Some Aspects of Diffusion Theory (C.I.M.E. Summer Schools, 42)

A. Pignedoli ( E d.)

Some Aspects of Diffusion Theory Lectures given at a Summer School of the Centro Internazionale Matematico Estivo (C.I.M.E.), held in Varenna (Como), Italy, September 9-27, 1966

C.I.M.E. Foundation c/o Dipartimento di Matematica “U. Dini” Viale Morgagni n. 67/a 50134 Firenze Italy [email protected]

ISBN 978-3-642-11050-4 e-ISBN: 978-3-642-11051-1 DOI:10.1007/978-3-642-11051-1 Springer Heidelberg Dordrecht London New York

©Springer-Verlag Berlin Heidelberg 2011 st Reprint of the 1 Ed. C.I.M.E., Ed. Cremonese, Roma 1967. With kind permission of C.I.M.E.

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CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C.I.M.E.) 40 Ciclo - Varenna - dal 9 al 27 settembre 1966

"SOME ASPEC'I'S OF DIFFUSION THEOR yn

Coordinatore : Prof. A. PIGNEDOLI

V. C. A. FERRARO

: Diffusion of ions in a plasma with applications to the ionosphere pag. 1

p. C. KENDALL

: On the diffusion in the atmosphere and ionosphere

pag.81

F. HENIN

: Kinetic equations and Brownian motion

pag.155

T.KAHAN

: Theorie des reacteurs nucleaires methodes de resolution perturbationnelles, iteractives et variationnelles. pag.349

C. CATTANEO

: Sulla conduzione del calore

C. AGOSTINELLI

: Formule di Green per la diffusione del campo magnetico in un fluido elettricamente conduttore pag.487

A. PIGNEDOLI

: Transformational methods applied to some one-dimensional problems concerning the equations of the neutron transport theory. pag. 503

A. PIGNEDOLI

: On the rigorous analysis of the. problem of the neutron transport in a slab geometry and on some other results. pag.519

G.SESTINI

: Principi di massimo per Ie soluzioni di equazioni paraboliche.

pag.485

pag.539

CENTRO INTERNAZIONALE MATEMATICO ESTIVO

(C. 1. M. E.)

v. C. A.

FERRARO

DIFFUSION OF IONS IN A WITH APPLICA TIONS TO

PLASMA

THE IONOSPHERE

Corso tenuto a JJ.arenna dal 19 al 27 settembre 1966

DIFFUSION OF IONS IN A PLASMA WITH APPLICATIONS TO THE IONOSPHERE by V. C. A. Ferraro (Queen Mary College, University of London)

1. Derivation of the diffusion equations in plasmas 1.

The term 'plasma' ,was first used

of a gas

by Langmuir for the state

which is fully ionised (for example, the high solar atmosphe-

re) or only partially ionised, (for example, the ionosphere). Our main interest in this course will be the diffusion of ions in such a plasma, arising from non-uniformity of composition, of pressure gradients or electric fields. We begin by considering the simple case of a fully ionised gas and for simplicity restrict ourselves to the case when only one type of ion and electrons are present. 2.

The velocity distribution function We make the familiar assumption of molecular chaos, in which

it is supposed that particles having velocity resolutes lying in a certain range are, at any instant, distributed at random. It is therefore most convenient to

use six dimensional

the resolutes

of the position vector

the plasma can then fat

(t,~,~)

component

,where 0(.. ,

space in which the coordinates are r and velocity

v. The state of

be specified by the distribution functions t

is the time, that characterise each particle

for example,

the ions or the electrons The quantity

(1 )

then represents element

dr dv.

the number of particles in the six dimensional volume In the simplest case, the plasma consists of single

-4-

V. C.A. Ferraro

ions (

a.. =

i) and electrons

(~=

e). In more complicated cases, the

plasma may consist of several ion species in addition to neutral particles (4'= n) such as atoms, molecules. exited atoms, etc. The total number of particles of constituent

in the element dr

DC<

is obtained

by integrating (1) throughout the velocity space. This number is. by hypothesis,

n", dr ....

and

thus

n«

=je(t, ...

r. v ) d v (2). The beha-

-

-lit

-C(

viour of the ionised gas is described by a system of equations (Boltzmann equations) which can be derived as follows. Suppose that each particle of

mass

time

in which

dt

is

acted on by force

mIIC.F"" ...

the particles of constituent

suffer

Of,

the same

particles that occupy the volume of phase space

at time

t

at time

then in a no collisions, dr dv

--fIl

would occupy the volume of phase (.r + v dt)(v + F dt) -

-II(

r;&

-

+ dt . The number in this set is (,,(t + dt, r

+ v dt, v + F dt)

....

Cit-I/.

and the difference E (t + dt, r + v dt, :reJ, - -(II.

v + F dt) - f (t, r, v ) dr dv

-01 -f(.

c¥;

-

-«. -

~

therefore represent the difference in the gain of particles

by collisi.ons to

this final set and the loss of the particle to the original set dt. This must be proportional to

Crt dr. dv III. dt . Taking the mann's equation

f ~

(3)

where

for

..

\J -v

f~,

dr

dt -+ 0 , we arrive

viz

-d. V

f "

+ (F

stands for the gradient

in velocity space.

dv dt; and we denote it by

~

limit as

+ (v .M)

in time

-(II.

.'Vv-Ill ) fol

operator

= C It

'0

at Boltz-

-- 5-

v. C. A. Ferraro 3.

Charge neutrality and the Debye distance In general a plasma will rapidly attain a state of electrical

neutrality; this is because the potential energy of the particle resulting from any space charge would otherwise greatly exceed its thermal energy. Small departures from strict neutrality will occur over small distances whose order of magnitude can be obtained as follows. The electrostatic potential

V

(4)

V V =-47C(Zn.1

Here Ze

satisfies Poisson's equation.

2

is the charge on

- n )e e

an ion and

-e

thermodynamic equilibrium , the number

that of the electrons. In

densities of the ions and elec-

tions respectively are given by (5)

where k is

n. = n.(o) exp(-ZeV/kT.), 1

1

n = n(O) exp (eV/kLt) , e e ~

1

the Boltzmann constant,

are the ion and electron e n~o) and n(o) are the values of n. and n temperatures and 1 e 1 e for strict neutrality so that n(o) = Zn~o) . In general, departures from e ~ neutrality are small so that we may expand the exponential to the first power of the arguments

only. We have (0)

Zn. - n :Zn. 1 e 1 and hence (6)

where (7)

T., T 1

approximately

ZeV (0) eV (1 - - - . ) - n (1 + - ) uT t e uTe

-6-

V. C. A. Ferraro

The quantity

D has

the dimensions of

a length and

is called

the Debye distance, The solution of (6) for spherical symmetry is (8)

where

e"

is

from the origin

the charge on the particle. For small distances r (r < < D), (8) reduces

to the pure Coulomb potential

of the charged particle. For large distances(r > > D), V -+ 0 exponentially . Thus in a neutral plasma in thermodynamical equilibrium the Coulomb field of order

of the individual

charge is cut

off (shielded) at a distance

D. Hence, we may aSbume that the particles do not interact

in collisions for which the impact parameter is greater than D. The Debye

shielding is not established instantaneously; oscillations of the

= (47rtl2/m J(SinCe the diso e e) placemente of the electrons (or ions) bodily by a distance x gives rise space charge will have

to

a frequency

w

an electric field of intensity

47Cn ex lending to restore neutralie ty) . Thus the time required to establish shielding is of the order

~_1 w o

4.

Diffusion of test particles in a plasma A particular particle, which we call 'test particle',

will suffer collisions with

in a plasma

the other particles in the plasma, which we

call 'field particles' . Electrostatic forces between the particles have a greater range than the forces between neutral molecules in an ordinary gas. Consequently, the cumulative effect of distant encounters will be far more important than the effect of close collisions, which change comple-

-7-

V. C. A. Ferraro

tely the particle velocities. We shall

therefore suppose that the de-

flections which the test particles undergo are mostly small. The motion of the test particle is most conveniently descibed in the velocity space, i. e., a space in which the velocity vector! is taken as

vector and the apex of this vector is

the position

called the velocity point of the

particle. Referred to Cartesian coordinates the coordinates of these points will

be denoted by v , X

V

,v .

Y z

As the test particle changes its position in ordinary space, its position in velocity space changes either continuously or discontinuously due to encounter with fixed particles. In

general the disp;lacement is

complicated. (Fig. 1) It is clearly impossible, and indeed futile,

to trace

the motion

and we are forced

of a single particle

to consider a statisti-

cal description of the motion. In this, instead of a single particle, we consider an

o

~

assembly containing a large number of test particles which

have the same velocity v --

initially. FIG. 1. Suppose these are concentrated around the point space . At subsequent times the cloud will size

and shape,

as

v

-0

in the velocity

spread, changing both its

a result of successive encounters.

-0

-8-

V. C. A. Ferraro

.1fo 'IIi

(1=0)

We now require

to

find quantities which

will adequately describe the process. One

" ~,

such

IS-jj?l

tl!.

quantity is the change in velocity of a

test particle produced by the

encounters. Suppose 0')-_ _ _ _ __

to the

z-axis

Jjv ,Av ,L1v

x

y is

that

v

is parallel

-0

and consider

the resolutes

of Ll v . Suppose that

z

produced x by the ith encounter. Then after N (llv).

Xl

FIG.2.

the change

in Llv

encounters, N

IJ.v = X

We assume that dy seen, we

all

L

i=l

(Av).

x

I

the encounters are random, but as we have alrea-

cannot predict the change

Ll v X

However, we can define an average value of

for a single test particles. _

t1 vx,

say

Ll v for the

x large assembly of particles under consideration. If the distribution of cities _ is isotropic, then But /Jv

z

!l vx =0,

need not vanish

by symmetry, and likewise .dvY.= 0.

since the assembly (or cloud) has an initial

velocity in the z-direction. However not vanish . This and

mean value will

the mean square

).(dv ). : If the collisions xJ that successive collisions will produce,

x1

all

as

equal. But

averaged over all

are

particles

un correlated . Hence

Jj}

products

will

x~

(~v). X 1

small we may expect

on the average, the

the first collisions. Thus the the mixed

of

.

contam terms of the form

(/J v

rage change

velo~

N terms

s~ave-

rAv)~ are Xl

(Llv ).( Av ). will vanish when x1 X J considered since successive collisions are

-9-

V. C. A. Ferraro

Lh 2 x

(9)

The dispersion

-IN,

of

----z

= N(Av).

X 1

the points in Fig. 2 will

therefore increase like

but not, in general, equally in all directions. But the centre of

gravity

may be displaced by an amount proportional to

N. (Fig. 2)

The dispersion of the points in the velocity space produced by collisions

of

the test

particles

to the diffusion of particles diffusion in per unit of

be denoted

x

<

To measure the rate of

the average

value of (9}

value of ~ v 2 , measuring the increase x

x

• If the velocity distribution of the field particles is

The encounters

< !:J. v

x

isotropic,

> and vanish identically, x

y

between test and field particles whiCh we are

considering are assumed to ~

ordinary gas.

of dispers ion of a group of particles per second, will 2 by Av > and called a 'diffusion coefficient', a term due to

the diffusion coefficients

Let

an

v direction, we consider

time. The resultant

velocity

Spitzer

the

in

with the field particles is analogous

be the velocity of a

be binary

encounters only . (;fq

field particle relative to

a test particle.

Then there will be only three independent diffusion coefficients, namely, 2 2 vII and v.L are measured . and <.6.v,L >, where respectively parallel and perpendicular on the

to

v. Their

values will

velocity distribution function of the field particles.

The justification

for this will be given in Section 7.

depend

-10 -

V. C. A. Ferraro

5.

Binary encounter of two charged particles (Hyperbolic orbit)

Consider the motion of charge e 2 relative to charge

e 1 ; let

position vectors to a

of

£1 and

e 1 and

£2 be the

e 2 relative

Newtonian origin. Then the eEluation of

motion of the charges are respectively e e r e e r .. 1 2•• 1 2mr =+--- mr =3 1-1 3' 2-2 r r where £

=£2

- £1 and

are the masses ••

r -

Let p

m

1 2

v

00

II

of the charges. Hence 1 1 -e e (- +-, I 2 m1 m I 2

...

- r -1

r

3

that

is,the relative motion is the same as that

of a

particle

under a central

is

the reduced

force at

be the relative

mass. (Fig. S).

velocity of the charges

the impact parameter. The energy integral

tion,

whence (10)

The Dolar

r ......

A varying inversely as the square of the ele2 distance whose strength is - - }\There m 12

FIG.3. In

=-2 r

m 1 and m2

eauation

of the orbit is

at infinity and

is, with the usual nota-

-11-

V. C. A. Ferraro

.t

r" 1 + e cos1

(11 )

-t

where

is

r -+ 00, gJ'" '" - w

the semi-latus rectum and so that

the eccentricity. As

(11) gives

cosw Also

e

e

AC" ae ; hence sinw = .Eae 2 2 I " sin w + cos w

Thus

2 _p_

1

2

"2' + 2 2 e

or

a e

2

+ P2 gi ving a

cosw=R 1

1+

e =1

sinw

p

=

tanw

=E a

L a

2

or using (1) ( 12)

6.

Calculation of diffusion Consider

particles

(~)

the scattering of test particles (<<,) by a flux of field

• The spatial

density of the latter

dnp"

where

v'is the velocity

function •. Ct

of the field

with a field

coefficients

is

f, ) (~'

d~'

of the particles

and

fp

the distribution

particles. Consider the collision of a test particle

particle

test particle is related to

P of this

flux. Then the velocity ~ «.

the velocity

v

-g

the two particles and their relative velocity

of the centre of mass u

by

of the of

-- 12 -

V. C. A. Ferraro

-tt

=-g v +

v

is unaltered oy the encounter,

v

hence, since

-g

m ctfJ

J1v-(i ,,-L1u ma..-

(13)

where

u

m OltfJ

the change

in

is

the reduced mass

u

produced

Also, in taking test particles in over all

the

particles

by

(14)

m p' and l1u

the encounter.

the average of tre change of velocity over the

assembly, the summation reduces to

of the flux incident on

The number of particles moving through 1L perpendicular

m~ and

of

to

in unit

1:1

dnp

I~

and integrate over all

the

of the field

we find

particles

< Ll v k >

(15)

of

a plane

is

f~ (v') d;'.' u dA

of the vector

plane

fixed scattering centre.

an area dA = pdpdj'

time

J d A "

Multiply this by the components

a

a summation

and

'l(

A -::c/,

then

given

by

(13)

over the velocities

"Sfp (-::') w k d-::'

(k = x, y, z)

where

(16)

where

kl =

w

m (

~)

2

r

Jll uk l1 ut

ud A

(15) and (16) are the diffusion

~t....f. coefficients . It will be convenient to com-

pute these

to a

integrals

relative

coordinate system in which the z-axis

-13-

V. C. A. Ferraro

is along

~

.

For

referring to

Fig. a . and Fig. 4b .

X

;:.

e

ac.

3-

L

FIG.4b FIG.4a

we have

Aux = usin 6 cos ~ l1uy = usin 8 sin, .l1u =-u(1 -cos 6) z

Also from (12)

and the

fact

that

6 = 11: -2w

we

have

6 •.. pJ.

tan 2

(17 )

P

ect ep

where

Pj,

=-2 mtl~u

Then

.1

PPJ. u = 2u - x 2 2 P +pJ.,

Integration

with

w

z

LJ

J

sin 0

.r

6('(' ~ -1~ J.

respect

to

=0

=w

w

whilst

"PP,l. u = 2 u -2-2 Y 'I. P +P,l..

cos ( f ) ,

x

=moLp met

p

t'+~l. and

1

Y

) Au plane

z

ud A.

over

the plane gives

-14 -

v, C. A. Ferraro If the limits

of integration ,for

diverges; however, we individual charges

is

pare

and

the integral

00,

have already seen that the Coulomb field of cut off at distance of order

stance, Hence we can take D as We then

0

the upper limit for

D, the Debye dip

in the integral,

find me(,

1 +-

mp

--241tu m

(18)

1+~

mp

=-:\--2 4'lru

where

1 =)

( 19)

'D

=log e

pdp 2 2 o J.. p +p

It has

been

is likely

=

to

1 ke V (

tacitly assumed

with (.!9a)

illustrate that this

i

2

D

D» P.l. ; to

be the case in general, let us take T 70 5 -3 10 K), n. '" n = 10 em ,and Z = 1 , then 1 e 1 D = ( kT 2)2 1 -3 81( fII; - -2 X 10 em

and

so that

that

P.J,.

Dip

-

in

the

:.L

1 -10 "2-'(10

10 7 , Hence in numerator )., -

log

(19) we

and

t

EPI ...

em may neglect

write = log [_3

l

= Te =

2e

3

PJ. compared

- 15V. C. A. Ferraro

Likewise w

m

XX

2

= (~)

)

mill

r

m 2 PPJ. 2 2 (flu) u dA = ( ;p) 2u "T2 COSI) updpd1 x CL plane P +p

J (

..L.

(4'ICQK ~2 - 4 '" u mol .

~D

_ _1_

1

m"p

"zz = ( - - )

21

l'I\t

~)

P

J

....

p = 0,

neglecting terms of ordert

a. f~:

mot

2

(tJ uz )

~2

(4mft'~ot~\2

4"mJ

wyy = wxx and Woe

Again

o

1

( A- 2)

P3 dP ( 2 + 2. 2

u dA = (m

p)

Finally 2

""

p1 2

(

J (- 2u -2-2-) updp d~

plane

and the integration with respect to

p

from

is finite ;

0

to

00

since the

integral

we find w

Since

this

to zero. We sor

is

A

zz

times

can

now

respectively. In

W

k

.L

in

fact,

49t(~) m u at

smaller than w

xx

express

wk and

or

w it may be set yy wkf, as a vector or ten-

fact,

=

-A.

'"-t

"¥

41tu 2

u

--k... u

Wkf (: : }(: : :)- (:' : :) (21)

= A(S kl

be carried out

t.1... 2

=

1+ (20)

can,

p +p

- \

l )

2u

u

-

16V. C. A. Ferraro

A -(A_ ~ )4'7tU _1_ (4 7Ct~12 m

where (22)

2

Finally

< l::l v > = - (1 + -

(23)

k

cl

m~

mp

w,

)Q

) u

~

u3

f

~

(VI)

-

d

VI

-

(24)

u

where

=v

-

VI

(25) It can

be

shown

that

the third

and

higher diffusion coefficients

are smaller than the first two diffusion coeffients by a factor

.A . This

of

be visualized as which

means that the motion

a diffusion

in

of Coulomb particles can

velocity space, The approximation in

only the first two diffusion coefficients are considered is called

th e Fokker- Planck approximation.

7,

Justification

of the assumption of binary encounters in

The assumption the interaction smaller thatn where lume

n

is

is certainly justified

the theo-

for short-range force. If

range

d(effective diameter of the molecule) is much -1/3 the mean distance between the particles, n the density will

of the gas,

contain

only a small

is Nd

the sphere

3

= nd «1.

of action , of vo-

number of particles N ,that d

-17V. C. A.Ferraro

Under these. conditions the probability of multiple collisions, involving three or more particles simultaneously, is very small. A description in terms of binary collisions is adequate. Coulomb forces acting between particles of a plasma are

not

short-range forces. The potential energy between two such charges e 1 and e 2 is

e e 1 2 r - - exp (--) r 0

(2 fi)

where

r

is

the distance apart of the charges. Thus the interac-

tion between them extends at least as far as the and for conditions in which re of action

we

Oebye distance

are interested 0 » n

contains many particles,

-1/3

0,

and the sphe-

i. e. ,

3 NO = nO »1

(27)

In this case

a

given particle will

interact simultaneously with many par-

ticles and the results derived earlier on the basis of binary collisions is suspect. A rigorous analysis shows that the formulae derived yield logarithmic accuracy. However, a non-rigorous, but plausible discussion can be given alonl5 the followirg lines. Let us consider suppose

that

a test particle moving through the plasrraand

it is so massive that its velocity can be treated as

r:onstant • Draw a cylinder of radius (Fig.

p

with

~1

()

() FIG. 5.

the trajectory as axis

-

18--

V. C. A. Ferraro

Collisions of the test particles with field particles for whtich p

> > n -1/3 will be many-body collisons. Those characterized by im-

mact

parameters p < < n

the method

-1/ '3 are

used to treat binary collisions need not

collisions with impact to parameters

»

n

-1/3

parameters

e 1e / r so

p <
D.

be restricted to

,but can be extended

potential energy between the charges

tha t the presence of other particles has no effect

on the interaction between two particles than

-1/3

.

Now, when r «D, the is simply

binary collisions . We shall show that

separated by a distance smaller

Thus results derived on the hypothesis of binary collisions

apply for all impact parameters smaller than the Debye radius, i. e. , P « D. Because

D» n

-1/3

,in the present case the collisions can be

regarded as binary interactions even when Accordingly, even if tion formulae which lomb interaction,

p» n

-1/3

as long as p

p - D, the difference between the exact takes

account of other particles, and

« D.

interac-

a pure Cou-

is small (by a factor of order 1) . Thus, cutting off the

Coulomb interaction for the impact parameter

p

=D

provides an appro-

ximate method of taking into account the effect of multiple collisions for which p »n

8.

-1/3

.

Diffusion in velocity space. From a microscopic point of view , the change of spatial coordi-

nates of a

particle during a collision can always be neglected. Hence, as

far as the spatial part particle:

of the phase space is concerned, the motion of a

corresponds to On the other

a

continuous point to point variation.

hand, collisions have

nuity of motion in the velocity space. The

a marked effect on the contivelocity can be changed abruptly

-19 V. C. A. Ferraro

by a single near collision:" essential in a vanishingly small val. Hence, a particular velocity point ! in a cloud velocity space can be

'annihilated' by a

time inter-

of particles in

collision and 'recreated' at

some remote point without passing through intermediate points in

the

velocity space. Thus in general the effect of collisions cannot be expres~.

sed in the kinetic theory by introducing a ten describing the divergence of flux in velocity space. But this will case

1/"

certainly only be the

for near collisions

in

which the

J--------- velocity of the particle is changed abruptly. 1). In the case of coulomb forces, the change in velocity, characterised by the quantities

<~ vk > and is due to the

1'x

FIG. 6

effect of remote interactions example,

if

.A = 15 ,

and the cm nges in velocity are small. For

then the relative

/L1! I

P;l.

v

p

= e -AD _ .... p

change

10

in particle velocity

-6 D

(~,

P

and so very small. If these interactions are referred in velocity space, the whole

process may be regarded as

a form of diffusion. The motion can be re-

garded as nearly continuous.

9.

Calculation

of the diffusion coefficient

for a Maxwellian

distribution of veloci ties. The expressions (23) and

(24) may be expressed

niently by introducing the super-potentials. In fact, since

more conve-

- 20V. C. A. Ferraro

we have

Hence (23)

and (24)

can

be written

(28)

(29)

where (30)

(31)

which

have

been iermed

'super-potentials' by Rosenbluth et al. For

a Maxwellian distribution function m 3/ f(v') = n( - - ) 2 21(n T

e

- 2kT

Chandrasekhar found that (32)



1 m~ mpv = - -1Ita Q (1 -I-- ) G ( - - ) 2 .. I¥P m 2kTp

p

- 21 -

V. C. A. Ferraro

2

2

= <6vxx > + (33) (34) where

is the usual error function and

s

(x) =

(35) Values of

10.

~(x) - x~'(x) 2x2

G and ~ - G are given

by Spitzer and others.

Relaxion times. (Collision interval) The term "relaxation time'! is used to denote the time in which

collisions will alter the original velocity distribution; or again, the time that the ions and electrons in a gas

will

attain, through collisions. a

Maxwellian distribution. Various relaxation times can be defined ; the time between

colli~

sions (collision interval or the reciprocal of the collision frequency) may be defined as the time in which small deflections will deflect test cles through 90 0

•

More precisely, if

parti~

't' 0 is the 'deflection time', we

have

< L\v~ >'to

(36)

Substituting from

(33)

2

=v

we find

3

(37)

1:" 2v o npQcLp(~~-

An energy exchange time

't' E

Gil)

can likewise

be defined

by the relation

- 22-

V. C. A. Ferraro

(38)

the change of

energy

l:l E =.! m(2v~v

(39)

2

If only dominant terms

11

+4v 2

11

+.:1

2

'J.

are required

< AE

2

2 2

> =m v


2 11

>

and (38) gives

v

(40) An important special case

is

3

that of a

group of ions, or a group of

electrons, interacting amongst themselves. If we consider such a group whose velocity has

(;;,; f

root

mean square value for the group,

then

1. 225.

In this case a

the

we find

measure of

both

1:D / "t'E = 1,14 so that 'tD

that

"" ~

and

is

the time required to reduce substantially any ani-

sotropy in the velocity distribution function and the time for the kinetic energies to approach a Maxwellian distribution. We shall call cular value of and will

"t' D the 'self-collision interval' for a group of particles

1: c

be denoted by

From (37) we

1

'tc

(41 )

where

T

is

m Z (3 k T)

31 i

in degrees

have

2

= --'---.~~-

4 4

'l.

5.7 1 it'ne Z log ...'" K, m

the group, It may be written ton

this parti-

AmH

is

the mass of a

typical particle

where m H is the mass

of

of a pro-

For electrons, A = 1!25 so that the self··collision time for elect1 . rons is 43 that for protons, provided the ions and electrons have the

-23 -

V. C. A. Ferraro same temperatures. We consider next the approach to ponent plasma; to fix tuents are ions and

our

equilibrium of a two com-

ideas we consider the case when the consti-

electrons. There are three stages involved in the

process. First, collisions between ions and electrons lead to an isotropic,

velocity distribution of electrons, and the same time collisions bet-

ween electrons themselves establishes a Maxwellian distribution. Secondly, collisions between the ions themselves establishes an isotropic velocity distribution amongst the ions. Thirdly, the ions and electrons which have already attained Maxwellian distribution, but possibly at

different tempe-

ratures

T. and T ,will be brought to the same temperature by colli1 e sions between the ions and electrons.

using (43)

ltv, Av, = v, v, - v. v, 1

1

1

1

1

1

Using (28) (44)

since (45)

2

= - 2Q AV flJJA = - 2Q (D 1 1 ~r T,.. M(lJP

Since the distribution of velocities are Maxwellian, this may be rewritten

- 24-

V. C. A. Ferraro

(46)

where Also

c:

(47)

where

(~)

olp

(..!. m )'Z

~~p

1[~

2" 2 2

=-=

010

€ =k(Tcc + -mot T, ) . After

3/

2

'"

e.p ~log )..

some

m~

0(

t

1

4'l(.v 3

algebra, (46)can be redu-

ced to

It is

7:. Otp

~

=

dt

where (49)

T

dTa.

(48)

-T QI

't'

",

3 mit. mpk J. 2 8(2 7t) 2. ~ to(

easily verified

3/2 2

lop log ~

where

T_ ..... T~ and

mQl.

3/ 2

+....:.! ) mp

that

*

7:"".... : '7::I. : t' :tr. : 'z::. ee

Tn

Trt

(-

11

el

where

le

= 1: /fnMM -: - : -

M is

m

the mass

m

m

of the ion

and

m the

electronic mass. Equation (48)

was first given by Spitzer; it shows that if the

mean square relative velocity, which change appreciably, ,

*

~ot~ is

is oc.

(-mT~

"

T(4)

+-

nearly constant and

rnA

,.

,does not

departure from equi-

partitions decrease exponentially.

11.

Relaxation towards the steady state The solution of

Boltzmann's equation for non-uniform gases is

found by successive approximation. We write

-- 25-

V. C. A. Ferraro

f = f (1 o

+

e ),

is small f is the Maxwellian distribution function and o compared with unity . We have seen that in a plasma of two constituwhere

ents each constituent:

will

approach its Maxwellian distribution in

-

a

time equal to the relaxation time 't" and the two constituents will attain equal temperatures

in

relaxation time

a

. As a first approxi-

mation , therefore, we can take the collision term C

De

f - f

(50)

so

to be of the form

o

-~

that

if

f

is the

distribution function at time t

Maxwellian distribution function, then f - f

o

-+

12.

0

with

time

Equations of

as

=0

the o departure from a Maxwellian state

continuity and motion for

and denote their

and their velocities

by

(51)

n i

where and

v. and v

-1

=

-e

Jrf.1 dv. J.

number

a fully ionized gas

densities

by

ions (i) n. and n 1

e

respectively. Then ,

n = e

Sfe--e dv

f. and

f denote the velocity distribution functions for the ions e electrons respectively and dv. and dv denote an element of volu1

-1

--e

me in the velocity space for ions and electrons, their masses by gas

f

e -t/,,; .

We consider the plasma to be a mixture of positive and electrons (e)

and

by

Pi

and

1

Pe

and the densities e respectively, we have

of the ion

and

electron

Pe =ne m e

(52)

Denote

m. and m

respectively. Denoting

by -1 V. and

-ev the mean velocities of the ion

and electron gas

- 26-

V. C. A. Ferraro

in a volume element of the plasma, then

='v

n v f dv e-e J-e e -e

(53)

It is convenient to introduce the total number density nand

Po'

total mass dens ity

'0

defined

as

o

::in

i

+n

e

Po "Pi + fe

(54)

and the mean

velocity v

of the plasma element defined by

-

-0

.v. + Pe-e f 1-].

(55 )

Let

n

V and -i

electrons,

V

be the peculiar or thermal -e respectively, defined by V

V. = v. - v •

(56)

-1

-1

_0

V

-e

=v

-e

velocities of the ions

and

-v

-0

Then it follows from (55) that (57)

The partial pressure res defined in a v

-0

for the ion and electron gases, am

frame

total pressu-

of reference moving with the mean velocity

are respectively given by

(58)

The hydrostatic partial

Pe

pressures for

r

= e-e V Ve' ions

and electrons

by

(59)

and the corresponding mean

kinetic temperature by

are defined

"- 27-

V. C. A. Ferraro

p "kn T e e e

p, " kn,T"

(60)

1

1 1

Boltzmann's equation for

aftt

~ + (v •'V)

(61)

11

whe"re

-0£

t

f ..

""

the two distribution functions

+ (F

-It

f and fare i e

• 'Vv ) f = C • Q! " i, e -tJ. /J. lit.

m, F" and

m F are the forces acting on an ion and electron e-e respectively. If these are produced by an electric field E and magnetic 1 -1

field B , then

ei

(62)

F, " (E -1 m, 1

where

e. and 1

e

e

+

(E

v,)C B) ,

-1

-

-

are tIE charges

form the moment equations;

lecular properties for the constituent plying equation (61) by

Jill.'

,eB)

-e-

carried by an ian and

respectively. We next

+v

if

«.

t

(~at)

electron

be any function of mo-

of the plasma,

then

by multi-

integrating partially and remembering that

(63)

we

find

(64)

1 t ~)" +y.(n.. -Cb v ) - n ... Jot. -a(, 0(.

3(naL

The right-hand side due to

(65)

represents

collisions. This

~ nat

"S "'C.

F. -0(

V ~"

the change

vanishes if

-r:+V·(n v) + 'I. (n at oc, -0 Ol

1oe." 1

tD C dv Jilt... - '"

of the mean value and

of

$Jot

(64) gives

Snl(

V)" ~ +'V.(n .. val) = 0 -at ot ... - ..

which is the equation of continuity for the component

c/, •

the equations

(65)

of continuity for the ions and electrons

Multiplying by

m. and 1

- 28-

V. C.A. Ferraro

m respectively and adding we e plasma as whole,

~fo +t7

(66)

-

j>ct = mlt :::411.'

set

of continuity for the

(Ov • 10-0) = 0

~ tV'

If we

have the equation

then

• after

some simplification

and using

(65), one obtains dv

fDt

d(

d~+ (V· POl- rlt~lI.) +

;: Jmct:::1JI, Cct

+ Pflo Vel V'!.o Adding the equations for the ions

by collisions,

of the ions

+f.(fi1):::o

dt

(67)

total momentum

p" Y.)

and

and

d::: et

electrons and noting that the

electrons in the element is unaltered

we get

r

dv

.

~=-V.p 0 to.F·+P F o dt 11._1. e-e

(68)

which

is

the equation

of mass motion. Equation (67) refers to an ele-

ment of either constituent following the mass-motion of the plasma. An equation can

also be obtained referred to

constituent, ::: et' Denoting by do( / dt se,

the local mean velocity of the the time derivative

so that dIlL

(69)

dt

we find after (70)

r.cl.

some

d

v

=~ + V ot

-Ii

.V

rearrangement

It -oCt V.(p - P. V dt at It -.(

of terms that

Y.l- e.. ~

K

=

r

mit ! . Co( d!"

where (71)

Pet = plit -~V

V

III -Il

rcVV-fcVV « -It -Ill /I. -01-111

in this ca-

- 29-

V. C.A. Ferraro is the relative pressure tensor. It is easily shown that this is equal to

u:u

v)

p = Q (v (v -v) = IJ.. AI " r A -CIt -at -.l..... II - ... -"

(72)

when u

-cL

= -c(. v - v is the velocity of a particle relative to the mean -til

velocity of the element. Thus (70) can now be written

(73)

p

laC

which

~"+ \/. p

dIt dt

is the equation of

mean velocity

_ p F

I" --.

CIt

Jrm

=

I(

v C dv -Ill III

-"

motion of the constituent ()( referred to the

v of this constituent. -~

13.

Approximate calculation of the collision term Since particles of one constituent can collide with each other and

with particles from another constituent, the collision term. in Boltzmann's equation (3) may be written (74)

where

~,giVeS

for particles

r,.

of contituent f«.

=fc""( It}p)

CGC.

the change

per unit

of the constituent

p,

time in the distribution function

due to

or;

particles

C., depend on the respective distribution functions

Certain properties of the collision terms are immediately obvious

and do not depend on the explicit form of the (CadV

(75)

collisions with

j

ILr

-&

Smvc "

neglecting processes which

at

11(.

C,,~

. Thus

=0 dv

-&

=0

may convert particles of one constituent into

that of another, e. g., ionization, dissociation, etc. We have alredady noted in section 11 that as a rough approxi-

- 30-

v. C. A. Ferraro mation we may write

c

(76) (0)

where f 0( lation

is the Maxwellian

distribution. On account of the second re-

in (75) , we may take

from now

=

"

to

7f

't"et-P

be

't'.:p

equilibrium are small, we may treat

evaluate approximately the collision term

(77)

v

where

v is the

mean

-0

-0

mass

as

constant. We

can

in (73) . Write

+ V .. ,

=y

-~

and that, if departures

- ...

velocity of the

two

Sm

r

corstituent plasma.

Then

J

m

(78)

C dv rJ. =

y

«-II II

Since

v

is a

-0

virtue of

the

constant first

-

in

equation

J

(79) (76)

C dv.. + m V C dv Q( - ... j ot -c III! -#II

the first integral, • Hence

in (75)

m V C

Substituting

v

1Il-o

crt -lit

III

this vanishes by (78)

dv

-.

in (79) and noting that

dv

= dV

-

flo)

JmCIt V ~ t'

(80)

But

-tI.

ren

J at.

v f(o)dV

CIt

II(

-~

vanishes

d V -

-rJ.

Jm

• this reduces

to

-(J(

f

V..!!. d V "i:" -f/

01 -"

identically; thus (80)

reduces

to

V 17:

(81)

Here

reduces to

-n m It C/t-oL

denotes effectively the electron-ion scattering time and

we may interpret

this

red velocity with

respect to the ions in a time of the order 't;

hence lose momentum

result as

m 0(.

~c£. per

follows. The electrons lose their orde-

particle Gl waich

and

is communicated to

-31V. C. A. Ferraro

the particle

~

tional force erted on to (81)

. This implies that the particles are subjected to a fricnJ m

... CIt

V It

-rt/' •

This is equal and opposite to the force ex-

P . In

the particle

fact, since

the corresponding equation

Ie

it-lit

+n

,.

we have, adding

= 7:po'

for the particle

n m V Go

,.

't'aifl

P

rnA V Il = 0

r-r

\

Using this relation, (81)

may be rucpressed

in

terms of the mean

relative velocity, namely, (82)

since

=V

v - v

-ce.

-~

-CI.

- V

A

-

,.

•

Hence

equation (73)

can finally be writ-

(83)

14.

Rate of diffusion of the two constituents Dividing this

corresponding

equation

equation

for

sion for the differential (84)

-

-

-"

-~

!

v - v = -t

It is convenient

at

f

0

and

the ~ -constituent

substracting from we obtain

it the

an expres-

or diffusion velocity

dV (1(4 dt

this

by

stage

dv-p t -1 V. P.., - -V:PJ\ 1 E -(F dt p . . . p" ,.. """ to

introduce the coefficient

~

F II)

-r

of diffusion

of the other two constituents, namely. (85)

where

m =m

o

oc.

t rnA

r

•

•

We find, after some algebra, that (84)

can be

- 32-

V. C. A. Ferraro

written

v - v =

-,J.

-p

n (86)

+

where

n on

it .~

Po = ~ + P~ is

7·

not (F

n

-0(,

0

the

total pressure,

- F ) -~

and we

have written

d lit ~I(

etc. The four terms inside the bracket (86) correspond dt to components of the relative velocity of diffusion due respectively to (1)

1. c(

for

th e relative acceleration, (2) the pressure gradient, (3) a concentration gradient, and (4)

external forces. Note that gravitational forces do not

contribute to the velocity of diffusion. These component velocities of diffusion tend to have the

following effects: (1) and (4)

have indeed the sa-

me effect and tend to separate the constituents in the direction of the relative acceleration

or forces. (2)

tends to make the composition uniform

and (3) tends to increase the proportion of the heavier constitution in the regions of higher pressure.

15.

Three-constituent plasma. (Partially ionized gas) . We shall consider a

partially ionized gas consisting of electrons

one kind of ions and one kind of

v , v., v respectively. Because of their -e J. -n the momentum of the electrons may be neglected

constituent will be denoted much

smaller mass,

neutral particles. The velocity of each

by

in defining the mean mass velocity (87)

where

v

-0

=nm +n m 1 inn

v , which

-0

is thus approximately

1 - (n m v + n m v ) , i i-i n n-n

po

- 33-

V. C. A. Ferraro

However, as

in

the case of a two-constituent plasma, it is

more convenient to derive the moment equation of the Boltzmann equation of each constituent relative to axes moving with the mean velocity v

of that

component. The equation of continuity (65)

before , and the equation of momentum before, except for the collision term It

w ill

hold as

will likewise be the same as CO(.,

is clear that the collision of particles of

each

constituent with those

of the other two constituents will yi,eld a collision term of the form (82); however, we can no longer drop the suffixes so that denoting by

Pi ' Pn

the m;:;ss derivatives of the electrons, ions, and neutrals,

fe' we have (88)

J

f ePi

m v C dv ~- - -

e-e e -e

- /7:--Fe r n

(v - v.)

-e

fi+fe

e~

-1

Pe+f'f'l,

~ (v - v)

-e -",

I'!

I.'"

(89)

(90)

S

The

"l:'

m v C dv n-I\ n. -/I.

~

--v' )/'""'-_ _ P".f,:

pII.fe

(-v

rn+fe

- n - e

n.e. f~+r~

s are called the 'collision intervals'

(v

-~

- -v.)

-"

/ ~.

n.e.

by analogy with what has

="

,1:. =?:., 't el ie 10 m en ne that there are effectively only three 'collision intervals' ; writing been said previously. We have

7:: . =?:

'C"

(91)

the equation of

(ct,

P=

so

e,i,n)

of

motions for the ions, electrons and neutral particles are

respecti vely d·v·

/,.-/,.

(92) Pi

dF" = -V'Pi

+ Zrt-,;e

(~ + :::i)<.~) +Fi~~- 8L'e(~-:::e)

- 8'-71

(~i-:::II)

-34V.C.A d v ( 93)" e-e I e dt

(94)

r

n

= _\loP

d v n.-I\."

-v)

-n e:E + ~ .>
- 8. (~- ~) - 9 (v

-flop +OF

dt

Ferraro

""

~jI.

IJ.-JI.

-I\,

ell -

-,;

n.

-

v) -.,

where

n., n ,n are the number densities of the ions, electrons and 1 e n neutrals, P is the relative partial pressure of the neutral gas, and n

Ze, -e are the

16.

charges on a positive ion

Diffusive equilibrium in

and

electron

respectively.

a fully ionized plasma in a magnetic

field The equations

of

motions

for the ions (assumed to

be si ngly

ionized for simplicity) and electrons may be written respectively (dropping the bar over the velocities) , dv

Pe

-.:i + ..l \l.P - F. - L p,;

dt

i

(95)

d~e

-+ dt

where

W. =

1

ft

f;7

yo P

e

E - v xW . = --(v.-v )

m~

-1

- F

-e

+-

-~-"

e

-

for the ions

0

and

I P0~-

equations

(97)

-II

m (m. e

1

and

D./ D I 1 I 0

n./n ~ 1. Also m 1m. ~< 1 so 1 ee 1 ~ 1. With these approximations the -

may be rewritten

dv

(96)

v.)

w = eB/m c are the cyclotron frequencies -e e electrons . Now a plasma is electrically neutral to a

I

high degree of approximation so that Ie

W

eB/m. e and

-1

that

fi

-e

= - - (v -e -eo fo'?: -eo

E + v ')(

mt -

fot'-~

o

-L

_

"Le.

-+dt

V·

mr~-~

me m~'2:'

v - V.j(W. U = -

-e.

-L

m~

/,-

m

e

e.

(E - _l_v.p.) - (lie 1

+ F. sG

.

1-(.

- 35-

V. C. A. Ferraro

where u is

a

F

and F

are extraneous forces

tic

forces or pressure gradients.

-i

-e

unit

vector along

An inspection transient part tially with

order of the relaxation

terms

in (96)

for

v, and v

-1

which decays exponen-

time. The plasma therefore attains

diffusive equilibrium in

and

that there exists a

-e , a t'1me as e -t/7;, , 1 . e., 111

approximately

termed a state of

other that the electric and magneT

of these equations shows

of the solution

time

= I-1 w, J ,we = J -e wI, and

B, w.1 -

can

(97)

which

0

f

what

t he is

the acceleration

be neglected, yielding the approximate

diffus ive e qua tions (98)

v, - v - v, x -1 -e -1

(99)

v - v -e -j

where

(100)

(101 )

where (98)

n = w 't' e v • =

-"

A,

1

/,-

It.e,-u

v -e

and

and (99)

v x nu + -e

Solving

).. u

Ae

m nu = G, _i -1 m e

these

+ ~2 rl

?:

=G 't

-e

vector

equations we

m'

[(-'

me.

G . + G ) 't' +

-e

-~

+ --1...2 [( miG, + G ) t"t rl me" -"

rl

-e

~ -e G Y. uJ -

are arbitrary parameters. However, multiplying

scalarly

by

n

and

adding

the

resulting equations

have m,

( 102)

1

( - G. + G m

Using the approximation ( 103)

obtain

-'"

n

1

-(

) . u

In:. 1 e

this

=

0 reduces to

+fF).u=O e-e -

- 36-

V. C.A. Ferraro

Thus , along

a

magnetic

line

of force,

the pressure gradient in

that direction balances the total external force in the same direction. In the case of the protonosphere this implies that, along a line

of for-

ce, the pressure decreases exponentially with height, the temperature being approximately constant in this field depends

roughly on the magnitude of

w 't', the ratio e If the electrons

n»

1

of the

cyclotron

are able

and we see

ions and electrons

n -t 00

;

from

frequency to

(lOO) and

to the magnetic field due to

is, the product

the collision frequency.

-1

n

collisions, then

(101) that, whilst motion of the the component

and thus becomes vanishingly\ small

that the drift of the ions at right angles the pressure gradient is greater than the

for the electrons.

These results, of course, are to deration

that

along the lines of force is unimpeded,

we note also

corresponding drift

n,

to spiral many times between

at right angles is of order as

region. The effect of the magnetic

be expected from general consi-

of the motion of charged particles in

a magnetic field.

- 37-

V. C. A. Ferraro

II. Application to the Ionosphere

17. The atmosphere The sllientific study of the upper atmosphere is nowadays called aeronomy - a term

due to Chapman. The discussion of the atmosphe-

re is based largely on chemical composition and temperature and the various layers into red

to

which the atmosphere can so be divided are refer-

as !.spheres' • The upper boundaries are referred to

as

'-pause'. Thus the troposphere denotes the layer extending from ground l'evel

upwards

in

which the

this layer is the stratosphere in the temperature remains

temperature decreases with height. Above which. for many kilometres

constant. The tropopause is the upper level

of the troposhere and separates it from the the tropopause varies with

the stratosphere. The height

of

latitude, being about 11 km in mid-latitu-

des. Above the stratosphere the temperature increases

and

the region

of higher temperature is the mesosphere; the temperature then decreases again and reaches its

lowest value (about 180' K) at a height of 80- 85

km. The temperature rapidly rises above this level (mesopause) to about 300

km where it attains a temperature of over 1000· K. This region is

called the thermosphere. Above this level the atmosphere is maintained in isothermal equilibrium. This is the exosphere and here collisions between the molecules are so rare that they move in free orbits under gravity. The various regions are illustrated in figure 7 .

18.

The ionosphere, heliosphere and protonosphere Sunlight of wave lengths less than

about

2900 A ( 1

A = 10

-8

cm)

is capable of ionizing oxygen and nitrogen. This radiation cannot be obser-

- 38-

V. C. A. Ferraro

ved at ground level owing to strong absorption by atmospheric ozone. Its emission from

the sun has been detected by satellites and other space

vehicles and, pri.or to

this, its presence was indicated by the existence

of several ionized layers in the atmosphere. Radio methods of observations of the ionosphere have revealed the existence of two main layers, the E

and F

about 120 km

180-300 km. The

height

and

the other

at

F

layer is

thicker than the

E

the lower called

F 1 and the upper F 2 . At night they partly merge and

become

layer and separates into

layers, the former at

indistinguishable. Below

two parts during daytime,

the E-Iayer there is

another at

about

70 km called the D-layer . The thickness of the F-Iayer diminishes rather slowly with height and has no the heliosphere where neutral

well defined upper boundary. It merges into and ionized helium are present to

of about 1000 km and above this we have tons. This

highest part

many ionized hydrogen or pro-

of the ionosphere is called tre protonosphere

but its limits are difficult to define. It seems li,\cely that in diffusive equilibrium. The various

19.

Processes We shall

in

a height

this region is

layers are illustrated in fig. 8 .

the ionosphere

here deal only with the large- scale structure of the io-

nospheric layers, and particularly our attention will be dIrected towards the effect of

diffusion

on the distribution of ionization

of the F 2 region.

The ionization in the ionosphere is due essentially to tion of diation

ion-electron at least in

the produc-

pairs by the absorption of solar U. V. and X-ray ramiddle and low latitudes, At higher latitudes, iom-

zation can also be produced by collisions between high-energy charged particles , precipitated in the atmosphere, with neutral atmospheric molecules

-39-

FIG.7·. U.S. STANDARD ATMOSPHERE (Government Printing Office, Washington, D.C., 1962). Vertical distribution of pressure p, density p, temperature T and mean molecular mass M to 250 km. The composition is assumed constant up to 100 km.

