e d i s t r i b u t i o n f u n c t i o n .

These f l u c t u a t i o n s have a r a n g e much

l o n g e r t h a n t h e one of t h e f l u c t u a t i o n s d e f i n i n g t h e c o l l i s i o n i n t e g r a l .

In order

to d e s c r i b e t h e f o r m e r , we may c o n s i d e r t h e Boltzm ann e q u a t i o n a s a L an g e v in eq u a ­ tio n w ith a given source of f l u c t u a t i o n s : [25].

t h e l a t t e r was f i r s t s t u d i e d by Kadomtsev

The d e velopm ent of t h e k i n e t i c t h e o r y o f t h e e q u i l i b r i u m and n o n - e q u i l i b r i u m

f l u c t u a t i o n s i n g a s e s i s a n o t h e r pro b lem of ou r book ( c h a p t e r 4 ) . i n g t h e o r y f o r p las m a s i s s t u d i e d i n c h a p t e r 11.

The c o r r e s p o n d ­

CHAPTER 1

The Method o f Distribution Functions and the Method o f Moments EQUATIONS FOR THE POSITION AND MOMENTUM DISTRIBUTION FUNCTIONS IN A GAS OF MONATOMIC PARTICLES The m i c r o s c o p i c m e c h a n i c a l s t a t e o f a monatomic p a r t i c l e ga s a t tim e d e f i n e d by t h e s p e c i f i c a t i o n o f t h e p o s i t i o n s

t

is

and o f momenta p ^ ..........p^

f

F o r c o n c i s e n e s s , we i n t r o d u c e t h e n o t a t i o n s : x . — ( r. , p . ) , U t t a s i x - d i m e n s i o n a l v e c t o r d e f i n i n g t h e s t a t e o f t h e p a r t i c l e l a b e l l e d i ( I < £ < N) , of a l l the

N p a rticle s.

and x = (x^j . . . j x ^ ) 9 a 6N—d i m e n s i o n a l v e c t o r d e f i n i n g t h e s t a t e o f t h e c o m p le te sy s te m . The d i s t r i b u t i o n f u n c t i o n o f t h e v a r i a b l e s

x i s d e n o te d by f ^ ( x 9t ) .

e x p r e s s i o n f ^( x 9 t ) dx r e p r e s e n t s t h e p r o b a b i l i t y t h a t , a t tim e and momenta o f t h e p a r t i c l e s have v a l u e s w i t h i n a r a n g e dx i s norm alized as fo llo w s: Let

/ dx

The

t 9 the c o o rd in a te s

aro u n d x.

The f u n c t i o n

Kx , t ) - 1 .

rj \ ) ~ ® d e n o t e t h e p o t e n t i a l e n e r g y o f c e n t r a l i n t e r a c t i o n o f

th e p a i r o f p a r t i c l e s

L e t a l s o m d e n o t e t h e mass o f t h e a to m s.

t,J .

Then t h e

H a m il t o n ia n H o f t h e g a s c an be w r i t t e n a s :

H= where u(r .)

& - + u[r. j ) +

2

y

9. .

( 1 . 1)

i s t h e p o t e n t i a l e n e r g y o f an atom i n t h e e x t e r n a l f i e l d .

The d i s t r i b u t i o n f u n c t i o n

11* + 9t

f ^( x 9t ) obeys t h e L i o u v i l l e e q u a t i o n :

T l< i< N

9r,

dH 9 f =o dr. dp. )

( 1. 2)

We now u se H a m i l t o n ’ s e q u a t i o n s f o r t h e p a r t i c l e s :

P. —k- = v . m i

. _ 3H ” 3fT

5

(1.3)

KINETIC THEORY OF NONIDEAL GASES AND NONIDEAL PLASMAS

6

2

^

P , r . *)

(1 .3 )

E q u a t i o n ( 1 . 2 ) can t h e n be r e w r i t t e n

3 // „ l f +

^x* 2 1 ^ i, ^ j

v.•

93 // , .

We i n t r o d u c e d h e r e th e e x p r e s s i o n F ( r . , t ) p a rtic le lab elled

(1.4)

+F(ri } '

t

r e p r e s e n t i n g th e f o r c e a c t i n g on th e

i : (1 .5 )

(ri - ^ ‘ - S 7T < \J < N % +F^{ri ' ' t 11 < where F = —du/ dr . i s t h e e x t e r n a l f o r c e . 0 ^ We now c o n s i d e r s e v e r a l d i f f e r e n t forms o f t h e L i o u v i l l e e q u a t i o n .

We

i n t r o d u c e th e f o l l o w i n g o p e r a t o r n o t a t i o n s :

'

=

6

= ! !ii

T

(v • ’ *ri

'* 1 ..........* s

E(S 0 ) ....... *« Using t h e d e f i n i t i o n s

+

+

— +2 34 (1 .6 ),

_L

+ F • —- ) ° a,V

( 1 . 6)

e ..

2

(v. . ^ - + F . — \ 1< « S ^

3ri

°

(1 .7 )

K i< j (1 . 8 )

1

( 1 . 7 ) , t h e L i o u v i l l e e q u a t i o n can be w r i t t e n a s :

( Vi ' 3 ^

+F° ' * P l ) ~

^

" °

(1 .9 )

....... / We s e e

th at

th is

eq u atio n

is

p a r ti c l e d i s t r ib u t i o n fu n c tio n .

2

*!

1

J

not

clo sed

V 2 ( * i » * 2 »*)

2 3 rl for

( 1 .1 9 )

dpx

, as

it

a l s o i n v o l v e s t h e two-

S i m i l a r l y , i n t e g r a t i n g Eq. ( 1 . 4 ) o v e r a l l t h e

v a r i a b l e s e x c e p t t h o s e o f two p a r t i c l e s , we f i n d , f o r

N > 1, t h e e q u a t i o n f o r t h e

fu n ctio n f 2 :

= « | d x 3 ( 0 J 3 + e 2 3) ^3 Here we used t h e d e f i n i t i o n s

(1 .6 ),

Thus t h e e q u a t i o n f o r fu n ctio n

.

.

(1.20)

(1 .8 ).

f 2 i s not c lo s e d , as i t in v o lv e s the d i s t r i b u t i o n

The e q u a t i o n f o r

h ie r a r c h ic a l chain of eq u atio n s.

in v o lv e s

f ^ 9 and so o n .

We th u s o b t a i n a

For t h e d e t e r m i n a t i o n o f t h e f u n c t i o n

to know a l l t h e d i s t r i b u t i o n f u n c t i o n s o f t h e s y s te m , up to

.

we need

T h is c h a i n o f

e q u a t i o n s i s c a l l e d t h e Bogolyubov h i e r a r c h y , o r t h e BBGKY h i e r a r c h y (B ogolyubovB o rn-G re en-K irkw ood-Y von).

(2 < 3 < N) c an be w r i t t e n

The g e n e r a l e q u a t i o n f o r

as fo llo w s:

J