10000 6370 ---- EARTH RADIUS

3000

PROTONOSPHERE MAGNETOSPHERE

1000

~

300

.. 0

100

~

30

. ::>

OZONOSPHERE

10

3

500

1000 TEMPERATURE TI'K,

1300

10 5 0 ELECTRON DENSITY NIIO"

ni",

FIG. I. REGIONS OF THE ATMOSPHERE, SHOWING CONVENTIONAL NAMES DESCRIPTIVE OF LEVELS, PHYSICAL REGIMES, AND CHARACTERISTIC CONSTITUENTS. The temperature profile is taken from the U.S. Standard Atmosphere and the electron density profile represents average daytime conditions for middle latitudes, high solar activity,

- 40 -

M. Z.v. Krzywoblocki

where

1 1

*" I ( 1'-) =

(2.3.61)

n

2 n E(I', 1') (l-t) d't.

-1

Bergman considers a partial (2.3.62)

(j 3

't'

+ F (y,

z)

'Y =

We introduce the variables: and express the function of Z and Z

:!t-

differential equation of the form: O.

X = x,

Z = (z+iy) /2,

,If-

•

Z = -(z-iy)/2 ,

F(y, z) appearing in Eq. (2. 3. 62) as a function

; we also use the symbol

F for this new function ..

The equation (2.3.62) then assumes the form: (2.3.63)

,If-

"'XX - 'YZZ~ +. F(Z, Z)

't'

=0

We proceed to obtain particular solutions of (2.3.63) which are polynomials in X, as follows. Let

'" ~ (Z,

Z ~ ) be any solution of the

equation:

(2.3.64) and let the polynomials

P

(N, k,o.)

p(N,k,k-2)))

(2.3.65)

N=O, 1,2"

Let the functions (2.3.66)

(2.3. 0'7)

""

N

k=O, I, 2,' . "

N-k+V

*V

Z,

2N; \)=k, k-2, . . . , k-2

[~1.

~

(Z, Z ) satisfy the equations :

- N v Z~ o '

p(N-l,k,k)

- N 'Iz

P (N-l,k,'I»

ft

k-'I>

== ( k-)' ( .),) ) Z

.. ;

IT (N, k,d.)

be defined as follows:

-

rr(N,k,k). ~ + ZZ N

-

'i z

F rr(N,K,K)

p(N-l,k-2,\»

-

-

0

+

+ (',)+2)(\)+1)1T(N,k'Y+2)_lT~~!'V) + FlT(N,k,)I)=O, ',)
-41-

V. C. A. Ferraro

Negati ve ions may be formed in the lower ionosphere by attachment of elec trons.. The important losses of ionization arise from (radiative) recombination,

atomic ion .. electron

molecular ion and electron

(dissociative)

re~

combination and , in the lower ionosphere, by the attachment of an electron to a neutral

molecule.

Ionization may also

be affected by transport processes; the ions

and electrons (plasma) in the ionosphere may be thought as stituent of the atmosphere. It is acted gradients

a minor con-

on by gravity and by pressure

in the plasma. Unlike the neutral constituent,

the ions and elec

trons are acted on by electric and magnetic forces. As we have seen in section

14, the plasma tends to

forces acting on

it

are not

diffuse through the neutral

air if the

in equilibrium. Ions and electrons diffuse to-

gether, since any tendency to separate the positive and negative charges w: give rise to a large electric field opposing this separation. This is called 'ambipolar'

or

'plasma'

diffusion and proceeds rapidly in the

region but not in the lower ionosphere.

In

F

this region the plasma tends

to be set in motion by movements of the neutral air, which may be due to large scale

wind-system or to

temperature.

The various processes outlined above which

modify the ionization

in the layers of the ionosphere must balance and this balance can be expressed as

an

equation of continuity. If the transport processes (wind

and diffusion) result in a

net

electron density by

the rate of production and

by

(104)

q

and L

n.

drift velocity

.!' and we denote the loss

respectively. the equation of continuity is an + div (nv)

'Ot

-

=q - L .

of ionization

- 42--

V. C. A. Ferraro

In duces

the absence of

production and loss

to (65) already found in of section Various

of

these lectures to go into more than a few

ionizes

O2 to

O2+-, In

formed

+

being

0

10 n-ion

+

170-796Aand 791,1>.

the ionization, in this case the ions

processes we may note the following

"J1

recombination (coefficient X

which

N2

Amongst the loss (a)

details _ A major part of tre

the F-region the wavebands

-+

and

be outside the scope of

the wavelength band 911-1027 A

to 911Aare mainly responsible for

re-

from the sun are responsi-

ionization. It would

E-region ionization arises from

ionization (104)

12

wavelengths in the radiation

ble for the production

of

+ Y

-+

X

y.

+

(b) Electron-ion recombination (coefficient c:t )

e

(i) Here

denotes a

Nt

+ X

Three-body:

+ e

neutral particle

+

M~

which

X + M exchanges energy and mo-

mentum but does not take part in the chemical reaction.

Here

X

(ii)

Radiative:

(iii)

Dissociative:

:$

denotes

Process (i)

X+ + e ...... X*-i> X + h U • XY+

an atom left

+ e -t X 1C + Y 1C in

the

can occur in the lower

excited state. D

region but is rare at

greater heights. Process (ii) the uppermost regions the

is- likely to

be the fastest loss process

levels of the F regions. Elsewhere in the

dissociative

E

recombination process are important.

only

in

and F

- 43-

V. C. A. Ferraro

(c)

Ion-atom interchange

~

At t XY

b (iii)

re is

X/

Ion-atom interchange

(cl

is the princ ipal

process

still

loss

+ A

followed by dissociative recombination

considerable controversy

in the as

E

and

to precisely

F

regions. The-

which reactions

are important. The rates

for processes b (iii) and

the reaction constants

~

and K c ' by the

(105)

dn(e) - = - K b n(XY+) n (e) dt

(106)

dn(A ) dt

+

If we

suppose

that

the electrons q

the atmosphere is electrically neutral,

+

=nee)

positive

the ionization is

b

+ Eliminating n(A) and

1,

+

K n(A )n(XY) c

+

n(XY) by

using

2 Kb Kc n(XY)n (e)

If

q

K n(XY) e

equilibrium. Then if

per unit volume

+

(109)

in

ions are produced by incident radiation at

q = K n(XY )n(e)

(108)

expressions

n(A )n(XY)

further that

and

are given . in terms of

+

e

n(A ) + n(XY )

and we suppose

the rate

= -K

+

(107)

(e)

= Kc n(XY) + K}) nee)

»K n(e) ,this b

reduces

to

(107)

we

find

-44-

V. C. A. Ferraro

= ~n

q

(110)

which

corresponds

coefficient

2 (e)

to

quadratic

a

region where

Kcn(XY)«

If

q

(Ill)

=K

which corresponds

c

of recombination

(J,

2 n ,the

a, being equal to \ . This law holds

of recombination

very nearly in the E

law

~n(e),

0(,::

10

then (109)

-8

cm

3

reduces

sec

-1

.

to

n(XY)n(e)

to

an

'attachment'law

of the form

f3 nee)

with

an 'attachment' coefficient

/l= Kcn(XY)

(112)

If,

as

is

usually

Chapman's

20.

the

perature. Such if

h denotes

of gravity,

m

an

n(XY) decreases upwards, so

will

p

Theory

We consider the ochromatic radiation

case

in

simple

case of ionization by absorption of mon-

an atmosphere of uniform composition and tem-

atmosphere will be distributed exponentially; in fact

the height, the mean

n the number density , g the molecular

mass

acceleration

of the gas, the statical equa-

tion is (113)

dp - '" -nmg dh

(114) Also

p

where

k

is

temperature,

"knT

Boltzmann's

constant

(1. 38

x 10

-16

dgs)

and

so that eliminating p between these equations

T

is

we find

the

- 45V. C. A. Ferraro

where

H

is

(116)

=n

n

an

that is,

exponential

sity of the radiation

o

e

of

at

the height

:x

height

at

then

any time,

of length ds

h

we

is

the

-16Tl

I

denote the inten-

the intensity of the inci-

co

distance of the sun at the

the decrease in

by absorption over

I

)dh between the levels

(sec

absorption

h

+ dh and h is

X )dh

dl

can

be

integrated

cross-section

(on 0

to

I loge (I)

(118)

sec.x ) e

~

I

co

as

h

I

(119)

molecules. Using

absorption

is

dl/ ds = (dl/ dh) cos of

of

=I

= -(ern

exp (-tJ'n

'"

radiant

X

-h/H

dh

0

Hsec

X

)e -h/H

dh

co . Hence

~

The

absorption

the

of

give

co

I

,

have

T =-

since

and

denote the zenith

= (sec X

d I =-

where (j

which

gives

by

(117)

(116)

a length . called

distribution of density. Let I

radiation. Let

the path

(11 5) now

of

-h/H

dent solar h

= --

a quantity having the dimensions

the scale height . Integration

given

1 H

d logn = _ mg dh kT

(115)

0

He -h/H sec

energy and

if

unit quantity of energy.

X

)

per

unit volume

p

ions

of the atmosphere

are produced

the rate of

by the

production of ions

-

46-

V. C. A. Ferraro

volume is

per unit (120)

The total

q(h)

=

of

ions

number

I

r(J.

cr exp

produced

of unit area of cross-section diation is clearly ~

n

fXJO

h

m

IfXJ cos:X

Denote the values qo ;

h

of

e

(124)

h q

of

maximum

n 0'" H secX

and

m

h /H

In terms

a

the

qm for

overhead

sun

(;t= 0)

then

(123)

(125)

q has

(P100cos X )/He

q = m

(122)

whence

column of air

0

giving

and

a vertical

/"

.

h /H m

e

0

in

secY)

where

(121)

by h

-h/H

0

by the complete absorption of the incident ra-

The rate of ion-production height

(-h/H - n crHe

qo and

0

= n 0
=h

m

0

+ H log

= q cos

m

0

I =

o

e

PI

00

/eH

secX

X

h • we may 0

now write

(120) as

h- h

o

h-h

(126)

or measuring

q(h) = qo exp (1 heights in

h-h

(127)

I Io - e

terms of

writing o z=---

H

H

H

secfo ) as

unit

from

the level

h, o

- 47-

FIG. 9.

NORMALIZED CHAPMAN PRODUCTION FUNCTION

q(z.X)/qo = exp (1 - z - e- z sec X). Values at several reduced heights are shown as a function of zenith angle. The broken line 1s the envelope. q /q = cos X.

m

0

- 48 M. Z. v. Krzywoblocki

Once we know the relation between the density can integrate Eqs. (2. 6. 17) the physical

(x, y)-plane.

and velocity

q, we

and (2. 6. 18) to yield the sonic line in

r

For stream functions

f

defined by corresponding

X1(8), X2(8)

given in (2. 6. 16), the sonic line in the physical (x, y)-plane is g iven in terms of the parameter 8 by : (2.6.19)

x = (1 / 2) (6/ 5 )3[ a - (6b / 5

(2.6.20)

y =

(6/5)3~

D cos 2 8,

(1/2)[a+ (6b/5)] (8 -11/2) +

(1/4)

la- (6b /5)J

sin 2

e} ,

provided we take the origin in the physical plane as the image of the point q=(5/6)1/2,

8 =1r/2 on the sonic line in the hodograph plane.

The value of

is calculated

t=

q

by

M2 = q2/[I_

(¥-

1) q2/2] with

1.4 .

Example

(I) : sonic line in physical plane is circular.

If we consider the stream function

r

aefinea

by

(2. 6. 16Jwith.

a, b related according to : (2.6.21)

a + (6b /5)

=

0,

then (2.6.19) and (2.6.20) yield for parametric equations of the sonic line in the physical plane : (2.6.22)

x = a(6/5)

Elimination of (2.6.23)

e

3

cos

2

8,

y = (1/2)a(6/5) sin 28 .

gives:

223 x + Y - a(6/5) x = 0 ,

-

49-,

V. C. A. Ferraro

(called the reduced helght), we

q(h) = q exp (1 - z - e o

(128)

This

is called

esting

property

is shifted to the

have

the that

cos Z

'V

sec '" ) .

Chapman-function (Chapman, 1931) . It has the inter-

X

as

varies, its

shape is unchanged, its peak

= log (sec X) and its amplitute is scaled by m e This can be seen by writing the above equation in

the level

factor

-z

.

z

the form z -z q = (q cos.x) exp [1 - (z-z )- e m o m

(129)

The ratio

q/ q

o

is

shown

the actual

In

in

Fig. 9.

ionosphere

bly more complicated,

but consists

the

production

formula is considera-

partly because there are different atmospheric

gases, differently distributed, romatic

J.

and

the ionizing radiation is not monoch-

of a range of wave lengths and

a-

depends on thE:

wavelength • The above

theory neglects

the

curvature of He earth; Chapman

has considered the modifications introduced by taking this into account. This correction

is only important near sunrise and sunset.

The theory can not at

the same

portional

to

(131)

be extended to deal with gases which are

temperature at all:

the height, so

(130)

then

also

heights. If the temperature is pro-

that

H=K +yh

o

it

can

be

shown

that

q

m

=q

(125)

0

takes

(cosX )

l+Y

the modified

form

- 50-

21.

Plas ma diffusion The F2

No

V. C. A. Ferraro

peak

of

ionization

is

observed at about 300 Km.

mechanism seemed capable of causing a

a height

and Bradbury

red

a lower height (the

at

the upward increase in

(1938) suggested F1

peak

that

of production at such

the production peak occu"-

region, in fact) and he attributed

electl'on denSity to

a

rapi.d upward decrease of

a linear loss coefficient. The hypothesis is unsatisfactory, for even if varies

as

where

H P is

the scale

the production

peak

height,

then the electron denSity well above

is approximately

q/~ DC exp

[ +(h-h

)(~

a H"

-

~)]

,

Ha

where H is the scale height of the ionizeble gas. a Since Ha < H ~ ,then q/P increases indefinitely upwards, and we find some other

P

mus,

explanation for the peak in the electron denSity . We

require some transport process to

limit the value of the electron

ty at great heights. One such process is

densi-

plasma, or ambipolar diffusion

Attention to the probable importance of diffusion in the ionosphere was first directed ter detail in

by Hulbu rt in

1928.

1945 and showed that

I

considered the problem

diffusion was unimportant in

and

F1

region of the. ionosphere but tbat it might

the

F2

region and above it.

Yonezawa first discussed in

in greathe E

become important in

Mariani (1956) drew similar conclusions but detail the problem

of

the formation of the

F 2 region and showed that diffusion could provide an explanation . In deriving the equation of diffusion foil' the ionization, we shall neglect,

for

the present, the earth's magnetic

forces acting are gravity ces due to

g, the electric field

collisions. The full equations for

field, so ~,

that the only

and the frictional

a three constituent plasma,

for-

- 51-

V. C. A. Ferraro

(92-94), , have been derived in section 15. However, in any external electric fields, the slightest separation

the absence of

of the ions and ele-

ctrons will give rise to large electrostatic fields opposing any further separation so that we many set Also, if we assume more,

that

n.

1

= nand v. = v in these equations. e -1-e = 0 . Further-

the neutral air is at rest, v

-n

the relaxation times in the F2

region

are small , so

that the

steady state is quickly attained. That is, we may neglect the acceleration of the ions and electrons

in

(92-94). Writing

11. = 1/1:. , etc. , 10

ill

for the collision

frequencies,

(132)

-\lpi +

(133)

-'VPe - neE-

(134)

-,¥Pn + nnm n-g + nm.1 '"in41 v' + nme II v = 0 en-D

where we

have written

these become

=0

ne E + nm g - nm V v ii in-D

+ nm fI --~2

-

nm e Venv D

=

0

~D for the common velocity of diffusion , of the

the ions and electrons, and n for their number density. Further simplifications

can

be made by noting

that m

e

«m., 1

m. l/.

»m 11 (i. e. collision with the neutral particles are important 1 10 e en for ions but not for electrons). Also, if T. and T are the temperature 1 e of the ionic and electronic constituent, we have (135)

Using

P.

1

= kn.T. , 1 1

P

these equations, on adding (132)

the velocity of diffusion

e

= kn T

e e

and

(133)

and

solving for

of the ion-electron component we find, if h

denotes the height at any level,

- 52-

V. C. A. Ferraro

(136)

MV

where

vD is

= -1 -

m,1I,

D

m

1

t

-1

n

an

a-[nk(T' + T ) + m,g] 1

measured positively upwards, In the

electron and neutral air temperatures may all likely that T.

1

F

J

region, the ion,

= T, the temperature of the neutral air.

air scale height and

1

be different. It seems

Then , if we

= T / T,

e

H = kT/m g,

and

n

introduce

introduce the neutral

write u. = m./2m , 'Z: = TIT. =

the coefficient

r~

of

1

n

e1

diffusion of ions through

the neutral gas = kT/m, lJ. 1m

D,

(137)

m

equation

becomes

(136)

(138)

IfT =T

at

=T

e

-v

(139)

where

all

D

we

have

heights, 1:a n n ah

D

aT +..c) u. Oh H

1

+_ -

= D(- Il.!!

written

'l: =1, and

T

= 2D

in

The contribution of diffusion to the continuity equation (104) is to add

the term

d(nvD I'd h

to

the left

hand

side

of equation. In gene-

ral (140) where

iJ

is

a

differential

The value of D is on the temperature. It is

operator of the second order. of the form

now

hln , where the factor b depends n

believed that in

the

+

gas is mainly atomic oxygen and the ions mostly 0

F2 layer the neutral . The value

of b is

- 53V. C. A. Ferraro

affected by charge exchange and derived from

its value not exactly known. The value

kinetic theory, taking account of the effect of electrostatic

induction by the charged ions on the neutral ions, gives n D of 19 order 10 . But this may well be too large by a factor of 10 beca(h-ho)/H use of charge exchange. Also DOC 1( n OC e ; if we aSSUme that the temperature of the gas

is

uniform, then

(141 ) and the diffusion side

takes

term in

the continuity

equat ion

on

the right hand

the form

(142)

(Ferraro 1945) . The first term in this equation is characteristic of diffusion formulae. The second and third arise from the height dependence of

22.

The equation We shall

the effect of gravity and

D.

of

diffusion

for the

F2 region

ignore for the present the geomagnetic field; we shall

also assume that we are considering a locality on the equator so the time in

radians measured from

sunrise ~ It

is related to

that

tre

X~

actual

time by the equation ( 143)

t ~ k

where k will

~

1.37

be taken

x 10 to

4

sec

-1

1'

. The rate of

be represented

production

by the Champan function

h - h o h-h - I{ [ Q,=qoex p 1- HO-(cosecJ)e (144)

of electrons q

J

p,

-54V. C. A. Ferraro

=0 where h refers to the level of maximum rate of production of ions q. o 0 Electrons are assumed lost in the F2 region according to an attachment-type law (119) with a coefficient tional to

the dens ity of the neutral particles. Thus

(145)

L = Kn

where

h-h

K

Then the equation (146)

(147)

which decreases at a rate propor-

=A exp( ___0)

ro

H

of diffusion to be solved is

~

~2 h +

h-h

~

o~ = q - ~ on exp (- H 0) + D(a ~ 2~ 0 ~ + 2~2) where

D

=bin

n

and

n n

=N

0

h-h exp (___ 0 H

)

Since D = (bl n ) increases exponentially upwards, whereas both q and L n decrease exponentially with height, it follows that at some level the diffu'sion term will

be the dominant term in equations (146) which then reduces

to (148)

Solving this equation

we

find

(149)

and from

equation (141). it follows

diffusive equilibrium ,

with

v

D

that the first term corresponds to

= 0, in which the ionization

.

has

a

scale

-55 V. C. A. Ferraro

height twice

that of the neutral gas. For the second term vD

is positive if A2

>0

. This represents

flux of ionization at is gained or lost

h

==

at the top

type must be included in 23.

Solution of

a boundfl.ry condition of

which is upwards

00,

f 0,

of the ionospher<e,

if

A2 then

j

>0

and

a finite

. If ionization

a term

of this

the solution.

the diffusion equation

Writing h (150)

h

o

x "-H-

and expressing written

~

in

the

in

terms

g>

of

.by (143), equation (146) can be

form

non~dimensional

(151)

where

N H2

and ",,__ 0_ 1- Kb

(152)

are non~dimensional parameters. The solution of lyon these two as

parameters

we have already seen

ionization

a

solution

q

is given

of (151)

The method of solution

thus

that

n

on~

~

of the flux of

0 as z..,

-00

•

by the Chapman function (144), we

which is periodic

of this,

depends

conditions. One of these,

above, depends on the value

at z = +00. The ,)ther requires

Assuming that require

and the boundary

(151)

in"

with

period

21[.

and related, equations has been given

by Gliddon (1959) and consists in determining a suitable Green's function for this equation. The analysis is involved and reference must be made to the original

pap~rs

. Typical solutions indicating the variation

at various heights from

of the ionization

the level of maximum ion production are illustra-

- S6-

V. C.A. Ferraro

ted in Figs. lOa and lOb. Fig. 11 illustrates the variation of the level of maximum electron density for two distinct cases. From such solutions, the following general daytime characteristics of the behaviour of the ionization may be deduced. (i) The F2

maximum electron

density occurs at a level where 2 diffusion and loss are comparable , i. e., where K D /H and m -

the subscript m

refers to

(ii) At the maximum and below

ximately given by n :. q/K,

m

the maximum. it, the electron density is appro-

that is , balance between production and

loss, as in the absence of diffusion. (iii) Well above the maximum, diffusion be comes important and n -(1/2)z varies as e as already noted in section 22. (iv) The level of maximum ionization falls rapidly at sunrise because of the rapid production of ionization in the lower

F regions. It reaches

a minimum height of one or one and a half scale heights above h

o

the level

of maximum ion-production which remains at about the same level until

late afternoon. Therefore, the level rises again steadily to

a height of a-

bout three scale heights above h after sunseqSee Fig. 11) o

The night-time decay of the ionization has been studied by Martyn (1956) , DUncan (1956) and Dungey (1956) . This is also

affected by verti-

cal electromagnetic drifts .

24.

Effect of a magnetic field on diffusion .in the ionosphere We now consider

the effect of a magnetic field on diffusion of ions

in the ionosphere. We are here dealing with a thernary mixture of neutra:!. molecules, ions and electrons, of which the ionized particles form a minor

- 57-

.i+

,,1

.,.

""'ot••

90"

s.....I ..

Fig.IV b

Fig. 10 a

..

"-

1

•

• •

SUII..t

, MI" •

n',lot

Diurnal variation of electron density at intervalu of one Beale hei~ht.

Cane II

(~" 10,

Y .. 20)

•

- 58 M. Z. v. Krzywoblocki

below (2.7.22) : ('V-l/2~-1/2)

(2.7.24)

fn(r,t)=P n

'

t = cos 29 , r2 = x 2 + y2

where

Polynomials (see [161

(t)(kr)

and P (ct, n

-f-v

~)

~~+Y+2n(kr),n=0,1, .. ·,

stands for the Jacobi

):

=

p(v- 1/ 2,}l-1/2)(t) (krt"'v- J (kr) f (r t) = n 'J fA t)l + 2n - n '

, r I~+n+ 1/ 2l[r (\»)[' I1/2f In+ 1l] -I[rlk''''~ -~ Jf +Hin Ik~ l} .

. t (
x = rcos9,y = rsin9,

1 2 2 ,2,[, "\ = - k Y sm 't /4,

~1

= _ y2

sin2~

/(4 xO").

k'f')}'> 0J~= 0, 1, 2, ....

An arbitrary solution of the class S may be represented in a series

[39J :.

form

00

w(r, 9)= (kr('-l>

'i a 2n nt(n+)H 1/2~

-1.

n=O

\ P (V- 1/ 2,p -1/2)(COS 29) J (kr) n . l'+})+2n

(2.7.25) and an

even analytic function regular about the origin may be expres-

sed as (2.7.26)'

00

f (cr') = rr -fA -v , ' \ a 2 J

4r

n

,l

~+v+

2 n

W).

Hence for r sufficiently small it follows that the class of analytic functions (2.7.26) is mapped onto the class of solutions (2.7.25) by an rator of the form :

ope~

~+ tt

Jr

• ~3 ~ 0'. II"l

~ _

•

I

2.

~

.-!j. o

u ~

..s:;E

7Mao1

S

.g

Fig.

0

~nri$e

"',ht 11 •

.1

&

Noon

Comparison of height of two distinct

max~um

....

G

6

Sunset ,s ..... s.t

electron density for

cases: case I for, case II for.

T

Mid

"i~h.

- 60 M. Z. v. Krzywoblocki

j

+1 ( 1 _ ~)V-1/2(1+t) 11+1/2 P (V-1/2"M-1/2)(f) P (');-1/2' jll-1/2) ~

-1

'n

m

;>

.

(£)

(2.7.30) Thus, if we define:

Kn(~' r,

E)

2

~n(kr/ka')

.

(2.7.31)

where

-(~+'V)t

a 2n (kO"')

J~+\)+2n(kll') (Jr +V+2n(kr))

Pn('Y-1/2,~ .. 1/2)(f) (1_E)~-1/2(1

bn = (2n +fA +V) nn +}H»)

we have

f'4+ p

-1

.

+$)'''-1/2,

W(n +).4 + 1/2))-1,

-(~+») J fA +'1> + 2n(kO")

+1 - [ - 1 Kn (IS', r,

s) .

(2.7. 32) Hence

+1

(2.7.33)f(k")"L KI
where K((j',r,~)=2

. l: 00

- ( \1

'

~)u[r((!+S)/2)1/2,

+\) )

(2n+f'+»

(rid)

lr

,\' + \)

(1-5)

d~,

r((1_Sl/2)1/2

\? - 1/ 2

(n+V+I") (f(n+fl+'V))-

AI -

1/ 2

(1+~)". 1

:1 -1 ('V-1/2 f- 1/ 2) J~+v+ 2n (kat) [ J fH+ 2n(kr)J Pn '

(S ) .

To verify that these formal calculations are justified and that

K

dS

- 61-

v. C. A. Ferraro constituent. The relevant equations of the problem have been given in section 15, namely equations (92-94). However, for much the same reason as stated in section 21, we

may neglect the

acceleration

of the ions

and electrons, and assume that the neutral air is at rest. It should be stated, however, that Yonezawa (1958) and Dougherty (1960) have suggested the possibility that the neutral atoms and molecules are accelerated by the flow of the ions through the neutral air,

in the comparatively short

time of 20 minutes. Nevertheless we shall restrict ourselves to the case when

v - 0; Again quite apart from the fact that the collision frequencies

n-

for encounter between ions and electrons exceed those for collisions of the ions or electrons with the neutral atoms, the differential velocity v. -v -1

-e

between the ions and electrons must remain small otherwise large electric fields would develop because of the consequent large separation of charges of opposite sign. Likewise, we must have, very nearly, overall charge neutrality, so that

n. - n . Assuming also that the ion and elec1 e are equal , equations (92) and (93) then reduce to

tron temperatures (153)

-\lP. + ne(E + v. X B/c) + O.g -

(154)

-'iJpe -ne(E+v XB/c)+f g - -e e-

1

where

n

is

-,

-

'1-

p.110-1 V. v. = 0 -0

I

the number. density of the ions

tic field intensity, V.

e

JI v =0 en-e

or electrons,

~

the magne-

the collision frequencies for ion-neutral en atom and electron-neutral atom encounters. Again, in the F2 region, 10

m.

and)/

11 .'

»m 11 so that collision with neutral particles are important e en for the ions but not for electrons. In fact. we may neglect this term and 1

10

also the weight of the electrons in electrons. 'Ilis now (155)

(146)

because of the small mass of the

becomes approximately,

- 62-

v, C, A. Ferraro Also P =knT(- P.l and e

-

if, as is legitimate. T is assumed constant,

1

this equation eRn he written kTV log n + e (E + v )C B/ c) = 0 - -e -

(156)

Hence, curl ~+curl (~X~/c) = 0

(157)

or, using Maxwell's equation,

o~

~

(158)

at

showing that

curl E = ~

1

~

a~ (ft' this becomes

= curl (v )C B)

-e-

the magnetic

field

lines are frozen in the electron gas. Sin-

ce the magnetic pressure of the geomagnetic field greatly exceeds the electron

gas pressure in the F2

region, the electrons can only move fre-

ely along the magnetic lines of force, but not at right angles Thus v is parallel to -e

Band (148) now gives the electric field kT e

(159)

to them.

E :: - -

'iJ log

as

n

or n

(160)

where

V is

the

oc:

e

-eV/kT

electrostatic potential. This implies that the electron

density attains its thermodynamic equilibrium ting (159) in (153) now

a

each instant . Substitu-

gives an equation to determine the velocity of

diffusion of the ions, namely,

(161)

The solution

-2kT Vlog n + ev.)C B/c + m,g -m ,., V :: 0 -1 1iin-i' of this vector equation

is

-- 63-

V. C. A. Ferraro

(162)

~p

2

+ n ) = ~ + (~

+C X n

~)~

where eB (163)

mcV

i

in

and

'V log

C ,,_1_ (-2kT m V i in

.!:

being a

unit vector

of diffusion directed

along the downward vertical, and D the coefficient

V log

defined earlier, In general,

along

the vertical so

Equation (162)

shows

that

n will also be a vector

C is vertical.

that the velocity of diffusion

has three components, one along cular to

n + m.g) = D('tJlog n + _1 _k), 12H

~

and .the other two along and pEr pendi~

the magnetic field. Again, in the F2-region,

being of the order of 200 magnetic equator

where

of the ions

Inl is

large,

at a level of 300 km. Hence, except over the and

C

are in general perpendicular, the

~

largest component of the diffusion velocity is along the magnetic lines of force and the smallest, that along

~.

Hence, very nearly; we may write

(164) where

b is a

unit vector along

preted; a large value of

n

!!

(mr

implies that

of force many times between

~D.

This result is easily inter-

the ions can spiral around a line

collisions and hence its velocity is along the

line of force except when changed abruptly by a collision. The net results of such collisions is to give rise to

the 'Hall' component of the velocity

of diffusion given by the second term in (164) . The equation of diffusion tion of continuity for the

ions

can now

be obtained by using the equa-

-64-

V. C. A. Ferraro

g; =

(165)

and substituting for For a plane

~)

q - L - div (n

v. the expression given

-1

stratified atmosphere, I

sion term in (165) can be

in

(164).

showed in

1945

that the diffu-

expressed in the approximate form

D.9?tt~

where ( 166)

I

is the inclination

tic dip tion

of the lines

angle) and

h

tic equator

This

the vertical

above

the horizontal (magne-

height; (168.) is

the same as equa-

(Ha) except for the extra factor sin 2I, This approach is incorrect

if the neutral air is in

and a

of force

motion, as may well be the case. Near the magne2

the 'correction factor'

more careful derivation was derived

sin I fails to

give the correct result,

of the diffusion operator D is required.

by Kendall (1962) and by Lyon (1963) . Assuming that

the geomagnetic field is that of a centred dipole, Kendall found tha,t one form of the operator is given

Q~ = sin 2I

~

2R

2

n+

3 cot 8

by 1

2

2 2 2 a sin 8(1 + 3 cos 9)

aR(1 + 3cos 8) 4

2

+ ( 15cos 8 + lOcos 8-1) 2.

2 2

n ,

sin 8( 1 + 3cos 8) where

8

point from

is the magnetic colatitude and a the distance of the highest the centre of the earth.

-

6S-

V. C. A. Ferraro

The geomagnetic field in effect reduces the coefficient of

diffusion from

D to DSin 2I and this diminishes the veloci2 ty of diffusion in the ration 1 : sin I. The 1 correction is small, amounting to 2' for () = 45 0

;

but near the equator, where I is

small , vertical diffusion is clearly negligible. One final remark needs to be made; although we have been able to determine the velocity of the ions, that of the electrons cannot be determined. However, as

we have already mentioned, the velocities of the

ions and electrons at anyone point cannot differ greatly, and their components along a magnetic line of force must be very nearly equal.

25.

Diffusive equilibrium in the upper ionosphere The equations of

are also

diffusion of ions and electrons, (132) and (133),

useful in discussing the eqUilibrium of charged particles in the

topside F region. At tbese heights photochemical processes are negligible and the collision frequencies 11. ,V

so small that they can also en the ions are stratified horizontally, that m

be neglected. Assuming also that

they are singly ionized and all at the same temperature T .• 1

of equilibrium for the jth species

(167)

d(n,kT.) J 1 dh

(168)

d(n kT ) e e dh

of ions and

= -n,m,g + n.eE J J

= -n m

e e

J

g - n eE e

the equatior

electrons are respectively

(j

,j,

1, 2, ... )

- 66-

V. C. A. Ferraro

where E

is the elctric field and T the electron temperature. Since there

must be very nearly overall charge neutrality we must have further

1:

(169)

n, = n

J

e

so that adding the equations (167) and (168) we can eliminate the electric field

Defining the mean ionic mas s m+ by the equation

(170)

Enm =mn

+e

, j j J

and negle'ting the electronic mass, we find after el imination of E that d log ne (171)

dh

m+g

=--kT 2(1+'t')

where 't' = T / T, . Substituting this equation in (168) determines E which

e

1

when inserted in

(167,) gives the equation for the distribution of ions as

d log n, (172)

dh

J

If there is only one species of ion present, and T = T. (so that 't" = 1) , the e 1

mean ionic mass is equal

to

m, so that the effective scale height is twiI

ce that of the neutral atomic mass, as before. However, a light atom for ' h wh IC

It mj < 1+i

m+ actually has a negative scale height, so that nj

increases upwards, as first pointed out by Mange (1957) . The solution of the equation (171) and (172) has been effected by Hanson (1962) taking account of the variation

of

g with

+

height. A typical equilibrium distribu-

+

tion computed for a mixture of 0 ,H reduced scale height z refers to trations of

t

t

0 and H are equal. e

t

,H

ions is shown in Fig. 12 . The e the level z = 0 at which the iomc concen-

- 67-

a

,

,, ,, I

.......

........

,

".

I

.

11-...... .......

/+ z0

,,I

\

,, , ,

~,

~)o... ..........

r..... ' .......,

I

I

/ /

I

,. ,/

I

,, I

I

-8QQo3

,

o,H

0.03

(1,1

I

I

,I

~

I

I

03

,

" 3

""-

~

10

JO

IovllCl""' FIG. 12 IDEALIZED DISTRIBUTIONS OF ELECTRONS (e) AND OF 0+, He+ and H+ IONS, COMPUTED BY SOLVING EQS. (III-59), (111-60). Electron and ion concentrations are given in terms of the electron density Nea at the level z = 0, at which height the ionic composition is taken to be [0+] = [He+] = 49i, [H+] = 2%. The level at which the He+ and H+ concentrations are equal is near z = 17. The unit of reduced height z is the scale height of neutral atomic oxygen.

Fig. 12

- 68 M. Z. v. Krzywoblocki

f =a 1

m

n+mp=O ;

p=-nm

-1

P -1 g=x ; n+mp=O ;p=-nm

f = exp(n a) 2

P -1 f1= exp (ima); f2= exp (ina); g=x ; n+mp=O; p = -nm

(3.4 . 4)

where in the last proposition only the rea 1 parts

Re { fJ should be

taken into account. One may construct easily more complicated functions

f. ( i = 1, 2) and 1

(3.4.5)

x= rna + x; y=yexp (na) ; y=yexp (pi),,= yexp (px) ,

which results in (3.4.6)

g

p = - n/ rn , or :

x = ma + x; y = yexp

which results in

p = - m

-1

(aexpn) ; y = yexp (pi) = yexp (px) ,

exp n. The absolute invariants of the group

are :

1. = y

(3.4.7)

exp (px) .

In a similar way we deal with the dependent variables ; thus , as an example one may choose :

For illustrative purposes assume a function

g(x) equal to

r x ,

say . Then it should be : (3.4.9)

f

l

3 1

1

The procedure is identical with the one, solute invariants of the group (3.4.8) are : (3.4.10)

gl =

Hence by virtue of equation :

,etc explained above. The ab-

- 69-

V. C. A. Ferraro

ATMOSPHERIC DYNAMO 26. Introduction Balfour Stewart in 1882 first put forward the hypothesis that the daily variations of the earth's magnetic field could be ascribed to electric currents induced in conducting regions of the upper atmosphere by its motion across the geomagnetic field. The motion

of

the atmosphere

was attributed to tidal forces due to the sun and moon. The theory, generally known as the dynamo theory, was developed mathematically by Schuster in

1908 and later by Chapman. However, their theory encounte-

red certain diff iculties connected with

the reduction of the electrical con-

ductivity, by the geomagnetic field, the required value appearing to the too large to be reconciled with

theoretical values of the

estimates of

the tidal motion. The difficulty was eventually resolved by Martyn and Hirono, almost simultaneously, who showed that because of the horizontal and vertical variations of the inducting polarization electric charges are set up which tend to restore the full electrical conductivity. We shall begin by calculating the electrical conductivity in a highly ionized gas .

27.

The electrical conductivities These were derived in formal manner by Baker and Martyn

(1952-3) and reviewed by Chapman (1956) ; we shall denote, as before, all quantities referring to ions, electrons ani neutral particles by suffixes i, e, n. Also the region of the ionosphere in which the dynamo currents flow ::Ire now known to lie at a height of about 110 km, that is, in the E-region . In

this region, the number of

ions and electrons

per cc. Thus, in the equations (92H94) we have

Pn »Pi

5

is about

10

»Fe

so that

-

7U-

V. C. A. Ferraro

eie ...-

fe ~ ie'

e.'" ". Y. , een ...- fe 'I en Ie - '1 In

and these equations become

approximately

Pl'

(173)

dv. -d1t = -'i/P. +n.e(E +v.X B/c)+D.F1

(174)

1

-

-1

-

11-

r

II. (v.-v )-pY. (v.-v) e Ie -J. -e 1 In -1 -n

-vp -n e(E+vXB/c)+p,F-p JI. (v -V.)-DJ) (v -v) e e - -e ee Ie -e -1 I e en -e -n dv

Pn 2dt

( 175)

= -VP n +

0 F

In -n

-D. Y.In (v - v.) 11 -n -1

In the dynamo region it is legitimate particles; it is also

-pe Yen ,) (v-n - v). -e

to neglect the acceleration of the

convenient to introduce the mean mass velocity v

and the electric current density Dv r 0-0

=r i-iv

1 as

-0

new

+pv e-e +

variables, where

fV' n-n

(.176)

-

where

j = ~ (n.v. - n v ) , C

1-1

e-e

Il 0 is the total

mas:; density. The variables are thus v ,v and -0 -n n'" n = ri and neglecting the ratio m / m. , we i - e e l

,

_j ; assuming

that

find approximately n

m

v. = v + - n (v - v ) + - e - c J. -J. -0 n -0 -n m. ne1

(177) v -e

n +...E. (v _ v ) __c_ j . -0 n -0 11 ne -

=v

From (173) (175) we then

find the approximate equation of mass equilibrium

-71 -

V. C. A. Ferraro

u = -Vp+jXB+ p,F+m (V

(178)

-

whilst it ca n

-

1-

be shown

that

e

en

-Y,

in

El(V, e

+V m~'b (v -v en

in

ITIl

I0

-0

-n

1

the approximate equation for the electric

current density 1 is

V

B mec e - P +' JX_ o=-ne~+jXB+-e-(Yie+Yenli-~( m.Y, lx~

(179\

1

where we have

taken

m

~

n -

E

(180)

-

-

-0

-

1

1

in

»m y . Here e en

= E' + E"

-0

where E' = E + v >(B/c is

m. and m,)J,

ln

the electric field following the mean motion and

lin ,the 'equivalent electric field produced by the electron pres· e e sure gradient. Equation (179) can be solved for 1 as a linear vector E" = (V'P

function

of

E

-0

and

'Yr.

Writing

eB

w

(181 ) so

-i

that

b is

eB B=Bb

=-

m c' e

w. and ware the

ion and electron cyclotron frequencies -e a unit vector along~, and defining the conductivity -1

Y

(182)

e

=

Y.

1e

+)1

en

we find

222

(A +D )j = AO"'E + c:T"(AC+D HE . b)b + crDE )( b -

-0

-0 - -

(183)

AC Qp X b + CD (\/p X b) X b B B where

0-

an,

-72V. C. A. Ferraro

w w, e 1 C =)1 'JI, '

A = 1 + C,

(184\

e

In the case

when

E

(185)

where

-0

E

11 parallel and

and

=..! lie

be simplified. In fact, if

we write

=E + E

-..L

-11

E J. are the components of the total to

~1l = (~. £)£ (83) can

In

(173) can

perpendicular

(186)

so that

V p = 0,

w D

electric field

E

-0

, we find

B

-

E

'

-

= b X (E )I. b) -

-0

-

now be written

(187)

Here 0-0 ' crl ,

<72

are

respectively the direct transverse and Hall conductivi-

ties and it is easily verified that

(HI8)

(T=a-

o

(189)

{190\

where by

0""2 =

c:r is

given by (182)

2

D

A +D

2

. These

formulae

agree with those found

Chapman by a somewhat different approach . If B = 0, then C = D

and 0'"'1 = ~ and 0'"2 = 0 as

should be the case, Equation (IS7) can

be then written,

(191)

(T

-

j "(T E

0 -0

=0

-73 -

V. C. A. Ferraro

the usual form

of Ohm IS Law. If

f

B

0, then

(181}, shows

that the

electrical conductivity is anisotropic, and if we take cartesian axes O(xyz-1 with Ox may wirte

£ X~,

along

(IS7}

-


E

and

Oz

(the right

hand side

of rr and

~).

conductivities

we

j=(f.E

--0

1

--<J'2

0

<72

Oi

0

0

0

~

of (192) denoting the contracted product

Formulae (189). (190) show that the transverse and Hall

are reduced respectively

A

(194)

If the magnetic field is

in the ratio

D

large,

A

and D will

be large

tion of the electrical conductivities t:r 1 and 0"'2 will and the electric

currents

will

flow mainly

any further direction of

also

bXE

--0

so that the total

is a constant. Substituting in

electric field (181)

be

large,

of force.

which . prevents

flow of the Hall current. Such a· field must be -

the reduc-

I

along the lines

Let us suppose that an electric field is set up

A.

~11

is the second order tensor

(193J

where

along

in the form

(192)

where

Oy along

is now

in

the E +)p)( E , --0

-

--0

we find that the Hall current

-

74--

V. C. A. Ferraro

(parallel to b X E) -

-0

vanishes

j = cr (E . b)b

(195)

-

"A

= 0-

2

/(1"1 ,

and that

+0- b X (E X b)

3-

0-0--

where

-0-

2

(196)

is

provided

C-=<J+ 3

1

<7'2

-

0"'1

called the Cowling conductivity. In a fully ionized gas,
that is,

the conductivity is the same as in the absence of

field. In

a partially ionized gas; however, although C' 3 exceeds both 0-1

and (f" 2 •

28 •

it is in

general smaller than

~

o

Numerical illustrations A review

of the electrical conductivitie:iin the ionosphere has

been given by Chapman. Values were given for cool

a magnetic

(850~)

ionosphere

and

in

a hot (1500 -K) and a

the table below

the values of

II . • 11 . • V are taken from a corresponding table in Chapman's Ie m em paper. In the table below we also give numerical values of the conductivities for the higher temperature. The condUcting

cr

o

increases upwards

over the range considered (100- 300 km ) and the conductivities <1"j' ~,CT3 each have one peak

29.

in

this range.

Effective conductivities in Baker and

the ionosphere

Martyn have shown

that because of the limited vertical

extent of the conducting layer of the E-region, the flow of current is nearly horizontal. In fact, if

.1

accummulate on the boundaries

contains a vertical component, charges will of the layer because this current cannot

flow in the region of low conductivity. These charges are called polarisation

Hei~ht

90

JJ ie

125 7.63><10 2

6.66 .... 10 2

4.84 )1'10

1.83)(10

4

2.81,.10 3

8.80,,10

6.83"10 5

1.61><10 5

)I

in

4.12 "10 4

8.59)(10

IV e

1.26:1(10

5.32)<10

w/Vi

3.92 v10

0"0

cr 1 0"2 0"'3

175

9.35xl0 2

en

r-

150

2

}J

we tn

1.43)C 10

100

in KIll

-3

1. 88x 10

3

6.16;( 10 2

2

3.

200

250

74~102

3.08><10 2

2.68xl0 2

4.50xl0

3.

3.57 .... 10

2

300

1.21lC 10

2.90

1. 08

2.48~103

6.33 )(10 3

1. 181(10 4

2.14.10 4

2.82XI0 4

1.67

6.10

1.69 .. 10

7.691(10

2. 29.)otl0 2

1. 62,..10- 11

4.14.10- 11

7 _72.1C10-11 1. 76»(10- 10

-15

2.88 8 C10- 15

1. 04XI0

2

.282 2.91,..10- 12

1.12"'10

4.12)(10- 15

1. 74x 10- 13

2. 70 Jel 0 - 1 7

1.23)(10

3.25 >cl0- 16

3.27)(10- 15

4.54)(10- 15

1. 72'c10- 15

1. 7hdO- 15

2.29)(10- 17 1.38wl0- 18

3.93)(10- 15

B.70XI0

-14

1.73)(10- 14

3.91 ')(10

-15

1.07 )(10- 15

3. 88x 10

-16

1. 29 )(10

74~10

3.20,..10

5.92",10 -2

2

2

-15

3.87" 10

-16

-16

1.

07~

10

1. 07)c 10

-16

-16

Table I Collision frequencies, electric conductivities, and the ratiO! of cyclotron to collision frequencies for a Illogel ionosphere at a teIllperature of 1480 K.

2.79)('10 4.

31~

10

-10 -17

1.87~10-19 4.31."10

-17

- 76--

V. C. A. Ferraro

charges. Because the flow (J'"

by a

components

(197)

(7*

2 X 2 tensor depend

to use coordinate tions. Then

is horizontal we can replace the 3.x 3 tensor

the magnetic dip angle 1. It

on

x, y

we can

write

:xy]

-a-xy

yy

where (J'

()o 0"1

,

-

<7i . 21 sm

0-0 (1"2 sin

a-

(198)

0' sm . 2I + a cos 21 o 1

xx

will be con venient

for the magnetic southward and eastward direc-

=IO"XX

(1"*

representing the layer conductivity whose

a- sin 21 + o

xy

C"2 2 (T cos I

sinI

1

2

2

<1"2 cos 1 (S'

yy

=

. 2 2 sm I + a- cos I o 1

The approximations given

on

oro- 2'

valid

Here

but they are we

(199)

+ 0- -

1-

(j

not

the right near

C

1

arise generally the magnetic

since

equator

0"

o

>xr

1 where 1=0.

have

(j

xx

=(j, 0

0"

xy

= 0,

0" = 0- + yy 1

Two consequences follow immediately from

2 0'"2

cr1

=0-

3

(199) ; firstly .. the high

conductivity 0are

along the magnetic lines of force ensures that these o very nearly electric equipotentials • Secondly, the east-west conducti-

vity 0-

yy

at the equator

is very large, being the Cowling conductivity 0"3

-77V. C. A. Ferraro

which is comparable with, though smaller than, cting strip

along

the magnetic equator carries

known as the equatorial es in

cr o . a

This highly condu-

large current, a belt a few degre.22 sm 1« cr cos 1.

electrojet and is confined to

widths about the magnetic equator, where

(j

o 1 Outside these belts the electric current falls rapidly. The simple model

of the dynamo region is therefore a relatively horizontally stratified layer. The magnetic variations observed at the ground which are produced by these currents

are best calculated by conSidering the layer as a

current-sheet, with

integrated conductivities,

r; 2 " Sa-:2dh where the integrals are taken If

"l7t: denotes

the tensor

over the thickness of the horizontal layer.

)O'*dh, we can summarise the electrical

equations as

*-t , L:.E

J "

(200)

J

"J1 dh

where E "w X B -

(2011

w being

-

-t

-

Vp,

the velocity of the neutral air and ~ the

hal . The first term electrostatic field ly.

-

in (201) ropresents

-V cp

forces

electrostatic poten-

the induced field, whilst the

the electric currents to flow horizontal-

- 78 M. Z. v. Krzywoblocki

cannot be zero . If in the last equation of (3.6.2),

4 f = 0

then ti) = 0, '0x ou U(x, y, u) contrary to the original assumption. Therefore it is necessary that

() U = '0 U = -UU = 0 -

" WhICh case m

uy

f4

f

·Iu (x'

y -'

0 . Similarly, the subgroup:

_

x

S

2

. = f1(x y' a) Y = f (x, y ; a) , A . '" 1 must have an inverse and therefore neither the Jacobian determinant

(3.6.10)

associated with SA 1 (3.6.11)

Ii

"0 (x, 5')1

=

() (x, y)

ux ux ux

'OY

nor the Jacobian determinant:

I'll

(x, y) (3.6.12) I u(x, y)

1--

ux UX

'0xuy

I

~ 0Y VX -2,y associated with the inverse transformation From (3.6.12) , it follows that not both equal to zero .. (3.6.13)

ox uy

JU Ifu x f

0,

may be equal to zero.

~ _yX

and

~~

are

then from (3.6.8)

=0,

and therefore (3.6.14)

uyY r./.

()

O.

Eqs. (3.6.6), (3.6.7), (3.6.8) and the above remarks imply that:

-79 -

16. D. F. Martyn, Processes controlling ionization distribution in the F2 region, Aust J. Phys. , ~, 161 -165, (1956). 17. H. Rishbeth, Further analogue studies of the ionospheric F layer, Proc. Phys. Soc., 81, 65 - 77 (1963).

18. T. Yonezawa, A !Jew theory of the formation of the F2 layer, J. Radio Res. Lab. , 3, 1 - 16 , (1956). 19. T. Yonezawa, On the influence of electron-ion diffusion exerted upon the formation of the F2 layer, J. Radio Res. Lab., ~ , 165 - 187 (1958).

- 80 -

M. Z. v. Krzywoblocki

and

(3.6.21)

where /-I,

R

C and k are constant! v The above system of six partial differential equations in six un-

known functions of

x, y,

z

and t will be referred to as the

system "A" . In order to find the conditions on a group that

"A" is conformally invariant under

find the system corresponding to

Al

such

Al ' it is necessary to

" A" and A

l'

A is a transforma1

tion group defined as : AI: -x =f 1(x, y, z;a) , -y=f 2(x, y, z;a), -f3 z= (x, y, z;a) , _

4

5

6

u =f (u;a)::: f (u;a) u + f (a) . Since no new techniques are involved in finding the system in

question, this problem will be left to the reader. The present paper will confine its attention to one such group under which " A " is conformally invariant. It is the following group:

i

= t - 64a ,

u = u,

v = v, w = w ,

CENTRO INTERNAZIONALE MATEMA TICO ESTIVO

(C.1. M. E)

. P. C.

KENDALL

"ON THE DIFFUSION IN THE ATMOSPHERE AND IONOSPHERE"

Lecture 1 : Ambipolar diffusion in a uniform magnetic field 11

11

2 : Diffusion in a

plane stratifiea atmosphere

3 : Numerical and analytical methods "

4 : The diffusion operator with curved magnetic field lines (neutral air at rest) 5 : Equilibrium solutions

n

11

6 : Time dependent solutions of the F2 region diffus ion equation

n

II

7: Motions of the neutral air induced by ion-drag

Corso tenuto a Varenna dal 19-al 27 settembre 1966

- 82 -

M. Z. v. Krzywoblocki

(3. 6. 24)

" B "

(3.6.25)

(ii)V 5

...

+ ... ) +...

=0 ,

i) Y3 + 2 tl 0iYl; + ... ) +...

=0 •

+K2 )) 1 ~

01Tl3

VV3

(1ll))5 ( ~ 3 ~\

(3.6.27)

~))2

()V2

(t 3

r

(i v) \) 4 = R \\ \) 6 '

~ Y5

(v)

'I 3 ~ +'t2 i> "h +~2

(3.6.28)

(3.6.29)

V(» 5 )\ )

c »

(vi)

v

~ ()\ Y 3)

() (V 5 Y 3) Q"rl2

5

(~

OV6 3

O~3

+t4

Q.l) + ~ ~3

VV6

t "0th 2 1 ~1

=0,

+ ... )+.'..

=0

.

The system "B" is absolutely invariant under the group:

(3.6.30)

~

j

=\l

j

( j = 1, .. " 6) ,

whose invariants , as in the previous case, may be chosen as follows:

- 83-

C.1. M. E. lectures by P. C. KENDALL Introduction

(University of Sheffield)

This set of lectures is not intended to be a comprehensive survey of the ionosphere. It is intended to lead the student with great speed to certain specialized research areas of possible interest to applied mathematicians . I apologise to the many ionospheric physic ists (both exper imental and theoretical) whose work is not mentioned here; the short list of references included at the end reflects current theory. Present day achievements have only been made possible through the patient experimental work

and international collaboration of many scientists over decades.

-

84-

P. C. Kendall

Lecture

1

In his lectures Professor fusion

Ferraro will show how dif-

of a plasma (a gas composed of equal numbers of ions and elec-

trons) takes place through a neutral atmosphere. The electrons ar p often very mobile and would in absffice of electrostatic forces disperse

to

infinity in a very short time. The ions are bigger and heavier and so cannot move as quickly through the neutral atmosphere. Thus, the tendency of the electrons to

"boil off" is prevented by the electrostatic forces

between ions and electrons, and an electric field is set up. This has the effect

of increasing by a factor of two the pressure gradient which cau-

ses diffusion; thus causing faster diffusion, The purpose of this lecture is to

known as

ambipolar diffusion

establish the equations of the problem

including the effects of a uniform magnetic field H . In general these o equations have not yet been fully solved. I will describe what can be done, and also

mention the difficulti es. Practical application

may be made

in

the upper ionosphere,

of the equations

in diffusion of meteor trains and

in the laboratory, wherever the plasma is a minot constituent.

Solution of the problem (ion-neutral collisions only) The main problem

is how to work out the electric field arising from

attempted charge separation. In the case when the electrons are perfectly mobile (i. e. we c?n ignore collisions of electrons with both ions and neutral atoms) this can

be done in full (provided that certain other minor simpli-

fications are made) . In particular, it will be seen that the ex1ernal electric field gas.

gives rise to

complicated drift motions

of the joint ion-electron

- 85-

P. C. Kendall

The equation

of motion of the electrons is dV

(lj

-e

N m

dt

e

'" - V p - e N (E + V

-

e

--e

It

~)

Here, m '" mass of electron e pe '" electron pre ssure N '" electron density

'" ion density

'" electronic charge

e

V :: electron velocity

-e

;)/at + (V .V)

d/ dt = mobile operator E

=total

electric field

-e -

in

fixed axes

The magnetic field is assumed constant and ions and electrons is so small

that

uniform (Le. the number of

their associated electric currents

may be neglected) . Thus the electric field is a potential

Neglecting terms

which

o i. e.

(3)

So, in constant

=0

curl E

(2)

an

=-

Vp ... e

Vp

-

e

e

-

m (which is small) e

- e N (E+ V X B),

-

at

-e

-

uniform temperature T, if

that p

=- V n

-e NVn=-Ne V)( B _ -e-

atmosphere

such

involve

E

field, i. e.

=Nk

T,

k

is

Boltzmann

- 86-

P. C. Kendall

equation

(3)

becomes ,,_ (keT

(4)

\ = - ::::e X B log N - rl}

Write rl = ~ e

(5)

log N + rl

H

Then

~. ~ ~

(6)

showing

that

the

line. Further,

if

potential

rlH is

- V 11

E

-H

we

= 0,

-

constant

along

a

magnetic field

H

obtain

v

(7) This

is

known

electric field sent

=E

-H

-e

~H

as is

)( B /B2

the Hall at

right

must be regarded as an

The electric potential

kT

drift or Hall motion of electrons. The angles

(log N) /

ask what happens

mes

infinitely large. The answer

dV -i N ml' -dt

Here,

when

N

becomes invalid

The equation

the magnetic field and if pre-

applied electric field, external in origin.

well

neutrality, which

to

of motion

e is

~

0;

internal in origin. One might

apparently the electric

lies in our assumption for s mall values

field beco-

of electrical

of N.

of the ions is

+ N m.». (V. - V ) 1 la -1 -a

= -

V p. + e N (E + V,lC -1

B)

--1-

- 87-P. C. Kendall

m = mass of ion i

p, = ion pressure 1

N = ion density

= electron density

V, = ion velocity -1

V = neutral air velocity -a

~

ia

= collision frequency ions - neutral atoms

In the diffusion approximation it (The student should find out

N m,

Adding (3)

and (8) to

N m,1

(9)

1a-l

-a

= -

eliminate

Y,

-1

V p, + e N (E + V,)( 1 -1

B) -

E.

(V, - V ) = -

1a

to neglect the acreleration.

why). Then

~,(V, - V )

(8)

1

is usual

-a

V (p.1 + Pe ) + e

N V. X B - e N V X B -1

-

-e

-

Writing

p. = p = N k T ,

(10)

e

1

8.nd (11 )

we see

by

N m,

~. 1 1a

e V N--- V X B -e m i )I ia

-

that

(12)

This

2kT

F =V -a

V -i

must be

+

solved for

using the continuity

e

=F

m.1 ",1a V .. Then -1

the diffusion

equation, which

.N

-;;;-t = - div (N V.) v

-1

is formed

in absence of production and

loss is (13 )

equation

= -V ,(N V,) -1

- 88-

P. C. Kendall

J. components parallel to and perpendicular to B.

Denote by" and Then

V."

(14)

-1

=F"

and

=

(15)

There

are many ways of solving this

thod is to to B

introduce coordinates

z

(Fig. 1).

F J.

equation. A quick convenient me-

x, y ,:or

y

x Fig. 1 Then if

J. V.

-1

we

= (V , V 2, 0) 1

have

It follows that if

we introduce

j=J(-I)

and write

with the

z-axis parallel

- 89-

p. C. Kendall J.

V

-i

=V + 1

J'V

2

we obtain

So equation (15)

becomes VJ. = FJot 1 + e B -i - {1 m. 11.

(16)

1

la

where .L .L F=V

(17)

-a

Equations (14) (16)

.L

2kT

V

Nm.». 1 1a

and (17)

-

J. e B 'V N+ - - - J-e m.v. 1

1a

describe the motion completely and with (13)

w.

give an equation governing N. First"discuss two special cases. These will help to

clarify the general

Special case

I

B

=0

Equation (12)

(no magnetic field) gives at once

V. = V -

-a

-1

This is

the case

problem.

2 k T N m.». 1

1a

of ordinary ambipolar diffusion. We define the coeffi-

cient of ambipolar diffusion in this

D =

2kT

a

so from

(13)

if

-VN

T

and

~ la

case· to 2

cm /sec

are constants,

be

-- 90-

P.e.Kendall ~N

-

(18 )

at

This led

is

2 " - div (N V ) + D V N -a a

the isotropic

a transport

diffusion

term. Its presence indicates

place relative to the wind D

equation. The term

V in the neutral -a

div (N V ) is cal-

-a

that all

diffusion

takes

air. Note the factor 2

in

of

given

a

Special case

eB/m. )) . ..,

II

1

la

co

Physically ~. la

= collision frequency

e B 1tm.

= Larmor frequency (*)

1

(*)In absence

of

electric

by

fields

the motion

a

single

ion

is

dV m.

1

dt

=eV)(' B

i. e. dV

-

w )t V

dt where

w = - eB/m.

-

-

1

Thus in absence of other forces the velocity vector rotates with angular velocity w = eB/m. in a certain sense. The motion is one of spiralling 1 about the magnetic field lines.

-- 91-

P. C. Kendall

Thus case II corresponds to many spirals between collisions and magnetic effects dominate. Clearly V lI is unchanged from (14); but from (16) and (17) -i

The diffusion equation then

a

a- N = - div

( 19)

2N I J. (N V' ) - div (N V ) +D -a -e aaz 2

ih

where z

is

becomes

the coordinate along

the only effective neutral Hall drift, caused

wind

'

the magnetic field. Note that

component is

along a field line

by electric fields is at right angles to

Rearranging equation (17) ,

.a.

.a.

4. Da ... - V - VN -a -e N -

=V

eB

+( 1+

m i Via

Thus from (16) j)

Also from

(14) II

V.

-1

So, letting

s

=e B/m.1 )/.1a

D

-1'"

J.

-e

2., D

...

(V - V N) -a -e N-

II

VN = V-a" _2 N "'" we have

.a.

j) V

(b) the

a field line.

In general

F

(a)

- 92-

P. C. Kendall aN

-

at

II

.

=.

1

s

.L

l.

dlV (N V ). - - div (N V ) + - - 2 div (N k Ie V ) -a l+s2 -a l+s - -a

• div (N V

1 . .L s .1. ) + - - 2 dlV (N V ) - - - 2 div (N kI' V ) -e --e 1+S 1+S

.t.

-e

+ D

a

where

~

= (0, 0, 1) .

~~ (!. !,I.N) =!.n~ ,,! N) i. 0,

Thus as

eN

-

=.

ch

div

II 1 (N V ) . - - div -a l+s2

we have

J. s .1. (N V ) + - - 2 div (N k X V ) -a l+s - -a

2 s ~ s ~ - - - 2 div (N V ) - - - 2 div (N l( lrV ) l+s -e l+s - -e (20)

D 2 2 2 + _a_CaN + aN )+D aN 2 a ".!I. 2 1+s ax 2 ay 2 vZ

=0

Note how on putting s to (19) on putting

s

=

IX)

first one remains when

•

this equation reduces to

The terms in

s ..

IX)

•

'"

~HX

Ji/B2) and winds

Ya

V J. disappear when s

-e

They might

te also how complicated the effects

be called

of electric fields

become when

s

~ffective

=V

-a

+

=0

it reduces and the

RHall terms" . No-

EH (such that V =

-

-e

..... 1 . The effective wind is

not V but is -a II

(18) and how

.L J. V - s kXV -a --a 1 + s2

-- 93-

P. C. Kendall

Asymptotic solutions of the ambipolar Consider equation (20)

in

diffusion problem

i. e. diffusion in a stationary atmosphere with other than

V =0 -a no electric fields present

the case when

=0

E

=II

and

internal ones. Then aN = _ _ Da_ (aN 2

(21 )

~t

..

1

+s

2---'2"

ax

Write

= z/JD

a

'If) Thenl\ becomes (22)

Suppose

that diffusion takes place

directions, a liven cloud plasma is released as

over

a region of infinite extent in all

of plasma being released at time

a point source such

that

if r

t = O. If the

= /(X,2 + y,2+z ,2),

then it may be shown that

N(r , t} =

3/2

x'

exp-

2

2 2 +y' +z'

4t

(4,t) It follows that

for

a general cloud of plasma such that N

(*)G(r)

~0

if r 1= 0

and

= N(x',

y', z', t)

is so defined that

IIf &'

(r) dx' dy' dz'

=1

-

94-

p. C. Kendall

we

have

1ff

N(x', y', z', t)

l

2 2 2 N(x ,y ,Z ,0) (x'-x) +(y'-y) +(z'-z) o 0 0 0 0 a 3/2 exp 4 ( 4'1't) t

J

dx dy dz . 000

So the asymptotic shape of any plasma cloud as N I N=

where back

0

(4Wt)3/2

= total electron content o we see that if

t.., ao

(1

N=

shows

cloud will

N dx

the asymptotic

(23)

This

~i~

that

take

the direction

a

is

2 2 + y' + z, 4t =

dy

~ao

N dx' dy' dz' . Converting

dz •

forom is obtained, namely.

3t N

(41fD t) a in

up

+ S2)

,2

Iff

N'

No = as

x

exp -

t

2

exp

r

-

2 2 2 2]

(l+s){x +y)+z 4D t a

uniform

magnetic field

the shape of an

a

discrete plasma

ellipsoid of revolution, elongated in

of the field.

,.

Dispersion of meteor trails A" meteor

"

trail" is an

infinitely long cloud of plasma left behind

as the meteor passes through the ionosphere. time t =0 one might suppose the trail to

~see

Fig. 2 for notation) . At

be of small cross-section.

meteor

- 9S-

z

P. C. Kendall

B

z*

(earth's magnetic field)

y

~--""y.

Fig. 2 Clearly the problem is in two meteor train . Assuming that an x

dimensions, at right angles to the

equation

of

~..

= x y = y cos at + z sin. z

and N= N(x

It

w

,y

'rI

type (21) hplds we may put

= - y sin fI. + z

COS"

,t)

giving . 2

( sm

(24)

Q(

2 cos ot

",,2

Q

N

+ --) -2 1 +s ay~2

Unfortunately we have omitted electron collisions. Nevertheless, this serves as useful illustrative problem. Exercise

Show

that

a solution

of the equation

a N _ ~2N

"it - ~~2 is

Hence

2

n a O N = --::;:""""-2 exp -

obtain

(a

+ 4t)

a solution

of

( Xl

2

(a

2

+ yl

2 ) -

+ 4t)

equation (24)

and discuss its nature.

- 96-

p. C. Kendall

Note . When

electron-ion collisions and electron -neutral atom collisions

are included the ambipolar

diffusion equation can

not be solved. A re-

cent paper by Uolway (1965) J. G. R. 70 3635 is knuwn as

curl

~

F0

to

be incorrect

- 97-

P. C. Kendall Lecture

2

Diffusion in a plane stratified atmosphere The diffusion equation

Figurt 3

z

L

- -horizontal

I • anglt of dip

-

=angle bttween B and horizontal All variables

are assumed to be functions of

z = height above some fixed reference

z

only, where

level (Fig. 3) . Roughly speaking we

choose

z

=

base of ionospheric

layer

the F2 layer, SOO km above the ground )I

ia

-

1 sec

-1

Therefore 1.6 It 10

26 !CIa

giving

s It follows that

F

on ambipolar diffusion may be used to simplify the

The preceding results derivation, for in

a at

»

1

-20

-24

Jt

110.3

1

=:s 200

- 98-

P. C. Kendall

(i) The only relevant equation along a

field

line

of motion of the ions is that

i. e. parallel to "§.

(ii) The motion V~ at right angles to

the field lines is deter-

-1

mined by the Hall electric field. Thus J.

~ " ~H X "§./B For a self

2

The physical meaning of (i)'Cii,is that the ions collisions

with

motions across

J. take V to be uniform. -i

consistent model we must

can spiral

freely between

neutral atoms, and do so many times. Their random field lines are therefore considerably reduced (Fig. 4) .

Figtrt 4

Schematic view of motion of ion However, the Hall drift is
V.J., -1

though the field lines were themselves moving

giving a transverse velocity

VJ. to -i

the motion of the

- 99-

P. C. Kendall

Figure 5

With cL /flJ. = 0 the equation of

motion

of

the plasma along a field line

is (Fig. 5)

2NkT (V,II - yll) = _ V· (2NkT) + Nm. gil D -1 -a 1_ p. + P = NkT + NkT) . 1 e temperature of all constituents is the same and is

(The partial pressure of We assume uniform

that the

the plasma is

and constant. Here D =2D

and from

gas

ia

= 2 11' coeff. diff. ions ..... neutrals

theory

D 0I:1/n (n = neutral

a

The neutral

a

atmosphere

is

in

air

density)

equilibrium. Thus the neutral air density

is n = n exp(-z/H), a 0 where H = scale We shall

assume that Thus if

~

m

height = kT / m g a a

=m

T denotes

i .

a

unit

tangent along

a field

line

-

100-

P. C. Kendall

- - --g" g

VII = ~ (t.V) and

giving

H

V. = V

-

N.)I .T.

\IDA - - ( T .V N + N - 2H

ct

-1

The diffusion

A

si nIT

=-

equation is

iN

-

at

=

formed

Sin

from

-

the continuity equation

Q - L - V (NV )

-'

-i

Here Q = rate of electron production (by sun)

L "loss rate (by recombination) The equation

could be written

iN ~N - = Q - L - W cos 1 - - V cos

at

iz

H

a

I

I~ Iz

sin

+ D~ N,

(1)

where WH = Hall drift

.l

r

to

field lines

(assumed uniform) V a

"Horizontal

velocity of neutral

air

(assumed uniform)

~

= diffusion

Df)N =

operator

defined by :

Y{D (! .Y N + 2~ A

= r:!:.~)

{

D

~ (!.YN+

N 2H

sin sin

!} 11

I)

-- 101 P. C. Kendall But in

this case ~

~

'!: .~ ;: sin D«':\ N

Thus

1;1

giving

,

as

D

This equation

is

,2

I

= sm C(

exp:(z

I

ch

~ 'lI.N D( CI ~ "'iZ

N + - 2H ).'

! H)

(2)

We have to

due

to

solve equation

production

may be taken

(1)

to

where

X

night

Q = D. The constant

is

the

(1 -

zenith angle

from

noon

(ell

t

i>

~

is

given by (2) . The rate of

function

- e -z!H sec'i. ) ,

of the sun, during daylight

q is the rate of o

~ = Northern declination of

sured

, where

be the Chapman

Q = qo exp

(3)

Let

Professor V. C.A. Ferraro.

sun,

= D~ noon).

+ I

Then

hours. At

maximum production. = local

time in

radians mea-

spherical trigonometry gives

(Fig. 6) cos

X = cos

(~- i-l

cos

e + sin (~ - & ) sin e cos

+ I


sun - - ... e Figure 6

-

102--

p, C, Kendall

=

The length of

day,

o= Thus time can

Then

we

sin ,

\\I

is

&

sin

be

cos

+ cos

(J

given

cos

in

sin

cos

(J

q,'

the equation

S

+ cos

(J

measured

by

S

sin

radians

(J

from

cos

t IjI

sunrise

b) writing

have sunrise

at

sunset and

4»

=

\II

= sin' cos

(J

at

X

cos

~ =0

+ cosi sin

(J

cos (; -

it)

(4) = function

The

loss

rate

is

proportional

of time, to

N and

is

given

by

(5)

~

The exact value of

for

~ =

and

numerical

has

calculations are possible in

dimensions

illustrative

layer is believed to

sec

be

1. 75 ,

range of analytic solutions

Ii o

the F2

this case, However, a wide

ls available in the case

,,= 1,

The constant

-1

solutions

The problem

may

be formulated in dimensionless variables

follows, Note that for simplicity's W

H

=0

sake V =0

a

we consider only the case

as

-- 103 -

P. C. Kendall

Write

(6) where

"'r = number of sees

in a radian = 1.37 X 10

4

{)='1'(1o

(7)

"

(8)

H2 TD o

Q

. 2

S1l1

D D z/H I ( = 0 e )

Then

C)N =T q exp ( 1 ~.

Also

-z H

0

-z/H

e . . _l-(1Nt-).z/H cosX

put

~

= e- z/

2H

and

N=

"!,

V

Then 1

+-

(9)

4~

The solution (i)

of

equation (9) is

~ a 0) ? tr

Solution is periodic

(ii) There are no (iii) " . . 0 as Gliddon

required under the conditions

(l9~)8

a, b) has

sources

X = O.

which

appear

provided

under the

There are in

00

(~ = 0)

Z+-oo (~ ... oo) solutions of this equation in full

the case ~ = 1 following earlier work case

at z =

by Ferraro and

possible generalizations in

the Journal of Atmospheric and

authorship of Gliddon

elegant mathematics are due to

..Ozdogan for forthe the case

A= 1

Terrestrial Physic

or Gliddon and Kendall. The analysis and Gliddon.

-

104 --

p. C. Kendall

Exercise

Show

equation (9)

by direct substitution

with"

= 1 and cos

that a periodic solution of

X = sin~

is

1

V (x,,

1

) = 2 l'

3

2

-4J' xlA( " • 1-2 r r(.,fo)l q e (~'&) J exp ~ - 4A (. ' • J f> S

0

x

0)

0)

d 0

o 1

00

1 4 +2'T q o e ((\1S) where

Z.

n=l 1

1

f1 (eto'.)

=

1(1,,)

2 cosec'" COSh{

1

(~) '2 (+0 -+1 + Sinhi(t )2('0 -+~ 1

1

~ (~o'+) = !(\~)-2

cosec

~

sinh

f (~)"2

1

f! -.1

('0 -;)) + cos h

)2(4'0

and

Note This solution

does not satisfy condition (ii);

rected this in his later There se

is

scope

however, Gliddon cor-

pape rs.

for

more analytical work

on

equation (9) for

the ca-

~ = 1. 75 (or possibly X= 2) .

The equilibrium problem At noon and for

about

3

hours,

and, choosing an

in

the

so

early afternoon

the sun

nearly overhead

that near-equilibrium may be reached. Then

overhead sun for convenience, cos

is

X~ 1

.

-

IOSp, C, Kendall

Thus (10)

One boundary condition

may

be

found from

the expression

for

If V =0 -a

. + N) - sm 2H

Thus

and we require (i)

i

v/

r>!> = 0

'S

at

=0

Also (ii) v~ 0

as

'!> .... <Xl

,

These are called two point boundary conditions, There solution (a) the method techniques, In

of

describing how to

Note

methods

integrate

that l' q

from the calculations

o

and

two methods of

binary splitting and (b) the use of lathce

the next lecture I

describe the numerical

are

shall give recommended formulae and used

in

detail, and briefly, before

numerically equation

e is

a

scaling

(9)

factor whIch

inserted afterwards.

with

may

8

la + f

0 .

be omitted

-

106-

P. C. Kendall

Lecture 3 Numerical and analytical methods The binary splitting method We

present

two methods

of

integrating the equation

(1 )

in preparation

for the

equation

(2)

The first

is

the binary splitting method

Briefly, we

used

by Rishbeth and Barron.

observe that equation (1) may be integrated under

the initial conditions

v= where tion

v

v

is a constant,. However, o will not satisfy the condition v ... 0

excer)t

for

tlOns

fall

types

of

a into

'!, = 0

at

o

as

in

~.-"

general we

solution

are

see

that

the

solu-

00

v ,say , v = v" . In fact soluo 0 0 either v..lt 00 or v.... - 00 • The different

particular value of two types:

,

shown in

Fig. 7 .

-

107-

p. C. Kendall

Figure 7 v

v*o conditions

In practice ded

as

assume

N

as

soon

as

> 0 always we v

becomes

that a solution

start with

is

k now that

-ve i. e. when

such

that

v

-It 00

solutions corresponding to v o

two

for

v

a solution

< C.

v

when "V

1

may be dis car-

We

also may

v reaches and

v

0

"V

10. We 2

such that

v.,.-oo

1

for v 2 Then

test the solution

for 1 - (v 2 I

v3 If

it

is such

that

If

it is such

that

In this

way

the computer to

we

v.., v~

refine

+

-00

we

replace

vI

by

+00

we

replace

v2

by v3 .

the value

permits. The two

the exact solution, and it is

approximation (Fig. 8);

v ) 2

of

final

v3

as far as the accuracy of o solutions give an approximation v

obvious where they fail to give a

good

-

108-

P. C. Kendall

v

Figure 8 1\\

4-9ood ...., approx.

,

, \

v = 0 and v = 9 (say) . The meo 0 Runge-Kutta· method for starting the inte-

A suitable starting pair

thods

recommended are the

is

gration

and Hamming's method for continuing it. These two integration 5 formulae are equally accurate (of order h) but the Runge-Kutta method is unstable when

v

starts getting large. This can cause convergence: to

a false value of

v~. The reader should note that there are many o

other integration formulae, including Runge-Kutta formulae of different orders. Runge-Kutta formula To integrate

the

system

of d. e. 's

dy.

1

dx we have for

n

=2

I

i

= 1,2, ••• , n

-109 -

p. C. Kendall

The Runge-Kutta formula for the increments ding to an

increment

b.

h in

x

in

y1

and Y., correspon-

"

is

y.

1

where 000

K . = h f.{X , Y1 , Y2 ) 01 1 K

Ii

= h f (x i

o

K2i ~ h fi (x K

To use

this in

3i

1 0 Y 2' 1

+ -h

o

Ih

+ '2

o+h

=hf(i X ,

integrating (1)

n Then

=2

~ =x

'

1 0 1 K Y +- K ) 2 01' 2 2 02

+-

olK

Y1 +2'

o+K

Y1

we

0

11' Y 2

1

+ '2 K12

o+K)

21' Y2

22

)

(i

= I, 2)

have

y

1

=v

dy

1 dx

= Y2

dy

2 -=

dx

The procedure then start

is quite

using

straightforward,

Hamming's

method {with

Hamming's method {see Ralston and Wilf To integrate

numerically

the d. e.

but we only make 5 steps, same steplength ! .

-

110-

P. C. Kendall

t let

=

= 1,2, ... ) be

x. (i

f(x, y) a

sequence of values

of

spaced at equal

such

that

y(x.) (0

require

to

find

simplify

the

x;

1

intervals,

h,

known. We To

and

the

values

notation, dy , ( dx)

1

~

i

~

n) are

write

=f x=x.

1

then

Yn+ 1

is evaluated by the

following

method:

(1)

(2)

Modifier

(3)

Corrector: C 1 n+

(4)

F inal Value: Yn+l = Cn+1

=

.!.[9Y - Yn-2 + 3h(m n+1 +2fn-fn-l )] 8 n I

9 ( Pn+I + W

- C n+1)

Lattice techniques Assume

at

S = 2h

is

that

the

solution at

v 2 and so

n-I

-

2v

n

h

equation

(I)

at

V

n

is

=v n

then

is

v , at o

S = h is

on. Then

V

Suppose ~ = '!,

'!, = 0

+V

n+I

2

• The finite difference approximation to

vI'

-

111 --

P. C. Kendall

v

0-1

- (2

2

+ 41l't h ~ '"

2~

)v

n

+ vo+1

n

(3)

At

'!I = 0

we

introduce a

have

av/~'!,

= O.

value

v

spurious

The correct outside

-1

procedure

the

here is

boundary. Then

to

on the

boundary!, = 0,

v - v 1

-1

~::-:--

2h

v

The d. e. is in

n "0

satisfied

(3)

(~

o

on

(1)

the boundary,

we

is

no

easy way

observe

Thus we choose

that

for n

for

large

large

N

we could choose ) N

as

a

values

1-2~ _~2/ e

enough

v=;)

this

we obtain from (4) with

of exactly ensuring that

of.

(6) impose

so

o

- v +v o 1

v::::::'!,

and

1

= 0)

(5)

There

" 0

N

of

v ... 0 as

!> .. 00

•

Frou

'S ('S» 1)

(!.

(= N , say)

1-2>' e -~: /

II.

,.

boundary condition. In practice

(Jy

~

30 and

= 1 as a remarkably good approximation to

00.

-112-

P. C. Kendall

Equations -v

o

(5) and (6)

(3)

+v

give

us

the

set

N-l equations

= 0

I

=F

v-Av+v

o

of

I

I

1

v-Av+v I

,

t

=

3

F

I

2.

(7) v

N-3

v

-A

v

N-2 N-2

+v

N-l

=F

=F

-A v N-2· N-I N-I

where An = 2 + 4A¥h ,.

This system

of

equations

straightforward enough given

to

is

2

'S n2A

F

called

solve

N-2

0(3

- J. _W I-2A e;N N-I ~N

2

n

= -4 "ISh "S ne

v0

•••

vN _ I by

using an algorithm

by Richtmyer.

Suppose

the

general

(8)

c.:~

Set (9)

v. = r v. 1 + H

(10)

v

v

1

i-I

i-I i

=r

system

form

(see Richtmyer)

is

+ Bi v +A v - K i i i+I - i 1+

i

v +H i-I i i - I

n

a tridiagonal· system. It is

for

Algorithm for solving tridiagonal

_12

-113-

p, C, Kendall

Then Cr v+CH +Bv+Av =K i i-I i i i-I i i i i+l (11 )

or

K-CH -Av i i i-I i i+l

v. C. r. 1 + B. 1

1

1-

1

Then comparing (9) and (11)

K - CH i i i-I C.r. 1+ B.

-A r

( 12)

Now

=

i

i C.r. 1 1 1-

+ Bi

1 1-

st

1st equation ;

from

(13) H

i. e, 1

Bv+Av o 0 0 I

=K

boundary

1

condition

0

or

(14)

so

Method

of

(15)

Solution

H

o

K

o

B

o

H., 1'. for i = ', ... ,"-1 using (12) - (1

(1)

Calculate

(2)

Usinr (10) repeatedly calculate v,

1

1

1

2

2

aN/at = ~ N/Ch (Note that --------------------~~~--~~~-Numerical

solution of

here are only two of many possibilities)

Well formulated problem (Fi«. 9) h = distance t

between

t

lattice

h

between

x

lattice

x

= distance

the methodes given

-114 -

-,""'--"----'-------_ ____ _________ •

I

I

_~J_J

p. C. Kendall

~

t •• ,1~L~-----t~-----x • ~I

~lLTI-----------I I Fig. 9

In

a

well

as

a

function

later

times

on x = x

of

and

x

under

given

N

it

is

be

will

required

given to

at

find

time N

boundar'y conditions: one on

=t

t

o

= N(x, t) at

x =x

o

all

and one

•

n

Wrong

formulated problem

numerical Denote

method (i. e. wrong for

by brackets

{ -

Jt

slow computers)

the value

of

any function at

time t.

Then

and

+ N1 -

giving

on

substituting

into

2NJ

2

aNI at = ~ N/ ax h

t + -2 h

Note It as

is

common

origin, thus

at

practice

x we n

t

to

2

{N -1 + N1 - 2N ~ 0

x

refer suffixes

to

the

central

have the situation shown in Fig. 10

point

-115-

p. C. Kendall

--~-- I I Xn --,--0 • -~ .. I-- ~--I ,• I •

••

i.e. N_I is Nn_I,No is Nn and NI is Nn+1 Fig. 10

This t

+h

ters

gives t

quick

and unstable

. The method

where the

is

integration

from

time

t

to

time

not recommended (except for very fast compu-

steplengtn can

be

made small

enough to aChieve stability.

Crank-Nicolson method (See Smith)

t+h Evaluate

all

lattice

the two

derivatives

points

at

{NoL+h

aN at

~

~2N ~~

have

no

at

x =x

a

n

point

0

midway between

•

-{No} t

t ht

{ N_1 + N1 - 2N0}

t+.! h

h2

~x

But we

t

2 t

x

known

values

approximation

f

t-t.!h 2 t

of

N at

t +

1

2' \'

so we use the

-

116-

p. C. Kendall

Then, substituting

into

a2NI

aNj at " s = h 12h t

ih

2

using the

abbreviation

2

x

gives

that is

= s {N

Thus

the

system

1

-1 t

+ (1 - 2s) {N

integration proceeds of

of the set

tridiagonal

from

1 + s (N \ 01 t II

time

t

to

t

t+ht

by solving a

equations like (7) . The first and last equations

are formed by introducing spurious

if necessary, and by assuming that

points outside the boundary,

the differential

equation is satisfied

there. n

Consider

This

the

may

mations

be reduced

given

~ f.(x l , ... , x ) & lax.

operator

i=l

to

below,

the

n

I

form

a f1

/tI.'1

I

by suitable

thus making soluble the generalised

transfordiffusion

equation n

i

which

can

then

be

=1

f. I

aNI ch.I

subjected

to

automatic numerical procedures.

-117 -

P. C. Kendall

full

In

Consider

the

( ... ) becomes

system

of

ordinary

differential

equations

f

_ 2 -~

dx

f

n

n

dX 1 - ~

with

x

2

=x

20

This

,

x

3

=x

defines

30

a

, ... , x

series

n

=x of

no

at

x

1

= 0 .

functions

x

no

x

)

no

)

-

118-

P. C. Kendall

Consider

the

variables

transformation from

X~ ,

x 2 ' •.. x 'n

X ' =

2

x

x ' =x n

Then

and

we

so

know

( .. )

defined

by

20

(x, x 2' I

no

(x, x , ... x ) 1 2 n

x ) n

that

becomes n

=

Note

that

all

variables xl' x 2' ... xn to

the

~

i=l

the transformations

numerically, even if

exact

f

~

ix.

1

of

integrals

variables can cannet

be

carried

be found. The

out

variables

x 20' x 30, . . . x no retain the va:ues of x 2' x3 ... xn at the start of the integration. At any subsequent point we obtain from (*.)

N

The values

of

N(x ' I

'

x 20 ' x30' ... xno + other coordinates) which

this

is

the

solution

are obtained

-119-

P. C. Kendall

by integrating the would be needed

system in

of

equations. Note that in any case

the computer

at

the correct

stage

these

of the

calculation. Exercise which

. Use

reduces

the above the

-

a

~t

to

analysis

+ W sin 0

where

wt +! w

W

a'

derive

the transformation

operator

the form

Answer t' = t,

to

=

a +-.!. cos

w

,

~

aa and Ware o

constants.

-120 P, C, Kendall

Lecture 4 The diffusion operator with

We choose

W is

a constant

drift operator

1

e2

is

the diffusion

diffusion

equation in

1

2

dimensions

of

velocity. ~ 1 is

by

V. (NV . .L

N = -

~-1

operator

)

defined

by

Dt)2N = - V,(NV.") -

Geometry of geomagnetic

the form

+ wt) N + D~ N

with

defined

W"

(2)

(3)

the

~t

where

and

write

aN = Q _ L

(1)

Hall

to

curved magnetic field lines (neutral air at

1

dipole field

lines

(Fig, 11)

, \

e

\

, I

.,;~

I

the

-121 -

P. C. Ker:da:l

Field lines

are

family of curves

r =a

(4)

sin

2

e

Thus

e/ A

sin I = 2 cos

(5)

cos I

= sin

e/ 6

where

II = J (l + 3

(6)

cos 2 e)

Also (7)

tan

I

e

= 2 cot

A

Thus

the

tangent

T to

a

field

1\

(8)

1: = (sin

I,

-V11

.'1)

- -

= "'!:. ( sin Form of

cos

-

'" = "T (T

line

is

I,

O}

I~ 8r +~~) r ae

£>2

Thus as 2NkT

(10)

D

II

~

•

- - V_ (2NkT) + Nm.g" 1.

we have (11)

Hence

NY II

= _ D; (sin

-i

-

~.NV .• J.

-1

I aN + cos I aN r ~ (}

rr

+ N sin I 2H

=-_(t.V)D(SinI'N+cOSI 'aN +NsinI) ... ir r' &e 2H

( 12) _ D(_VoT_) (sin

I aN

ar

+~ r

aN + N sin I) 2H

To

-122 P. C. Kendall

Using the fact is

D

that

ex exp

of the F2

the base

CIt. N 2

1')

'=

z = r - rand r

(z/H). where

layer. this

o

0

gives

a ) ( sm" I a; ~ N + cos I 'I N a; + -cosr -I iB - r - 18

(sl"n

=r

I &

+ N sin I) 2H

(13)

It may be verified

!.!~

(14)

= ( 9 cos e + 15

cos

3

(J)

I r6

3

i> 1

Form of We

that

have .I.

V. = E )t BIB .;..t, 1I -

(15)

Also , the electrostatic potential

2

is constant

along

a

field

line so

that

nH = -

(16)

where a

•

radian

is

the

longtitude (t = 1. 37 X 10 4) . Then

(17) and so

~H

in

spherical

+,

a) ,

=,.. .

= - grad

1

=(-2-' sin

+ (0,

(J

o,

where T = number

flH

polar coordinates

~H

(18)

F(

•

(r, 9,

2 cos (J 3 sin (J

r sin

0) ~F

cla

aF

1 (J

+)

a+

of seconds in

- 123 -P. C. Kendall That

is

A

E = -3-(cOS -H

- sin I,

I,

0)

sin B

(19)

+ (0, Also

if

M

" dipole

MA ( .

-

Thus

e ~+

r sin

moment,

-3 sm r

B

(20)

aF

0 ,

B " M

I

MA2 3 3 r sin 8

E )( B -H -

(21)

MA

)

A /r 3

cos

,

and

0)

I,

~F a;

(0, 0, 1)

3F

- -4-- - - (cos I, r sin B

- sin I, 0)

a+

yJ~Hl'~)

But (22)

= ~ .(! 1I \ ) &

-

iF

Ta

~H" (!Jt~ )

0

So

-MIl i.E..

(23)

3 3 r sin (J

MA

aa

aF

-+-4-- •• (cos r sin 9 v

And

(24)

as

r

=a

sin

we

2

1

(J

N

1

r sin

I

(J

a sin I a ar - -r- "j9){M2fl2r6

N)

- 124-

P.e.Kendall

Finally to

make

this

dimensionless we

F =

(25)

WM 2

r

where

f

is

f (a,

2 ~N= __a_~ 1 2 3a r o

~),

giving

aN

+

a~

r

2

r 2

o

~f/~

b

2 + 6r(1 +cos 0) 2 ro

Finally,

we

make

substitute

o

dimensionless

(26)

may

( cos

sinO

If

I ~ _ sin I &r r

..!E) U

N

at

A4

the substitution

(27)

to obtain

the

dimensionless

av

a+

= 1'Q 0

form

(29)

.,

i>

namely,

+ W' "

z/H V + _e__ ~I 1 ~ 2

f

L_ +r2

2

I

a f aY V = -1 r 2 ~..~. o I

(1),

-z/H e exp ( __z___ e _ _ ) _ (?> e- >.z/HV 2H cosX

(28)

H

of

2

r

0

~f

V

/ •• (cos I L

Asin

0

c!)

V

r

- sin I aV - r - ii)

-

125-

p, C. Kendall

+ (~~

(30)

2

H

+v.4 )(sin ... -

I...!.- -'-

h

~~ ~

ae

r

)V

where WI = ,W/H ,

(31)

and

(l

'If •We tial I

and

'¥

are

as

1 before, except that a factor sinllI is omitted fro

wish

to

integrate equation (28),

equation

in

3 independent

first

outline

which is

variables. This is

a

partial

now

differen-

possible, and

the difficulties,

Difficulties

lines

....a_"'-..L.__ Z-o

_ I o -........_ _

Fig, 12 Roughly ~

speaking we

are integrating outwards along lines

(Fig, 12) . The line z = 0

is

not

system (Fig, 13) .

1. See Lecture 2, equations

7

and

8.

a

natural

J.

r to

boundary in our coordinate

-

126-

P. C. Kendall

Fig. 13 Also

as

infinite need

r .. .., the

and

the

total

length

involved

a 'ong

a

field

line bec.omes

region of integration becomes infinite. We therefore

a coordinate transformation which

A suitable

(a)

transforms

(b)

transforms

transformation 2

(32)

x

"'

-

e

-hr

0()

into a

0

- e

-ha/H

!H-e -ha/H

.hr/H ·hr IH _l+ e ·e 0 -.:-hr-o""'lH~~-h:-a-"I=H e • e

(33)

where h = dimensionless

constant

>0

and x

> 0 in Northern hemisphere

x < 0 in Southern hemisphere Ifa»r

e

-hz/2H

natural boundary

into a finite

is

-hr/H

e

=0 z= z

point.

- 127-

P. C. Kendall

On equator

r

=

a , x = 0 •

Also

at r :: r

o

,

x = 1 (Northern ) or We thus obtain

the

mapping

shown in

-1 (Southern)

Fig. 14

z-OCN>

X-I

X-O ------------ equator

x··J

z-0(5) Fig. 14

Choose as

(34)

independent

x

2

variables

e

-hr/H

- e

-hro/H

1 ,,~ 1

= 1 + ~-..,--"'::""--- :: 1 + - -hrjH -haiR .A2 e

-e

and a

=

r

-.-2sm (j

Then (35)

&>

&r

=_l_L

- sin 2(j

ea

-hr/H

+(_£ _e_ _ H

A2 -ha/H

(36)

1 2 cos (j ior &9 . 3 sm 9

..!.. _(£ A_l_e_-:--aa H A 2 2

- 128-

p. C. Kendall

Thus or

'() Sm

'!'\

(37)

1.\ Q

2h e- hr / H H A 2 cos ()

_

2 cos () +- ;;; ir r ~()

":.\

~

c»(x 2)

giving (38)

"\ !\ -hr/H ~ sin I-~- +~ _~_ = -h sin Ie __ ) r r f)(). H( -hr /H -ha/H ~ 2

0

e

Similarly sin

a ()-

2 cos r

~r

() i)

~

;,

- e

A2 -,-3sm 8

) g(x )

~

hAle

(a;- H- A2 2

h sm . () e -hr/H H.I\. giving cos

I~

sin I

ar - - r -

a-

a()

A

W Sin 3()

-ha/H

.L)

~ (x 2)

~

2

~(x )

2 ~

6a

(39)

~

H

{(X2_1)Ae- ha / H A t sin 3 ()

+

Therefore

(40)

f

2 A -ha/H + _.....;.s....,in,.....;8_e_--r-_ -hr/H ~f / " (x -1) e ~ 2. A. 3 -hr IH -ha/H 2 r Asm() sm () C/e 0 - e ) t>(x )

_h r

2

o

...

\:

-

129-

P. C. Kendall r 2 cos I ]

(40)

3f

V

afJ

2r2 A sinO o

and

i)2 'v

=

(41 )

which

simplifies

Check: then

£>2'

equation We

shall

If

a

1

(14))

and

o

=1 ,

h

~

-z/H~2 / 2 cVax in

e -z/2H , J

= 90 • ,

agreement with Lecture

2,

take

Hall

drift

=

(r

2 2 o

cos;

/a )

which

is

independent

Ij) .

The full

x~

9 (~" x) .

f

giving a

(see

»r

V~4e

(43)

sin

to

diffusion eqrntion is

thus

of

a

at

the equator and Q(

- 130-

P. C. Kendall (1 _ 2 WI H cos a

4> )

aV + WIH sinp.l.Y ~+

-z/H

+ ,. qo e exp (- 2~ - ecosX

(44)

We see numerically.

from lecture

(3)

that

this

&a

) can

be now integrate

-- 131 -

P.C.Kendall Lecture 5

Equilibrium solutions Diffusive equilibrium (and comparison of theoretical and satellite results) When the velocity of the ions is zero, i. e.

v-i

(1)

=0

we have a situation known as
surprised that

however all

this means is that the ion gas is in hydrostatic equilibrium.

Production Q and loss

Q= L

I

but this

electron gas

is

L

such

of

a simple situation could occur at all;

electrons

could only be occurring if

unlikely to occur simultaneously with (1) . The ion-

has half the mean molecular weight of the ion gas and so

if the temperature is uniform

N CC exp (-z/2H)

(2)

where scale

z

is

the

above some fixed reference level and H is the

height of the ions. It follows

of the previous Note

height

that this

Thus, what

and

(2) should satisfy

equation (44)

lecture. The student should verify this as a.n exercise. is

also

a useful check of complicated diffusion equations.

we are in fact obtaining when we use (1) is a solution of

equation (44) of the previous lecture, time)

that

no

loss

((3

for the case of no production (night-

=0) . It follows that in investigating solutions of th

full diffusion equation we first study diffusive equilibrium. There is another reason for doing this. We can include the effects of a height varying temperature field quite easily. Consider a constituent characterised by ionic mass

m., partial pressures 1

field E . For equilibrium of the dip

angle

P.I p 1

for ions and electrons, and electric e ions along a field line where I is thE

-132P. C. Kendall

dp, _1

ds

where a

s

denotes

=~

arc

N IT!

g sin I + Ne E

i

length and

field line. For equilibrium

of

s

E is the component

s electrons,

of

~

along

neglecting the electron

mass, dp _e :: ~ Ne E ds s Thus, as

Pi

= NkT =Pe' where T = T(z) is the temperature, z being

the height, we have

along a field line

p, sin I dp, 1 1 -ds-- - - - 2H(z)

where H(z) = kT(z)/m,g 1

Fig. 15 But along a field line

(Fig. 15)

dz = ds sin

I

Thus dp,

1

dz

p, =-~

2H(z)

- 133-

p, C, Kendall

giving p.

1

exp

OC

-J

dz 2H

-

Thus along a field line

T

If

cal

-f~~

NCX! exp

(3)

polar

ot. is

the latitude,

coordinates, r, r

(4)

=a

e

,+

,

the eqUl hon of

= colatitude,

(B

a

field

line

in

spheri-

~ = longtitude) is

.22 sm e = a cos 0(

we see that N = f(a)

(5)

where

f

is

some

g(z)

function

of a only and

1

g(z) = if exp

(6)

-

r

dz 2H

'

On these assumptions, (i. e, ignoring variations of H with respect to latitude) we can compare theory and experiment, .

if

SatellIte Aloutte I results appear shows electron

in

the form of

density versus latitude, each curve

height above the ground (shown in Km) ,

~

Taken from a private communication from

Dr

J.W. King

Fig. 16 , which

being for a fixed

- 134-

P. C. Kendall

14.---------------------------------------, MEASURED CURVES (Kio~l 10>00 LMT 10/3:62

12

10

Figure 16 4

THEORETICAL CURVES

Figure 17 8

6

4

710 - - - - - - - - - - - -_ _ _ .

~.

OL--~~~3~O--~W~~IO~~O--~IO--~W=-~~~~40~~~O~ -N

·S Ma,nllic Dip An,l.

A clue constant and

as

to

the

procedure

f(a) = any function with

is

seen

a single

when

we

take

maximum. Then

K=

we

obtain Fig. 17 (see Goldberg, Kendall and Schmerling, 1964, Journal of Geophysicul Research 69 , 417-427) • This is

so

similar

is worthwhile. Baxter

to

the previous diagram that further investigation

and Kendall, 1965, Journal of Atmpspheric and

-135-

P. C. Kendall

Terrestrial Physics

~,

129-132 have made quantitative comparisons bet-

ween theory and experiment. Thus, if the lowest of the constant height experimental curves is assumed

to be given together with the equatorial

profile, we can deduce all the other curves using this theory. As the experimental curves are also available this is an interesting comparison between theory and experiment. Thus, at fixed height

,cc.)

N = N (z1

Z

= given function,

where 01.. = latitude . Thus , putting r 1 = r e

+ Z 1 ' where

radius and using (4), (5) gives

giving

We also know that

at

the equator

N = N (z,

0) = given

function.

Thus from (5) g(z) = Nez,

01 If (r)

= N (z, 011f(r

If

follows

that

at a

general

N = f(a)

= z1 (say) we have

e

+ z)

point

g(z)

N(z1,cos- 1

j(r{os2oC./r~NCz,OJ

N[zl ' cos- 1 /(r/r}]

r

e

is the earth's

-136 p. C. Kendall

The results lines

of the investigation are

shown in

Fig. 18 . The solid

are observations, the broken lines are theory .

.. 'eu

• 12

~

¥

,,-ii

..

'Q 8

"i o

~ 4

i '" 20

IS

o

10

·s

5

M.gnelic Lah1ude

10 'N

All we can conclude without. further information on the temperature and composition is that mathematically the distribution N is almost separable in the variables

z

and a.

Effects of electrodynamic drift on the topside ionosphere By expanding in a Journal

of

Atmospheric

power series Baxter, Kendall and

Terrestrial Physics

have studied

the disturbance of

upwards and

outwards

Hall

1965 27 , 1263-1273

this type of diffusive

equilibrium by an

drift. They find that the ionization adjusts

itself so as to compensate for the Hall motion at field lines. The diagrams will

and Windle,

ri~t

angles to the

not be reproduced in these notes.

The ad

hoc methods they used to solve the diffusion equation show that there is room for mathematical work

Equilibrium solutions

and

in this field.

the effects

of electrodynamic drift

Bramley and Peart (1966) and Hanson and Moffett (1966) have both integrated

the equilibrium equations with

terms. The word "equilibrium I is taken

realistic production and loss

to mean

a Iat! 0 .

Thus we take

-137p. C. Kendall

a /a~. 0 in equation

~ = constant and put

(44) • Then, using obvious

abbreviations

av

-

aa 0

= f (x, a ) + f 2 (x, a )V

1

0

0

+f

aV ax

'iv

(x, a ) -+ f (x, a)-----.. 0 3 0 ~ x"

3

where a = a/H o It should be noted that these authors did not use the mathematical transfor-

mations of Lecture 4. The problem ly. The calculation is

is clearly readily tractable numerical-

started from the field line

a =(6550 km'YH o and continued as far as may be required. The boundaries of the system are shown in Fig. 19

dipole field lines (oaconstant) ~~~~~~~~~~~~~Ie~velz-o N X-I X--I S

Fig. 19 The problem in

the a ,x plane is straightforward (Fig. 20) o

I

I I I

0

I

-I

hQ

I

I I

,

~ 00

direction

X--I

h a steplength of numerical process a in 00direction

Figure 20

-

138-

P. C. Kendall

z = 0 , we put V = - f 1(x ' are sket ched in Fig. 21.

On the boundaries (which happen also to be a ) / f ( x, a ) . The o 2 0

results

obtained

Nm<electron density)

•I I

I

I I

.......e uator

ISeN 15·5 latitude Observed (Appleton 1947) (noon)

Theoretical (symmetry)

Figure 21 A drift upwards of about

lum/ s is needed, with typical F2 layer

parameters, to produce a suitable "Appleton anomoly" in the case of symmetry. Hanson

and Moffett find that

a SN wind of

60 m/ s could cause

the observed asymmetry, but do not claim this as explanation.

-

139-

P. C. Kendall

Lecture 6 Time dependent solutions of the

F2 region diffusion equation

(lVl~""

Dr. R. G. Baxter) Using obvious abbreviations, the full diffusion equation of Leoture 4 (equ.44) becomes approximately, if W'H/a c 1, ~V

-

().p

tlV +f(;)-

h

(1) = f (x, a,.p)

1

The operator on

the right

t)V ~2 V + f (x, a,; )V+f 3(x, a,4» --;+ f (x, a, ~ )-2

2

'q

hand side

may

be

x

reduced

4

Ox

by integrating

the equation da = f(+)

d+ Thus,

as

the integral

is

J• d~

a = ao + where a

o

is

a

constant of

+=,

f

o

integration, the required transformation

I

(2)

cia

a'=a-f

o

That

Also

this

,

algorithm works can

-c -;)

~al

f

d~ be

seen

at

once,

for

is

- 140-

P. C. Kendall

Thus t)_!) () -:a-+f-

a~

at' Using

~a

variables x, aI, ~ )

so

that

f.(x,a,~) = f.{x, 1 1

a' +

I+'

fd
+')

i = 1,2, ... 4,

o equat ion

becomes

(1)

(3)

av

This is the form

of equation

~~ I

by numerical

=f + f V +f

1

2

2

~V + f i v

3 ~x

4 ~x 2

dealt with successfully in lecture 3

methods. Mathematically we simply choose a

then integrate

in

4>

I

as

value

of a',

usual, using the Crank-Nicolson method to

advance

X.I~-"""---

..direction of x- O~---· - -------cquator integration Xa-I

4IN' Fig. 22

the integration

from.'

v =The integration in the

, ,

to

~ +S~(Fig. 22) . On

f/f2

is started at

4'

I

the boundaries we put

(boundary condition) = helo"

where

h

+"

is

the step length

~' direction. This a voids difficulty with the singularity at ~ I =0 .

-

141-

P. C. Kendall

VVe can

thus develop the values V

= V(x,

aI

,

,

Hence V = V(x, a,cp) Then the electron

density is N= e- z/ 2H V,

where

z

denotes

the

height

,and

If

is

the constant scale

Physical explanation

N

5

Figure 23

height.

-

142-

P. C. Kendall

Fig. 23 shows

how

the Hall drift

rotation so that a magnetic

~

l(

~/B

2

combines with the earth's

field line moves up and down and also rotates

about the axis of symmetry. We have simply chosen coordinates moving with the field lines. The diagram shows the surface described by a movin~

field line as it passes round the earth.

Processing of results The results appear as x for

a

tables of

given <J>' and a' . This

N, the electron density, against

immediately gives us

4>

a = a'

f d;,

+

f

o

so we then know

which

field r

we are on. Knowing

a,

ture

value

4

gives

one

line 2

= a sin ()

to

each of

value

of

x

equation

(33)

of

lec-

r. Thus we can make the computer print

out x

N

r

()

o 0.1 0.2

(say)

1.0 The problem can thus be readily converted back into cal polar because to

coordinates t = '" ~ ) • As 000

() = 90 , 89 ,88

r, (), cf>

• (of course

a final grace

... and

also

i

4'

the original spheri-

lIeasures

the time,

we can make the computer interpolate = 0, 0.5,

1. 0 ... This enables us to

-143 p. C. Kendall

draw curves Physical

of N at

fixed latitutle or fixed height.

data The measurements

made

of electron density in the ionosphere are

by ionosonde from either the ground, thus dealing with the 'botto-

mside' of the ionosphere, or 'topside'

from

ionosphere. A radio

flected at a point where

a

satellite, thus dealing with

wave travelling into

the ionosphere is re-

the electron density has a given

depends on the wave frequency. Thus, in a layer with

the

value which

a single maximum

of electron density there is a critical frequency f0 F 2 ( in the ordinary mode of propagation) beyond which radio waves pass through without reflect ion. There is another mode of wave

propagation, the extraordinary mode,

with . the critical frequency fEF 2 . [ See

Thus an analysis

using

many ground based stations (1947) (Fig. 24) and

Marty~

Ratcliffe's bOOk]

the critical

gives results

such

frequency as those

f F2, from o of Appleton

(1959) (Fig. 25)

Nm (maximum electron /I density>

N

15° 0°

ISO S

Appleton a~!ftOly (1\ OOll. )

Figure 24

-

144-

P. C. Kendall

Fi~lnJ

N

-------?j , ,

60"

:§

\

2

I

I

30·

\

E

!

1

~5

I

j

I I I I I

D" 30"

I

':::'j\ \

\

_ -2 5-"

GO· .~

\I

" S

I

I

I

\

0

/

,I

3~

----, \

! !t1'

I

-~;--'

{j .2

2.5

I

\

\,

,

,

00

24

12

Local time,

06

hr

World (lUl'vell of f.F2 for the equmoJ:. 8unapot minimum year IH~19« (after MABTYN).

In lecture 5 we

fixed al

height as a

saw some curves (resulting from a satellit<) of N at function

(1964). There are

tables e. g. from

of latitude. These may be found in King et

now many of these

the Canadian

Ottawa. Ground based results

results available as printed

Defence Telecommunications Laboratory, have been given by Croom et al for N

at fixed height.

Results. Figures 1&. and 2'fshow results for typical F2 region parameters and drift amplitudes of 7.3 ment

of the

m/s and 73 m/s respectively. Note the develop-

Appleton anomaly. These results

are only preliminary.

-145 -

..

1 .J

,

to

,

,..

.t

IS

I..~ "'".

If

"

~

to

''''

e)

PIG 26 Nmax world CUMes - bolines of NIDIlX (100/ 'l'~ ..

drift amplitude

= 7.3

m/I

,o~----~----~----~~ ~

11

I'

,"OCAI. 'tlfte

FIG 27 Nmax world

1~

ttR CUrTes

- Isolines of NIDIlX (100"'.1)

drift amplitude .. 73 m/s

-0 N M, *

It-

,

1"1

,..

'lO

tit

PIG 29

Diurnal variation of Nmax /-r,.t

at equater

-146-

p. C.

Kendall

Clearly 73 m/ s is too large, as the trough is too deep. However, the results cio show that Martyn (1947) had a good idea as

to the real cau-

se of the Appleton anomaly . He proposed that Hall drift would transport ionization away from the equator, thus producing the trough.

Near the dip equator, at nal variation

of N

m

has

stations such as

Huancayo

• t he diur-

a double maximum, as sketched in Fig. 28

('bite out' ~---

6 12 18 24 6

6 • 12 18246 sunrISe timeChrs) timcChrs) moderate latitUde Humcayo (say) (sketch) (sketch)

sunrise

Fig. 28 This is reproduced by our theoretical results for a drift amplitude of 73

mls

large

as shown in Figure 29. It shoUld be stressed that 73 m/s is too and that these results a'e only preliminary ~

-

147-

P. C. Kendall

Lecture 7 Motions of the neutral

air

induced by

ion-drag (Nith Mr. W. M. Picker-

ing) The ion-drag problem Cowling (1945), HirOllO (1953), Baker and Martyn (~953),

Hirono and Kitamura (1956) and Dougherty (1961) have suggested

that the motion of the plasma (in the F2 region) could itself set the neutral air in motion. Consider the case of moderate latitude, with stratified atmosphere,

A

as in lecture

a horizontally

2 (see Fig. 30)

A

~1

z V

horizontal

I. dip angle Fig. 30 Let u be

the diffusion velocity ( along a field

line) and v be the spe-

cified Hall drift. Then if the neutral air has achieved zontal velocity as

the ions, v a

But along

a field

the same

= u cos

I - v sin

line the plasma equation

of motion is

~{Pi

2NkT . -V-(u - vacos I) = - sin I But p. = P = NkT ( we assume) and 1 e

+ Pel

az

- Nm g sin I i

H = kT/m.g. Thus 1

hori-

-

148-

P. C. Kendall D aN (N sin I ih

u = - v cot I and if

t

shown,

the continuity equation

is the unit

tangent

aN

Cllong

"n

Band

is the unit normal

becomes ~

it

N 2H

+-)

= Q - L - y.N(u.!

A

+ v~)

1\

/I.

--

-

= Q - L - t.V(Nu) - v n. V N A

But!. = (cos

I, sin I) and

:~ = Q =

A

~

= ( - sin

L - sin I

a

a;:

Q- L -

Thus as D

r, cos

:z

-

I) . Thus

(Nu) - v cos I

t)N D(~

~~

+ 2"Hl N

ex exp (z/H)

we have

c-.*

~N () t

-=Q-L+DcFJN

'

where

fiJ" ~ (~22 ~z

+ _3_ 2H

~ + _1_) ~z

2H2

Looking back to lecture 2 we there obtained ~

We also see

that the

completely caneelled. may thus be serious.

. 2

sm I

electrodynamic

(Hall) drift

The consequences of

term v has been

including neutral air coupling

-

149-

P. C. Kendall

The temperature gradient problem Apart from the problem of ion-drag there is another problem. King and Kohl (1965) have suggested that as there is

a temperature

gradient in the upper atmosphere (see their references) it

might drive a

neutral wind through the pressure gradient. 1ley showed that in the F-region the major force

on the neutrals, apart from this pressure gradient,

is due to their collisions

with

ions. It is clearly of interest to investi-

gate the coupling between the plasma and temperature gradient would certainly set ver, the velocity would be coupled with

neutral

air

motions. A given

the neutral air in motion; howethe ion

velocity; giving rise to

a coupled system of equations. If the ions were at rest, the problem of working out the neutral air velocity v would be relatively simple .The -a

ions

are, however, free to

move along the field lines. In

,he 11 layer,

for example, diffusion is regarded as unimportant and Geisler (1966) has shown how the

problem

of

neutral air motion can be

treated. We he-

re consider the F2 layer where Slipping between the moving plasma and the neutral atmosphere is will

believed to occur. The configuration in Fig. 3

be used. The magnetic field

axis is also

.II

horizontal. The y-axis

of x and y

is taken is

to

be

horizontal. The x-

vertical. All variables are functions

only, being, in particular, independent of lime. The plane

x = 0 is taken

to

be the

equatorial plane,

and there is supposed to be

symmetry about this plane.

y

__--~~?--------x f. equator

Fig. 31

-150 -

p.e.Kendall

Notation n = equilibrium neutral air density a = n exp(-y/H)

o

T

= equilibrium temperature , assumed uniform

H

= scale

height

= kT/m.g 1

m. = mass of ion ( = mass of neutral

atom)

1

Boltzmann constant

k

=

g

= acceleration due to gravity

v = velocity of neutral -a

air

v = horizontal velocity of neutral air al va2 = vertical velocity of neutral air v.

= ion velocity

u

= ion velocity along

v

= Hall drift

-1

a

field line

.l r to a field line

= constant

T' = artifiCially maintained small temperature perturbation D =twice coefficient of

diffusion

n' = perturbation in. neutral a

Q = rate L

=rate

In the model

we

and their effects We also

air

ions through neutrals

density

of production of electrons of loss cf electrons assume

that

all

temperature gradients, velocities

are small. ignore the self viscosity of the neutral air.

.- 151 -

P. C. Kendall

Equations of the problem The horizontal and vertical linearized equations of motion for the neutral air are

~ (v D ai

(1)

- u)

= • ~(n' kT) crX a

-

-?-

(n kT') a

crX

and 2NkT D

~ ~ -v) = - - ( n 'kT)- (n kT') -m,n 'g ay a ely a 1 a

(2)

--(v

The linearized

horizontal equation of motion

(3)

2NkT ~ - - (u-v ) = - (2NkT) D al ax

We

no te

plasma, as

that

a2

there is no

vertical

it moves vertically with

of the plasma is

equation

the given

of Hall

motion of the drift

v,

assumed

assume

that

constant. Adding (i) and (3) gives

Thus n' a where

F is

an n' a

+ 2N+n

T'I T :: F(y)

a

arbitrary ~

0

function

N'" 0

and

of y. We T'-'IIO

giving (4)

Substituting back

n' + 2N + n T'/T = 0 a

a

we

obtain

(5)

v

::

U

D +N

(6)

v :: a2

V

Dn T' D D +_a __ +- ~N +N i)y 2HNT H

ai

aN ~x

shall as

x ... oo ,

-- 152 -

P, C, Kendall

Assuming that N « n , so that production and loss of neutral air a

molecules are negligible, the linearized continuity equation for the neutral air iEO div n v "0 a-a

v from (5) and (6) ,

Thus, substituting for

-a

~

nD d ~ nD~ d nD (_a_~)+ (_a_~)+_(_a_) N ~x N ~y ~y H

Cl

- ( n u)+ax a C1x

+ Whence,

1Y

~ Dn: T' d 1Y (2HNT ) + "iY (n av) " 0

under conditions of symmetry (u " 0 at

x= ('), and assuming

that DCXn We obtain an expression

for the horizontal velocity of the ions, namely,

f

x

u" - D

(7)

-1 a

l.

V log

N dx +

o

o" Q-

a

-

D( o

This enables us to form equation for electrons,

~

the diffusion

~

L + - ND ~x

equation from

the continuity

- L "div (Nv.) , Thus

rX 'V l. log 0

-1

~N

1

~

N dx - v ( - + - -(xN) ) ~ Y H ih

(8)

Thus, even from diffusion

equation

a

very simple model

we

have produced a non-linear

- 153-

P. C. Kendall

3. Conclusion It is

of interest to compare equation

pler diffusion

equation

which might be obtained by making earlier ap-

proximations. For example. if of calculating

we had argued that

pressure gradient might as well be would have

well

the ions might as

v

-a

(8) with the the sim-

be

for

the purpose

assumed stationary. and the be (~ /~ ~n kT'). we a

assumed to

obtained

(9)

Then the

diffusion

equation would

~2N aN~2 Dna T' L + D - - - v - +{--} ~x2 iy~} 2T

o =Q -

(10)

have become

This equation is linear,

anti

has

terms correspondiIg in turn

to

term in (8). It is, nevertheless,

incorrect. We note, however, that

a temperature which

the

give

rise to

terms

transport of plasma

is. higher at

still

between the

away

questions

plasma

alsewhere

would

of the same sign in (7) and (9) ,corresponding to from

We conclude that although there are

equator than

each

and

to

be

neutral

the equator. current F2 layer theory looks promising answered

concerning the coupling

air motions.

-154 -

REFERENCES Appleton. E. V. 1946 Nature •

.!!!.....

691

Baker. W.G. and Martyn. D.F. 1953. Phil. Trans. Roy. Soc. A 246. 281 Bramley. E. N. and Peart. Margaret.

i965a, Nature 206, 1245 and 1965b

J. Atmos. Terr. Phys. 27 , 1201 Cowling, T. G. 1945, Proc. Roy. Soc. A !,E, 453 Croom, Sheila, Robbins, Audrey and Thomas, J, O. 1959 , Nature, Lond.

-

184, 2003 also 185,

-

902 (1960)

Dougherty, J, p. 1961, J. Atmos. Terr. Phys. 20 , 167 Ferraro, V. C. A. and e1zdogan L. 1958, J. Atmos. Terr. Phys.,!!, Geisler, J. E. 1966, J, Atmos, Terr. Phys. 28 •

140

70~

Gliddon. J .. E .. C. 1Y59a, b. Quart. Jour. Mech.~ Appl. Math.,!!, 340 and !,!, 347 Hanson, W.B. and Moffett,R.J. 1966, J.Geophys. Res.!!., 5559 Hirono, M.

1953, J. Geomag. Geoelect.

~,

22

Hirono, M. and Kitamura, T. 1956. J. Geomag. Geoelect. 8

d

9

King, J. W. and Kohl. H. 1965, Nature. 206, 699 King,J.W.-et al. Proc. Roy. Soc. Lond.

(A)~464,

1964

Kunz, K. S. 1957" Numerical Analysis, McGraw Hill Martyn, D. F. Proc. Roy .Soc. Lon!I. (A)

~,

241, 1947

Martyn. D. F. Proc. Inst. Radio Epg. N. Y. , 47, 147. 1959 Moffett, R.J. and Hanson, W. B. 1965, Nature 206, 705 Nawrocki, p. J. and Papa, R. Atmospheric Processes 1961 Geophysics Corporation of America Ratcliffe, J. A. Magnetoionic Theory, Cambridge Ralston, A. and Wilf, H. S. 1964, Mathematical Methods for Digital Computers, Wiley Richtmyer, R. D. 1957 , Difference Methods for Initial Value Problems, Interscience Rishbeth, H. and Barron, D. W.1960 • J. Atmos. Terr. Phys.

!!.

234

Smith. G. D. 1965, Numerical Solution of Partial Differential Equations, Oxford University Press

CENTRO

INTERNAZIONALE MA TEMA TICO ESTIVO (C. I. M. E.)

F. RENIN

"KINETIC EQUATIONS AND BROWNIAN MOTION"

Corso tenuto a Varenna (Como) dal 19 al 27 settembre 1966

KINETIC EQUATIONS AND BROWNIAN MOTION by F. Henin (Universit~ libre de Bruxelles) Introduction. In this series of lectures, we shall deal mainly with the mi-

croscopic theory of brownian motion. Brownian motion owes its

name to

bert Brown, who noticed in 1827 the fact

an

English botanist, Ro-

that small particles suspended in

fluids perform peculiarly erratic movements. The origin of this phenomenon is of course quite simple: we are dealing with molecular

a manifestation of the

motion.

The first satisfactory theory of brownian

motion was produced

by Einstein in 1905 who derived the diffusion equation. This result has been particularly important because the

expression he obtained for the

diffusion coefficient D allowed a determination of Avogadro's number N by Perrin. Thereafter, the phenomenological theory has been widely developed. A very

good presentation of the ideas used can be found in a re-

view paper by Chandrasekhar 1). We

shall give a brief summary of these

ideas i:t chapter I. The starting point is the Langevin duces as basic

assumption the

fact

equation which intro-

that the interactions of the particle

with the medium have a twofold effect; first, an

overall

dynamical fric-

tum, then a fluctuatin.r! force. Intuitive assumptions about the statistical properties of this fluctuating force lead to the Fokker-Planck equation for the time evolution of the probability distribution of finding the particle at a given point in space with a given velocity. One of the most interesting features of the Fokker-Planck equation is that it is an irreversible equation; it predicts an irreversible evolution towards an equilibrium distribution. The stochastic theory has

been widely used and proved success-

ful in the study of a great variety of phenomena. Neverthele , it requires a good deal

of intuition to reach a phenomenological description of the

- 158-

F. Henin

effect of the medium on the

particle. However, intuition can

sometimes

be misleading; therefore, there has been much effort to understand this phenomenon on a in several

microscopic

ways;

it will

level. Such an

effort can

be rewarding

clarify the conditions under, which the phe-

nomenological theory will be valid, it will give us the phenomenological constants ped that for

in terms of molecular parameters and finally, it may be hoit

will

show us

the

a microscopic

ded by the laws

fulfilled.

understanding of the brownian motion on

level necessarily requires the consideration

N-body system. It

cept

follow when the conditions

the validity of the phenomenological theory are not An

must

to

path

is

of an

quite obvious that the detailed description provi-

of mechanics

cannot be used directly and that one

resort to the methods of statistical Il!echanics. A most useful conis

the idea

tistical mechanics

of an

ensemble introduced by Gibbs. In classical sta-

such an

ensemble is characterized by the

N-particle distribution function which

obeys the Liouville equation.

The Liouville equation has been

the starting point for the study

of non equilibrium many-body systems by Prigogine and his coworkers. An extensive presentation of the

basic ideas can be

found in the mono-

graphs by Prigogine 2) , Balescu 3) and Rl!sibois 4) . This method emphasizes strongly the role played

by the correlations

distribution function. We really deal with

a

in the evolution of the

"dynamics

of correlations".

This formalism is particularly well suited to take account of the characteristic features of macroscopic systems: large number

of degrees of

freedom N,

these

large

volume

n,

finite

concentration;

allow the consideration of the asymptotic case: (1)

N ~

00;

n4

00,

N/n

=C

finite

features

-

159-

F. Henin

which brings in

several

important simplifications.

Moreover, in general, we are interested in the asymptotic behavior in time of the behavior

the system. Then , it can be shown that

of the system can

in many cases,

be correctly described by the

led kinetic equations. (A simple example of kinetic equation tzmann

so cal-

is the Bol-

equation for dilute gases) . The derivation of the kinetic equation

for the velocity distribution function will be discussed The kinetic equation is which such

an equation

distribution

which

an

irreversible equation : systems for

holds tend asymptotically to an equilibrium

is a function of the hamiltonian only.

Once we have equations for the description behavior of the

in chaper II.

of the asymptotic

N-body system, we can introduce the special features

of the brownian motion problem. There, we are interested in the motion of a

single particle

in a surrounding fluid. The simplest case will of

course be that of a particle moving in a fluid at equilibrium . This is in

fact the problem which , in microscopic theories, is often referred

to as

the brownian motion or

that the

fluid

is

at equilibrium introduces

in the kinetic

equation: all

All

but one, are in

of them,

the fact that

test particle problem. The assumption

the particles

an enormous

no

simplification

longer play the same role.

the eqUilibrium state

(strictly speaking,

one particle is out of equilibrium prevents the others to

stay in the equilibrium state; however, this departure from the equilibrium state is of order

-1

N

There are two to

and can

be

cases where

neglected). the kinetic equation, particularized

the brownian motion problem, can be shown to lead to

Planck equation. The simplest case which will

a Fokker-

is that of weakly interacting systems

be discussed in chapter III. There,

a Fokker-Planck equation

-

160-

F. Henin

is obtained whatever case

is

that of brownian

range forces where particles ce,

the mass of the brownian particle. The other

the

motion in brownian

systems interacting through short

particle is much heavier that the

of the fluid. This problem, in the absence of any external for-

will be studied in chapter IV . In chapter V, we

it to the case

shall

generalize

where the brownian particle is charged and acted upon by

a constant external electrical field. The interest of all these not

only lie in the fact that they allow

us to state the conditions of va-

lidity of the Fokker-Planck equation. They also

enable us to

expressions

enters into

of

Fokker-Planck they

show

equation

the

diffusion coefficient which

equation

us

when

the

can

chapter

way

obtain

required. This will

be

classical

systems. The same

we

shall show in

in

the work

one special a fluid

be

which class

at thermal

will

of the fluid) . Less

be described in chapters

of brownian motion: that of a parequilibrium (in chapters IV to

VI, the brownian particle is supposed to ticles

briefly discussed in Chapter V.

made with the results of recent ex4 3 the mobility of heavy ions in liquid He and He

VI concerns

ticle moving

the

corrections to the Fokker-Planck

only considered

VI. Here contact can

All III to

to

be extended to quantum mechanics, as

periments on

obtain

in therms of microscopic quantities. Moreover,

So far, we have ideas

problems

be much heavier than the par-

specialized situations could of course be consi-

dered. We could for instance consider the motion

of a

test particle in

not at equilibrium . In this case,

the problem is

a medium

which is

much less

simple; we can

no

longer

obtain

a single closed equatio.l

for the distribution function of the test particle . However, in all systems

-

161-

F. Henin

where the kinetic equation

is asymptotically valid, the basic features

are preserved. The distribution function of the particle will an

irreversible equation. Whenever, in the brownian motion

obey in a fluid

at equilibrium, we can derive a Fokker-Planck equation, the same kind of equation can

be

obtained if the fluid is

out of equilibrium

but

the coefficients appearing in the equation will be functionals of the state pf

the system. A simple example

of this

is

given in chapter III

for the case of weakly coupled systems. More details can be found in a paper by Balescu and

Soulet 5) .

However, there are

systems

where the kinetic equation

val :d, even asymptotically. An important case

is that of systems inte-

racting through gravitational forces. For such systems, new approach

seems necessary. We first

will, in

case, play the same role as

this

systems with

is not

an entirely

have to derive an equation whid the kinetic equation for

short range interactions. In the last chapter (VII) we shall

briefly describe a

recent

attempt by Prigogine and Severne 6) to

obtain such an equation. This equation predicts a behavior which differs in many important aspects from the behavior predicted by the kinetic

-

162-

F.. Renin

REFERENCES

1. S. Chandrasekhar, Revs. Mod. Phys.

~

1 (1943) reprinted in

"Noise

and Stochastic Processes", ed N. Wax, Dover Publications, New York (1954) 2.1. Prigogine, Non equilibrium Statistical Mechanics, Interscience, NewYork (1962) 3. R. Balescu, Statistical Mechanics of Charged Particles, Intersc ien ce, New-York (1963\ 4. P. Resibois, in "Many Particle Physics", ed. E. Meeron, Gordon and Breach (to appear) 5. R. Balescu and 6. I . Prigogine and

Y. Soulet, J. Phys. 26 , 49 (1965) G. Severne, Physi ca (to appear)

-

163-

F. Henin

1. STOCHASTIC THEORY 1.1. Introduction The stochastic theory does not make any attempt to describe in detail the interactions between the brownian particle and the particles of the fluid. Rather, it describes the effect of the medium on the heavy particle in a phenomenological way. From the beginning, one assumes that the influence of the medium on the particle can be

split into

two parts.

First, we have a systematic friction effect. Secondly, we have to account for the random motion. This is done by assuming that the medium exerts a fluctuating force on the particle. It is obvious that this force is not known exactly and that the best thing we can do is to make gues_ ses

about its

statistical properties. The main question will then be: given

the statistical properties of the fluctuating force, what is the probability that, if the brownian particle at t = 0 is at the point r

~o

it will

be at time t

at

the ·point!:.

with velocity

with velocity u , ~o

~?

The assumptions

of the stochastic theory lead to the Fokker-Planck equation for this probability distribution. We shall vin's equation of

first discuss the assumptions which motion for

the heavy particle

ke some further assumptions about tuating force (

~

3\ which will

distribution in

velocity

We shall then show how tion can be reduced

to

space

2) . Then we shall ma-

the statistical properties of the fluc-

enable us to

(&

(~

lead to the Lange-

write down the probability

4) .

the problem

of finding this distribution func-

the solution of a differential equation

the Fokker-Planck equation in velocity

(§

5) ,

space. The Fokker-Planck equation

for the complete distribution function in phase space, with or without an

- 164-

F. Henin

external field acting on

the particle, will then be obtained by means of

an easy generalization of the previous problem (

§ 6)

. Finally, we shall

consider the case of an inhomogeneous system where the density gradient is small
of the mean free path. Then, we

shall see that for times much longer than the relaxation time, the spa-tial distribution obeys a diffusion equation. All this discussion will follow quite closely

the excellent

review paper by Chandrasekhar 1) . An extensive bibliography can be found there.

L

2. Langevin equation The first step

quation of motion for

in the stochastic theory is

to write do wn an e-

the heavy particle. From the beginning , one assu-

mes that the influence of the medium

leads:

1. to a systematic slowing down effect ; the friction coefficient assumed to

particle of mass

M

(1. 2. 1)

~= 'YJ

is

be independent of the velocity of the heavy particle . Usual-

ly, one also assumes that it

where

~

is

radius a, 6T

motion

we

then

have:

a"J

M

the viscosity

2. to the random assume that,

and

is given by Stokes I law. For a sperical

of the fluid.

of the particle;

to

account for

this,

we

besides the dynamical friction, the medium exerts a

a fluctuating force

~

(t) on

the particle. This fluctuating force is as-

sumed to depend only on the time t. It

is

of

course not known hut

-

165-

F. Renin

plausible assumtions can be made about its Comparison with validity of these If we now

statistical properties

experiment will have to decide a posteriori of the assumptions.

suppose that these two effects are additive, the motion

of a brownian particle in

the absence of an external field of force is

given by the Langevin equation: dJ,L

(1. 2. 2)

dt where

~

is

the velocity of the particle

(1. 2.3)

u =

dt

! being its position. If an

its

effect

external

has

to

be

that (1. 2.1) has to be

field

force!S (£, t) acts

of

included in replaced

on

the particle,

the equat ion

of motion. This means

+~K ~ (r

(t)

by

du

(1. 2.4)

..:::. = - {\ u dt jr ~

We may notice an

, t)

important feature of the Langevin

The motion of the particle at

time

t

is

entirely

motion at previous times. Whatever happened to does not matter to corresponds

to

+A ~

independent

of its

the particle in the past

determine its future behavior at

the assumption that

equation.

ttdt. This clearly

the collisions between the brownian

particle and the particles of the fluid are instantaneous.

-

166-

F, Henin

I. 3, S tat i s tic alp r 0 per tie s d the fl u c t u a tin g for c e Our next problem of the fluctuating force

is

~(t).

now to specify Of course,

nomenological theory, this amounts number of a priori

in

to

the framework of a phe-

the introduction of a certain

assumptions, These

a very intuitive feeling

the statistical properties

assumptions will

of the phenomenon

be

based on

of brownian motion. Their

justification and limitations certainly require a description on a microscopic level

of the whole system,

First of all, me for which than

we

the variation of

know

empirically that the characteristic ti-

the macroscopic quantities

(i. e. the quantities

we measure, as for instance the mean velocity) is

much longer

the time interval between two successive collisions of the Brownian

particle with particles of the fluid (which sec, in a normal

liquid), Therefore,

always find time

intervals

pic quantities

change

by

<~(t + (1.3,1)

a

~

and

(t)

we

such

negligible

of the order of 10

shall

-21

assume that we -can

that during

6 t

macrosco-

amount

o.t)) - <~(t) <'~ (t)

while ~

~ t

is

«< 1

(t) undergoes a large number of fluctuations , such that

~(t+

6 t)

are completely uncorrelated. This assumption is quite reasona-

ble if we take into account the fact that the brownian particle is much heavier than the particles of the sunounding fluid. Then, during the collisions with the fluid particles, the ,elo, ily of the brownian particle changes by a very small amount. During red by

fl. t, the net acceleration suffe-

the brownian particle because of the action of the fluctuating

force will

be :

-

167-

F. Henin

(1. 3.2)

We assume that this net

/.l

time interval to

t

and

not on

acceleration depends only on the

the time t

at

which

we start

compute it, i, e, we again neglect memory effects, We

shall

occurence of

Il. t is

now

make an assumption about the probability of

different values for

due

to

the

~

A. t) . The

superposition of a

net acceleration in

large number of random

accelerations. This is very much analogous to the situation one encounters when

discussing random

flight problems. There, one looks for the

distribution function of the increment of a

particle

which

has

/:!). ~

during

At in

performed a large number of random steps.

If each displacement is governed by a probability distribution which is

the pos ition

spherically symmetric, one

~ (I r\ 2 )

shows that : (see appendix J)

(1. 3.3)

where D is the diffusion coefficient which depends on the average length of the step and on the time interval between steps (see A. 1.1. 17)Using the analogy between these problems, we shall assume that the probability distribution for

~(At)

W[,12 ( (l t)1 = (h qb t)-3/2 expl- ~

(1. 3.4)

where q

is given by :

is

~( ~ t)\ 2/ 4qAt1

a constant . The specification of this constant requires so-

me additional assumptions about the equilibrium properties of the velocity distribution function (see

~

4)

-

168-

F. Henin

1.4. - Velocity distribution function. Before dealing with the general plete probability

at

r

with

r , u , we shall

....... 0

""0

shall try to find the probability locity

u

if

at time t The formal

e-

-0

Both sides

consider

of

l ""wO r ,

u , 0)

""0

~

at time t

a

simpler problem. We

given the ini-

W~, t \ ~o ) that the particle has a ve-

initial velocity is the

u .

~o

Langevin's equation is :

~t = e - Pt (

t

e~~ ~ ( ~ ) d ~

L' e(\~ ~( ~)

= e -ft

CI.

we may also

velocity

....... '"

have the same probability distribution. Now, if

of (1. 4.1)

(1.4.5)

d

~

write:

~ =

(1. 4. 3)

if

its

solution

-u - u

(1.4.1)

W(r , u, t

distribution in phase space

to find the particle tial condition

problem of finding the com-

we divide the interval

N

L

j=l

eXPl-r(t-jdt~~(At)=

(0, t) into

of the kind defined above (i. e. such fluctuations while all

N intervals that

A

II t

N

L

j=l

~j

where b. t is

suffers a large number of

other quantities , such as

e - ,.,t, remain practically

constant) . With our of

~(

obtain

At) , i. e. of the

assumption (1. 3.4) about

d..." we ..... J

the probability distribution

can, using the theory of random flights,

distribution function

of

~

(see appendix A . 1. 2) :

-

169-

F. Henin

W(~) =

(411' f t q

dt e 2

'>

0

(~~t)1~3/2

(1. 4. 4)

Therefore, the velocity distribution function is :

(1. 4. 5)

x For long times

exp \ -

1), we obtain asymptotically:

(~t»

W(u, t ~ oo;u ):: (211' q/A)

(1. 4. 6)

--

..... 0

Therefore, we have an

-3/2

exp

(-/h

2

/2q)

"

irreversible evolution towards a gaussian

distri-

bution, independent of u . The system has forgotten its initial condition . ..... 0

A priori, nothing implies that this asymptotic distribution is the MaxwellBoltzmann equilibrium distribution . If we add this condition as a further requirement, we must choose the diffusion coefficient in

velocity space

to be: (I. 4. 7)

q = kT

Therefore,

with

the

~

/M

following set of assumptions

1. Langevin equation 2. characteristic

time

for the variation of

that the characteristic time

~(t)

much smaller

for the variation of macroscopic

quantities 3. net acceleration between t only

and

t+

At

depends

on II t

-

170-

F. Henin

4. distribution function for the net acceleration during

11 t

is gaussian 5. asymptotic distribution for the velocity

is Maxwell-Boltz-

mann distribution the distribution function in is completelY

velocity space for the brownian particle is

determined by (I. 4. 5)

and (1. 4. 7)

1.5. - Fokker-Planck equation in velocity space. So far, we

have

obtained

the probabil ity distribution function

corresponding to a well defined initial condition: u

~o

at t=O, the velocity is

. To do this, we have introduced quite specific assumptions about the

statistical properties of the fluctuating force. We shall

now show that

the problem of finding the distribution function can

reduced to

solution

of

a

partial

tion. In

fact, this method will require less

about the properties as

differential

equation,

be

the

the Fo kker-Planck equarestrictive

assumptions

of the fluctuating force. When the same assumptions

above are made, the general Fokker-Planck equation takes a simple

form

and its solution reduces

this method

to (I. 4. 5) . Another interesting feature of

is that when further restrictions on the problem are impo-

sed, they can be expressed as

boundary conditions for the solution of the

Fokker-Planck

this equation will appear as

equation. Also,

adequate tool for

the

comparison

with

the most

the results of the microscopic

theory. Again , we assume the existence of time intervals macroscopic quantities whereas

do not

Il t

such that

vary very much during these time int ervals

tte fluctuating force has

changed several times.

If we consider brownian motion as a Markoff process,

i. e. if we

-- 171 -

F. Henin

assume that the course

which

the brownian particle will

take is enti-

rely independent of its past history, we expect that the probability distribution function W(~, t

+ A t) will satisfy the following integral

equation:

W(!C.I+ At! " }(A!C)W(!C - A!C. I)

(1.5. I) where se

i'(~;A~) is the transition probability for a velocity increa-

~~

in

At.

Let us and

't'(~ -A~; A~)

the

now

expand

integrand

in

the

With the notation : (I. 5. 3 )


this equation can

be

Taking into account

rhs

Ihs in

in a

a

power

series

po'Wer series

of

of

/J. t

A~

Id(.6 ~.(j'(~ ;A~)

rewritten:

~~ At (I. 5.4 )

the

+ 0 (At)2 = -

~~i [w
i) ]

+

~ ~U,~~Uj [w
fact that in

matic effects are accounted

for in

the

Langevin

equation, all syste-

the friction term and that the flucua-

172

F. Renin

ting force

is

random, we have:

<J

'oAt

(1. 5.5)


Oot

dop ..

d'C1.('C)=

10 At

d orr A. 1

0

(~)

A.('t r)' J '/

(1. 5.6)

Therefore, if we

b.

take

the

limit

Fokker-Planck equation

for

the velocity distribution function:

~w = _

~t

(1.5.7)

-L{ ")u.

1

(A u) _ At

From the Langevin equation, we

If we

further

ration

~(At)

the

w'fJ

+

obtain the general

J ~ ,,2 f
1

2 'lu.i)u. 1 J

At

have:

A~ = - ~ ~ II t

(1.5.8)

t -i'D, we

+ ~ ( A t)

assume that the probability distribution for the net acceledue to the fluctuating force is given by (1. 3. 4)

and that

asymptotic distribution must be the Maxwell-Boltzmann distribution

the transition probability becomes :

t(~ ;A~) =(41rPkT At/M)-3/2 (1. 5. 9)

X

exp [- M ,

A~ + ~ ~ A t \ 2 /

Then, we have:

(A u)= (1.5.10)

and

)(

- pUi

(A u.1 6 u.) = J

4

~ kT 6 t

At

(2 A kT/M)

1-

1

S..J 1,

+ 0 ( A t)2

we obtain the special form of the Fokker-Planck equation:

- 173-

F. Henin

+!!

(1.5.11)

~2wl 2

M '}

u,

1

One verifies easily that its

fundamental

solution, i. e. the solution which

reduces at t=O to a

delta function

:

(1.5.12)

W(u, 0; u ) =

S(u -u

is

.......

given

by (1. 4. 5)

........0

. . . ""'0

. From it , it

solution corresponding to an

is

)

of course trivial to derive the

arbitrary initial distribution.

1. 6. -Fokker-Planck equation in phase space. The above procedtlre can be the complete distribution function (1.5.1),

we

generalized to find an equation for

Wet,~,

t) in phase

space. Instead of

have now:

HW(!:.- 6 !:., ~ - 4~,

Wet,~, t +At) = (1.6.1)

d(£1 £

t) f(!:. - A!:.,

~ - A~; a:., 6~)

) d (A!!)

From the Langevin equation (we directly' consider the case where an external force

is

present

), we obtain:

Ilr

= u

At

(1.6.2) Therefore, we have : (1. 6. 3)

We shall (1. 5.9),

take for in

into account

the transition probability '" (~ ; A~) the assumption

wh ich we

add

a term

-K A t in

the exponential to take

the effect of the external field. This assumption will lead

-

174-

F. Henin

us

to

a

generalization for the complete phase space of equation (1.

5. 11) . If we do

not

follow the arguments lization of

make of

(

Il t, t:.

~W +

u

~t

the

,

~

previous

';)W

i

A

r

and obtain a genera-

and expanding again

i)r.

)Ot+0(6t)2 = -

1

a

power series

computes

t ..., 0,

+

_---:-_--=1_

1u.

1

~u. 't)u.

J

1

If one

in

~W
2 ~ W(flu. Au.) 1 J+

1 2

II

paragraph

we can

one obtains

(1.6.4)

limit

t (~; d~) ,

special cho'ice for

(1. 5. 7) .

Integrating over of

a

one

the

various

obtains

averages

and

the Fokker-Planck

then

takes

the

equation:

";)W +u ')t

(I; 6. 5)

1\ ,.

~W_ + [_

1.7. - Diffusion

Let we

have

us

a

u

') u.

(kT/M)

"\.2) _,,_W_ "

uu.

1

2

1

equation.

now consider

certain

i

a

spatially inhomogeneous

number n of

brownian

system

in which

particles. We assume that

the dilution is such that we may neglect all interactions between these particles. Therefore, the probability distribution function W(lr\ ,I~J, t} factorizes into

(1.7.1)

a

product

W

of

n

(\r\, \~\

factors:

n ,t)=TI W i=l i

(!:., u.,t) 1

~1

-

175-

F. Henin

where

W. is

the one particle

distribution function which satisfies the

1

Fokker-Planck We

equation.

shall

In such a have collisions

also

assume that the density

system, we

with

gradient is small.

have two kinds of processes : first, we

the particles of the surrounding fluid which insure

that the velocity distribution function approaches the Maxwellian distribution ; next, we

have

tial uniformization

a diffusion of

of the system. The time scale for the first process is

1\ -1

given by the relaxation time me scale for interested in

the particles which will lead to spa-

,"

; this is

the diffusion process. As times long with

respect

much

smaller than

a consequence,

1 "'1 "'1

= n.(r., t) (M../21r kT) 1 -1

(I. 7. 2)

3/2

1

+ ~ W (r., i

if we are

to the relaxation time, we may

expect that the distribution function for one particle will

W.(r., u., t)

the ti-

"'1

be of the form:

e

2 -MiUi /2kT

u .. t)

-1

The first term describes

the local equilibrium distribution which is

reached

longer than

for times much

The second term

is a

small

existence of the diffusion

the relaxation time.

correction which takes

process~

it

into

account the

is of the order of the density

gradient. We shall ction

now

show that, under

n .(r .' t) obeys a diffusjon 1 ..... 1

If we integrate the

locity, we

equation.

Fokker-Planck equation (I. 6. 5) over the

obtain )(du. W.

(1.7.3)

these circumstances, the fun-

)

..... 1

~t

1

+

~ ,,1:.i

.{dll1·tI..w. =0 } ........ 1

ve-

- 176-

F. Henin

If we first multiply both sides

by

uil( (II(

"X •

y, z) and then

integrate over

~'dU-1.W.1 u.Ie

';)

(1.7.4)

') t

If we combine

of the Fokker- Planck equation

+- f)r . -1

the velocity, we get:

J .

du u W u = .... i-i i ie

~ Jdu-i

wu

i iC

these two equations, we obtain:

(1 7.5)

Now , using

(1. 7.2)

obtain the diffusion

and keeping only lowest order terms, we easily, equation: )n.(r .. t) 1 -1

(1. 7.6)

')t

D v2

r.

-1

n.(r. t) 1 -1,

with D = kT/ ~ M

(I. 7.7)

The density of the particles at a given point

~

of

space will

be : (I. 7.8)

Again, if

(1.7.9) we verify (1.7.10)

we

keep

only lowest order

terms,

1. e. if we

n.(x, t) 1-

easily that this also obeys the diffusion equation

take

-177-

F. Henin

Appendix I. 1 - Proof

of (1. 3. 3)

In the problem of random flights, one considers a particle which performs a sequence direction

of

all

of steps r .... r .... The magnitude and .........1

the different

ones. One chooses a priori a the probability

distribution that

a given

1

~l

step r. lies between rand .......... i

........ 1

to find the p;robability W( A~; .6 t) that

~l-

the particle has travelled a distance b. ~ in We

shall

one dimensional

give

a priori

fore,

if the particle is be

at

a

random

equal

will

"C .(r .) which gives

distribution function

r. + dr .. The problem is then

~l

""" 1

steps are independent of the preceding

proof

of

the time

walk with all

steps of the same length and with

probability for a step to the left or to the right. Thereat the origin

the point m

after

at

N steps

t=O, the probability that it (-N.::: m'::: N) , is given by:

W(m,N)=21 W' (m-1,N-1)+21 W(m+1,N-I)

(ALI.2)

W(l, 1)

= W(-I, 1) = 1/2

Using Fourier transforms: +00

r::=

(AI. 1.. 3)

m = -

from

(AI. 1.1)

and

W(m, N)e -ilm 00

(Al.l. 2) : N

(AI. 1. 4)

p (t) 1

"COS

1

Therefore,

(ALL 5)

A t.

(1. 3. 3) for the simple case of

(AI. 1. 1)

we obtain

interval

P N(l) = (cos 1)

N

>1

(N)l)

- 178-

F. Henin

and hence inverting

(AI. 1. 3) :

= (2 TT)

W(m, N)

(AI. 1;6)

r

-1

(coslt e ilm dl

-If

Now, with (AI. 1. 7)

(cosl)

N

-N =2

N

r:.

N! p!(N-p)!

p=O

e

-1(N-2p)~

we obtain easily: W(m, N)

(AI. 1. 9)

The last factor v'lnishes W(m, N) = 0

-N

=2

N

Lp=O

p!(N-p) !

sin T (N-m- 2p )) 1f(N-m-2p)

unless N-rn-2p = O. Therefore:

if N even and m

odd or vice versa (AI. 1.9)

W(m, N) = 2- N N!\[(N-m)/2) !~N+m)/21! \ -1 if both Nand m even or The first

result is

of

course

odd

(AI. 1. 10)

obvious. The second could have been

obtained using combinatorial analysis. However, the nlethod involving Fourier transforms can be generalized to more complicated problems and although exact results for arbitrary values of N cannot always be obtained, expressions such

as

(AI. 1. 6) are often useful to obtain an asymptotic

result. For

our

present problem, for N4 <Xl and

m

finite, using

Stirling's formula: (AI. l. 11) we obtain:

1 1 log n ~ = (n +"2) log n - n + '2 log

2'D'

(n ~ <Xl)

-- 179 -

F. Renin

log

W(m, N) = (N

l I N log N - 2" (N + m + 1) log

+ '2

)

(2

+ m)

1 N 1 - 2(N -m + 1) log (2" - m) - 2" log 21f - Nlog 2

(AI. 1. 12)

::::e log (2/N'I') 1/2 _ m 2 /2N and

hence the asymptotic

(AI. 1. 13)

each

If

two

expression:

step has

steps,

a length

introducing

1 and

= ml

x

the

W(x, A t)4x that

probability

is

~

the time

lapse between

At = N"l;

+ A x after A t is :( Am W(x,

if

2 exp (-m /2N)

the variables:

(AI. 1.14)

x

1/2

W(m, N) ~ (2/lfN)

=

the particle lies

between

x

and

Axil) :

btl Ax =

C

W(m, N)

mt. Am = (1/2)

(AI. 1. 15)

W(xll, A tiT:)

L::.. 1

mE Am = (1121) W(x/1"

where

the

factor

(1/2) takes into

Atl T )

account the fact that for N given

(odd or even), only one half of the values of m are odd or

contribute (those which

even).

Therefore, we obtain: (ALL 16)W(x, At) = (HJ D ~ t)

-1/2

exp (-x

with (AI. 1. 17)

2 D=J/2"C

2

I

4DAt)

- 180-

F. Henin

(AI. 1;16) is nothing else

than

(I. 3. 3)

for this

simple onedimensio-

nal problem. This asymptotic formula can be generalized for several three dimensional random flight problems. For instance, for a gaussian th probability distribution for th~ j step: 2 -3/2 2 2T .(r.) = (21f 1. /3) exp (-3 \ r.\ /21.J

(AI. 1. 18)

J ~J

J

"'J

J

one obtains: (A.I.1.19) with

N

<12)= N -1

(AI. 1. 20)

L.

12

j=1

j

The same expression is obtained if is identical

for each

step

is the average displacement of the

index

the probability distribution 2 spherically symmetric. Then (.1 )

and in

each

(the

1. ·s are independent J

j).

Appendix

I. 2. - Proof

We want

the probability

-

(AI. 2. 1)

step

N

L

j=1

of

(1. 4. 4)

distribution

of the quantity

~. -J

with (AI. 2. 2) when the distribution

function for

is again

flight

a

probability

random

distribution

for

~

( 0. t) is

given

by (I. 3.4) . This

problem, the steps being the each

c:Jt is -j

a gaUSSian and corresponds

- 181 -

F. Henin

to the function

"t j given

in (AI. I. 18) with :

.r J,2 = 6q WT J~

(AI. 2. 3)

Therefore, for a

large

6t

number of steps , using

(AI. I. 19) , we have

(AI. 2. 4)

With

Ji ~

(AI.2.5)

2 -2 A. t I, = 6q Ate ,..

J=1

If we

use

(1. 4. 3) ,we

J

the same

N I:.

A. A

e 2 ,- j

t

j=1

approximation

that

led

us

from (I. 4. 2) to

may write:

2

(AI. 2. 6)

3

Inserting (AI. 2. 6)

into

(AI. 2. 4)

, we readily obtain (I. 4. 4)

References 1.

S. Chandrasekhar: Revs. Mod. Phys.

~

J

1 (1943)

reprinted in : Noise and Stochastic Processes, ed. Nelson Wax, Dover Publications Inc. N. Y. 19, New-York (1954)

- 182-

F. Henin

II. KINETIC

II. 1

~

I n t rod u c t ion.

In this tion of an

EQUATIONS

chapter, we

N~body

shall consider the microscopic

system from

the point of view of statistical mecha-

nics. We shall restrict ourselves to geneous

classical systems which ar!,! homo-

in space, although the formalism can

be

extended to include

quantum systems and inhomogeneous situations, As this will most

useful for us, we

descrip~

shall

consider the case

be the

of a gas interacting

through binary central forces. On the microscopic level, a complete description

of the system is given

by its hamiltonian, i. e. here:

2

(II. 1. 1)

~+ 2m,

H=

vij

J

(Iq, - q.\) = H +~V "'l ....J 0

,th , m, is the mass of the J partIcle J tion and momentum. ~ is a dimensionless

a, and -
where

Once we evolution

of the

have

However, for is

not

hamiltonian and

system is

Hamilton IS equations

trons

the

very

of

of course

the initial

completely determined by

a large

system, a set

practical. Moreover, we

of

use the idea

never have, even

of

ce. We imagine a large number tonian but differing by their

6N

differential

equa~

can only measure a few

information about the positions and momenta of all

large set

conditions, the

motion.

macroscopic quantities and we

fore, we shall

p, its posi"'J constant.

at t

= 0 , a detailed

the particles.

There~

a representative ensemble in phase spaof similar systems, with the same hamilM

initial

states. If we take a sufficiently

of equivalent systems, the ensemble will be characterized by

a continuous

density in phase space

f (\~ \ ,is. \' t) . As

all points in

- 183-

F. Henin

the ensemble move ction

f

satisfies

in time according to Hamilton's equations, the funthe

Liouville equation:

~ = (H ,,1 = - iL f

(11.1. 2) where [H ,

1

p

is

the

Poisson

bracket

and, hence, the Liouville operator

L

of the Hamiltonian

and

P'

is

(II. I. 3)

From this equation, one verifies

easily that :

(II. 1. 4)

If

we

choose

this

normalization constant to

be equal to unity:

(11.1.5)

then, a

f (\ g,\, \2,\.

representative

t~dl!.d!l. \

point

in

N is

the probability of

the volume

element

findin~

\ dg, d!l, \

at time

t

of phase

space, A basic postulate in

statistical mechanics

is

that

all macros-

copic quantities may be computed by taking the average value of the corresponding microscopic dynamical quantity over

the distribution function

of a suitable ensemble : (11.1.6)

This description has behavior is (II. 1. 2) .

the advantage that

given by a single

the whole mechanical

linear equation ,the Liouville equation

-

184-

F. Henin

To the decomposition (II. 1. 1) of the hamiltonian into un unperturbed part H

raction) corresponds

a

(II. 1.7)

L=L

This

feature, as

equation (II. 1.2) will

,\ V (inte-

(kinetic term) and a perturbed part

o

similar decomposition of the Liouville operator:

well

o

+ ~iL

as

the strong

.,

and the Schrodinger

analogy between

equation in

the Liouville

quantum mechanics

enable us to develop easily a perturbation technique to study the

time evolution of the distribution function. In centrate ourselves on

this chapter, we shall

con-

the evolution of the velocity distribution function:

(II. 1. 8) Essentially, we shall solve formally the Liouville equation formal solution as of

a power series of the

and write the

perturbation. The introduction

a diagram technique to represent the various contributions will

us to

rearrange

the terms and to

write the equation

enable

of evolution in a

form suitable for further discussions:

(II.).9)

First, we have a non-markovian term which relates at an

earlier time t' . G(t) is

of the collisions which

occur

an in

feY) to

operator which describes

interval tion of

that

of the first con-

the collisions last over a finite time

l' The second term gives the contribution to the evolucol existence of initial correlations in the ott) due to the

"C

r

the fact

the effect

the system on the evolution of the ve-

locity distribution function. The non-markovian character tribution is due to

its value

·- 185-

F. Henin

system , these being described by We shall then short range forces at

show

and

such

that the initial

N ~ ao,

(0 : volume

of the

(0).

that, for systems interacting through

t=O are over molecular distances,

(II. 1. 10)

r:L}k

the functions

correlations present

in the limit of a

0 ~oc',

N/O

large system;

= C finite

system)

and for long times (II. 1.11)

t

the second

f

0 (t)

term

satisfies

pseudomarkovian

in the rhs

~f ')t

'\' (z)

is

its

derivatives

for

(II. 1. 9)

may be neglected and that

which may be written

in

a

form: . 0 1---

'I' (0)

and

of

a closed equation,

(II. 1. 12) where

» 't' colI

the Laplace limit when

= 0 ,\,(0) transform

z ~O. 0 is

z ~ 0 and takes into

f

0

of the collision a functional account

of

operator

f

G{t)

and its

the finite duration of

the collision. In this chapter,

equation

Wf'

shall

show

in detail how the kinetic

(II. I. 12) can be derived. We shall the n sketch briefly how

the same formalism can be extended to discuss the evolution of space correlations in the system. We shall also indicate the necessary modifications when

an external force is present . The equations so obtained will

be our basic tools for the next chapters. We shall be able to give here only a very short outline of the theory. More details, as well as references,to the original papers can

-186 -

F. Henin be found in the monographs by Prigogine 1)

Balescu 2) and

R~sibois 3).

II. 2. Fourier analysis of the distr-ibution function. Let us expand the distribution function in a Fourier series with respect to

the position variables:

(II. 2.1)

The factor

g-N is introduced to

fl e 9.1 d d

(II. 2.2)

N

allow the normalization of

f

to

unity:

ret) "ft
The formal expansion (IL 2.1) is very interesting. Indeed, it is easily verified that the Fourier coefficients

r

k

k

have a very sim-

-1"'-N

pIe physical

meaning. First of all,

we notice that, in a system which

is homogeneous in space, i. e. such that the distribution function is invQriant with respect to

space translations :

(II. 2. 3) only those coefficients such that:

~

(II. 2.4 )

i=1

are different from (II. 2. 4) is

k =0 -i

zero. Therefore, Fourier coefficients such that

not fulfilled are closely connected with the existence of spatial

inhomogeneit~s

in the system.

-

187-

F. Renin

Also, we shall,most of the time,be interested in the average value of microscopic quantities which depend only on a small, finite number s of degrees of freedom. To compute these, all we need are the reduced distribution functions: (II. 2, 5) In such reduced distribution functions, the only Fourier coefficients

which play a role are obviously those which have at most s wave vectors a'ifferent from zero, One of the most important coefficients is that w:th all wave vectors equal to zero. It is the velocity distribution function:

Po({gJ ,t) ={td~ \ Nf( tg,t,t~\, t)

(II. 2. 6)

.

From (II. 2. 2), we notice that this function is normalized to umty. To find out the meaning of the other Fourier coefficients, let us consider for instance the average rlens ity,


(II.2.7) =

N/n[l +

In this way, we see that different from mean density

L ~

ei}t

·;It({dg,\ Np (\g,i, J' i '\2.\

t)l

the Fourier coefficients with one wave vector

zero are connected with the local deviations from the

N/ n ,

As another example, let us consider the binary correlation function

- 188-

F. Henin

,)~(x'- -q-JJ\fdP ,q ,p ,p ,t) J -1,dp-J,f2(q-1-J-1-J

S

g(x, x', t) =rfdq ,dq, (x -q - ,.1' -1 -J - -1 1J

-JdD ,f 1(q ,p, t) (dP, f 1(q ,p, "'1 -1-1 J -J -J-J

(II. 2. 8)

t)}

In an homogeneous system, this reduces to :

(II. 2. 9)

Correlations among s

particles

ficients with at most the spectrum of

~

therefore depend on the Fourier coef-

s indices different from zero. For a large system, becomes continuous and the rhs of (II. 2. 9) vanishes

forl~-~'I-3JCO if r~, _~ is sufficiently regular.

Another interesting feature of (II. 2. 1) is that this is in fact an expansion in

terms

of the eigenfunctions of the unperturbed Liouville

operator. Indeed, from

(II. 1. 7), (ILL 3) and (II. 1. 1), we have:

(II. 2.10)

=--i

If we

L

o

1: ~ . ~ ,m,

J

J

')~.

J

use the same notation for eigenfunctions as in quantum mechanics:

(II. 2. 11)

we have (II. 2. 12)

-

189-

F. Renin

These eigenfunctions are orthogonal

and normalized to

unity:

(II. 2. 13)

From these properties I the time dependence of the Fourier coefficients

r\

~, in a system of non

interacting particles (~= 0) is easily found:

When the particles are interacting , the time dependence of the is

of

course

much

more

tial factor corresponding to a further (II. 2.14)

complex. Besides

ft

~~t)

the oscillating exponen-

the free propagation of the particles, we havi

time dependence in the coefficients

r

l~\( \ ~\) in the rhs of

because of the collisions occuring in the system.

II. 3. For mal sol uti 0 n

0

f the

r.: i 0 u v i 11 e

e qua t ion.

Resolvent operator The formal solution

of

the Liouville equation is of course very

easily written: (II. 3.1)

From this, we bution function :

=e

obtain for

-iLt

f(O)

the various

Fourier coefficients of the distri-

-- 190-

F. Henin

l-iLt exp ( -i ~ ~ k.,q. e . "-J -J J

(II. 3. 2)

As L elements in

is

an

operator in

the complete phase space,

It

the matrix

of (II. 3. 2) are still operators in velocity space. -iLt The operator e can be expanded formally in a power

series

the rhs

of the interaction:

=e

e -iLt

-iL t

-IX

0

ft

dt e 1

o (II. 3. 3)

+ (-I).)

2

ft 0

dt 1

It

-iL (t-t ) -iL t 0 1 SL e 0 1

01

dt 2 e

-iL (t-t ) -iL (t -t) -LL t 0 1 ~L e 0 1 2 ~L e 02

+ ... This equality Is of both sides

most

of

easily verified if one takes the time derivative

(II. 3. 3) •

However, the behavior of the system can be discussed much more easily -ILt if, rather than the operator e ,one considers its Laplace transform, the resolvent operatoF

(II. 3_ 4)

R(z) =

R(z):

JOO

-.)0

dt e

izt

e

-ILt

1

+

~ (z~S )

From (II. 3, 2) and (II. 3,4) , we obtain after an inverse Laplace transform:

-

191-

F. Henin

where the contour C is a parallel to the real axis in the upper half plane, above all singularities of the integrand. (11.3.5) can be easily verified for a finite system. Indeed, then, the operator L is a hermitian operator and

all its eigenvalues, although unknown, are real. For an

infinite system (Q -!I) 00) , the properties of the Liouville operator are not known. We shall however assume that (II. 3.5) remains valid when we perform the limiting procedure (II. 1. 10) . The resolvent operator can be expanded in a power series

of the

perturbation (II. 3. 6)

R(z)

This result. can

=

00

r:.

n=O

lIn

(~L;-:--L)

z _L 0

0

also be obtained

from (II. 3.3) through a Laplace tran-

sform, using the convolution theorem. (II. 3. 5) and (II. 3. 6) will

be our

basic equations for the following

discussion. Of course this means that we assume that perturbation theory up to an infinite order is valid. Whether this is true or not is an unanswered question and we shall not discuss it. The unperturbed resolvent ore rator is diagonal in the ,\~ representation. Its matrix elements are very simple:

<\~t I Ro(z) \ \ ~J) ::
z _lL

o

(II, 3, 7)

=

1

z- Tk .. v.

~""J -J J

where (II. 3, 8)

v.=p./m. -J .... J J

\\~tl

l)

-

192-

F. Henin

is the velocity of the As to

the operator

(II. 3. 9)

If

- i

.th . J partIcle.

SL,

SL = 'L-

we obtain from (II. 1. 7), (11.1. 3) and (II.I.I):

~V..

~

';jV ..

+ ---..!l

(---2l. i <j 4 Sli . j) £.i

we expand the potential

in

C)~j

~

.~)

".E.J

a Fourier series

(II. 3. 10)

we have:

-i

SL = (8V 3i/O) L. Lk i<j ik

(II. 3. 11) )t.

It is easy to verify that the

e

Vk

~. (~D.

"--1

-

~D.) II

)(

"--J

• (q . -q .l -1 -J

only non vanishing matrix elements are tho-

se where the initial and final states have only two different wave vectors, the total wave vector being conserved: ( k ... k .... k ..... k N -1 -1 -J -

I ~Llkl···kl -

....

-1

k~ ••• 'N k ~ :/

'J

(II. 3.12)

The fact that only two wave vectors are modified and that the total wave vector is conserved is of course due to our choice of binary central forces. An interesting consequence of the condition of conservation of the total wave

vector is that the Fourier coefficients are divided into subsets

corresponding to the different values of the total wave vector. Each subset evolves in time independently of all the others. Therefore, for instance, a system which is initially homogeneous in space will remain so in the course of time.

-193 -

F. Henin

II. 4 - D i a g ram rep res e n tat ion

0

tion of the Liouville The classification

f the for mal sol u -

equation.

of the various terms in the series (II. 3.6) is

best performed if one uses a diagram technique. Let us associate with each state set of

j \ ~ \) =l!!.l

n lines

1ed with an shall

... !.N} with

running from

n

non

vanishing wave vectors a

the right to the left. Each line is label-

index corresponding to

the particle; when necessary, we

also indicate the wave vector An example

is

given in

fig. II. 4. 1.

1

Diagrammatic representation of the state O\k k k ' " - -i -j -1 f

Fig. II.4.1 The matrix vectors

elements

of

SL

provoke

a modification of two wave

k. k ~ k'. k' .. Taking into account the fact that among these, -J -1 -J none, one or two may correspond to the wave vector Q., we have 6 ..... 1

basic diagrams (see fig. II. 4.2) .

~~

/ •. k ... k. \

,

-1

-J

SL' .. -1 k'... k' ...) -J

k + k· = k'+k' ..... i-j

-i -l

(a)

i .. k .•• k ... , ~

,

....... 1

""'J

q .. k'... o.. ) --1

k +k = k'

-i -j (b)

-i

......

-

194-

F. Henin

<..

k , .• 0 • •\iL \ .. k', .•

"'1

""'"

k

""i

=-i k'+k' ..... j

--1

k... J ( .. -1

k""J

1.<".

""J

.\~L\ .. 0.. 0 ..'

k +k =0

-

-

'/

...;. -j -

(c)

.

(d)

9

d

I .. k, .. 0.. \ iL\ .. O.. k', .. ) '"'1_._J

o =k'+k' -i -j

k = k' ""i -j (f)

-

(e)

Basic interaction vertices. Fig. II. 4. 2. Taking into well

defined

account the fact

correlations

clearly what changes in

in

that

the

states \ \ ~ \ ) describe

the system, the diagrams indicate very

these correlations occur as a consequence of

the interactions. The present formalism thus appear as a description of mechanics in With to

terms of a dynamics

of correlations.

the diagrams,

easy to represent any contribution

it is

the formal solution of the Liouville equation. To obtain

contribution

I \ ~ \).

to

Then, we go to

combinations of

r

the evolution of

the

six

the right

rt

~t

·h

the n' order

(t) , we first draw the final state

through n vertices, using all

possible

basic vertices which conserve the total wave vec-

tor. As an example, the second order contributions to the evolution of

~ (t) are given in fig. II. 4. 3).

-- 195 --

F. Henin

lei.)

t~)

l':)

(~)

~

«

Q

i..

o ""

c:=

.t

~

t

G:

(.t)

(

~---.:ct___ · ~ l~)

~

t

'"

C

~

~ to)

1ft

(

n (et>

Second order contributions to

r

k _ (t)

-1

Fig. 4.3 Also, it is very easy, once we

have a

given diagram, to write down its

analytic contribution . Let us for instance consider

diagram (g) in fig. II_ 4.3.

-196 -

F. Henin

Reading the diagram from the left to the right, we obtain (we do not write explicitly wave vector

equal to 0) :

(~L_l_)2 hk'~t =(k~\~L I -i~1 k~k.lklk~, k'l) '(
(II. 4.

g)

0

0

I

1)(~~, ~'I z-~ o ~'i ,~~) (~'i' ~ I~ LI~'i '~"j ,~)

•

"
-1

1

-1

-1 -1

-J -1

In other words, we write a sequence of matrix elements of $L, each corresponding in a well defined order to the vertices; in between these matrix elements, we sandwich propagators (see II. 3.7), which are matrix elements of Ro(z) for the corresponding intermediate state; we also have such a propagator for the initial and final states. II.5.Classification of diagrams. We shall now discuss the topological structure of the diagrams which appear in the solution of the Liouville equation. In the most general diagram, we may distingUish three different

regions. Let us denote by (at the right) , by

It ~'t) the

1\ ~ "l) the

initial state of correlation

intermediate state where we have the mini-

mum number of lines .(which may of course appear several times in the diagram and b~/~~) the final state. If s is the minimum number of lines in the diagram, it may of course happen that we have several different states with that number of lines. We then choose as from

the right . As

an

I\!' ~he

last one starting

example, in fig. II.5.1b we have two different

states with one line : ,~~ ,\

£.\) and \ ~j' \.£ \).

one when we start from the right. we choose

As the latter is the second

it as

our state "~' \).

Another example (with two lines) can be found in fig. II. 5. Ie.

.- 197-

F. Henin

In the most

where \\~" \) F\\~

general case

'\) f U~ \) we

have:

1.

a destruction region, i. e.

te H.t"~) to the state \l~ 't)in te is

identical to \\~'n; in

with given correlations to 2.

a diagonal region,

\\ ~ '\) back a succession

a region where we go from such

such

;

in

sta-

way that no intermediate staregion, we go from a state we

i. e. a region where

have

less

correlations.

we go from

the state

general such a region contains

of irreducible diagonal fragments. By definition, an irredu-

cible diagonal fragment is such that that

a

a state where

the state ,\~ I t)

to

a

the

we go from a given state back to

state through a path such that no intermediate state is identical to

the initial state. 3.

a creation region, , i. e. a region where

lt~~) to the final

state to

te

I\.t' t)

of correlations

Examples of (diagrams (a) and diagrams

H~') in such a way that no intermediate

'\~'t)

state is identical

to

in this region, we go

this decomposition

(b) contain the

H!l)::. l!, '!J .-~.)

IRD

region

\\~'t):ll\!f>

,01.)

are given

in fig. II. 5.1

three different types of regions while

only one

IRD

diagonal

from the sta-

a state of higher correlations.

(c), (d) and (e) have

creation region

we go from the state

or two of them) .

destruction region

- 198-

F, Henin

creation region

IRD diagonal region

destruction region

(b)

<S> IRD

o IRD

diagonal diagr'3.ffi

(c)

IRD

·.- 199 -

F. Henin

.

"

creation diagram

(d)

IRD diagonal region

destruction region

(e)

.- 200F. Henin

IRD =: irreducible diagonal fragment Examples of decomposition of diagrams

in creation,

diagonal and destruction regions. Fig. II. 5. 1 As we shall tions will

see later on, the time dependence of the various contribu

N

be closely related to this decomposition.

II. 6

N

In i t i a 1 con d it ion s .

We shall always consider initial conditions such that macroscopic properties like the pressure, density etc..

are finite at every point

of the system, even when the limiting case of an infinite system is considered. The interest of this class

of initial conditions is obvious from the

physical point of

be shown that once the existence and

view;

it

can

finiteness of the reduced distribution functions for a finite grees

of freedom is imposed

ry later

at

t=:O , it

number of de-

will remain so at an arbitra-

time. This choice of initial conditions introduct>s mathematical restrictions

on the class of functions

r

we consider. It can be shown that this initial

condition requires the following volume dependence for the various Fourier coefficients: (II. 6. 1)

where V

is the number of independent non vanishing wave vectors which

- 201-

F. Henin

appear in the set

k ... k

"'1

• (By this, we mean the total number of non

",N

vanishing wave vectors minus k .+ ... +k. = 0 "'J '"

the number

of relations

of the form

which they satisfy) . For instance:

"'1

3

= (811" In)

r

2,..

~1'~2

(II. 6.2)

r!E.-~

The coefficients or

N.

f

3 ,... = (8lf 1n)

!E ,-~

-f

k do no k 1.. · N

althoug h t~y might still

n NI n .

longer depend explicitly on depend

on

the

With these assumptions, although in the

ratio

formal solution of the

Liouville equation, we find terms growirg more and more rapidly 2 (N. N .•• ) , all contributions to the reduced distribution functions for a finite number of degrees of freedom remain finite. The proof of these theorems is rather lengthy and cannot be given here . However, we shall illustrate them with

two examples. We shall

of the two diagrams of fig. II. 6.1

consider the contribution

to the one particle velocity distribu-

tion function : (II. 6.3)

o

(a.)

Lowest order diagonal and destruction contributionsto

r

ott)

Fig. II. 6.1 The contribution of the cycle is (see

equ.

(n. 3.5)

(fig. II. 6. la) to the evolution

• (II. 3.7), (II. 3.12) and (II. 4. 1))

of

fo

(t)

- 202-

F. Renin

r

Q (t)l

II "

0

'jo

= __ 1.

211

~2

1

f

dz e -izt

C

1:. < i

/Ol~L'k.=k,k.=-k)
~

L '0)

L.
j

0

\ _1 Ik.=k,k.= -k z-L -1 - -J -

It

>"

o

V.

e - iz t 2 -2-( k)

k

z

I)

"

1

')

k.(~ - - )

-

.,0 1, ~

In the limit of a

')P. z-k. (v .-v.l

-J

-

-1

C)

'C)

-J

k. ( - - - )0(0) ")n. ')0. 10 ~1

""'J

large system (see 11.1.10) ,1he summation over

~

becomes an integral :

(II. 6. 5)

Rence

fr o(t)Jo =~2(8lf3 In) L< i

j

I

e 3 d k (1/2" i)/C dz

~izt

z

2

(II. 6. 6)

=

if we take

into

O(NC)

account the fact that the sum over

the particles contains

N2 terms. Similarily, using (11.3.11) , the contribution of the deJEitruction diagram

(fig. II. 6. Ib) is :

- 203-

F. Henin

L I I

rfo(t~C' -A(81r'jn)'

i

(II. 6. 7)

"(o!z ,,<

_lL

o

10 )(01

<j

~

dze- izt

C

>

S q~i::~' ~j:: - ~

-

)c

k.=k, k.::-k 1-1-lk::k,k,::-k")rk::k,k,::-k(O) '" ~J ~ z - L -, - ~J ::..t -1"'" ~J ....,

~1

o

1

dz e

"

1 z-k. (v ,-v.l ...., -l-J

For large systems,

r-

V k

z

(:L_l)

k -' le., ~~. 1 J

k,::k, k. :: - k (0) ........ 1 --

....... J

....

this becomes:

Li<j (II. 6.8)

- izt

')

(ik (1/211' i)

-

)

l

dz

(

e

-izt

C

z

" k, '" k, k, :: _k(O)

k.(- - - ) , '}OJ' z-k. (v . -v J "" 1 .... ...., - l ...... J

- ') °

,

........ 1

-.

--J

....

:: D(NC)

Let us now introduce these integrations over

two results

the

. Because of the

the velocities, all contributions vanish except if i :: 1 .

This means, that, amongthe we only keep

in (II. 6. 3)

N(N-1)diagrams

(N-1) diagrams

such

that

a

or

b

of

fig. II.6.1,

i:: 1 . Therefore we

obtain: -izt

y

lVk 12

~

P ' .. dp-N -k . ., 01 \d-2 hw

dz

~

z-k. (v -v, ) ...... -1 -J

"

z

"

X

- 204-

F. Henin

(II. B. 9)

= O(C)

r..

j> 1

\d 3k (1/2 tr

1 z-k.{v -v,) ..... ..... 1 -J

(II.B.IO)

i)1

-izt dz e z C

If

-r-1 k =k k " _k(O) - ' - j .....

= O(C}

The general arbitrary ber of

time for

mechanism

(II. B. 1) (which for

factor

of the contributions of all

but a finite

In many problems we shall

over

in (II. 6. 7)), then

nUlmber

of degrees of freedom.

further reduce the class of initial

of initial conditions

distances of the order

assumption

many diagrams once we perform

conditions we consider . For instance we to the class

0

-1

an

a finite num-

is thus twofold; first , our

instance introduced a

the integrations over

ves

convergence at

the reduced distribution fuhction of

degrees of freedom

the suppression

which insures

shall

where

often restrict ourselthe correlations are

of molecular distances. This will

be discus-

sed when necessary. The property of functions

finiteness

plays a very important role

equations for

the macroscopic

ascertained with respect to behavior

of the reduced distribution in

the

obtention of irreversible

quantities. Indeed, once finiteness

Nand

0, we can further

of the system and find out that

in

look at

is the time

the long time limit, some

terms may become negligible. This is not the case

for

the complet distri-

bution function because of the divergencl's with respect to

N in

the

- 205F. Henin

limit of

a

appears

often

and to

large

system. However, for the sake

convenient

not

to

worry about

write down asymptotic equations for

legitimate

provided we keep

tions

shall

we

in

degrees of

N divergence distribu-

procedure is perfectly

mind the fact that all asymptotic equa-

derive are valid only when.

computation of average quantities

this

simplicity it

the complete

§ 9) ,. This

tion function (kinetic equations, see

of

which

they are used for the

depend

on

a finite number of

freedom.

II. 7 - Tim e d e pen den c e In order to get

some feeling about the simplifications \\!t;,'h

arise when we discuss the long time behavior of the system, let

UI;

llI~,y (, T,'

sider in detail some simple and typical contributions which we meet In the evolution of homogeneous systems. To make things even c]eart::l', let us choose a special form of a repulsive potential which will enable us to perform all calculations completely :

V(r) = V e o

(II. 7.1) -1 l( is

here

sform

of this

the

range

potential

(II.7.2)

intermolecular force. The Fourier tran-

is:

V

k

Let us now

of the

-Itr

= V

o

811'1( 2 22

(k +f( )

first investigate the time dependence of the contri-

bution of the simplest diagonal fragment ty distribution and

to the evolution of the

veloci-

function: the cycle (fig. II. 6. la) . Using (II. 3. 6) ,ilL:3. 11)

(II. 7.2), we have:

- 206-

F. Renin

(II. 7.3)

r. (t~Jcycle = -( 1/ 2tr

IIf

0

i)l

c

dz e -i;t

z

i' 2(z) r (o) 0

with:

~2(o'lLZ_\ "LlO)

(II.7.4)

o

= _ ~2(8"" Vo}2(81J 3/0}

z:::..ld\ i

(k 2+ tC}-4

<J ,

~ ~ 1 ~ ~ k (- - -) k (- - ) ';)n. z-k.(v.-v.} ~·~o. ')0. "-1 "-J ..... ~1 -J J...1 "-

~. ,)0.

(II. 7. 5)

where

I

k2J3 ktlk~ (z, £.i' ~., ) - (81I'1(V o) d k -2-.--24-(2----4/\ J (k + K ) z - k . (v. - v.) I· ..... .....1 "'J

+

1

+

(z t S )

(II. 7.6)

the z

Using cylindrical coordinates with

axis

ty: (II. 7. 7)

one obtain!': easily:

+

1

A(Z.D .• £..,~=(U"/3)(81tkV)

.,_

"-1

J

0

2 ('

cJ

111..

f!

'Ie

along the relative veloci-

- 207-

F. Henin

~ ScC,x + ifIl,y ) (1/4) J L(

+00

J(

dk"

-00

(+00

+~

(II. 7.8)

", z

dk

) -00

"

As ca.n be easily seen when performing the kIf integration,

+

regular in S

and has poles

The quantity

this

S

at

z=-il(g

(11.7.9)

des

in

i

(I(g) -1

and

represents

the

time during which the two parti-

are interacting, i. e. the collision time

l' Using co 1 integration in (II. 7.3) .

we can easily perform the z

result,

1;

We obtain:

fir o(t] cycle =-it "'2(0)+ "'1(0) + Res [ 1 T2

(II. 7.10)

e -izt

t/

z

)

Z

j

- .kg

Z--l

Therefore, we have three types of contributions: one proportional to 1;, a second one which is of order

Tcoll/ t

when compared to the first and

finally an

exponentially decaying contribution proportional to exp( -t/t= coll)'

This last

term becomes quite negligible for times much longer than the

collision time. If we do not

make a special choice of the intermolecu Jar poten-

hal, it is

eas ily verified (see an example in chapter III J § 2) that the

function

I d.~ has

(II. 7.11)

+

the form of a Cauchy integral :

t '" ~ I'}

" \::

d"

,'~"~

(z

t S+)

- 208-

F. Henin

regular ties

satisfies

f(w)

Provided

+

in

and can be continued analytically

S

S

in -1

are at

. If we

of 't' co 11 (a lthough

a finite distance

assume these smgularities to

result (1.7.10) is valid

,,,ke a sum over

the residues

can also

'e'l

il1fin;te system, we

at all poles

ith

~l(Z)

(811 3/11)

oX

Ir;

C'llntrast with

tion operator

r ,

tential

:--1

ille

-J

<J

-r

be discussed 3)), the

in

S).

of the simplest

t

depend not

i. e: on the initial

';)";)

k.(---) k '}12.. ~~, 1 J

)(

~

-J

~

2(z), the Singularities

of the destruc-

only on the type of intermolecular pok dependence of the function

co~ditions.

Let us denote

by

lC~~rr

range of the initial correlations. We may for instance suppose binary correlation

function

(II. 7. 14)

g(~, ~' ,0) = g

Then we

have

destruc~

k. rk,k.=_k(O)

-1

we choose, but also on the

~,-~

be simple poles

-izt _e-~1 (z) z

d,

dkV

..

the operator

~ l(z)

axis, of the order

ott) (fig. II. 6.1b) . In the limit of

L{3

z-k.(v.- v,) ~

. Its singulari-

S

(see (II. 6. 8) ) :

1

(11.7.13)

r

of

[fO(t~C=41/2"

(11.7.12) \1

have

, this function is

(instead of a single residue, we must

Let us now consider the contribution lion fragment to the evolution

in

of the real

other types of singularities

!eneral

4)

some general conditions

g(~: ~', 0)

(see II. 2. 9)

q ~ -~' I

is of the form -I(

,0) = e

that

(Ix -x'l)

corr

~

-

the the

-

209-

F. Henin

""

fb

(11.7.15)

l( carr 2 .2 2 (k + I( ) carr

_t(O)

Therefore, besides the pole at cient

of

the potential, z =- i

(II.7.16)

in

the rhs

of

we

now

have

the

poles

Fourier coeffi-

at:

I(corr g . We

(II. 7.12)

~

z = -i Kg due to

thus obtain:

[_e-_izt_~ (Z)1

(0) + Res 1

z

1

.

z = -lKg

(II.7.17)

'" ] + Res ( _e-izt __ d.) (z) z

As that

~

1

for is

(z)

the diagonal of the

form

1

Z=-l.

"

carr

g

fragment, taking into account of

this result very easily for the case

the fact

Cauchy integral, one generaliles

a

where one does

not

aSSU1M"

!':U"

ticular form of the interaction and the correlation function. For times time and

which

are

the characteristic time

in the rhs

of

(11.7.17). In what

much

longer

than both

g) ,olll,v the first reJ:'l:~ remain corrfollows we shall restrict ')Uffet"·:,;,,, (t(

situations where the initial correlations are due

to

mule cula]!' interaction

tions. Then the range Df the correlations is of the urder' the interaction and Let

both

of the range

characteristic times are .cll:rl'.lC'L

us now cons:der the simplest creaLor; fl'agment;

==:=>

(fig, II. 7. 1) Simplest

the cu 1 lh i:m

-1

creation fragment Fig. II. 7. 1

- 210-

F .. Henin

It is a contribution to the evolution of

f~, _~ (t)

. Analytically, we have:

-izt dz e Z with

.,0.

(~-~) ";lg".

C1(z) = (S .... 3/ rl ) Vk k " z-k.(v.-v.l -'

(II. 7.19)

-

-1

C1(z) ro(O)

-1

J

~1

The main difference with the two preceding cases is that we

have no

longer a summation over the wave vector. However, our aim is to compute average values of dynamical quantities in phase space. If we compute the contribution of (II. 7 .17) to the complete phase space distribution function

f ({g,,\ ,ls}

t), we have:

rr(t~\'I1I' t)].J' -(I/'1\'

(11.7.20)

it

,-:'t r I (,) foro)

d,

where (II.7.21)

= fd\

exp rik.(q.-q.)] C (z)

L~

~J

-1

1

With our assumption (II. 7. 1) for the potential , we obtain:

(II. 7.22)

[r (\ ~ H~ I .t)]:>

The last term

is proportional to

r = a . -q .' It will

-

'l.l-J

exp [- k(

t» we shall

1' ~ik

I!:, - ~tl)]

become negligible fo,r times

(II. 7. 23) Later on,

r 1(0) + Roo ( ' -i:t((,) t

g

where

such that :

r/ g

only be interested in the value of the distribution

function for relative distances of the order of the range of the intermolecular forces. Then the characteristic time

r/ g

will be of the order

- 211 -

F. Henin

of

t

will

r

reduce to

quantities

»

,the asymptotic contribution coll 1(0) . This means that, for the computation of the

the collision time and for

"C

defined above, we may take the asymptotic express ion:

The contribution of other types of diagrams inserted

(diagonal fragments

on a line , free propagating lines, destruction diagrams

involving exchange vertices (fig. II, 4. 2 a, f) can be discussed in a similar way but shall not be considered here (see ref. 1))

II.S - Evolution of the velocity distribution function. From

fo(l)

(II. 3. 5) and (II. 3. 6) , we have:

~-{1/2lf i) fe

.

d,

00

e -lzt ' [

L. <0 _1 (i L _1 ) z-L z-L

{~\ n=O

(II.8.1) If we

f

0 (I)

0

n

Ilkt) '" ~

0

" fiU(O) separate out

~ -(1/21i)

(II. 8. 2) -(1/2 lfi)

~c dz

the diagonal part, we obtain:

Ie e

d,

-izt

t <0 _1 (SL

n=O

z-L

0

_1 z-L

)n lO)f 0

0

(0)

-izt

It is quite obvious that all

of (II. 8. 2) will

e

contributions to the first term in the

rhs

be successions of irreducible diagonal fragments (see fig.

II. 8. 1) . As to the second term, we shall start from the right with a

-212 -

F. Henin

destruction region until we reach the vacuum of correlations

to t.

Then

we can again go on towards the left with a succession of irreducible diagonal fragm.ents n=1 n=2

(see fig. II. 8. 2)

0

6

-+ CX)

C> +0 + C:x:::J + ~ -+ c:J:)() ~ 0 · <S> +0+00 n=3

OC>

-t

First diagonal contributions to

fo

(t)

Fig. 11.8.1 n =1 n =2 n =3

C CYCOC

+

c= -+

0

-t

C

~ -+

+~

c= C>

-- 213 -

F. Henin

+~

+

+

+ First. non diagonal contributions to

f (t) 0

Fig. II. 8.2 If we use the diagrams

(a) the sum of all

of fig.(II. 8. 3) to denote

irreducible diagonal fragments whose initial

(and final state ) is the vacuum of correlations (b) the sum af all destruction fragments whose final state :s the vacuum of correlations we easily obtain a regrouping of

all terms in the rhs

of

(II. 8. 2) in

terms of diagrams (fig. II. 8. 4) (a)

(b)


+

6

C ~L

+

OO-t

~CC ~

c=

Diagrammatic representations of diagonal and destruction ope rators. Fig. II. 8. 3

-+ ...

- 214-

F. Henin

Diagrammatic representation of the evolution equation for the velocity distribution function Fig.II.8.4 Let us introduce the following operators co

m

:;72 (0 \ (II. 8. 3)

(IL z_1L ) o

I O)irr

z m

(11.8.4)

where

z-L

o

) 'i~') irr

the index irr. means that only terms such that all intermediate

states are different from the vacuum of correlations ,\ 0 into account. This condition means D\~~ (z)

l) must

be taken

that all propagators in f(Z) and

are different from z. With these operators (II. 8.2)

fo(t) = -(1/2 11' i)

f

C

may now eaSily be written as ;

e -izt dz - z

(II. 8. 5)

Differentiating with respect to time • we obtain;

~ fo(t) = 'It

- 215-

F. Renin

+(1/2")

(11.8.6)

Ie

dz e- izt fo(O)

+

(1/211")

Ie

dz e

-izt

x

)( lttfl2.t L... DJ~'(Z) rtU(O) 1

The second term vanishes because -its integrand has no singularities. As to the remaining terms, let us perform the Introducing time dependent operators tively the inverse Laplace (II. 8. 7)

transforms

G(t) = - (1/211 i)

fe

G(t) and of

t

z

~, ~\(t) (z)

and

integration. which are respec-

l

D ~~Z) :

dz e -izt '" 1 (z)

(II. ~.. 8)

and using the convolution theorem as well

as

(II. 8. 5) , we obtain:

(II. 8. 9)

This generalized "master equation", which has been obtained in a straightforward way from a rearrangement of the terms in the formal so-

-

216-

F. Henin

lution of the Liouville equation describes the exact behavior of the velocity distribution function for any time. It may seem much more complicated than the original Liouville equation. However, as it will appear below, it has the great advantage that for a very wide class of initial states, it has simple properties in the long time limit. Let us first notice that we have decomposed the time variation

f (t) in two

of

we have

f

contributions of

0

0

a very different

kind. First of all,

a non-markovian contribution which is expre&sed in terms of

only

; this

contribution describes scattering processes; the inte-

gration over the past corresponds to the physical fact that the scattering processes have a finite duration (collision time) . On contrast with the first term, the second term on

r

0

but on

the initial

in the rhs

of (II. 8. 9) does not depend

correlations present in the system. This

term describes the destruction of these initial correlations.

II. 9 - Kin e tic e qua t ion. Let us now consider the case of systems interacting through short range forces and such

that the initial correlations are over a mo-

lecular range . For such systems, the duration of a collision is very short and many simplifying features appear if we consider the asymptotic behavior of the system, i. e. its behavior for times

t»'t'

(II. 9. 1)

colI

Generalizing our discussion the operators t(Z) properties: 1. they are analytical

and

such that

1:

of

§

7 , we shall

DJkt(z) fikl(O)

\~\'~

functions of

assume that

have the following

~J

z

in

the whole complex plane except

- 217F. Henin

for a finite discontinuity along the real axis . These operators are analytical in

S+ and can

be continued analytically in

the lower half pla-

ne. These properties are consequences of the definition of these operators provided perturbation calculus converges. They have been verified in

r..

detail for the lowest order contribution to \lI (z) and

1 in ~ 7;

D k (z )

tk\

tJ

f.,_tkJz)

there it has been shown that both these contMbutions are Cau-

chy integrals. A

detailled discussion of the operator 'f'(z)

orders has been recently done for the problem of

at higher

anharmonic solids and

some quantum field theory problems 5) • 2. The singularities of the analytical continuation in S- are poles at a finite distance from red as

the real axis. This assumption must be conside-

a sufficient condition for the validity of the kinetic equation we

shall derive. We have seen how it can be realized for a simple type of interaction potential

and a simple initial condition in

§7.

For more

complicated interactions or initial conditions, singularities other than poles could appear and the following proofs must he amended but we shall not consider such cases here. With these properties of the diagonal the z

plane, our results of

t7

and destruction operator in

can be easily generalized for the di-

scussion of the two kinds of contributions

in

(II. 8. 9) .

Let us first consider the destruction term. Using (II. 8. 8) , we obtain:

L.. Res rL J

=0

(II. 9. 2) where

the

~j 's

D\k t(Z) e

l~ -k\ ""

are the poles in the lower half plane of

-iz~ k 1(0)1 I~ -' 1f·d J

the function

- 218-

F. fienin

~

D\!E\{z)

rt~\ (O)

9~ 2)

. (II.

corresponds to our result of 7 that the lo-

west order destruction contribution to the evolution

of

f

0 (t)

is asym-

ptotically constant; upon time differentiation, we obtain zero. Therefore, once (II. 9.1)

is fulfilled and the initial correlations are short range,

the master equation becomes:

(II.9.3)

while (II. B. 5)

reduces to

folt) . R" [, ~i't (II. 9.4)

ttl'» tJoIO)

(1/p~ q~) (-it) p\1ry(z)

t. t

p=O

~

dz P

q=O

where the function '¥(z) which has tical continuation

to

L'

1

be used for

of the function defined in

S+

z....., 0 is the analy1)

It can be shown, through some lengthy algebrliic manipulations

that this gives rise to

~~ (t)

(II. 9. 5) where

the kinetic equatlon:

-a°t

i

n

is

=

n '1'(0)

r

o(t)

a complicated functional of

t

and its derivatives

for

z ~O (II. 9. 6)

00

n

=

L- nG(

"'=:0(II. 9. 7)

n= 1

(II.9.B)

Q = lim

1)

°

1/

z""O

net

(z)

~

>1

Po (0) is a modified initial creation and is given by the expression between in the rhs of (II. B. 5)

-

219-

F. Renin

The

0 cl (z)

are given

by a recursion formula:

at-I

(IT. 9. 8)

= (lie()

The operator

1:: \

~ =0

'ilz

oc(

-1 _

~(z) t(Z)} Of' (z)

0 takes into account the finite duration of the collision.

We shall not give this derivation consid~rations

';)

here but rather use some simple

which will emphasize the meaning of

both the operator

f

and 0 The operator (see

t

has

the dimension of the inverse of a time

II. 8. 3) . As this operator describes the collisions occuring in the

system, this time is of the order of tre relaxation time. (for instance, in dilute gases (Boltzmann equation), the relaxation time is connected with binary collisions, i. e. those terms in Derivation with respect

to

z of

t

'¥ which

increases

involve only two particles). by one the power of one

of the unperturbed propagators. In our simple example of seen that this amounts to bring an extra factor contribution to rhe rhs of p and

q

(II. 9. 4)

~

colI corresponding to

+7,

we have

. Therefore, any a

given value of

is of the order

(t/'C

(II. 9. 9)

reI

) P (1:

coll

l-c reI )q

't'

1/"C l' i. e. let us cons ider the collis ions as col re instantaneous events. Then, we may restrict ourselves in (II. 9. 4) to the first neglect

Let us

term

q = 0 and we obtain:

(II. 9.10) This leads (II. 9.11)

f

00

ott) = ~ us to

(l/p~)

(-itt(O))p fo(C) + 0 (Tcolll'rrel)

-

220-

F. Henin

This equation

is a very simple generalization of Boltzmann IS equation;

besides two-body collisions, it includes collisions between an arbitrary number of particles. However, if we do not neglect collisions between more than gas

two particles ,it is not consistent. Indeed, for a dilute

of hard spheres for instance, where we can restrict ourselves to

two-body cOllisions, it can be shown that the relaxation time is given by: -1 2't'1=Cav re

(II. 9. 12)

where a is the diameter of the partiCles and ~ their average velocity. As the only dimensionless parameter we have is a 3 C , we must expect that, when we

take into account higher order collision processes, we

shall have an expansion analogous to the virial expansion : (II. 9. 13)

T

-1

reI

A 3C) 2 + ... -- Ca 2-v [ 1 + G( a 3C + ,.(a

1

Now, we also have :

-r

(H. 9.14)

coIl

=

al

v

and thus 3

(II. 9.15)

1:coIII r re 1 = 0 (a C)

Therefore, the procedure we have followed is certainly not consistent: we cannot keep higher order collision processes (i. e. corrections of 3 order a C in 't' 1) and neglect terms of the order of 't' liT 1 in re col re the rhs of (II. 9. 4) . In order to understand the general evolution equation (11.9.5)

let us keep in (11.9.4) the fira correction, that which

is proportional to ve :

1:

co

lIlT:

re

1 (i. e. the contribution q =1) . Then we ha-

- 221-

F. Renin OQ

+

L

p=O

OQ

L.

(l/(p+q)! )(-it ,\",(0) )p

q=O

(II. 9.16)

i' I(O)(-itf(O))q fo(O)

2

+ O'('t'cOll/ r reI) Upon differentiation with respect

to t, it

is easy to show that one ob-

tains :

The operator '1'1(0), which takes

into account, in first order • the fi-

nite duration of the collision, is precisely identical to the operator r2l in (II. 9. 6) .

II.IO-Evolution of the correlations in an homogeneous system. As all derivations are very similar to the derivation of the generalized master equation for the velocity distribution function, we shall only indicate how they proceed and what are the final results . The most general diagram contributing to the evolution of a given correlation contains all three types of regions defined in ~ 5. We shall now write: (II. 10. 1) where, by definition :

Hi, (t) contains all diagrams without creation region D.ic

This

(t) contains

If-t decomposition

all diagrams which end by a creation region

is performe.d in detail for all

contributing to the evolution

of f~,

_!

in

second order diagrams

fig;Il. 10. 1.

- 222-

F. Henin 12

Q

Q

:c

A Q

lb)

la.)

~Q

(t)

(

J.

t -t)

I

t h>

c

(

l~)

l\.)

c=

oc. l"ft\)

o ---1.(__

('1\.)

C

--c

C

If.)

(

--C

a

l, )

(0)

~

"

o

ri. -!,

.

•

(n~ diagonal fragment)

~ .

l~)

Second order

G) lS)

l"-)

Second order contributions

contributions to

lp)

.

\.

fi'. d

_~ (one diagonal fragment)

1

Q

:

- 223-

F. Henin o

--)

r-'k" -k-

Second order contributions to

Second order contributions to the evolution of the binary correlation f!,

_!

Fig. II. 10. 1 To discuss the evolution of

r~k

(t). one decomposes the relevant

1_\

diagrams into those which are diagonal and those which contain a destruction region (see fig. II.lO.l) . In this way, one verifies easily that these functions obey an evolution equation very similar to the general master equation for the velocity distribution function:

- [ d1: G

(II. 10. 2)

where

+

\!dt

-T)r

UP')

~:~\I!'l (t, fl!!.'1 (0) )

G \!\(t) is the in verse Laplace transform of the diagonal operator:

(II. 10. 3)

while ~~I!') (t,

1

IlO

ftU(Z) = ~~!'ISL rf!"

(z -L!44t!\)irr

(0» is the inverse Laplace transform of the

destruction operator :

The dash on the summation over

t ~"

in (II. 10.2) means that

1

states 1t!'~whiCh are such that the transition !\~'

only those

I describes

a

- 224-

F. Henin

destruction of correlations must be taken into account. The evolution of ft~ \(t)

is due to

the dissipation of the ini-

tial correlations through the collision processes. For long times, a pseudomarkovian equation similar to

(II. 9. 5) can also be derived from

(II. 10.2) .

Plf t \(t),

As to the evolution of if we have at the left

a given creation diagram, corresponding to a

'

transition \ t \t-\~"

the main point is to notice that

we may have at the right of this creation diagram

any of the

diagrams which contribute to the evolution

Iff tt'\=I.Q.\

'

evolution

we may have all

of

Pit, \("1: )

the diagrams which contribute

to the

of the velocity distribution function), if or is the time corre-

sponding to the first creation show rigorously that one

vertex. This remark makes it possible to

has:

(II. 10.5)

where

the dash on

se states

the summation

I\t' \)c0rresponding to

must be taken

into

t

C tH~'rt)

a

over \ t'

t means

that only tho-

lower state of correlations than B~t)

account. is

the inverse

Laplace transform of

the creation

operator: (II. 10. 6)

Equation (II. 10.5)

describes

the continuous creation of fresh

tions by direct mechanical interactions from

correla-

less excited states.

-

225-

F. Remn

II. 11

w

A P pro a c h toe qui Ii b r i u m

0

f the vel

0

cit y

distribution function. For weakly coupled or dilute systems, the approach to equilibrium is usually discussed by means of an

l£ -theorem.

More precise-

ly , one shows that the quantity (II. 11. 1) decreases monotonically in time and that the stationary solution (which is unique) corresponds to

the equilibrium distribution.

Unfortunately, this theorem cannot be completely generalized when higher order

contributions are taken into

account. We shall only

consider the case of systems where there exists a parameter such that a perturbation expansion in

powers of that parameter has a meaning (cou-

pling constant ~ for weakly coupled systems, concentration C for dilute systems) . As an example, we shall consider the case where an expansion in powers of

~ has a meaning . Then, with the following expansions:

(II. 11. 2)

fort)

= r(~) (t)

+~f

(II. 11. 3)

1'(0)

= }. 2 'l'2{O)

+\3 'l'3(0) + \.\1'4(0) + ...

(II. 11. 4)

r2 = 1 + ~2'1'2 (0)

the kinetic equation (II.9:5)

{II. 11.5)

(II. 11. 6)

gives

2)

(t)

+ ).2

pi!) (t) +...

+ ... us

a set

of

equations:

- 226-

F. Henin

~ ~ - , t ,(0) f (:) (t) - ''\'3(0) r(!)(t) - ''1'4(0) r(~) ~~2) ...

(II. II. 7)

- i'l'2 (0)'\'2(0) f (~)

(t)

(t)

etc ... The ~ -theorem for the lowest kly coupled systems, see chapter

III,

order approximation (for wea-

If 2 and

3) shows us that

r(~)(t)

decreases mo,10tonically towards its equilibrium value

r

For times

much

7»O(I)(t) I 0 ~ ~2t

where g

0

\ 2

)(

is

(0) r(l) (t) _ 0

now

'n

that

i'"

13

the next

one

(0) f(O) (H ) 0

can

show

that

'" (0) g (H ) = 0

is

0

an aTbitrary function

of the unperturbed hamiltonian. A

general method to verify this property can be 'found in is based on the discussion A more cumbersome method in a number appear in

f(~)(t),

by:

= _ i-"

The interesting feature

(II. 11. 10)

(H )

longer than the relaxation time for

approximation is then given

(II. 11. 9)

= f(O)

(O) (t ~ co) o

(II. 11. 8)

of

of an

3). This method

integral equation and rather formal.

1) 6) consists in the splitting of each

'I' n

operators according to the number of particles which

the diagram. For instance, in the operator

t

3(0) , we have

J[ThiS is valid for gases where the interaction is velocity independent. For anharmonic solids for instance, the situation is more complicated because of the action dependence of the potential and this property is not valid. This makes it very difficult to study the approach to equilibrium at higher orders than ~. 1) 5) 7)

- 227F. Renin

two diagrams

t(~)

(fig. IL11.1) : one with three particles (a) which we call

, the next one (b) with two particles which we callt(!) .

o A


Contributions to

3

Fig. II. 11. 1 One then shows that : (II. 11. 11)

n=3, ~

As an example, let us verify this for

= 3 (diagram

a, fig. II. 11. 1)

We have: d/(33)(O) g(R )

1

(i. 1.1.

=

0

lim

z:., 0

L

ijl

!(O\~L \ !!..=~, !EJ.= - !!.) " ~

1

12'rl 'Ilk .=k, .=k, k. = -k\SL\k =k, k .=-k) .... k. = -k ,_1 ·lk-1.=k, .........k .=-kVk -N-1 -J

z- L

.....

o

J

-J

-

•
-

z-L

-1 -. -J

o

-1 - -J

-

-1 ..... -J

O)g(H) 0

Using (II. 1. 1) ,(11.2.12) and (11.3.11) as well as : (II. 11. 13)

lim

z~

a/ (z - a) = - 1

0

we easily obtain

,\,Ii (0) glHr .~~

;. tv. \ {d 'k

Ill. 11. 14)

x

k. (v .-v l )

""

""1 "'"

z-k.(v ,-v J - -l-J

2

Vk

~ • I!I
-

IC

-

228-

F. Henin

Now, let

: i ~ j, j ~ i, ~ ~ -~

and take half sum of the rhs

of

(II. 11.14) and the term obtained through this interchange of dummy variables. We obtain:

=0

because tre integrand is an odd function of Let us now go back to (11.11. 5) , (II. II. 6)

the discussion of the s~t of equations

(II. II. 7) , etc ... for long times. Using (II. 11. 10),

J

we notice that (II. 11. 9) reduces

~ (1) (11.11.16)

k

fo

to:

(t)

=_idl(O)O(l)(t)

~~2t

12,

0

As only the lowest order operator remains in this equation , we again obtain the result : (II. 11. 17)

r~l)

It is then trivial

using again

(toi (0)

=

P) (H

o

)

(II. 11. 10) to show by a

recurrence pro-

cedure that:

in) (H )

(I1.11.18)

o

Therefore, we obtain: (II.11.19) which is the equilibrium distribution . The function (normalized to unity) as far tely determined from

as this proof

the initial condition

f(H) is arbitrary o is concerned but is comple(see ref. 1) ) .

- 229-

F. Henin Let us stress the fact not so powerful as the Indeed, the

t

g -theorem

that

this generalized ~-theorem is

-theorem for the lowest order approximation. for the lowest order approximation actually

amounts to proof that all eigenvalues of the hermitian operator -

it 2

are either negative or zero and that there is a unique eigenfunction (the equilibrium distribution) corresponding to the eigenvalue zero. In the case of stronger coupling, what we have actually done here is to verify that there exists one zero eigenvalue for the complete evolution operator, with the equilibrium distribution as eigenfunction. A true

)C -theorem

would require a proof that the eigenvalue zero is unique and that all other eigenvalues are negative. This is of course very likely, at least for systems where a perturbation expansion h8s a meaning, i. e. when the lowest order terms give the dominant features of the behaviour of the system.

II. 12 - A P pro a c h toe qui Ii b r i u m

0

f the cor reI a t ion s

in an homogeneous system.

r'~ ~

\

The asymptotic solution for the equation (11.10.2)

of the correlation can be discussed in

for the part

a way similar to the above

discussion for the equation for the velocity distribution function. The main result is : (II. 12.1) As a result, in the equation for fragments which start

(II. 12.2)

from

P"\ k\(t)

f"h. \ (t).

we only keep those creation

the vacuum of correlations:

.- 23U-

F. Henin

These results mean that the initial

correlations tend to dissipate. In

the long time limit, only the fresh correlations which are continuous· ly created from the velocity distribution function remain. For times such that the velocity distribution function has reached its equilibrium value, we have

j

(II. 12.3)

or, using the Laplace transform of the creation operator (see II. 10. 6):

(II. 12.4) :; (1/211'i)

Now , as fact that we shall

( -izt -1 }C dz (e - 1) z Ct ~W2.~Z) f(H 0)

in our discussion of

t

7, we take into account the

always be interested in average values of dynamical

quantities, i. e. in expressions which involve a sum over the wave vector. In such quantities, the operator r(z)

which is

t ~"2..\(Z)

C

a Cauchy integral (see for instance

have:

(1/2'1f i) {dZ (e

the

-izt

- 1) z

-1

11.7.21). We then

r (z)

e

(II. 12.5)

where

is replaced by an operator

~j s are the poles in

S·

of r(z) .

For long times, this becomes:

(II. 12.6)

z:

lim. t~

0()

j

res

rl (e -izt

)

f·:1

J

- 231F. Henin

This result

allows

us

take

(II. 12. 7)

(OJ\ L:. ( z-~ k)

=lim

z"" 0

m= 1

m

IO)f(Ho)

0

If we denote by

-

the set of all to

(II. 12.9)

where

V is

(f k \\ (_1 ~ L) m \

p (m) = lim I \ k\

(11.12.8)

z~ 0

'-

z-L

0

f !IE! ,contributions involving m

0'\ f(H ) '/ 0

vertices we have:

m

J::...lo) m!

(m)

FtU

the intermolecular potential

(see (II. 1. 1) ) . (II. 12.9) can

be provep. , using a recurrence procedure. For m

==

I, we have, using

(II. 3.9) : O(1)=k,k,=-k "lim
= (81r

3

=k,k,=-k\-l_~Llo) -

-J

, / fl) hm

..... 1 ..... J

"-

-

£.j

tj

0

o

}

o =(k,=k,k.=-klv-

x

'0

';)f(H ) C)Ho ~f(H

-

0

'i)

(8v 3/fl) Vk

-1

f(H)

z-k.(v ,-v.) Vk k.(~ ---U-)f(H )

z~o

(II. 12.10)

'" z - L o

-J"'"

,

d~

lo'/

Some care has always to be taken m the use of (II. 12, 7). The use of such an expression does not lead to difficulties when one is interested in average qunn tities which are linear functionals of the correlations. When non linear functio nals must be considered, one must~o back to (II. 12.4) as has been shown in recent work on anharmonic I,>olids .

-

232F. Henin

Assuming (11'.12.9) to hold for a given

r t~\

( +1)

h (

= ~i~ 0
=

(II. 12.11)

= lim

[~~\\

z _1 L

r<1~\ I

z 1_ L

O\~I

= lim (\ z~ 0 l~

r

(m+1)

v

0

6lJ

'O)f(Ho)

0 L\\~I\)r~~)'t ~

~ L\\~I\)({~I\ \

~ mf(H ) ~H

0

k\l

~

1 Vm -L ~ L zm 0

m

d

~1'''~\J z~O

z -

r: "

0, 0)

;)H

m

o

3

I Q-)

Vm

~mf(Ho)

m!

";)H m

"'1 ....... 1

N+1

lim

k"Y

i=1

[ exp

= i(81t

-i

3

! rl.) N+1

t= .qJ ~

1:

~

lim

0

m!

1=1

3

= i(8 n In)

N+1

d dH

o

m+l

lim z~

0

z-

L ~i' i= 1

m • .

fidQ.! N exp [_ i

L- ........ k " v, 1 ....... 1 i=1 m+1 f(Ho)

z-

Vm

:(,1

or

-1 1

~ ~m ~mf(Ho)

k -;)v -i -i ' - .... a . " 0 1=1 "~l ""'-1

i 1

z~

N

I0

0

z~O

N

>

~


k,.v,

i =1

= - i( 81{'

f

o

f(H)

-1

hm

X {ds. \

~

non vanishing wave vectors k .•• k , we obtain:

, =

,m+1

1

z-L o

z~ O\~'t

;,row, if I k \ contains

value of m, we have:

:(,

1

" Hm fJ

o

t ~i' S.i] i=1

l(

- 233-

F. Henin _1_ ~m+1f(H.) (m+1)! ;) H m+1 o = i(81T 3/f2) N+1

lim

z -

z~O

(II.12.12)

"

~ { ds.} N exp

L. ~ ... ~.l ~ "

[- i

II

i=]

Integrating by parts over

1 .•

~1'

,,\m+1

1

iJ t"i

d

f(Ho)

. d H m+1

(m+1)l

o

one obtains

easily (II. 12. 9) with m re-

placed by m+1 . Combining (II. 12.9) and (II. 12. 7) , we easily obtain:

(II. 12. 13)

which is

the cQrrect value of the equilibrium correlation. Therefore, once the velocity distribution function has reached

its equilibrium value, the fresh correlations which are continuously created from

f

0

are the equilibrium correlations. (II. 12.7) gives us a dynami-

cal description of the equilibrium correlations. A comparison with Mayer's cluster

formalism can be done

11.13. Response

to

but will not be oonsidered here.

an external constraint,

As an example, let us consider

a system of charged particles

which is at equilibrium at t=O: at .t=O, we switch on a spatially homogeneous extern'll electrical field

~

(t) .

To the hamiltonian (II, 1. 1) , we now have bing the effect of the external field:

to add

a term descri-

- 234-

F. Henin

(II.13.1)

is

where e i

L e.

=

HE

. 1

E (t).q.

1"'"

the the carge of the

-1

ith particle.

The Lbuville operator corresponding to

~ L + iL

iL = iL + i

(IL 13. 2)

o

with iL

ell. 13. 3)

E

=

L e.

this problem is then:

E

E.~.~i

1"'"

i

As we have assumed the system to be at equilibrium at t=O, the initial condition is : exp [- (H

0

=

(II. 13.4)

+ ~ V)/k1f]

If the external field is sufficiently weak, we can restrict oursel-

ves

to

a

linear theory

f (t) =f·equ.

(II. 13. 5) where

At is

(II. 13. 6)

linear

in

(L G

(II.I3.7)

in E; therefore, we have:

+ Ap(t)

E. Using : +SL)

r

equ.

=0

'r

0 equ /';)t =

the Liouville equation reduces to

(II. 13.8)

jAr 't

= -i (L + SL) 4

r- r iL

d O E

equ.

- 235-

F. Henin

when terms of order E2 and higher are neglected. This equation can be solved formally very

(11.13. 9)

4 fIt) • -i

[

dt' exp [i(L + o

easily:

~L)(t-t')J LE(t') f equ.

This solution takes into account the initial condition(II. 13.9 )) (6r(O)=O), Let us again expand the distribution function

in a

Fourier se-

ries of the position variables. Then, for the velocity distribution function

r

0 (t)

(which is the only coefficient we require if we want I for instance I to

compute the current in the system), we obtain:

(II. 13. 10)

Taking into account the fact

that E

is spatially constant, we have:

(II. 13. 11)

and therefore :

(II. 13. 12)

)( L (t') E

Any time dependent of oscillating fields

rn.\

equ.

field can be represented by a superposition

with various

frequencies. Therefore, we shall

restrict ourselves to the case of an external oscillating field:

(II. 13. 12)

E(t)

-

=E

-0

e

-iwt

Using the convolution theorem and (II. 13. 13) , we easily can write:

- 236-

F. Henin

where R(z) is the resolvent operator defined absence been

by (II. 3. 6)

in the

of external field. The matrix elements of this operator have

discussed in detail in , 8, where we established the generalized

master equation for the evolution of the velocity distrihution function in the absence of external field. Using these results, we obtain:

Il P (t)=-( 1/ 21f i) { o

c

r

L-

[1 "'(z)] nl _1 '" dz _e-izt 4-(, E ,~- equ . z n=O z 1 z-w I 1""'0 Cl E-i 0

(II.13.15)

where the operators

'Y (z) and

D. ~-,(z)

are given

by (II. 8, 3) a nd

(II. 8. 4) respectively. Differentiating with respect to

t and using (II. 8. 7), (II. 8, 8) , we

obtain:

(II. 13.6)

This transport

equation

is

valid

at

any order in

the cou-

pling constant' and the concentration. The term

,lLE f equ. 0

in the lhs is the usual flow

term

- 237-

F. Henin

which describes the effect of the field on

the unperturbed particles

(i. e. between successive collisions) . As to the non markovian term in the lhs of (I1.13. 16) it describes the effect of the external field on the particles To see

during the collisions.

this, let us consider the static case (w=O) . Then, for times long

with respect to the collision time, we have (see (II. 8. 8.))

(II. 13.17)

As

L [t

n.'

0

we have seen, the equilibrium correlations are created from the

equilibrium velocity distribution (see (II. 12.7)) D\k \(0) U~l ~

c

T

L p'equ.

L

and we can write:

EltU

=

{~\ D I ~\(O)

LE

equ. 0

·h. ,(0) f

(II. 13. 18) " lim

z-L

z~O

"r

L

o

_1_

E z-L

o

(h _1 )' JL I 0) ~ z-L 0

equ. 0

If we compare the operator in the rhs with the operator t(z) given by

(II. 8. 3)

only through the replacE'ment of one

, we notice that they differ

the unperturbed propagator S 1/ (z-L ) in t(z) by : o (II. 13. 19)

1 z-L

o

LE z-L

o

= z-L

0

-L

E

1 z-L

2

+ O(E ) o

In other words, the operator in the Ihs of (II. 13. 16) describes the corre-

- 238-

F. Henin

ction to a collision process count (to first order in

which. one obtains when one takes into ac-

E) the effect of the external field on one of the

intermediate state.

11.14 - Stationary transport equation in the static

case. After a long time, we may expect that a system submitted to an external, spatially homogeneous, constant, electrical field will reach a stationary state : (II. 14. 1)

where

fl

r :t.

is

time independent. As the contributions to the rhs of

(II. 13.16) come from

times"C such that: t - "t' ~"[ colI

(II. 14.2)

we also have, for

very long:

Taking also into account (II. 8~ 8), the general transport equation becomes: (II. 14. 4)

iL E

requ~I!\t!\ r. D

0

or

equ. (II. 14. 5) iLEV 0

+

r.

D

(O)iL

f~qu.= lim P equ·.

(0) iL

f!' {t\'

f

El!\t~aoO

E

t

d't' G(t- 't' )Arsot .

= -it(O)A

H!l

In chapter V, we shall use this equation as

r

at

0

~

starting point i1 a

discussion of brownian motion of a heavy charged particle submitted to the action of an external constant electrical field and moving in a medium of light particles.

- 239-

F. Henin REFERENCES.

1.

1. Prigogine, Non Equilibrium Statistical Mechanics, Inter science, London - New-York (1962)

1.

R. Balescu, Statistical Mechanics of Charged Particles Interscience, London - New-York (1963)

3.

P. Resibois, in "Many - Particle Physics" ed. by Meeron, Gordon and Breach (to appear, 1966)

4.

c..

5.

1. Prigogine and F. Henin, J. Math; Phys .

George, Physica 30 , 1513 (1964)

..!.'

349 (196 0 )

6. F. Henin, P. Resibois and F. Apdrews, J. Math. Phys. 7.

~,

68 (1961)

1. Prigogine, F. Henin and C. George, Physica (32, 1873 (1966))

- 240-

F. Hemn

III. BROWNIAN MOTION IN AN HOMOGENEOUS, 'WEAKLY COUPLED SYSTEM Ill. 1 - Introduction

We shall consider here a weakly coupled gas, i. e. a system such that the coupling constant is very small : (III. 1. 1)

\-2 The relaxation time of such a system is proportional to A (see the Born approximation) : (III. 1. 2)

"t'

reI

~

\ -2 J\

We shall consider the evolution of such a system for times of the order of the relaxation time, i. e. we shall take the following limit: (II.l.3)

As the collision time is independent of the strength of the interaction, we clearly have : 1:'

(II. 1. 4)

colI

/~

reI

~O

and the evolution of the system will be described by a markovian equation.

The weak coupling condition implies that with

we exclude all forces

a strong repulsive core. Strictly speaking, there are no known

intermolecular forces for which the theory of 'weakly coupled systems may be applied. In all physical cases, the interaction becomes too strong at very short distances to be handled in a weak coupling theory . Nevertheless, we shall consider it here because it is the simplest example where the brownian

- 241-

F. Renin

motion problem can be discussed starting from a microscopic basis. This model has been discussed by Prigogine and Balescu 1)2) and will already show up interesting features when compared with the phenomenological theory of chapter I, and in any ease, we can expect that it will give us a good description of the effect

of the collisions which are not

too close. We shall first

find the equation for the reduced distributi on fun-

ction for one particle assuming that initially there are no long range correlations. Then, we shall specialize to

the problem of brownian motion

where the particle moves in a fluid at equilibrium. The equation so obtained for the one particle velocity distribution function will be of the Fokker·Planck type. Roweveir» in

contrast with the assumptions of the

stochastic theory, the friction coefficient will appear as dent. In fact, this dependence will

velocity depen.

be important only for velocities equal to

or higher than the mean thermal velocity of the fluid.

1II.·2 - Equation of evolution

of the velocity distri-

but ion function for weakly ooupled systems. From the discussion of chapter II.. ~ 9 , it is quite clear that if we take into vian corrections

to

account

the kinetic equation, i. e. take

n

(III. 2. 1)

(III. 1.4), we must neglect all non marko-

= 1

in (II.9.5) When we do this, to rator

t

be consistent, we

must keep

only the lowest order contribution, of order

diagram which we have therefore to cycle (fig. II. 6 ta) . The evolution

in the ope-

A2 .

keep in the operator equation then becomes:

'I'

The only is

the

-

242-

F. Renin

(III. 2. 2) where

t2

is the operator associated with

discussed this operator

in

ch. II.

~

the cycle. We have already

7, using the simple case of an ex-

ponential interaction law. If we do not make any special choice of the potential, We have

(see II. 7.4)

(II. 2. 3)

i' 2(z) when

We have nvw to find out the limiting value of the analytical continuation of real axis. As

this operator when z

+

z~O,i.e.

approaches the

we have already said this can be done easily, in the limit

of a large system, using the theory of Cauchy integrals. Indeed, when the limit

N...,

(Ill. 2. 4)

00

,

11 -')

in taken, the spectrum of values of mation over

k

in (III. 2.3) (811"3/ 11)

(IIl2.5)

N/11

00,

t

=C

finite

becomes continuous and the sum-

bec'Omes an integral:

~~ \

3 d k

Then we have: - i t 2(z)

L.

(81t 3/ I1 )i

,

i(j

(III. 2. 6)

X

z-k. (v ,-v ~

If

we

-1

.l

~J

3 d k

tvk \2 k . (&) ~

d

'i) ~i

~

k.(- --) () p , C)p,

~

~1

~J

take as one of the integration variable the variable:

d

--) ,--~~,

J

x

- 243-

F. Renin

(II. 2. 7)

x = k . (v . -v.) .....

the dependence on z

.....l .... J

is of the form:

F(z) = J+:x

(Ill. 2. 8)

..B&.. z-x

-00 This is precisely a Cauchy integral. Its analytical continuation is :

I

'+00 +00 F(O'+) = - )-00 dx (((1/x)f(x) - 11 i -00 dx a(X) f(x)

(III. 2.9)

The rhs of (III. 2. 6) is an even function

of

~

. Therefore, the contri-

bution involving the principal part vanishes and we are left with the following kinetic equation

~p

(t)

_0_

~t

(III. 2. 10)

= (81f

for a weakly coupled gas:

4

X2/ n)

[

.<.

1

i

3

d k 'V \ k

J

2

d

;)

k. ( - - - ) ..... i)P. ;)P. -1 -J

)C.

)( ~ [k. (v .-v.)l k. (_d_ -~)p -J ..... oP. ,.(t) .....

~0 .

.....1

""1 .... J Let us notice that with this equation it is very easy to verify

Boltzmann's

}t!

-theorem. Indeed, with Boltzmann's

)f

-quantity defined

as : (III. 2. 11) the kinetic equation allows us

to write :

;~ ~ - (8ll '}.2/Ql (III. 2. 12)

" [k. ( L -:L)r ...

The function

?i Id\ IvS Sr~· (~;-:::)\

)f (which

~ £.i

~£.j

0

(t)1 2

J

(0

is related to the entropy) decreases monotonically

- 244-

F. Henin

to its equilibrium value. The rhs of (III. 2.12) vanishes once

r 0

has reached its equilibrium value :

f equ. = f(H

(III. 2.13)

The function

0

0

f cannot be determined by (III. 2.12) but is determined by

the initial conditions. We shall see assumption

)

in the next paragraph that the

of initial molecular chaos leads to the Maxwell-Boltzmann

distribution.

III.S - Kinetic equation for

the one particle velo-

city distribution function. Let us call

fs

the reduced distribution function for s momenta:

'fs(;::,I.· .. ;::,s.tl

(IlL 3. 1)

=fd.ES+1 .. ·d~N

Integrating (III. 2. 10) with respect to

fori ~'.t)

all momenta except;::, 1 and taking

into account the fact that the distribution function vanishes at infinity. we obtain:

(III. 3.2)

This equation gives us the evolution of the one-particle velocity distribution function in terms of the two-particle distribution function. Therefore • it is not a closed equation and if we

do not make any

further assumptions. we have actually to deal with an infinite hierarchy of equations.

- 245F. Henin

,t 9),

According to our basic assumption (Ch. II.

we consi-

der a system where all correlations are of finite extension. Now, the velocity distribution function is an average over the positions of all particles of the complete distribution function. When we do this, if we consider two given particles, it is quite clear that the contribution of those configurations where the particles are correlated is much smaller than that of

those configurations where they are uncorrelated. This

allows us to neglect the effect of

these correlations on the velocity

distribution function, i. e. to make the assumption of molecular chaos at t

=0

:

fo

(III. 3. 3)

(t=O)

=

nUl J 1'1

(v., t) ~J

Once molecular chaos is taken as

an initial condition, it can

be shown to persist for all times in the limit of infinite systems 1) 3) We shall

not give the proof here. With the initial condition (III. 3. 3) , we may write in the rhs

of (III. 3. 2) : (III. 3.4)

and we obtain:

I> lD

(v ,t) 1 1 1 t

o~

(III. 3.5) 't

= (8 tJ 4 A2( \1

SrL'"k

m 2) 1

I

i

\vk \2

dV , \ d \

,~J

J

m

. (v -v.)1 '" 1 '" J

A similar equation

1 m,

J

k .:

~

t>

"v -1

)I

~)18 (v ,t)'f (v" uV,Tl"'l l"'J

t)

'" J

can of course be eas:ly obtained for the re-

duced distribution function of any particle of the system. Therefore, we are now dealing with a closed set of equations.

-- 246 F. Henin

Let us now verify that the equilibrium distribution corresponding to the initial condition (III. 3. 3) is indeed the Maxwell-Boltzmann distribution. Using (III. 2.12) and the assumption of molecular chaos, at equilibrium, we must have :

~

(III. 3. 6)

C>

k . (~ - -:::;- )CD1(v ., t) 1,{)1(v ., t) '"

U

I!..i

U

I!..J

I·

"'I

I

,

"'J

=0

whenever k is such that:

k . (v . -v .l = 0 "'I "'J

(III. 3. 7)

'"

We verify easily that (III. 3. 6)

implies:

~lp'f1(~i't) ~.(

(III. 3.8)

• ~p.

~lnf1(~j't) ""J

"'I

Whenever (III. 3. 8)

)=0

i)D.

-

and (III. 3. 7) are simultaneously satisfied, we must

have:

)lnfl(~i) ~I!..i

~1~(V

.) "'J 'Sp. "']

= P.

I!..i - I!.. 0

""J

al and

I!..

o

=

ol

m. J

mi where

- I!..o

are constants .

Integratfug (III. 3.9) , we obtain : (III. 3.10)

In

'f1(v .l = "(P. - P "'I

"'I

"'0

)

2 .

12m. + In V . 1

01

where ~ i is a constant. This gives:

(nI. 3.11)

~l(v.) = v. , "'I • 1

exlel jp .-P

L'

"'I

12/2m.] 1

",crI

- 247F. Henin

The normalization condition requires: 3/2 (III. 3. 12) We also consider systems where the average velocity is zero; hence (III. 3. 13)

D

~o

= 0

~

Defining the temperature through: (III. 3. 14) one obtains eas ily the usual Maxwell - Boltzmann law;:

\D (v) = 41f(m/21fkT)

(rII.3.15)

3/2

11~

In this way, we

have

2 exp(-mv /2

kT)

verified our statement at the end

of

the previous

paragraph.

III. 4 -

B row n ian

mot ion ina

fl u ida t

We shall now consider the simple case 1 moves

in

e qui 1 i b r i u m .

where the particle

a fluid at equilibrium. We then have:

'" (v., t) =

(III. 4. 1) 11 ~J

41f (m./21f kT)3/2 J

jf1 Then, equ. (III. 3. 5) becomes to be equal to m) :

(III. 4. 2)

(assuming the masses of all fluid particles

- 248-

E. Henin

In this way, we have obtained a closed equation for ~ 1{~1' t) . The

~

integration

is

easily performed in

a reference frame where

the relative velocity (III. 4. 3)

[

is along the

z

= ~1 - v

axis:

3 \V \2 k,k.S(k.g) b, = lfB(a lb k 1J "'''' J xg x

n, (d k

1)

(III. 4.4) =1JB( ~ .

g1 b

- a .

1

3

0()

B=

depends only on the The last

0

dk

k

!

yg

b ) y

1

[3" f'~)

where (III. 4. 5)

+a

g

2

'Vk I

intermolecular potential .

expression in (III. 4. 4) is valid in

an arbitrary reference

frame . Using dimensionless quantities : (III. 4. 6)

we obtain:

4>

(III4.7)

w

x In

a

w w i

j

~ (

ml

( - - + 2 - u.l- UU,

m

is along the

z

1

1

i)u.

1

dw e

-(~-~

'"

1 woJ

~

reference frame where

~

axis, we have:

2 l(

- 249F. Henin

f

II = d~ e (Ill. 4. 8)

,+

roo -(w -u) 2 -,... (l/w) = 211 0 dw w

=~ /u)

1 d

-1

cos

0 -(w 2+u 2 -2wucos 0)

e

J0oo dW\ e (

- w-u 2 - w+u 2 3/ 2 ) - e ( ) 1= (If lu)

where

e

(III. 4. 9)

-x

4>l"'-)

2

is the error fUnction . We also have: 2 -(w-u) -3 a. dw e ,... w.w. w b. ~

I

-

1

(III. 4. 10)

I

2

=

I u.b) 1 - '"

I u . b)

2 .............

+1 2 -(w 2 + u 2 wucos 8) dcos 0 cos (J e 2111co dw w ,

10

-1 2 fCO

=

,...

1 b-a.u - n 1- '" '" u 4

- (1/2) (a. I 2b - 3 a. u ~ ....... ....... "'''''U

= err/2u) (III. 4. 11)

J

j

= (1/2)(a.I

dw w e

o

-2 ,~ ="..r 3/2 { U . ' (u)

2 - u -3 (l-u)

dcos

(J

e

2wucos

(J

-1

2~1+1

[2 d -(w 2+u ) dw - 2 w e o dw

r

+1

2

dw

2 )fCO

(If/2u)

d2

_(w 2+u 2 )

dcos

(J

e

2wucos(J

-u + 2e 2 }

-1

+l

(U)j

where

(II. 4. 12)

.lI(U) = d.(U) = (2/ du

'f

f1r ) e _u 2

Introducing (III. 4. 8) and (III. 4.10) into (III. 4. 7) , one can finally write (III. 4. 7) as :

~"l(!!.,t) ='t- 1 \ _-=-__ 'It

a(u)

12

- : - + b(u) "' 2

uu.

1

[~2 'I) (u. - - ) - u. 1

f)u.

1

1

~u.

1

1

.- 250-

F. Henin

where (III. 4.14)

b(u) = u -1

(III.4.15)

a nd where 't' has the dimension (III. 4.16)

'C

-1

= (3211

da(u) du of a time :

6 \2 ' 2 3/2 CAB/ m ) (m/2kT) 1

III. 5. Link with stochastic theory. Equation (III. 4; 13) can be easily written in the form of

the ge-

neral Fokker - Planck equation (I. 5.7) :

';) 'f =~_ (4 ui) 1) t

~u.1

At

'f

1

+ .!. 2

(III. 5.1)

with (III. 5. 2)

4t

=-

u 4-

i u

m 1 -1 ( 1 f-.-: ) g(ul ~ m

where (III. 5. 4)

g(u)

= '12 u -1 [ u -1

+

(u) -

t

I

(u)~

J

This equation has been obtained from first principles as an asymptotic equation describing the motion of a particle of mass at thermal equ ilibrium , with

m 1 in a gas

the assumption of weak coupling. The avera-

- 251-

F. Henin

rage values

(4 u.) ,(6 u. t. u.) 1 1 J

are such

that

the velocity distribu-

tion function reaches monotonically the Maxwell-Boltzmann distribution after a long time. No other assumption than the hypothesis of initial molecular chaos has been necessary to obtain this result. The relaxation time, i. e; the characteristic time for the evolution

of

'fC~.'

t) is given by (III. 4.16) :

(1II.5.5)

As

expected, it decreases when the concentration increases.

It is also a function of

the temperature

and the intermolecular forces

(seem. 4.5 for B) ; it depends upon the ratio of the interaction energy and the mean thermal energy of the particles of the fluid. It decreases whenever this ratio

increases.

If we compare (III. 5. 2) with the corresponding expression derives

from the

Langevin equation, we notice that the

introduces a coefficient of dynamical friction"y}

microscopic theory which

is velocity

dependent:

"J

(III. 5.6)

-1 = 4

't"

Let us introduce the following dimensionless quantities (1II.5.7)

Y= (m/m l ) 1/2

(III. 5. 8)

x = (m/2kT)

'( is

the ratio

particle ; x

is

of the

the ratio

its thermal velocity. With

1/2

VI

=u/r

masses of the fluid particle and the brownian of the velocity of the brownian particle and these quantities, we have:

- 252-

F. Renin

(1II.5.9)

For x «1, (and

r.:::

1) we have:

(III. 5. 10)

fu this case, the dynamical friction coefficient

If

is approximately

constant:

1x» 1, i. e. if the rarticle has a high velocity, the dynamical fric-

tion coefficient is

very small :

(III. 5.12)

Dependence of dynamical friction coefficient on velocity

For

tx

Fig. III. 5. 1 «1, we also have:

(Au i fl. U J.)

(III. 5. 13)

For "«1

=

r

' . . (8/3 1,J

Ii )T- 1

and x -I i. e. for a heavy particle moving with

o

-,

locity in a

medium

Planck equatioll takes

')1'

(III. 6.14) ~

of light particles at the simple form:

therlI\al, ve-

equilibrium, the .Fokkep

- 253-

F. Henin

analogous to (I. 5. 11) . The expressions (III. 5.2) and (III. 5. 3) for the average value of the velocity and square of the velocity have been obtained first by Chandrasekhar 4) • In an analysis of t
systems of stars, Chandrasekhar

first considered single stellar encounters idealized as two-body problems. If a mass

star of mass m

and

perpendicular

m l and velocity

velocity

~,

v

~l

collides

with a star of

the increments parallel (

~ ~111 ) and

(~v~1 1.) to its direction of motion can be easily writ-

ten. The net increments

6. v

such that v

~1

does not change

II and

fl v

I , due to a large number .. of successive encounters with field stars during a time interval t:::.. t

~1

~1

appreciably, are easily computed. Assu-

ming the velocity distribution for field stars

to

be a gaussian, one

obtains: (III. 6.15)

A ~111

(Ill. 6. 16)

l\.~I.L

=,

vI

0

(for more details, see Prof. Ferraro's notes in this volume)

III. 6. A p P li cat ion. These results have been used to discuss transport processes in fully ionized gases. A good account of this can be found in Spitzer's book 5) . However, the application is not straightforward. Indeed, the interaction law in this case is the Coulomb potential: (III. 6. 1)

2 V(r) = e / r

which Fourier transform is :

-- 254 -

F. Henin

(III. 6. 2)

Th8refore the coefficient B which appears in the Fokker-Planck equation is : (III. 6.3)

B '" (e

4

4

/41t )

!

00

0

dk(1/k)

We notice that B diverges logarithmically both at the upper and lower limits of integration. This is due to the fact that the Coulomb potential has an infinite repulsive core at small distances (hence the upper limit divergence) and

has a long range (hence the lower limit divergence) .

The long distance divergence is well known and appears also in the equilibrium properties. In fact, because of the long range of the potential, thE interactions in such a medium have a collective character: configurations involving many particles playa dominant role. Both in equilibrium 6)7)8) and non equilibrium properties, this problem can be settled by

a summation over a well defined class of diagrams. The result of this summation is to introduce a screening effect : in simple cases, the effective interaction vanishes exponentially for distances greater than the -1

Debye radius I( (III. 6.4)

:

2 -I(r Veff '" e e /r

with (III. 6. 5)

One way to take into account

these effects semi-empirically is

to introduce a cut-off at both limits of integration:

(III. 6. 6)

-

255-

F. Henin

The lower cut-off

K takes irito account the screening effect whi-

Ie the upper one eliminates the effect of the very close collisions. The theory of weakly coupled gases may then be used. However, this approach is not very satisfactory and the true way to solve this problem, although we shall not discuss it here, is, within a perturbation theory, to sum

first all relevant contributions.

From (III. 6. 4) , it is quite apparent that

the adequate procedure is not

to limit ourselves to first power in the coupling constant e 2 or the concentration C, but to retain all terms proportional to any power of 1(., 2 i. e. of e C, in the expansions. This summation introduces a dynamical screening effect, i. e. a screening which depends on the velocity of the brownian particle 9~However, if the velocity of the particle is such that (kT / m ) tJ ~< I) the dynamical effects may be neglected and the Debye potential is a good approximation. For more rapid particles, one can still write a Fokker-Planck equation but there appears a further velodty dependence of the coefficients due to the collective effects (excitation of plasma oscillations) .

III. 7 - Brownia n motion in a

fluid which is not at equilibrium.

In this case, we must relax assumption (III. 4. 1) and use equation (III. 3.5) . The main feature is that, whereas, in the equilibrium case,

we have a single closed equation for the distribution function of the brownian particle, in the non equilibrium case, we have a whole set of equations for the velocity distribution functions of the Brownian particle and the fluid particles. Following the same procedure as above for the integrati'on over the wave vector, we easily obtain:

= (8'"

5\2 (B/m 21 l ) dv- \'i 1\

~X.l

g

- 256-

F. Henin

(III. 7. 1)

This second order differential equation can be easily recast in the form of a generalized Fokker-Planck equation:

(III. 7.2)

where the transition moments are given by:

~f( ~,t) ')V. I

-3 -2g.g fI (v, t) 1

I "-

(III. 7.3)

(III. 7;4)

The transition moments are now functionals of the state of the fluid .

- 257-

F. Henin

References

1. 1. Prigogine, Non Equilibrium Statistical Mechanics Intersci.ence, New-York Longon (1962) 2. 1. Prigogine and R. Balescu, Physica 23

J

555 (1957)

3. M. Kac, Lecture on Probability Theory, Interscience, New-York (1959) 4. S. Chandrasekhar, Astrophys. J.

~

, 255 (1943)

5. L. Spitzer, Physics of Fully Ionized Gases, Intersci.ence New-York (1956) 6. J. E. Mayer, J. Chern. Phys.

!..!!. '

1426 (1950)

7. E. W. Montroll and J. C. Ward, Phys, of Fluids, 8. M. Gell-Mann and K. A. Bruekner, Phys. Rev

.!.'

~,

55 (1958) 364 (1957)

9. R. Balescu, Statistical Mechanics of Charged Particles, Interscience, New-York (1963)

- 258-

F. Henin IV.

MICROSCOPIC THEORY OF BROWNIAN MOTION OF A HEAVY PAR-

TICLE IN THE ABSENCE OF E4TERNAL FORCES.

IV. 1 - In t rod u c t ion In the preceding chapter, we· have discussed the brownian motion of a particle which is weakly coupled to a fluid at thermal equilibrium. However, as we have seen, there are no known intermolecular forces corresponding to the weak coupling approximation. Therefore, that problem was rather academic and we shall now consider a more realistic situation. The discussion which will follow will be valid for all cases where the forces are of short range (Coulomb forces can also be included provided the screening effects are taken into account in a phenomenological way). The case of long range forces with no screening (gravitational forces) will be dealt with in chapter VII. For this model we can use as a starting point the kinetic equation (II. 9. 3) or (II. 9. 5) . We shall consider the case where the brownian particle is much heavier than the fluid pc:rticles. For the case of a brownian particle moving with thermal velocity in a fluid at equilibrium at temperature T, we shall show that an equation of

the Fokker -Planck type

is indeed obtained for the velocity distribution function of that particle if one retains only the lowest order terms in the expansion of the kinetic equation in the ratio of the masses of the light and heavy particles • The method we shall follow enables us also to compute the corrections to the Fokker-Planck equation. However, we shall

not

consider this

problem here but we shall rather discuss it in the next chapter where we consider the particle.

same problem but with an external force acting on the

- 259-

F. Henin

IV. 2 - E qua t ion for

the red u c e d vel 0 cit Y dis t rib u -

tion function of the brownian particle. Let us start with the kinetic equation in the form (II. 9. 3) . Assuming the fluid to be at equilibrium. we have

r

0 (t)

(IV.2;1)

= =

f (y. t) (P

f 0 )

equ.

2

f cy. t) ~ (21f kT/m) 3/2 41'e -mv. /2kT N

1

where V is the velocity of the brownian particle (mass: M) while v. -

-1

is the velocity of the ith fluid particle (all fluid particles have the

sa-

me mass m) . Integrating both sides of (IV. 2.1) with respect to the velocities of all

the fluid particles,

we readily obtain an equation for the redu-

ced velocity distribution function

;t

')~v, t)

(IV. 2. 2)

=

It

0

"r Y' t) of the brownian particle:

L

d"C rd~' N G(t-'t)

1(Y' 1: )

(f

f 0)

equ.

which is valid only asymptotically. Let us introduce the operator

r

r (t-"t') = ~\d! \ N G{t- "C) (f ~)equ.

(IV.2.3)

This operator is of course a differential velocity

y

operator with

respect to the

of the brownian particle. (IV. 2. 2) now becomes:

(IV.2.4)

From the very definition of G{t) and (IV. 2.3) • the Laplace transform of

- 260-

F. Renin

r

the operator

(t) is obviously:

(IV. 2. 5) Following the same (II. 9. 5)

procedure as that used to go from (II. 9. 3) to

, the equation (IV. 2. 4) may be written in the pseudomarkovian

form: (IV. 2. 6)

")d)(Y, t) 1

where the rivatives (see

operator

+ 'f (Y"

,,~i'1:l'

(0)

is

'tj

given

t)

in

terms

+

of

(0)

and

~I(O) by the same relation that holds between fl and

its

de-

t

(II.9.6) to (II.9.B)):

The equation

(IV. 2. 6) will

be

study of brownian motion. We equation

of

in the mass

IV.3 -

the ratio

our

basic

equation for

shall now show how

(IV. 3. 2)

reduces

to

an

Fokker-Planck type when only the lowest order terms

m/ M

are retained .

Expansion in

powers of

the

As the fluid is at equilibrium at temperature (IV. 3.1)

it

a dynamical

mass

ratio.

T, we have:

(v.) " (2kT/m) 1/2 1


If the brownian particle moves

with thermal velocity, we

(IV. 3. 3)

V" 0(2kT/M)I/2"

(iv. 3.4)

P " 0(2MkT)

1/2

O(¥(v i ))

-1 "O(r


have:

- 261-

F. Henin

with

't

(IV. 3.5)

=(m/ M)

1/2

«

1

The unperturbed and perturbed Liouville operators may be written in a way which exhibits their dependence on

¥ . We

have:

(IV. 3. 6)

where

L

f o

is the unperturbed Liouville operator

while

r

N

f

(IV. 3.7)

L

=2: o i ~1

A

Lois that for

for the fluid:

'i) v --i· 'i)!:.;

the browni an particle

A:

(IV.3.8) Similarily, we may write: (IV. 3. 9)

with

(IV.3.10)

(IV.3.11) With

these

in powers

expressions, we can easily expand the rhs of (IV. 2. 6) of

~

. Using

(II. 8. 7) , we have

:

- 262F. Henin

Taking into

SL f is

account the fact that

a differential operator with

respect to the velocities of the fluid particles, we have: t(Z) " {\d!

\N (0 \(3Lf +r~L A) ~

(z_lLo SL)n1o)irr(f!)equ.

"Yi~ tr (0 \ h A 11';' !ci".!.'')f¥::l(1';. ~; ".!.' I~ (,: L ~SL)n 10/;"

(IV. 3. 13)

f 'X (fo)equ.

Using (IV. 3. 6) and (IV. 3. 9)

, we then immediately obtain the expansion:

(IV. 3. 14)

with

(IV.3.15)

L. n=1 00

2T2

o

Y

(z)

_1_ SLf n (z-L )' 0

°~ irr

N or =?: L-1o\iL A \ Fl K ~

-

f

}

(fo )equ.

k. = "'1

N (k." -K,Kl KK) f1Jdv\ -

-

-

-1

'" '"

1: (~~L,m '0)' (f) m=O z-L 1rr equ. f

00

o

In principle,

in the

first

term

in

f

0

the rhs of (IV. 3,16), the

sum over

k

- 263-

F. Henin

m shuld go from wever, taking into

1 to infinity instead of from zero to account the fact that

L

A

o

infinity. Ho-

has no non diagonal

elements and ir:reducible contributions only have to be kept (no intermediate

state equal to the vacuum of correlations), we may add the

term

=0 .

m

To the expansion (IV. 3.14) of expansion

of the

operator

+

(z)

V as given

corresponds the following by (IV. 2.7) :

(IV. 3.17)

+ ... Hence,

if

we do not

We shall now discuss the rhs

of

retain terms of higher order than

in details the various terms which

(IV. 3. 18) and show

how

r

2, we have:

appear in

this equation reduces to

a Fokker-

Planck equation.

Stu d Y

IV.4 -

This

operator

given by (IV. 3.1)

for

0

f the

0

per at 0 r

(IV.4.1)

z ~ 0 . Using

lim

~O

I

1( 0 )

is the analytic continuation of the operator

distribution for the fluid, we have:

z

+

R 11m . =(8'" 3If'l) d~ e -iK. ........ z~O

(II. 12. 14), for the canonical

+

1(z)

- 264-

F. Henin

where fle~~, is the Fourier coefficient

of the complete equilibrium

distribution function for the fluid:

r

2

11~

f =exp[ __ equ. kT 11

(IV. 4.

~R-r'llll1( J

mV i t [ V.. (lr,-r,l) tLV, 2 '<' 1J "'1 "'J ,1A '" 1 J 1

'I~\d!:."~ \N exp r-k; \2:(mv~/21 + ~ Vij((!".(!".jll+ Z>

iA

[

1

1

r\~~u,

Introducing

;; fidd

(IV. 4.1) into

N

exp [-i

_I

(I !".il' I II]] 2

, exp [- k~{~(rov~/'1 + VfJ]trd~N expF k~ (f ~vi

(IV. 4. 3)

"'1

+

VIJl'"

~ ~i' ~i] f :qu.

(IV. 3. 6) for z,O, we

obtain:

(IV. 4. 4)

Now, we have :

( 01 sLA ,

(IV, 4. 5) =V K . K '"

Taking into

account

K, K)'dR

k, = -

""1

................

J........

e

-~ ~ te~~. ........1

~(dR e-i~'IlJldr IN

?t J '"

the fact that

1 '" I

VK

is the Fourier transform of

we have: K '"

(IV.4.6)

Therefore,

we obtain:

"'If

iK.r, f e '" equ.

e iK. . . . . (R-r,) .............. ] . . .

'lV'A( , R................ -r,1 \) 1

ViA'

- 265-

F. Henin

ol-ViA(\~ - £'i P

f

l

(IV.4.7)

~

f equ . ~,E 1)

. d,E = 0 and the

equation of

evolution

the brownian particle

(IV. 3.19) takes the simple form:

')f(V,t) ')t

(IV. 4. 8)

for the velocity distribution function of

= _ iy2~

12

•

(0) U)(V,t) ~

\

non vanishing effects are proportional to the mass ratio

The first

m/M. IV. 5 - Stu d Y

0

f the

From (IV. 3.1) and

r

2

(IV.

N

2,nu

t2(O)=tz l N

OG

.. _1n \dR~

01

:

(IV.5.2f~ e-i~I.~ (IV.5.3))

L 00

n=O

1 (-f

z-L

+

2(0)

(IV. 4. 2) , we obtain:

5.1~\w:d <~i = - .5!Sl ~ )1

Now, using

per at 0 r

0

E~

~,tr.~'tLA\~i=-!S'~)

(1

f)n

t

z_Lf cJ L

1 -f

z-L

0

Rf equ {.tl}

A

A

(L 0 +~L )\i~'

o

-iK'. e ........

r[~~! i~'t lL

f)n

KI\

1

-f =

z-L

r~qUlo) nN+1 1 f

z-L - I L

)<

f

0 0 0

and performing the summations over

i ~,t, E,

~

, we

obtain :

k

\ E') )(

- 266-

F. Henin

(IV.5.4) Using

the explicit

t2+2(0)

forms

= (0)-

1

of

A r A Land f) L ,this becomes: o

')~ V'A(lR-r,1l ~ d!!~ 1 1')!!'" "'1 . 'i)t 1C

(N

i!.;OJ\d~t

1~V. (iR-r,l)

(IV.5.5)

J "\dr\

N

f

1

z-L -IL o

[

f

Y.

-

1 1A '" "'1

').\. ~ R +"-')-R---

'"

'"

.

~~

~fequ.

r

f (see IV .2.2) , we obtain easily: equ. ~ t.V, (IR-r. II ';) y 2 ..1-. 2(0) = 0- 1 lim \\d~\ N d!! 1 1A ; "'1 . lP ')( • T z~ 0 'f) '" '" (IV. 5.6) ~~ 1 f N 1 1 V'A(' 1 -R-r. -1 ') (C)- + v)r \d~ z-L f -SL f . 'i~ kT '" equ. 'l~ o

Using the explicit form of

"~ \

Let us introducing the diffusion coeffiCient: DO' = 0- 1 IJ

(IV. 5. 7)

j 1 '" .... \ J dvdr

N dR E )) " fl'(!t, - \-

'X

where (IV.5.8)

is the total force operator (IV.5.9)

t

exerted by the fluid

on the Brownian particle; the

2(0) becomes:

r

2

'i)

f2(0)

=Dij ~Pi

The evolution equation (IV. 4. 8)

~ 1 (- - + - V ) 'itp. kT j

J

may thus be written:

- 267-

F. Renin

(IV.5.10)

This is

indeed an

equation of the Fokker-Planck type. An explicit

expression in terms of microscopic quantities is now obtained for the diffusion coefficient. IV.6 -

Diffusion coefficient.

Taking into account: (IV.6.1)

the diffusion coefficient may be written as : Dij (IV. 6. 2)

=lim \00

z..., 0

dt (S1)-l

0

\\dtd~ \N

dg Fi

(~, \t})

)r

'" exp f_i (L f +SL f -z)t}F.(R,\r l ) rf l 0 J~ ~I equ.

Taking into account:

Lfp~ =(Lf+~Lf)rfequ.

(IV. 6. 3)

I~.

=0

0

and the fact that the integrand is a function

of relative distances only,

we obtain:

0
"

!':

0 \:
lId! "!: \ N f ~qu.

exp [-i (L f -Z)t]

= lim \: dt z",O

(Fat

F~

(g.

Fe @. Ir I) •

l!:i)

(g ,\t\) exp [- i (Lf_z) t1 FA (!!·

r

h.\)

The diffusion coefficient is thus the average value over the fluid equili-

- 268 -,

F. Henin

brium distribution of the time autocorrelation function of the force acting on the brownian particle for a fixed position

of the brownian par-

ticle. As an example, let us compute this coefficient in the case of a weakly coupled system. Then we have : (IV. 6. 5) 3N

r~qu.

(IV.6.6)

= (m/21fkT) 3N!2 (4111

Expanding the interaction' potential in \ 2

D

oI~

e

= -"

-1'( L f

0

(n)

-z) t

N

-N

e

exp [-

~

mV l 2] 2kT

_N

Q

lim L: i, j =1 z"" 0

a Fourier series, we obtain:

f

00

dt

0

(t drdv l N

JI ~ ~I

(m/2WkT)3N!2

ikl.(r,-R) ~

~J

.....

Now, we have: (IV.6.8)

e -1'~ t

ik I. r , ikl . (r ,-v ,t) e- ..... J = e..... .....J-J

With: (IV.6.9)

ik. r. {dr, e .............. 1 ~1

= 8lf3

S(!l

we verify easily that the only terms which contribute are those for which

i = j . This result is of course quite obvious . Indeed, the only

contributions to the evolution equation are those of diagonal diagrams, in which each particle must appear at least at two vertices. In the weakly coupled case, we only have to consider the cycle and have

the-

- 269F. Renin

refore only one fluid particle involved.

J ~l ei(~+k":')' ~i

Therefore, with (dr.

= 81T3

~ (k+k') ~ ~

(IV. w.1D)

we have:

D.~ = 321T4~2C(m/21fkT)3/2 z~nu ,: dt

(IV.6.11)

lC

\d\lvk \2

k

k

J. (!>

ik. vt e- -

e

2

exp (-mv /2kT) )( izt

We also have: (IV.6.12)

As the remaining part

!,

ction of we

are left

D,.f> =

of the integrand in (IV. 6. 11)

is an

the contribution involving the principal. part

even fun-

vanishes and

with:

5\ 2 3/2 , 2 3211' A C(m/211' kT) ) d~ exp (-mv /2kT)

(IV. 6.13)

Jd3k \Vk \ 2 kJ. kp

11

(' x b(~.!.)

One verifies (IV.6.14)

where

1 = (1 • 1 • 1 ) is the unit vector and B ,.., x y z

Performing the

is given by (III. 4. 5) .

::! integration, one obtains :

(IV.6.15) One verifies easily that with this value of the diffusion coefficient, the evolution equation (IV. 5. 10) is indeed identical with the particular Fokker-Planck equation (III. 6. 14) we obtained for a heavy particle moving with thermal velocity and weakly coupled to the fluid.

-- 270-

F. Henin V. BROWNIAN MOTION OF A HEAVY PARTICLE IN AN EXTERNAL FIELD V.I - Introduction. We shall again consider the problem of a heavy particle moving in a fluid at thermal equilibrium. However, now we shall assume that the brownian particle is charged and that at

t

=0 ,

we switch on an

external constant electrical field. The fluid particles are neutral and are not influenced by that field. After a long time, we shall reach a stationary state for the velocity distribution function of the brownian particle, corresponding to a balance between the effect of the external acceleration and the scattering by the fluid particles. Our starting point for the discussion of this stationary state will be the transport equation (11.14.5). Here again, we shall show that, when only lowest order terms in the mass ratio are kept, the equation for the stationary state is in agrel!ment with that of the stochastic theory. The calculations which

have been performed originally by Rl!si-

bois and Davis 1) will be closely parallel to

that of the preceding

-chapter and we shall go over them very briefly and rather concentrate ourselves on a discussion of higher order corrections. First, we have corrections to

the collision terms which are independent of the external

field and introduce fourth order differential operators in the equation of evolution. Then, we also have corrections which take into account the effect of the field during a collision. We shall

show that these correc-

tions may formally be incorporated in the Fokker-Planck collision operator. These results are in agreement with those obtained through a rather different method by Lebowitz and Rubin 2) . In order to make connection with this work. we shall show how the transport equation for this parti-

- 271-

F. Henin

cular problem can be recast in another form, which is precisely that used as a starting point for the m/ M expansion in Lebowitz and Rubin. This point has been discussed in great detail in

a paper by Lebowitz

and Resibois 3) •

V. 2

~

Steady state equation for the velocity

s t rib uti 0 n fun c t ion

0

di~

f the b row n'i a n par tic 1 e .

In chapter

II, ~ 14, we have obtained the linearized steady state

equation (II. 14. 5)

for the velocity distribution function of a system of

charged particles submitted to the action of an external electrical field. We have assumed that

at

t=O the system was

in equilibrium and that

the field was switched on only at t=O. Keeping only terms linear in E and restricting ourselves to the static case, we obtain:

(V.2.1)

iLE

r

where we have

equ. ,..

+ '- D (0) iLE C(k(O) ,~\ l!s\ ~

o

st f equ = ""(O)Ar . 1 i

0

0

used (II. 13. 18) to rewrite the second term

in the

1hs of (II. 14. 5) as an operator acting on po' As the brownian particle is the only charged particle we have here: (V. 2.2)

instead of (II. 13.3) . In (V. 2. 1) ,

f equ 0

is the velocity equilibrium distribution for

the whole system :fluid and brownian particle (V. 2. 3)

r

equ = ..;equ 0 1

V)

C...

rrN d)equ l'

(v.) -1

- 272F. Henin

(V. 2. 4)

(V.2.5) A

u

r

st 0

equ P o ' As previously, we

is the linear ( in E) correction to

assume molecular chaos for the velocity distribution function. Therefore, we have:

6

equ

N

f

st=

~ 'f(V) Tf

0

-

(V. 2. 6)

+

1

t{}

Q:i)

\

N

f:. ~\D(~.)

i =1

,

Tf \D equ(~.) ",equ(Yl jf=i

1

J

1

\

Therefore, (V. 2.1) becomes:

Integrating this equation, first over the velocities of all the fluid particles, secondly over the velocity of the brownian particle and the velocities of all but one of the fluid particles, one obtains easily a set of coupled equations for the two unknown functions can be proved

l) that,

\

'frv

and

~'f(v.) ~l

. However. it

once terms of order lIN are neglected the

velocity distribution function of a fluid particle remains at equilibrium in the stationary case:

~(v.) = O(l/N) ~O ,~l

(V. 2. 8)

We shall

not prove this

the probability of a given

here but it is a consequence of the fact that particle to interact w·ith

the

single heavy par-

- 273-

F. Renin

cle is quite negligible in the limit of an infinite system. This allows us to neglect all

vertices involving A and a fluid particle in the ope-

f

D \ k\' C\k\and

rators

The equation for &"

y. :(2' .. :( N

once we integrate over

'il(:( 1) then simplifies a

great deal and using

. argu-

ments similar to those which allowed us to establish the')( -theorem in chapter II, ~ 11 it is

easy to show

that (V. 2. 8) is its only

solution. Taking into account

(V. 2. 8)

,

the evolution equation (V. 2. 7)

becomes:

(V. 2. 9)

After integration over the velocities of the fluid particles, we obtain: (V.2.10)

where (V. 2.11)

(V.2.12)

,.....

~(O)

The operator +(0) is

.,- I.

N = ~\)id!\ D

identical to

N

f'j)

t!l (0) e~ . u~

C\,M0)

"IT equ

'tf

(:(i)

the operator +(0) given by (IV. 2. 5)

- 274-

F. Henin

V.3- - Expans ion in the mass ratio . Here again, we wri te : (V. 3.1)

~L=~Lf+¥SLA

(V. 3. 2)

Ii)

(V. 3. 3)

where

e~.

') f

the fluid operators

=

¥L E

f (' f

L ,iL

are given by (IV. 3.7)

a

respectively while the particle operators by (IV. 3. 8)

and

+(0)

The zero order term chapter

= yc¥l)(O)

2(0) =

(V. 3.5)

L 0 and

IV. The zero

~

j)

L

A

and (IV.3.10) are given

of

t 2 ~ (2) (0) + ... (0) +¥ 2 Z (2)(0) + ...

+

¥2(1)

(¥ 0)

Z (0)

~ (0) and

;

vanishes as

we have seen

'±

in

order term of 2,(0) vanishes because this

· h'IS operator involves the external field operator w hle Up to order

r

('

(IV. 3.11) .

We also expand the operators (V. 3. 4)

~

A

0

f order

,we therefore obtain the following equation

tion:

i LE~equ(~) + [¥2(1)(0) +~2 ,

+~ 4 2. (4) (0)) 'fequ (V)

,-, (2) &.:-a

= i [y
(0)

+

3 ,..., (3)

+ Y ..:."

r

2 +(2) (0)

(0)

+

't' of evolu-

- 275-

F. Henin

In chapter IV,t~ 4

and 5, we have shown:

(V. 3.7)

,(1)(0)

11 T .

(V. 3. 8)

2 ",,(2)

=0

(0) = Dij

~

'i) 1 'lP. ( "P. + kT 1 J

Vj )

where the diffusion coefficient is given by (IV. 5. 7) . An alternative expression in terms of the average autocorrelation function of

the force ex-

erted by the fluid over the fixed brownian particle is given in (IV. 6. 4) . As to the interference term between the flow and the collision, it is again easy 10 show

that:

2

(V. 3.9)

(1)(0)

2,(2) (0) = 0

=

Indeed the term of order ~ in

(V. 3.10) \' 2,(1)(0) =

•

V

0

(0) where Di~J(O) and

the

rhs

L. (id~\N D(~\)(O) \k\)

(0)

C1~\(0)

t~

of

(V. 2.12) is

(0) iLE C U(O)

f

1TNi 'fequ (~i)

are the terms independent of

ction and creation operators:

..

-~ )Ii~t) z-L

(V. 3.11)

o

(V. 3. 12) o Taking into

account

we easily verify

(V. 3. 9)

for the operator

y of the destru-

- 2i6-

F. Henin

As to the operator

V2

o

2(2)(0), we have:

2 (2) (0) =y2 [.~dV' N D(l) (0) ,~\II ~

(V. 3.14)

2.,.{£

+

¥

lt~l

N

Using the same

argume~t

easy to

that

show

iL e(O) (0)

E i~'

\U

(0)

(1)

Di~\(O) iLEe,~}(o)

Tf "equ(v.) i \

"'1

TT

I ~ ~equ(~i)

as above on the structure of

~(~\ (0)

the second term in the rhs vanishes . As

, it is

to the

first term, we have:

yD~~~Z) = (V.3.15)

to (O\XL

A

z_lLf(IL f

+f.. t
z_~~f\i~')

o

n=l m=O Again the second term

gives a

vanishing contribution when we integrate

over the velocities. Therefore, we have:

Y2 2(2) (z)

=V •

2

ttL ~\d~\ n=O

m=li~\\

Now, using (IV. 4.1) , (IV. 5. 2) and (V.3.17)

Y2 2(2)(Z)

N

(0\ ~ LA

z-L

f o

(IV. 5.3) , we easily obtain:

=~2 ~\d~\ N(O\dL A Z_L/_1L f fequ \0) iLE o

From (IV. 3.11), (IV. 6.3) and (IV. 4. 2) , we easily obtain:

- 277-

F. Henin

,tViA(\ (.i-~l)

(V.3.18)

2. (2)(Z) J 2 (1/z~\~ d!d~ \ N 1 "a~ =

. ;£r~qU

=0

(see (IV. 4. 7)) . Therefore, if we restrict ourselves to terms of order mlM, we easily recover the stationary Fokker- Planck equation in presence of an external field acting on the heavy particle: (V. 3.19)

The diffusion coefficient is not affected by the presence of the external field. V.4 - Higher order corrections to the collision operator. Role of the irreducibility condition. contributions are respectively:

- 278F. Henin 1

(V.4.2) X

z-L

f

o

n"

x

i

,

equ(v.) ~l

The first point we want to discuss here is the role played by the irreducibility condition. To do this, we will need a theorem first established by Balescu 4) : If we hrve a succession of diagonal fragments, the only contributions

to

the reduced distribution function of particle 0( which do not vanish

at the limit of an infinite system are those where the diagonal fragments are semi-connected, i. e. wher e with the preceding

they have a single

particle in common

diagonal fragments.

We shall not prove this theorem in full generality but illustrate it on an

example. Let us consider a succession of

we have three cases Q(

two cycles:

(see fig. V.4.1) •

0 6 \. .e. o Ol~) d

(a.)

~

- 6. C> \ d

~

'I

0 0 " el

.,

lb')

(e)

~

Possible connections in

a succession of two cycles.

Fig. V. 4.1 1) they are disconnected: no particle in common (a) 2) they are semi-connected: one particle in common (b) ,(b') • 3) they have two particles in common (c) .

- 279-

F. Henin Because of the integration over the velocities of all particles but

c( ,

o(must

necessarily appear at the first vertex on the left. We obtain: (al

0>

= trJld~~N-l
(V.4.1) iol&Ljllk.,k =-k)(k.,4: =_k.I~Ljll O>fr(O) "'J "'1

"'J

-J -1

-J

0

Now, the integrations over v. and v commute with the first two matrix ele-J -1 ments. Therefore, we have: ·1 (a) dv. dV l dLJ ... = 0 -J -

f

This argument may easily be generalized to more complicated fragments and successions of more than two fragments; whenever there is at least two disconnected fragments, the contribution to the reduced distribution function vanishes. Let us now consider the case of the semi-connected diagrams (b) and (b ' ) • We have:

(b)+(b~ = ~(i{dvltl-l(OIILic:(lk.=_k.J ,k;.' 1JJl _\ -1 -", { ij Lk.=-k ... ,k...... "" \bLi«,\O?.(oIJL I .......k'.,k.=-k ."/ ....... 1 ....... 1 ....... J . . . . '1 (v.4.3)~k!,k.=_k'.IJLijl 0>+Lo I6LO(j(k ' ,k.=-k -1 -J -1 -c( -J -0( ~

1

l

k' ,k. = -k IJLO(j I.. -0( "'J -01 l

10

)

>

It is easily verified that none of these contributions vanishes;indeed, in the first

case, the integral over v. does no longer commute with the first two matrix -1

elements; in the second case the contribution proportional to

d /ax'd. of J L

C{

.

J

in the third matrix element is non vanishing. As we have already seen «II. 6. 6)), the contribution of each cycle is proportional to

-1

n ;because of the double

summation over i and j, we obtain a contribution proportional to C2. Such
-1

n ; however, we

only have one summation over i;hence we obtain

a contribution proportional to C/ n , which vanishes for

n~

00 •

- 280-

+

Let us now suppress the irreducibility condition in the operator

(0) and discuss the error introduced in this way. We consider a

4> (0)

given contributiQn to

with m

vertices:

and suppress the irreducibility condition on a given intermediate state (say after r vertices) . In thi.s way, we add to

'" (m ) the following con-

tribution :

0( (V. 4. 5)

=}'\d~,N(O\XLA(z_~o 3L)m-r '0) irr

)(

" (0 ,(2iL)r '0). rr"equ (v.) z- L lrr . 1 1 o

As we have seen

1

such a contribution

limit of a large system

will

be different from zero in the

only if the two diagonal fragments are semi-conne-

cted. We have two cases: a) the semi-connection

is through one fluid particle; but. then ,A does

not appear in the last fragment on the right and we have:

(0 \ (z-~o ~L{ \0) ~~equ(Vi) (V.4.6)

=

(0 \ (~~Lf{ \0) ~ 'fequ(~i) =0 z-L

o

if we take into account (II. 11. 10). b) the semi connection is through particle A. Then, all the fluid particles in the first fragment are different from those in the last and because of the integration over the velocities of the fluid particles, the first vertex in the second fragment must necessarily involve A ; hence:

- 281-

F. Renin

ot

=

~\{d!\N-S(a'&LA(z_lL S L)m-r \a)irr ~

(V. 4. 7)

)(~id~\S (a'~LA(z_~ ~L{-l

\a)irr

o

if

"

o

{l dv l

Jl ""

s

TI''fequ(~i) 1

Jl dv"'1.... dv. ,where i ... i "'lIs

are the fluid particles involved

in the second fr1 gment s . W e easily recognize that the second contribu-

+

tion is a contribution to the operator

r

given by (IV. 3.13) . As we have 2 seen, the first non-vanishing contribution is of order and we have:

a(

=

Yhd~ \

(V.4.S),,¥2

N-s

(a, ~ LA {z-~ J

L)m-r \ a)irr

o

~\\d~\s
['tequ(~)1 N-s

)c

a)irrl,e qu(;(l1 s

o

With the same kind of arguments, it is again easy to convince oneself 2 that the fragement on the left is also of order 0 at least. Thus, if we supress the irreducibility condition in the operator ; of at least order four in add

(a) , we add a contribution

l . The contribution of order

r

4 which we

is: 00

L m=a ~

A

1

(SL A + LJ

(_~f

(V.4.9) X (dL

J[

A

+Lo ) - f z-L o iLr

10)

irr

o As a conclusion, we may suppress the irreducibility condition in If we suppress

it (which will

4> (3)(0).

appear convenient below) in +(4)(0), we

have to subtract the contribution

(V. 4. 9) . Because of the factor

we easily notice th at this f'ontribution will diverge at the limit

1/ z z ~.O.

,

- 282-

F. Henin

This is easy to understand. Indeed, a product of two irreducible dia2 gonal fragments brings a factor t in the formal solution of the Liouville equation as compared with the single t factor brought by one In the long time limit, the

irreducible fragment.

first one diverges. The role of the irreducibility condition in t(z) is precisely to suppress such contributions. This is well~known in the discussion of the three-body problem 5) . The operator

f (z)

brings

in the evolution equation only the contributions of genuine three-body collisions, i. e. of those collisions where the three particles interact almost simultaneously (i e. on a time scale of the order of the binary collision time) . The suppression of the irreducibility condition would amount to the inclusion of those three body processes which are a succession of 2 two binary collisions and would introduce a divergence. However, it is often convenient to write ble contribution

,\,(Z) as a difference between the reduci-

(which includes all three-body processes, whatever

the time ordering of events) and the reducible term (which describe those processes which are the result of succession of collisions). Both terms diverge but the difference is finite; the cancellation occurs only for the diverging parts. We shall see an example of this procedure below.

V.5 - Higher order corrections to

the collision operator.

Explicit evaluation. The above discussion shows us that we may forget the irreduci-

t

bility condition in the third order operator; it is then with the arguments we used in chapter IV, We obtain:

a simple matter,

5 to compute ;(3)(0).

- 283-

F. Renin

where the force

~

acting on

the fixed brownian particle is given by

(IV.5.8).

As the potential is spherically symmetric, one verifies easily that all contributions to the rhs of (V. 5.1) vanish for symmetry reasons. Therefore: (V. 5. 2)

~

3 +(3)(0) = 0

Let us now cOl1sider the fourth order contribution. If we denote by ¥4)(0) the operator +(4) in which we suppress the irreducibility condition, we obtain: (V. 5. 3)

- 284-

F. Henin

where

is given

01.

by (V. 4. 9) .

Following the same procedure as that used to compute +(2) and

~(3),

we have:

-

~

~R,

J

+ F,(R, J-

C) - + F 1(R ~R u

1

1

I> \r- \) ,}P. J

'X

'"a ,t r_lL ) -'i)P 1

1

)(

This is a fourth order differential operator with respect to the velocity of the brownian particle . It diverges as can be easily verified if one keeps only lowest order terms in the coupling constant. As to the operator

~ ="o4 z~ lim 0

TJ'f 1

equ

cI.

t

p=O

(~i

)

which we can easily rewrite as :

,we easily obtain:

- 285-

F. Henin

(V. 5. 6)X

ni'"

equ (~l' )~d ~ \N/, 1 ,0 ,L A f , z_Lf_ ~L o

It

(~LA+LA)rf II 0

equ

\0)

Using

r~qU' 0) (V. 5. 7) =

=

L.1nJ,~)(i~\' ~I f7 U 10)

\t~ ~

C

I\~\ ,~~~" ~lrefqUlo)+ lo)il~equ(~.)

{t\, ~ f{ 2-\

we have:

L

{t'}~~\ (V. 5. 8)

=2

{~'\~'

_)

<~k~\,~11

;

z-Lf-

~L

<~ t' \ ,~' I :

fl2-\ J.

U'\~' r

f 10)TT"equ(v i ) i f

z-Lo-~L

r

flo) equ

l~k~ 't, KII----':-lf \o~ ~ f '- f f equ ' I irr \0\ z-Lo-"L

Therefore, we obtain:

(V. 5. 9)

i

1

irr

- 286F. Henin

Using

(IV. 6. 1) and the convolution theorem we easily obtain:

'It =V 4 1

•

lim

z"" 0

roo

it

dt 1

0

1

0

f

K requ Again this is a fourth order differential operator with diverging coefficients (this is again easily verified if one takes into account only lowest order terms in the coupling constant) . In the contribution ol 2' because of the irreducibility condition in the diagonal fragment on the left, none of the propagators is identical to

z and this contribution is perfectly finite at the limit z

~

0 .

Introducing the operator: 1 f

t

z-L -elL o (V. 5. 13)

f

t A A (OL +L 0)

f

1

z-L - ~L o

f

f

Xrequ '0) lrr .

we have (V.5.14) if we take into

account

(IV. 5. 6)

The irreducible operator +(4), which does not diverge limit

z~

(V. 5.15)

0, may

thus be written (see (V. 5. 3) ) :

at the

X

- 287-

F, Henin

The infinite parts in

,.., +

(4) and 0( 1 cancel each other .. Introducing

the fourth order differential operator:

a sum of two finite terms:

The advantage of this form will appear once we calculate the corrections to the now term, Then shall see thatthese exactly compensate the effect of the contribution

~

2 .

_

One verifies easily that the operator ,(4)(0) may be written:

- 288-

F, Henin

(V,5.IS)

t dijkl VI vjH

:here = lim ijkl ~ 0 z-, (V.5.19\

Xe

1

_IF \

b iJ"kl

2

0

_i(Lft

It2 dt {IF -i(L

Fie

~Lf_z)(t

I

t

dt

0

0

_i(Lft e

I 2

f

3 Fk )

-t ) .i(L f 1 2 F'\k e 0 jr\ 1

11 dt 2 r 2 dt IF" 1 3:~

t~Lf_z)t

_i(Lft e

3F )

}

k

SL f -z)(t

-t )

1 2 F" v ,. J

0

1

f

+~L-z)(t

0

f

-t) 2 3

f

f

f

'"

-i(L +SL -z) (t -t ) r;) -i(L +k -z)t "" u -e 0 2 3 --e 0 3F

~~

x

(' f ten-Z)t

0

0

f: J

t

0

~Lf_Z)(t -t )

0

31' i

0

0

<Xl

-i(L

(V.5.22)

dt

i

f

e

Itl

-z)(t 2-t 3)

e

(V.5.29)

X

f

lim

z-=t

1

0

~L

-i(L of t

dt

00

~Pk t k~ Vk ,

~~

k

>

- 289-

F. Henin

where (V. 5. 23)

The form (V. 5.18) may be compared to the results of Lebowitz Rubin ; it

and

in complete agreement with their only correction to the

i.S

lowest order Fokker-Planck equation.

V.6 - Higher order corrections to the flow term.

The first correction due to the effect of the field during a collision is given

by:

¥ 38(3 1(01.(QU CD ~ N3~d! IN ?E D\~~(OIH 'E'j~ I(O)(f ~I 'qu'f,q\YI (V. 6.1)

+~ 3 (\dV\ N L.. D(:)(O)iLEC(~) (0) ,~ \~\ \~\ t_\

(ff) 0

~equ(V)

equ

~

The third term vanishes because of the integration over the velocities of the fluid particles. As to the second term, one verifies easily that one has:

(V. 6. 2)

=

I

~

01 irr

1k lj / f '1~1 z-L -AL o

(r

f) 0

\fequ) equ I CY

Pf 12. (~I.:'. + kTl y) I equ- 41 . _ -

1 0''I ",equ(y) 1-

=

0

- 290F. Henin

In the first term, we take into account the fact that L with

"f . I equ

(V. 6. 3) )l

E

commutes

We then obtain easily:

1

f

z-L -~ o

Lf

r

f

1'L E

equ

if we take (V. 5.13) into account. The fourth

order correctIOn vanishes for symmetry reasons.

V.7 - Stationary transport equation up to order

¥4 .

Summarizing the results of the previous paragraphs, we have:

(V. 7.1)

(V. 7.2) where the Fokker-Planck operator operators

--(4)

cp

(0) and

--(2)

f

~ (2)(0) 11

is given by (V. 3. 8) while the

(0) are given by (V. 5.18) and

(V. 5.13) respe-

ctively. Therefore , up to becomes:

(V. (. 3)

4 order ~ , the stationary transport equation

- 291-

F. Renin

Expanding

~~(y)

in powers of

¥ ' we have:

~~y) =Y -1 [~f(Y)l 0 + ~ [~(y)]

(V. 7. 4)

1

which leads to : (V. 7.5)

(V.7.6)

which we may rewrite as : iL E

(V. 7.7)

1eQu(y) ,,[y2+(2)(O) +t ¥4)(o)1Sf(V)

which shows that the corrections due to the action of the field during a collision can be formally incorporated in a modification of the collision operator. The corrections to the lowest order Fokker-Planck equation are thus entirely given by the operator 't(4)(O) . Part of these corrections may of course be incorporated in a modification of the diffusion coefficient D which appears in the lowest order equation. Equation (V. 7.7) agrees exactly with the transport equation derived by Lebowitz and Rubin.

V.S -

Alternative form of the transport equation

for the brownian motion problem. Through a rather different method, Lebowitz and Rubin have obtained the following transport equation for the velocity distribution function of the heavy particle: (V. S. 1)

~ ~J, t) () t

+ iLETeq\y)

r

=)

0

"rr d't'J\(t-'t

)~cry,~)

- Z92-

F. Renin

The collision operator

...,

(V. 8. 2)

where

)\(t)

f

X is given

= -(21f i)

by :

-If

dz e -izt K(z)

is a projection operator:

r

/.) ... =

(V. 8. 4)

U

f Ct dr dv l N equll - -J

The remarkable feature is that, except for the trivial flow term, all the dynamics of the problem has been incorporated in the collision operator 1(

; at

first sight this seems to be in contradiction with our ge-

neral result of chapter II. However, we have already seen that, up to fourth order in

~

, in this brownian motion problem,

the corrections to

the flow term

in (V. 2. 9) can indeed be taken formally as a modification

of the collision operator. We shall not give here the original derivation of (V.8.1) which can be· found in ref. 2) , but rather concentrate ourselves on the equivalence between the two transport equations for this problem. First of all, we shall show that the solutionsof (V. 8.1) are identica] to the solutions of the following equation:

~8 'f (V, t)

------+1 iL .,equ(V) + 1)t E1 -

rt 0

d 'C ~ (t- "r )iL "equ(V) E I ....

(V. 8. 5)

where

A (t)

and

X(t) are the inverse Laplace transforms of the operators:

-

293-

F. Henin

-

(V.8.6)X(z)

=

Jl

L

I (l-I)(L 0 + 0t L)I -i ldr..d~1 Nt tiL A Z-(1-I)(Lo+~L)

where I is a projection operator: f 1. .. = (r2) -N . (Q)

(V.8.8)

, 0

equ

f equ f

i \

tdrdv N ~-

(I involves the fluid velocity equilibrium distribution function while

~

involves the complete fluid equilibrium distribution function for a fixed position of the brownian particle) . Using Laplace transforms, one easily obtains the formal solutions of (V.B.I):

~

(V. 8. 9)

0'f(z)

=[ -

iz-K(z)

] -1

i(z+w)

-1

~.

y 1equ) (y

and of (V. B. 5) : (V. 8. 10)

~ ~ (z) =

[-iZ-

X

(z) ] -1 [1 +

li (Z)) i(z+w) -1lf.. YTeq\y)

(We consider here the case of an oscillating field as given by (II. 13. 12) )

In order to establish tre identity of these two functions, we have to show that the rhs are identical, or equivalently,that we have:

where

f(}~")

is an arbitrary function of

Y;

Now , using the identity, valid for an arbitrary function

- 294F. Renin

Ir.

=

t:> ~1)A(V,j rl,lvt) (U .... 1 .... J l.:.

(V.8.12)

=(1-1)

f equ

rf

- (n) ~N (f)f equl~........ r dv l N J

A

0

(\d:td~ \ N

A

equ)

as well as :

1

1

-z--(l--6''X')-(L---'-+'''''~L''''''') = z~(l~I)(L +~L)+(r~I)(L +JL) o 0 0

(V.8.13) =

=

1

z-(l~I)(L

o

+1tL)

1 z-(l-I)(Lo+iL)

I

1+1~I

(

I

1+

(

f,tdrdv ~ 1 L+dL )f equ . . -\ N(0 ) z-(l-Ii')(L + ~L) 0

1~1

f~ drdv

)fequ

....

tA J N(LA+"L 0

)

1 } z-(l-f)(L o+IL)

we easily obtain:

)C

(L

o

+ 1.L)~f

fLY)

I equ

(V.8.l4) +

i" 2tJd.Fdy\N~L p-"I

X f\drdy'\N Now, if : (v. 8. 15)

we also have:

\

A

1

z~( 1 ~I)(L + I L)

o

(l-I)ff equ

LA)c

Z-(l-~)(Lo+'L) (l-ij') (Lo+ 3L) f~qu f(~)

0

- 295-

F. Henin

(V.8.16) Applying the operator

f

to both siges , we have:

Jlr.zOf (J.drdvlNr=O I equ)' .... _I

(V.8.17)

which shows us that the second term in the rhs of (V.8.14) vanishes. Hence,

(V.8.18)

X(L +SL)ff f(V) o equ-

The Ihs of

(V. 8.11) can be written, with the definitions (V. 8. 6) and

(V. 8. 7) :

[X (z) -izA (Z)] f(~") = -i~dr.d~ \NlL A Z-(l-I)tL o+SL)

(1-1) )(

(V.8.19) 1C \

z + (L +SL)Ilr f f(V) o I equ ....

In order to prove identity (V.B.ll) , we thus have to show that the quantity 0( given by :

~ =f~d~~At\N~LA Z-(l-I)~L +~L) \(1~) o

(v. 8. 20)

S

" (l-l):L + L) I - Z(l-I)} o

f fequ

(Lo+&L)

f(V)

-

vanishes. N ow, using again (V. 8. 12) , we have: 1(l-
(v. 8.21)

(1-1)

{

f

(L+'L)(l-I)f o equ

-r

f

~

f(V) = .... N

f

f

Idrdvl (L+SL)f -zr If(V) equ I .... -I 0 equ equJ '"

-296 -

F. Renin

*

We also have:

f

. f Jdrdv l N(L t~L)r "'YL A~drdv\ N f + 1 - -I 0 equ 0 0 - equ (V. 8. 22) f =V LaA til• ~-r dv-Jl N!L A equ

f

r

o

(V. 8. 23)

r

rdv\ N~L A f - equ

vL

A

0

1\

f(V) = 0 '"

If we combine (V. 8. 20) ,(V. 8. 21), (V. 8. 22) and (V. 8. 23), we obtain

_[ (J

Q. - - J1drd~d

(V. 8. 24) )C

N (' A f ( N~ A 1 lC uL (l-I)r equ - J\d£. d~ \ cl L z- (l-I)(L +cl L)

(1- 1) 0

f,

I equ

Now, we have:

dr dv l - -I

N

1L

1

f.equ

A f

o

f( V )

'"

N A f f( V) = 0 ~ J1 dr- dv---1l AL f equ-

(see IV. 4. 7)

Therefore, we indeed obtain:

0(

(V. 8. 26)

and the identity

=0

of the solutions of (V. 8.1) and (V>.8. 5) is thus established.

Equation (V.8.5) has the same structure as the transport equation we obtained in Chapter II, (V.8.27)

~

13. In the steady state, we obtain:

iLE'feq\~) +

£\(0)

To establish the equivalence with (V. 8. 28)

iLE~eq\V

=XiO) ~'fCy)

(V. 2.10) , we have to show:

F. Henin

x

(V. 8. 29)

(0) ~~(~) =

f

(0)

~'fC~~)

Let us first consider the lhs of (V. 8. 29) . We have:

(V. 8. 30)

if we take into account :

IL of

V(8.31)

=

0

(V. 8. 32)

Through a straightforward expansion, we obtain:

However, we have:

<~!t, 'f\I(L: + &L)H !'~,!S ')F( \~ \' y)Sl,: k .+K, 0 SF t'tI£, Q. " Ll NdR exp [ . r k .• r.1e -iK.--- ---R n-N·(rd ~dr'dv'.l N~LA(R,V) =jIdr 1 ""1 .......

..............

(V. 8. 34) "

=

-1

i ...... 1

f

--1

equ

""

'"

0 ........

.......

S

k!+K' ,0 'Ti ......................

"VS!i' Q.~!S' Q. S~ t'i+!S', Q. (f:)equ~\d~"N
- 298-

F. Henin

A As a consequence all irreducible contributions involving ILo or I ~L

in

(V. 8. 33) vanish (see the product of Kronecker's deltas in the

rhs of (V. 8. 34). Let us now consider a reducible contribution made of a product of two irreducible contributions :


(V. 8. 35) 'Ie

q

Al--&\ (L f +(I-I)(L A+~L)"O). (rf) S,,(V) o z- 0 '0 0 lrr 0 equ I ....

(V. 8. 34) shows us that we may drop the operator I every where in the second

fragment except at the left. (combine the Kronecker's deltas in (V. 8. 34) and the irreducibility condition). Let us first consider the case q=O . Then, using (V. 8. 34), we have: (V. 8.36)

If q

is different from zero, we have:

(0 H[(1-1) SL-IL A]_l_ \q, Lf +(l-I)(L A+SL)\l~. (rf) \f(v) o z-L 1 0 0 'Ilrr 0 equ .... o

=


C

~~\l5 =I~\

(V. 8. 37) =(OISL

1

0

................

f

(I:: SL)q 10) . (P) lu»(V) z- 0 lrr \ 0 equ I ....

and we obtain:

0

(Y)

-

299-

F. Henin

~=(\dvlN<01~LA-1 \1(1-I)~L-ILA1-1 \P lO )(OlSL(_l ~L)q '" J ~ I · z-L L' 0 z-L z-L 0 0 0 10 ) irr (V. 8. 38)

-

X

~

d~\

(r~)equ1(Y) N A 1 (olh

z-L

A)

m1-I)~L-ILo

o

1

f

z-L \Plo)(ro)equ "Ie 0

{\d~ \ N
If no fluid particle is common

to both fragments, the rhs vani-

shes trivially; now, in our discussion of the role of the irreducibility condition, we have seen that when such a product of fragments has a single semi-connection, it is through particle A; if we have more than a semiconnection, we obtain a contribution of order

-1

N

. Therefore, at

the limit of a large system: (V. 8. 39)

p

=0

This means that all reducible contributions to (V. 8. 33) vanish. Taking a-lso into accour:t the remark following (V. 8. 34) , we obtain: N

i(z)~1rv= EoJ\d~\ (O\~L

A

1 m+l f (Z_L oSL ) \O)irr(fo)equ X

(V.8.40)

which establishes (V. 8.29) . (V. 8. 28) can be established in

a similar way, if one takes into account:

<\~\, ~1(1-I)r~qJO)iLE: 0 (V. 8. 41}

if

:(~" : iLE

l\~\, Ii):

10 )

Ii\rf \ 0) iLE equ

({~ \' I.tlf~qu I 0>

-

300-

F. Renin

This completes tre proof of the equivalence between our starting point and that of Lebowitz and

Rubin~

This equivalence shows us that quite

generally all corrections due to the effect of the field during a collision can, in this brownian motion problem, be incorporated formally in a modification of the collision operator

(+~1C).

References.

1. P. Resibois and T. Davis, Physica 30, 1077 (1964) 2. J. Lebowitz and E. Rubin, Phys. Rev. 131 , 2381 (1963) 3. J. Lebowitz and p. Resibois, Phys. Rev. 139, All01 (1965) 4. R. Balescu, Physica 27 , 693 (1961) 5. p. Resibois, J. Math. Phys .

.!'

166 (1963)

- 301-

F. Henin VI. BROWNIAN MOTION IN AN EXTERNAL FIELD. VI. -

QUANTUM CASE In t rod u c t ion

We shall

now consider the problem of brownian motion of

of a heavy particle

in a quantum system. Our starting point for a

microscopic discussion of the evolution equation of quantum systems will be the

Von Neuman equation for the density matrix. As we shall

see, provided we choose suitable variables, this equation can be written in a form which is very similar to the Liouville equation. The main difference

will be the replacement of differential operators by displace-

ment operators. This corresponds to the physical fact that energ'y transfers are infinitesimal in the classical case

while they are finite in the

quantum case. The similarity between the quantum equation for the density matrix and the Liouville equation for the distribution function enables

us to extend the whole formalism very easily to quantum systems. However, the problem of brownian motion in quantum systems

presents features which are quite different from those of the classical problem. In the classical case, we performed an expansion in powers of the mass ratio and showed that, to lowest order, the velocity distribution function of the heavy particle obeys a Fokker-Planck equation. At first sight, we might expect this to be true also for the quantum case, the quantum effects appearing in the diffusion coefficient. However, if we go back fora while to the classical problem, we easily notice that, what we did, was to assume that we were dealing with a particle moving with thermal velocity (i, e. a velocity of the order of its equilibrium velocity) in a fluid at equilibrium. We then had: (VI. 1.1)

(p) IP

"O( m/M}

1/2

- 302-

F Henin

Our expansion in in (p)

a power series

of ,

was actually an expansion

/p . It is easy to convince oneself that (VI. 1. 1.) does not neces-

sarily hold in the quantum case. Let us for instance consider the case of a heavy particle rmving in a weakly coupled Fermi fluid. At very low

temperature, the particle collides with fermions whose energy is

very close to the

Fermi energy

£ F' Then we have: (VI. I, 2. )

where ((: /kT)1/2 F and we may expect to find a Fokker-plank equation

(VI. 3, 3)

only in the region

where

~

S«

(VI. 1. 4)

1

This condition is much more restrictive than the condition we

met in

the classical problem. With this example, one might think that such difficulties will appear only if we consider

fermions, because of the exclusion prin-

ciple and the existence of the Fermi energy. However, we shall see that it is not the case and that the difficulty is more general. At very low temperatures, we must always expect that the Fokker-Planck equation will not be valid. We shall first show how the Von Neumann equation may be written in a form very similar to the Liouville equation 1) . Then, assupling that an expansion in

(p) Ip

is valid, we shall easily obtain

a Fokker-Planck

equation. We shall discuss the quantum corrections to th e diffusion

- 303-

F. Henin

coefficient in some simple cases

2)3) and compare briefly the theoreti-

cal results with the results obtained in experiments on heavy ions movin g in liquid helium. Finally, we shall discuss on somewhat more general grounds than above the validity of expansions in powers of(p}/P.4)

VI.. 2- Von Neumann - Liouville equation. Let us consider one heavy charged particle acted upon by a constant external field moving in a fluid of light neutral particles. If we use a second quantization representation for free fluid particles and a plane wave representation for the heavy particle, the hamiltonian operator is (we take" = 1):

H=H

(VI. 2.1)

o

+}.V+H

E

The unperturbed hamiltonian is a sum over the kinetic energies of the fluid particles and the brownian particles: (VI. 2. 2)

Ho

+

where a k

~

=

L;

2 (k 12m)

at

2 a~ + K /2M

a k are the creation and destruction operators for a fluid par-

ticle of momentum k . As in the classical case, the interaction is a sum of two terms: one which describes the interactions of the fluid particles among themselves and a second one which describes the interactions of the heavy particle with the fluid particles:

V,.(~/2n)

r= v(~ h£:

klpr

£)

............................

(VI. 2. 3)

+(~/n)

L kl

(k 1) u",,--.e

-i(k-l).R + ................... \ a1

-

- 304-

F.Henin

The contribution due to the action of the external field on the heavy particle is:

=e

~. ~

1.1 the mixed representation

\!5

(VI. 2. 4)

HE

,\n\) of the eigenstates of Ho:

(VI. 2. 4)

the Von Neumann equation for

where [H,

rl

the density matrix

is :

is the commutator of the two operators

Hand

r

Let us now perform the following change of variables: for the heavy particle: ~ (VI. 2. 6)

""

=K

- K' '"

for the fluid particles

"k'" =nk - n''"k '"

(VI. 2. 7)

N k

= (nk +n'k )/2

'" '" '" We also write any matrix element of an operator A in the following

form:

(VI. 2. 8) = A 1f

.\.,{E, \Nt)

To tm operator A, 'We associate an operator the relation :

iC",

which we define throu-gh

-- 305 -

F. Henin

(VI. 2.9)

where the ~ and bles P

'?

IS

are displacement operators acting on the varia-

and N respectively:

(VI. 2.10)

1

+It ~ 'P --f(P) = exp [+)( -'::".)....

2

f(P)

~~....

at ) =f(!P +:::. .... -

+ LV

(VI. 2.11) One can

k ~. 1,... ~ 1\ - .... f(\Ni) = exp [+ -L v ~ .... !

1 2 k

1

oNk

...,.

'2

1"'}

f(\N\) = f(lN~- ) 2

then easily write the Von Neumann equation

as:

(VI. 2.12) which is formally analogous to the classical Liouville equation. The Von Neumann-Liouville operator

n

can be split into three terms:

(VI. 2.13) The unperturbed operator)t

o

is given by:

<~ '\"'\)(01 ~' ,\v',) = S~ ,~I (VI.2.14)

X

[~ ~!Z + ~ (.',2m)"k] ....

The interaction operator is given by :

-r:r cr"k'Yi ........

)(

- 306-

F. Renin

1~I v.'.'+V'+yl ~NII ,

P

r

(V.2.15)fV, I +1 V P P "

k

S., 10'

+ .!.)(N ~) (N + .!.)(N 2

I

r

I 2

P 2

~)

r 2

1 )( 1/2

,<>I +1 r

x

The external hamiltonian

is given

by:

<~ ,\Yq1CE\~" \v't) = ie~ . ~E

)(

(v. 2.16)

The algebra which leads to these results is very simple. We give examples in Appendix VI. 1. The equation (VI. 2.12) can be treated in the same way as the Liouville equation. Whenever we had a matrix element of the classical Liouville

opera~or,

we must now replace it by a matrix element of the

Von Neumann Liouville operator . If we compare those matrix elements,

- 307-

F. Henin

we notice that the main difference between quantum and classical matrix elements is the appearance of displacements operators in the former while we had differential operators

in the latter. This is specia-

lly striking for the operators acting on the heavy particle variables: the classical operator ~ / ";)P

f"'''' operator,,) - -

is now replaced by the displacement

; this is due to the fact that we used a plane wave repre-

sentation in the quantum case which is the analog of the Fourier expansion in the position variables in the classical case. For the fluid particles, the similarity is also very striking when one compares the quantum equation to th e Liouville equation for a system of oscillators in action (J) - angle (

c( )

variables). The displacement operators acting on the

N

variables are the analog of the differential operators acting on the action variables in the classical problem (for more details

see 5 )6) ).

VI. 3 - S tat ion a r y t ran s p 0 r t e qua t ion. The quantum analog of the stationary transport equation (II.14.5) in a static field will be : (VI. 3. I)

itE

pequ(p,~N') + i J

0

(V. 3. I) = i

Here

,\,(O)~

L. D~,,v,L(O))(E,v Ct. \~(O) "equ (P,\Nt) \

~ \."

r

~

~

elements of the

Neumann matrix. At equilibrium, we have in the

(VI. 3. 2)

'"

0(£ '\ N\)

~ o(P, ~Nt) represents the diagonal


0

Von

I\ nt, IS)representation:

-(H +).V)/kT ISlfqUhnLIS)=Qn\,lde

X [

:£:::. <~n't ,IS' \

\n't,,!5'

0

hn\,IS) X

-(H + ). V)/kT e

0

Hn'tIS~lJ

rl

1

- 308-

F. Henin

Because of the non commutativity of H and V, this expression is o much less simple, if all

orders in

>. are

kept, than the correspon-

ding classical expression. It is only in the weak coupling case, where we can neglect the interaction that we obtain:

(VI. 3. 3)

or

<~"\~lf~~o~n\,K)' exHt 2~ "t + ~~1/k T\L [r -::, + 2~J/k )(

re~u(~'lN\)

(VI. 3. 4) This will

~~o X

=

r

\n\!5

ex p \

(L:. \NH~

-ff

2 :m

-[f~ l! 2m

TIY'

~

2 Nk + :M]

2

-

exp \

I }~ ~1

2

N

~

kT

+ ~J/ltTr 2M

our discussion of the validity of

equation in the quantum case.

The collision operator

(VI.3.5)

~

play an important role in

the Fokker-Planck

nk

exp \ -

lim z~o

t

is given by :

<0f1{ t(z _~ 'W)n 10) n=l

irr

0

while for the operators of creation or destruction of correlations we have: (VI.3.6)

(VI. 3.7)

D~~ "",(0) = z~ lim <0\ i: tv: z-Jt )n I~IV~rr 0 n=l r0

The index irr means that only irreducible contributions have to

be kept,

i. e. contributions such that no intermediate state is identical to the vacuum of correlations (diagonal elements of the Von Neumann matrix).

- 309F. Henin

As in the classical problem , we assume molecular chaos ; hence:

f e~u (~,\Nt) f qU (E)\r: (N)} equ

(VI. 3.8)

=

6fo (~,\N\) = ~~(E) \ f :(N)\equ

(VI. 3. 9)

if we take into account the fact

that the mo.dification of the density ma-1

trix of a fluid particle is of order N , hence negligible in the limit of a large system.

VI. 4 - Expansion in the mass ratio. Whenever the ratio (VI. 4. 1)

(p)/ P


1/2

satisfies the relation =O(t)

we easily obtain an expansion in the mass ratio if we follow the same procedure as in the classical case. We decompose the unperturbed operator

1t o :

(VI. 4. 2)

Similarily : (VI. 4. 5)

where (VI. 4. 6)

where

Y

represents the fluid-fluid interaction (first term in the rhs

- 310-

F. Henin

of (VI. 2.15) . \fA is the

f1uid~partic1e

interaction (second term in the

", .. ae

rhs of (VI. 2.15) . If one expands the displacement operators ~ -- in a

r.

power series, one easily obtains an expansion of ries of

For instance,

'If:

\T A

in a power se-

is obtained by the mere replacement of

the displacement operators by unity:

(VI. 4. 7)

(VI. 4. 8)

The external field contribution i,s, as in

i . The main difference with

the classical case, of order'

the classical case is the appearance of

higher order terms in the expansion of '\fA . However, one verifies easily again , that, to, lowest order in

¥ ' the equation may be written;

- 311-

F. Henin

e

. .J f -I(A> -z)t

x

where (VI. 4.10) Moreover, at this order, we can restrict ourselves to the lowest order term", equiP) in the equilibrium distribution function.

1

0

~

Hence , we have:

where (VI. 4.12)

f

(p. .I)

Uvte

equ

~ e -~'~N +"/2~\e -Hf/kTI\N_ v/2\) L ({N\le -Hlj kT I~N\) \N\

where Hf = HO + V f is the fluid hamiltonian operator VI. 5 - Fokker-Planck equation. Let us consider the rhs of the contribution involving twice given

(VI. 4.11) at t=O (the calculation of

"U ~

at a time t .

different from zero is

in appendix VI. 2). For the contribution involving

~~A

twice \) 1 '

~

we have:

"'When no confusion is possible , we do no longer use a special notation for vectors.

-- 312 -

F. Renin

Using (VI. 4, 8) • we obtain:

= ('>./n) ~

u(k-f) ~ DNk + Y~/2) (NI+1 + ,,; /2) ) 1/2 )(

(VI.5'2))(rr\v't"~k-l )C (N1 -

'''{ +1 •

~r-k+r

({Nt'· Nk -l

.Nl+1)~qU+ (Nk+l-"U 2))(

~ 1/ 2) ] 1/2[ff \"',', ~~-1 ,\I { +1, lC' -k+l (tN\' ,Nk+l, N1-l )] \ ')

(1/2)(I-k) , '} P !ilOing back to

the occupation number-plane

wave representation we

obtain:

where the force operator is given by : (VI. 5. 4)

Therefore, we have:

ot

=

2:

IN'

r.,vt L(ollf

+ f~

lC

A 1

\ltlyt)r I:<\N +" /2\\F. i

K

_KFi(K)\\ N- -I /2\)(1/2)

1

~ P, 1

(K)

~f l( -K

X

-313F. Henin

+({N-"t.-xluHNI)(iNt\Fi (K) rfae.

'6 2

')( (1/4)

~P.

1

,P. J

-K+

y~

_KFi

(K)"N_~\)})c

\C.

J

(VI. 5. 5) =

L

\ N\

r [. <~N~F~¥)F.(K)f~ _K+F~(lC)r~ ij

1t K

J

J

1

_KF . (K) 1

'It

+F.(K)f~ F:t~ +P~ F.(K)F~(lC)\\N~.!. 1 AI -K J , .. -K 1 J 1/4 Taking into account

the fact that we can interchange

~.K

and i and j

J

and that the trace of a product of operators the invariance for cyclic permutations of the operators , as well as the hermiticity properties o f the operators, we obtain:

As in the classical case, we see that we have here the average over the fluid equilibrium distribution of the tensor operator FF. As to the second term, at

t

= 0,

we have:

~ = \~t~ r. r. 2:g
=

L. ~ LIf (iN\ IF~)() ~.J p! I

i N\ .. t

1

+ ¥.

J

f~ F:(g)I~N')~; i 1

CJ

(P/M)

- 314 --

F. Henin

Now we have

, performing an integration by parts in the second step,

= "' ~ (In'i\F 1 j

(k')r f k+k'

'tnt)

and therefore :

L(O\'\fAl It Arf

(VI. 5. 9)

tNi

Therefore, (VI. 5. 10)

equ

0

at t

10)

=

(F~ F.) J

1

'C>p 'i)

( F+ i

F'~r~ j I () p. 'l P. J

1

is identical

to

where F .(t) is the Heisenberg 1

F.(t) 1

=

(VI. 5. 6)

P

+ kTM

In appendix VI. 2, we show that the first term,

(VI. 5.11)

j

= 0, the integrand in the rhs of (VI. 4.11) becomes:

1

from zero,

(P /kTM) i

j

1

at a time t

different

but with F. replaced by 1

F.(t) 1

representation of the force operator:

'H f l'Hft e- 1 t F. e 1

A similar proof can be given for the second term the Fokker-Planck form of (VI. 4.11) :

and we obtain finally

- 315-

F. Henin

where the diffusion coefficient in terms of microscopic quantities is given by (VI. 5.13) As it can also be shown that, to first order

in

~'

we

.t-

have :

'fe:u(p) = (M/'ZrkT)3/2 4lf' exp(-p 2/2MkT)

(VI. 5. 14) we have: (VI. 5.15)

This completes our outline of the derivation of the Fokker-Planck equation in the quantum case. Let us

stress that

this equation is valid

whenever the condition:

(p)/p

(VI. 5.16)

«1

is satisfied.

VI. 6 - Diffusion coefficient for a heavy ion in a slightly imperfect Bose fluid. In a weakly coupled Bose

gas,

the condition (VI. 5.16) for the

validity of the Fokker-Planck equation would always be fulfilled. When strong interactions are present, the zero point motion of both kind of particles starts to play a role and it is difficult to make general assertions. We shall assume that the interaction between the fluid and the • One can for instance use an expansion of must have : just seen,

+(O)~equ

;(2)(0) is

= 0 , we have

~equ in

,(2)(0)

powers of

'fe~u=

As we

O. But as we have

the Fokker - Planck operator: hence

be the Boltzmann distribution.

r. f e~u

must

-- 316-

F. Henin

heavy ion is very weak and that we can neglect it also restrict ourselves to that the temperature

in Hf . We shall

a slightly imperfect Bose fluid and assume

is sufficiently low to insure that

the Bose-Ein-

stein condensation has already occurred to a large extend. Then most of the bosons are (VI. 6.1)

where

in the ground state and we have: 1\ o

~

N

n

is the number of particles in o can apply the well-known assumption :

(VI. 6. 2)

a

o

'" a

the ground state. Then we

+ 0

The force operator becomes: 11- 1

[ (k) = (VI. 6. 3)

+ 11- 1

k

L.

I/O while

u(k) rno (a: -a k ) k u(k)' a; a l _k

, if we use a pseudopotential

(see ref. 4,with

U '"

41f' a/ m

(a.: radius of the particles) : V

= (Uj211) rn! + 2no

l

(VI. 6.4)

+n We shall and

+

bk

(VI. 6. 5) (VI. 6. 6)

(VI. 6.7)

use

o

L. p=o

L.. (/-p ap + ap-p \ ) p/O

(/a+ +a+ a p -p p-p

)1

the Bogoliubov transformation to phonons operators b k

(see for instance ref 4) : +

a=gb+fb k k k k k-

-317F. Henin

(VI. 7.8)

2 -1/2 f =t( (1 - ... )

(VI. 6. 9)

c( k = 1 + x - x(x +2)

k

k

k

2

(VI. 6. 10)

x

This transformation

2

2

2

= k 0/8 -rr an

o

=k

2

/c

2

does not change the commutation rules and we

have (VI. 6. 11)

Hf=E

(VI. 6. 12)

o

+Lwb+b k k k k 2

wk = (k/2M) (k + 161fano 0

(VI.6.13)

Similarly, the force operator can

bk, bk . In this representation,

compute the diffusion next paragraph oneself to

coefficient

)

be written in the phonon represen-

+

tation

-1 1/2

it is a matter

of algebra to

(see an example of computation in the

for Fermi systems) . One obtains 2) , if one restricts

small wave numbers (i. e. to the linear term in the dispersion

relation (VI. 6. 13)) :

f'

_ nr/o 2

"ij -

(VI. 6. 14)

I.

) kl

k k

i j

(f2 f2

1 l-k

+ 2 2

)

0

gl gl-k m l

(mo +1) l-k

~( \l-k\ -1)

m~ is the distribution function in a slightly imperfect Bose gas in the limit

of small

(VI. 6. 15)

momentum:

Or]

~ =lexp (wl/kT)-1- 1 =fexp (cl/kT) -

The integrations can be performed and one obtains,

11- 1

if one reintroduces

-

3181'". Henm

explicitly Planck's constant:

~

(VI. 6.16)

=

(21r3/45(kT)5 a 2 1'r- 3 c

The mobility of a heavy ion is given

5),

t" = eD/kT

(VI. 6. 18) where D seen

by

-4

is the diffusion

in chapter

I, we

coefficient

in ordinary space. As we

have

have:

(VI. 6. 19) (note that

D is

ordinary space,

the coefficient appearing in the diffusion equation in (M/kT~)

tion in velocity space " ;

-1

that appearing in the Fokker-Planck equa-

that appearing in the Fokker-Planck equation

in momentum space) . Therefore we obtain: 3 3 4 -2 -4 (45/2tt' ) e 11. c a (KT)

(VI. 6. 20)

In a measure of the static mobility

+ ,

T

-4

of (He ) m liquid He n

4

at low

temperature, Meyer and Reif 5) have found:

r '"

(VI.6.21) which

Tk with

k

=-

3.3 + 0.3

is in good agreement with the above result if one takes into

account the fact that

our model

VI. 7 - D i ff us ion

0

is quite rough .

f a he a v y ion ina we a k I Y co u-

pled Fermi fluid. We have seen

in the introduction that in a Fermi fluid, at low

temperature, the Fokker-Planck equation

is valid only if the condition

(VI. 1. 4) is satisfied. We shall discuss again this problem in the next

,

paragraph but presently we consider

a situation where this condition is

- 319 -

F. Henin

satisfied and compute the diffusion coefficient, assuming the fluid to be weakly coupled. As well known, in a Fermi system, the creation and destruction operators anticommute (VI. 7.1)

,a

1

k

+

= 1

and the only possible values of the occupation numbers are n = 0 k

(VI. 7.2)

In order

to

avoid difficulties with the subsidiary condition

Ln = k k

(VI. 7.2)

we shall

or

consider

N

the grand canonical ensemble . Then the equilibrium

distribution for a weakly coupled system is given by :

(VI.7.4)

where

f F is the Fermi energy . Thus, going back to

have:

the occupation number representation we

-

320-

F. Henin

Using (VI.7.6)

<

(VI. 7. 7)

\ 1n~ =

=l n k+1)

'nk)

nk -1

~ nk

0

Sn k, 1

one easily verifies that one must have:

1

(VI. 7. 8)

= 11

and one obtains:

X

~

exp [_i{k2 _(k_l)2 tt/2m)expt

)( 1[ \1 (VI. 7.9)

~

exp [- (k 2/2m -f F ) /kTl}-l

\2 /n

= ("

nk(k 2/2m - EF)/kTl

2,..Z ) L. iu (k)\ 2 k

2 2 ] X k.k. exp [. -1 (k - k' ) t/2m

kl

1

J

X exp (- (k2/2m-EF)/kTl\l+exP[-(k2/2m-tF)/kTl}-I)( )i

\1

+ exp [-lk,2/2m - t:F)/kTl}-1

At the limit of a large system, the summations over

k

and 1 become

integrals. If we also perform the asymptotic time integration

in (VI. 5.

13) , we obtain

~ ij = 2m 11'~2 (ik(d\' (VI. 7. 10)

2

IU(k)12 kik j

~(k2_kI2) exp[+ (l42m- tF)/kT JX

l-l J

)( {I + exp [(k /2m - tF)/kTjJ

11 + exp ~kl

2 1 /2m - fF)/kTJ}-

- 321 -

F. Renin

One easily verifies that this tensor is diagonal and that one has: (VI. 7.11) with

~

=(1611'3 m/3)

(VI.7.12))C

X~1

C

dkk 3

[

dk'k,2IU(k) \

2[~(k_k') + S(k+k ' )]

l(

t: F )/ kTl\l+ eXP [(k 2/2m-f F )/kT]}-1 X

expDk,2/2m -

+ exp [tk12 /2m - f F)/kT1Y 1

We integrate over

k' and approximate the potential by a constant; we

then have:

(VI. 7.13)

4 2

3

= - (128/3)1f m U (kT)

~(256/3)l!3m 4(kT)3u2

'0 dE

f

dII[1

+ ,( t

3 (<Xl

I:

where

~

(VI. 7.14)

=(

E

F At sufficiently low T, we have:

~

(VI.7.15)

(VI. 7 .16) , <Xl

t2

/kT) 1/2

»1

y-l dyy (1+e )

= ~4/2 +1T

2

/6

2

d \

'dE

1+e

-~2)rl

(t:

-

S2) lJ-1

~

322

~

F. Henin

where we

have neglected exponentially decreasing contributions in the

first integral as well as the last one whose integrand is exponentially small. Therefore, we obtain:

From this, we easily obtain the mobility (see VI. 6. IS). Introducing and the collision cross section 2 = 2m Tf u ) , we obtain: 311'

(VI. 7. IS)

cr in the Born approximation

(a-

11

=

eii 3

Davis and Da gonnier I } compared this result with

the experimental mo-

bility for a heavy ion in liquid He 3 at 1. 2 oK measured by Meyer and al. 6} • HoweV'er, the comparison is not

very easy because of the lack

of information about the collision cross section (radius of the ion) and the effective

mass

of the ion. With reasonable estim8tes for these

quantities, they find a good agreement.

VI. S - Validity of

the Fokker-Planck

equation

for brownian

motion in quantum systems. Let us first

consider

again the case

of motion in

a weakly

coupled Fermi fluid. We may distinguish three temperature regions: a) the temperature is so high that both the fluid and the particle behave classically. Then, if the particle moves with thermal velocity, we have: (VI. 8.1) and the classical

= 0 (m/M)

1/2

Fokker-Planck equation

is valid.

-

323-

F. Henin

At lower temperature, quantum

effects become important

for the fluid. Only those fluid particles near the Fermi surface interact with

the particle and we obtain:

<.P) /p = 0

(VI. 8. 2-}

(m IF/ MkT )

1/2

't

= 0 (~~)

We may now distinguish two cases : b) an intermediate temperature range where the more restrictive condition :

~~ < <

(VI. 8. 3)

1

is satisfied and the quantum c)

the case

of very low temperature where

~~ and where

Fokker-Planck equation (VI. 4. 11) is valid.

> 1

the Fokker-Planck description

does no longer hold.

In a discussion of the possibility of convergence of the {p) development that,

in

realized at

presented here,

general,

R~sibois

one must not

and Dagonnier

/P

3)8) have shown

expect this convergence to be

very low temperature, whatever the statistics. A very sim-

ple argument is the fact that at very low temperature, the average momentum P

is independent of the mass ratio. It is essentially determined

by the interactions with

the fluid molecules and momentum transfers

can become very large; the Fokker-Planck description is then no longer valid. Let us show briefly how this conclusion about the independence of the average momentum at very low temperature can be obtained. For simplicity, we consider the case

of Boltzmann statistics. The equilibrium

distribution function for the brownian particle is :

- 324-

F. Henin

f

equIP)

Tr fluid Tr

(VI. 8. 5)

r

equ

f

fluid, A equ ...

f

A

tr fluid expL.- ~(Ho + Ho + V +

U)]

where (VI. 8. 6)

Let us write: _AH = e ,-

(VI. 8.7)

Then:

One verifies easily that this quantity satisfies the Bloch equation:

(VI. 8.9)

~~f) =\exp[~H~ +' H:)JV + U) exp (-P(H~ + H~)n (.) (~)

Taking into account the commutation (VI. 8.10)

[V, e

_~HA] 0

relations

= 0

(VI.8.11)

and introducing (VI. 8.12)

'"

f

-PH of )

(VI. 8.13)

'"

f

f

this equation can

V = exp (~Ho) V exp (

U = exp (~Ho) U exp ( -~Ho)

be rewritten:

(VI. 8.14)

Expanding the rhs in

powers of HA , we obtain: o

- 325-

F. Henin

')o~(~) = \v + U+~IH~ , U1 + ~2 [H:,[I(,ul]+ ... \ w(~)

(VI. 8. 15)

After a rather lengthy calculation 8) , one can finally obtain the equ (=m/M) correction to the maxwellian ~o (P);

(VI. 8.16) ~

equ

where Q/, is unity. T Q is

equ (P) =~o (P) +

,2

~ equ 4 ¥2 r,~P 2/2MkT)(kT Q/kT) +~'fo (P)+ 0 (t )

a factor which guarantees the normalization of a characteristic quantum

'f equ

to

temperature which expression

is quite compicated but can be shown to have the following properties; (VI. §..17)

lim

T Q(tl, T) = constant

T~O

(VI. 8.18)

From

11

lim ~ 0

TQ (t;, T) = 0

the above equilibrium

,p /

(VI. 8. 19)

2)

distribution, we obtain easily

= 3m ~

-2(;

ikT +

At sufficiently high temperature, we

r

2

kTQ) + 0

;

(¥ 4\1l

have

(VI. 8.19)


(VI. 8. 20)

;;

kT

However, at sufficiently low temperature, we may reach a regime where; (VI.8.21) In

that case;

does no

longer

dependon

r.

-- 326 -

F. Renin

A more rigorous mathematical analysis is of course quite difficult in general because of the complexity of dense systems. A more elaborate discussion can be found in Appendix to the

3)8)

VI. 1 - Examples of the algebra leading

Von Neumann - Liouville equation (VI. 2.12)

In the mixed occupation numbers-plane wave representation the Von Neumann equation is

(A VI. 1. 1)

Let us consider

in

the first term in the rhs one contribution to the

fluid-fluid interaction: (A VI. 1. 2)

aI.

with

V=V(k,l,p,r)~kl oat + -p-r, k

(A VI. 1. 3)

We perform the following change

(A VI. 1. 4)

-J

=)t

n -h' k k

K"-K'

= )('

n"-n' V' k k k n tn' k k =N 2 k

2

ha'(e

variables:

K-K'

K+K'

Then, we

of

=

P

=

k

-

327 -

F. Henin

Using the notation (VI. 2.8)

.,.. r-~ - _~'

0( = L-

.I.

I

IV kl

lC,1

(A VI. 1. 6)

Vlj;

XP

'~I ~ V

"

1\

, this becomes: ~I

I

(P _ ~

2

VI ~

+ -2-' {N + -2-1) )(

I V _ V I " (P

+~

2'

{N _ ~

2

+~ \ } 2

Introducing the displacement operators defined by (VI. 2.10) and (VI. 2.11) we may write :

d. = L.

x' r~'

I". ~ ') k

k

~1\Vlk\

-)( ~I "

V" _ I" _.. ,,(P,\Nf) ~ ')(.1

...,'

11} K

k

)(

"J

(A VI. 1. 7)

(the displacement operators act

on everything that stands at their

right. ) Now, using again

(VI. 2. 8) and

V - 'l(. , ,v - v ~

I l

I, (P,

,

~Nf)

(AVI.1. 3), we have:

= v ( k, 1, p, r)

=V(l,lp,r)~k+l_p_r , OS~, ~I

(A VI. 1. 8)

X (N H/2)(N + 1/2)] 1/2

P

r

Sk+l-p-r,O

V

)C

i... "I~.,.I

i,hlpr" i'" i

..

k' "k

-1 X

-

328-

F. Henin

If we combine (A VI. 1. 7) and (A VI. 1. 8), we readily obtain the first term

in the rhs of (VI. 2. 12) . As another example, let us consider the external

field term.

One can also very easily obtain the following contribution to the Von Neumann - Liouville equation:

\ I ~ (P, \ N~)

z: L.,,,\vt!)CE l~vI\xI) r~, 1(,

I \v It

= Lt~I \vl!

-11:1

-~ =

.,

r. L

~I ,",vlt

J

", r", \ 'f' k k ) "l

~ T"

E

"

" f~k

HlG_~,,\V_Ylt(P,~Nt)~'

k

{exp

(~I ~p) exp[~ 2>~' k

-

';>P

\p - X-2}('

x rX. , \"I~

4(P +

k

-rk .. k

(- " 2

~rlt'~v't(P,\Nt)

l(

'1 Nk\

2...) exp 'j)P

[_l rv' ~1/p+ 'I(_~I 2

k k C) Nk \:

2

,\NI v -

[-

l2 r" L k k "i) Nk

2

I

2

2

QP

,

v k' ' \

{ dR ~

-,<,

.... ~.

'"

R

...

I..

,i( lC.

.12

I-

~ ). R

.,1

I N +'' -Vl '1 2

IN-~f)exp (~2 )exp[!.[~

'1

JC.I - It 2

1

t\

2-1lI

2 k k () Nk

x

II

_~I

2

(P,\Nt)

=l:!:TTcl ~I ~VlJ

k

~t\exp 2 ~

(-~ ~) exp 2

-lC

I

r"k E

(AYI. 1. 9)

~.~.

_

H}(_1I.Il"'_~ll(p,\t'rt)~-}

E. R' P - lC - lC ' iN '" ... 2"

- exp

v

X

- 329-

F. Renin

=i

E.

z:. L. 1f.

~P

~,\",\ k

g

,i

"k'''k ~

I

P~'lv'~ (P ,\Nt)

,'"

... ,: ,

J

which agrees with (VI. 2.16)

Appendix VI. 2 - Equivalence Let us discuss

=

0( (t)

between (VI. 4.11) and (VI. 5. 6)

the contribution

LL

involving twice

lfA1 ..

L L (o,\J~ \ \"he) 'X

~N) \,,\~ \"')~' \""t~"

(AVI.2.1)lc'(~"'~1 exp )(

~f~"

[- i

(I.f -z)tll1&'\v")(¥'\Y't

\Y"i( \N\)

Usinf (VI. 5. 6) , we have \ (see

where

the force operator is given

~ (t) =

r L!. i

§

5)

:

by (VI. 5. 4) . Rence , we obtain:

L L (o\ut \ \Y\ 'aC)

K \N~ \ ..., X

(AVI.2.3)"<\~tl(exp

I \ftllC.n~"n\) ')(

X

\".'t-'

(-itftll\v'\X') (\N+

+ rf F/K)I\N -

v2'

1, K)

T}

,}C'

1Fi(K) f

~P. 1

Now, let us differentiate with

respect to

t

the quantity

- 330-

F. Henin

with (AVI. 2. 5) hence (A VI. 2. 6) We have:

or , going back to

the occupation numbers-plane wave representation;

where (AVI. 2. 9)

H = Hf f 0

+V

(To obtain this result, we take into term

invol ving

~

11' Ao

, we'

have;

e-i(~-k+l-~').R

felH

(AVI.2.10) = JdR e-i)(R

account the fact that, in the

{dR'

e-i('\oC'-K).R' ('Jf(R,t)

ei(k-l).R~f(R,t)

>

e iKR =(,,(ei(k-l).RPf/ K

Therefore, we have

where

F. (K, t) 1

Therefore;

,

is the force operator in Heisenberg representation.

-

331 -

F. Henin

(AVI. 2.12)

+ ffF.(K, 1 From now, on ,

t~e

tll\ N ~ f-l' K)

1)

P.

1

derivation goes on as

in the text and leads to

the second order differential contribution in (VI. 6.12) .

References.

1. T. Davis and R. Dagonnier, J. (hem. Phys. 44 , 4030 (1966) 2. R. Dagonnier and P. Resibois, Bull. Acad. Roy. Belg. Cl Sci. 51.., 229 (1966)

3. P. Resibois and R. Dagonnier, Physics Letters (1966) 4. A. Abrikosov, L. Gorkov and 1. Dzyaloshinski, Methods of Quantum Field Theory in Statistical

Physic~,

Prentice Hall, New Jersey

(1963)

5. L. Meyer, and F. Reif, Phys. Rev .

~,

727 (19til)

6. L. Meyer, T. Davis, S. Rice and R. Donelly, Phys. Rev. 126

, 1927

(1962)

7. P. Resibois and R. Dagonnier, Bull. Acad. Roy. Belg. Cl. Sci. (to be published)

- 332-

F. Henin

VII. GRA VITA TIONAL PLASMAS. VII.l- Introduction. In all the preceding chapters on brownian motion, we have used the kinetic equation (II. 9. 5) as the starting point. However, when we derived this equation, we stressed the importance of time scales. This equation fails to

describe correctly the asymptotic behavior of a system

where there is no net separation between the collision time and the relaxation time, or where the initial correlations are over long distances. Now, long collision times or long range correlations must obviously be expected in systems interacting through long range forces. As examples of such systems, we have immediatly in mind, on the one hand, systems interacting through electrostatic forces and, on the other hand, systems interacting through gravitational forces. In both cases, the interaction between two particles is inversely proportional to their relative distance. A great deal

of effort has been done to understand the situation

in the case of electrostatic forces. If we consider a charged test particle moving in a plasma, it polarizes the medium: the charge 'distribution around the particle is no longer uniform. The medium screens the interaction between two particles and we are no longer dealing with a pure Coulomb force. This idea, in its simplest form leads to the Debye Huckel theory. Out of equilibrium, it has

been shown that, if one sums

in the operator '\' all contributions proportional to of this

summation is to

ning introduces

and a long time (VII. 1. 2)

introduce

a dynamkal screening. This scree-

a short time sc ale :

scale t

r

(e 2 C) m, the result

= (e 4 c) -1 m 1/2(kT) 3/2

c

-1

-

333-

F. Henin

We are again in the conditions where an asymptotic kinetic equation may be written

equation).

(Balescu~Lenard

What is now the situation with gravitational forces? The essential feature is that the interaction is now purely, attractive, whereas when we deal with electrostatic forces we have a mixture of attractive

and

repulsive forces (with a condition of overall neutrality). In a discussion of equilibrium properties in the framework of Mayer's cluster integrals, one notices very easily the importance of this difference. Indeed, there it appears clearly that the most divergent contributions can be eliminated because of the electroneutrality condition; the next dominant terms (which diverge also) can then be summed and lead to the screening. This

pro~

blem arises always as soon as one computes average quantities, but, in the derivation of the

Balescu~Lenard

to an electron plasma with

equation, one can restrict oneself

a positive background; one notices then that

the background plays no role in the derivation. In this case , we are dealing with purely repulsive forces. The first idea is then to perform in the case of purely attractive gravitational forces, the same summation that worked for purely electrostatic repulsive forces, or for a mixture of both attractive and repulsive electrostatic forces. In simple models, which leads for electrostatic potentials to the Oebye potential:

e -1Cr/ r

(VII. 1. 3)

one obtains, for gravitational

systems an effective potential of the

form: (VII. 1. 4)

This partial

V

G

summation

iKr

=e

/r

does not lead to a screening; the "effective

interactinn " has the same range as the gravitational interaction. The intuitive ideas which lead

to the Oebye potential for an

elec~

-

334-

F. Renin

trostatic plasma can obviously not

be extended to the gravitational

systems. As we have seen, more elaborate techniques did not succeed to bring anything which ressembles whatsoever to an effective short range interaction. In view of the failure

of all those attemps to

justify the use of the kinetic equation for such systems, one may try to use

a completely different starting pOint . If we cannot use the kinetic

equation, we must go back to the exact equation, the generalized master equation (II. 8. 9) . One can then try to discuss

the new features

introduced by the long range character of the interaction. This is what Prigogine and Severne 1) attempted very recently. T hey considered a very idealized model of

a gravitational system: a weakly coupled, homogeneous

system, with no correlations at t=O. Although the first condition c:m 1)2) be justified with the conditions p.revailing at present in our galaxy the other conditions of the model vitational

are certainly not realistic. A real gra-

system is an inhomogeneous system. However, the findirg

of a proper treatment for this idealized model , which is the simplest model of a gravitational system one can find, is certainly the first obvious step one has the much

to take if one wants to achieve an understanding of

more complex actual systems (or at least of realistic simplifi-

cations of the actual systems) . We shall first show on a very simplified collision operator what is the basic difference between systems with a very short collision time (as compared to

the relaxation time ) and systems where the collision

time is infinite . This will enable

us to convince

ourselves that, even

if we require only an understanding of the asymptotic behavior of the

latter systems, it is necessary to start with the generalized master equation

We shall then discuss the characteristics of the long time evolu-

tion as described by this equation in the limit of infinite collision times.

-

335-

F. Renin

More details as well as general considerations about the following dithe paper by Prigogine and Severne 1) as 5) well as in a paper by Prigogine scussion can be found in

VII. 2 - Simple model for the collision operator. In order to see what are the difficulties which arise when one deals with long range forces, i. e. long collision times, let us investigate a simple model for the collision operator 3)4) :

(VII.2.1)

'I'(z)

The collision time in time

is given by The non

c( =-

( II(

z +i ~

,~

this model is given by

AI

'"

-1

)

j\ -1

0) while the relaxation

A. J"

markovian equation

for this case , if we do not take

into account the destruction term, is

(VII. 2. 2)

= (2 1f i)

-1

rt

J0

Provided that we can neglect terms of the form exp(-t/"t 11)' an ex2 co pansion of (VII.2.3) in powers of ('"t"coll/"t"rel) =cl~- ,leads to the pseudo-markovian equation (see chapter II,

~ (VII. 2. 3)

r

0

= -i

!"l(0) '1'(0)

r~)

= -i \ t(O) +,\,'(0)'1'(0) +H'I'''(01f 0 )

+ t,2(0~t(0) + ..

=-

\1

+,~ - 2 + 2

t 9) :

·\r

(cl

ott)

~ - 2)

2+ ...

1(" / ~ )f (t) 0

- 336-

F. Henin

This expansion has a simple meaning for small values of "I: On the other hand, if we times

(p ... 0) ,

IT:

l'

eoll re take the extrerre case of infinite collision

it is meaningless; the operator t(O) is no longer de-

fined. However, in any case the non markovian equation predicts a well defined behavior of the distribution function. I ndeed, from (VII. 2.1) and (VII. 2.2) we have (VII. 2. 4) and we easily obtain the second order differential equation: (VII.2.5)

The solution of this equation is very simple and perfectly well defined whatever fo(t)

J\

(note that

[if 01 u

t]t=o = 0) :

=(l/2)fo(O{1 - , (f\ 2 -411()-1/21 exp

(VII. 2. ')

+(1/2)

f (O{1+ ~(I' 2-4~)-1/21 0

r- (f + (~2;4,,)1/2J t] exp

r-If -(i _~.)1/2J t]

When (VII.2.7)

't coIl / T: reI

='"A,-2-,

4

we have a monotonic decay of 0 (t).

If"~

-2

«1,

I

0

neglecting terms proportional

to

e

-t/ i: coIl

(e

-

rft t ),

we obtain: (VII. 2.8)

fo(t)::

which is the I behavior that

flO) exp (-

-1\-1 t)

is given by the pseudomarkovian equation if

we restrict ourselves to the first term in the rhs of (VII. 2.3)

- 337-

F. Henin

For (VII. 2.9)

i. e. for large collision times, we obtain damped oscillations. In the

extreme case of infinite collision time, we obtain a purely oscillating behavior: (VII. 2. 10) In

other words, for long range interactions (small

due to

1\ ).

the corrections

the non markovian character of the equation are quite impor-

tanto VII. a - Non mar k 0 v ian e qua t ion i nth ewe a k c 0 u pI i n g approximation. Let us consider a gravitational plasma: N 2"" 1: {l/2)mv. + ~ L.

H=

(VII.3.1)

i=l

1

i<j

, r.-r .1-1

1

"'"J

where

~

(VII. 3. 2)

= _

Gm 2

where G is the constant of gravitation. The characteri.stic parameters of such a system are (numerical values correspond to conditions prevailing now in our galaxy and in the vicinity of the sun

2)): -8

G=6,7.10 m

{VII. 3. 3)

= 10

33

(constant of gravitation)

cgs

(average mass)

g

v= 3.10 6 cm/sec C = 3,4.10

-57

cm

(mean -3

star velocity)

(number density)

- 338-

F. Henin

With these parameters we can form one non dimensional quantity prow portional to

G:

r=

(VII. 3. 4)

C

1/3\ _2~1 ,,(mv)

~6

= 0 (10

)

The smallness of this parameter justifies a weak coupling

appro~

ximation. In our discussion

in

chapter III,

asymptotic weak coupling operator presents

t 6,

we have seen that the

'1'(0) for a potential in 1/ r

a double divergence. We shall

show here that the

non~mar~

kovian corrections, even in the weak coupling approximation ,permits to remove the long distance divergence. The short distance divergence, due to the close collisions cannot be removed in this way; however as close encounters are not very frequent, we shall neglect them in the evolution equation; in other words, we shall cut-off the potential at some short

distance R. In the following analysis, we shall find -two time scale;; :

the "nominal" relaxation time: (VII. 3.5)

t"r =C

-1 \-2 "

2.3 -1/3 m v =C

r -2_-1 16 v : : 7 .10 years

the duration of close encounters: (VII.3.6)

_-1

't' = Rv c

=C

-1/3

r --1 v

7.10

if one takes for R the distance corresponding to

-2

years

a mean 90 Q deflection

in the two body scattering problem. We shall, as usual, consider (VII. 3. 7)

N

~

00

,

n ~ 00,

the limiting case C finite

339 -

F. Henin

We shall also neglect all time variations occuring on the close collisions time scale t:" • c

We shall assume that we are dealing with an homogeneous system in which there are no spatial correlations at t=O. 'Ben, the destruction term in the non markovian equation vanishes identically and this equation has the form (VII. 2. 2) . In the weak coupling approximation, the collision operator is

given by (see IIJ.2. 3) : , 3 1\ \ 2 elm) 2 \v(z) = (8111'

I

(z)

I

~ - - -C)) -~-1 d 3, k V \ 2 k. ( )( k ~v lv k.(v -v )-z -1

(VII. 3, 8)

f

-2 -

-1 -2

() ~ " k.(-----) ')~1 'i.'(,2

In order to treat in a proper way the divergence, we shall consider the gravific potential as the limit of a screened potential: (VII. 3,9) Following the technique of chapter III, (VII. 3, 10)

U1(z)

I

~

4 , one obtains easily:

,3\2 2 'C) It. elm ) ~ T

= (32111

tJ

gr

~

~ rs "gs

with (VII. 3, 11)

T

= Til g g g II r s

rs

-2

+ T.L(g

2t'

c)

r, s

-

g g )g r s

-2

where (VII. 3, 12)

iJ'

R.

= -1-2 V -v

is the relative velocity. T,l and se

T.1,. are

the

parallel

2) 1\ z+iKg K --

- --z+i g

components:

(VII. 3. 13)

,

11m 4" It = K.., 0(

T

0

rr \

and

transver-

-- 340-

F. Henin 2 (VII.3.14) T.= lim (4,,) ... K-,) 0 where R is

-If -In i z+iKg - ,g

z+lr- g

1 II} - - --2 - , 2 ta- R z+l .... g

the short distance cut-off and

(VII. 3. 15) For instance, in

a reference frame where the z

using cylindrical coordinates,

(VII. 3.16)

(z

£

S)

[,

have:

+00 2 -1 2 -1 2 2 -1 2 2 -2-1 } " (41') lim \ dk k (k g-z) \ (k +K) -(k +K +R ) z Z Z z z KO ~ -00

The last integration is

+

we

axis is along

and

leads

easily performed by the method of residues to (VII. 3.13) .

Also: 3 -1 = (411" )

+00 lim \ dk z K.., 0 -00

IR

3 2 2 2 -2 -1 dk.1. k.1. (k z + k.,L +K) (k zg-1)

0

r

k2 + 2 z2 dk 2 - 2 2 2 g-z) -1 K':'O -00 z k +K R (k +f" z z z The second term is easily computed, using, the method of resi-

(VII. 3. 17)

+00

= (811 3) -1

lim \

\ In

~

dues and gives the second contribution in the rhs of (VII. 3.14) . As to

the first contribution, we complete the real

at infinity in

S

axis with

and avoid the two branch points at

-iKby making a cut. (see fig. VII. 3.1)

-i

t"

a half circle and

-

341 -

F. Henin

Lnz

Integration contour for log. term in (VII. 3. 17) Fig. VII.3.1 One can then easily obtain the first contribution in the rhs of (VII. 3. 14). The most important feature of the operator the limit

K -) 0 , the transverse part T.L

larity at z = 0

t

(z) is that, at

has a logarithmic singu-

The collision operator may be rewritten:

(VII. 2. 18) with

if one takes into account

the fact that any function of g

commutes with

A

the differential operator '" . We have singularities both at rities at

z = -i

rg

z = -iKg and z = -ir g. The singula-

are related to the close collisions and are at a

finite distance from the real axis. As the time scale for the close collisions is much shorter than any time scale we shall meet, we treat the close collisions as instantaneous. This means that we consider values of z

such that:

(VII. 3. 21) We then obtain the approximate form

-- 342 -

F. Henin

(VII. 3.22)

(VII. 3. 23)

Now, in the case of electrostatic plasmas, K is finite and if we ne-1 glect effects proportional to (Kgt) ,we obtain the asymptotic form: (VII. 3.24)

J

R-

(VII. 3. 24)

which

1

=0

r

3 2 2 -2 "'" dkk (k + K)

is eaSily verified to be identical to (III. 3.5) when one takes as the

interaction the screened

Coulomb potential. However, in the limit

K .., 0 , this procedure is meaningless and we obtain: (VII. 3, 25)

with (VII. 3. 26)

The solution of the non markovian (see II. 9. 4) :

r

ott)

f

equation (VII. 2. 2) may be written

-izt

dZ

e

1 ( )dZ

e

= - (2'0' i)-l

(VII. 3. 27)

=-(2l1'i)-

-izt

_-:-1--:-

z- t(z)

r( 0

0)

-

343-

F. Henin

VII. 4 - Tim e

e vol uti 0 n

0

f the vel 0 cit Y d is t rib uti 0 n

function.

t "'"

The operators have a common set of

and

+

do

not commute; hence they do not

eigenfunctions and the analysis of the time evolu-

tion is not very easy. However, the problem will be considerably simplified if one of the contributions turns out to be dominant. Let

us first

investigate the type of behavior determined by the second term. A

f

The characteristic time involved in hon time

1:'

r

is the nominal

given by (VII. 3.5) . The operator

- i

"'Y '

relaxa-

as we have

already seen (see the discussion of the )(, -theorem, chapter III,

f

2)

has real, negative or zero eigenvalues which define a spectrum of relaxation times. We may in a qualitative discussion introduce the following approximation: (VII. 4.1) Also , we replace close encounters (VII. 4. 2)

r

Rg

-1

-r c .

by its average value, the collision time Then

we obtain:

o(t) = - (2 1f i) -1

f

dz e -izt [ z-x (z)

where

+ In (-iz/r c )

(VII. 4.3)

X(Z) = -\(1/2)

In order to

understand the evolution

the singularities

o

Let us introduce : (VII. 4. 5)

of

i(i"t

r

0

1-

1

r )-l

(t), we have to discuss

of the integrand; i. e. find out the roots of the equa-

tion : (VII. 4. 4)

for

z=-w-i'<

-

344-

F. Henin

Then (VII. 4. 4) and (VII. 4. 4) give us: -xtiy = (1/2) + In r-1(_x+iy)

(VII. 4.6) with (VII. 4.7)

=ct"t r

x

y =wt'

A detailed discussion of the dispersion

fI'=~/-r

r

r

c

=10

18

equation (VII. 4.6) can be found

in ref. 1) . The main point is that the equation can be very much simplified if one takes into

account the largeness of ...

. A whole spec-

trum of solutions is found. For consistency, the range of the spectrum is restricted to frequencies such that:

w «w

(VII. 4. 8) The

frequencies

W"",

."L"

-1 c

are essentially the odd harmonics ~ (2n+1)11' n -

(VII. 4. 9)

w

In the useful

part of the spectrum

the damping is such (VII. 4.10) In fact, in

max

0(

n

of w = "C

o

-1

r

/1: r

(frequencies less than 10

-2

w ), max

that:

)4"C"

-1

r

the major part of the spectrum, the damping is found to be

of order: (VII. 4.11)

0(0 =t'

-1 In r

r- 3

= 40t'

-1

r

The time scale for the oscillations is of the order of the nominal relaxation time 't' by a factor

-1 r

the dampi.ng it is

much shorter (at least

4). The essential feature is that we hi ve now an oscillato-

ry relaxation of relaxation

while for

po (t) on a time scale much

time and given by :

shorter than the nominal

-

345-

F. Henin

(VII. 4.12)

t

re

1 = 'C /In('C'

r

r

/7:c )

The consideration of the complete spectrum of eigenfunctions and eigenvalues of

i"

avoided in the approximation ( VII. 4.1),

would only

lead to further complications of detail. However, we still have to examine the effect of the operator

f

which appears in the complete equation (VII. 3. 27) • The time scale for the effects due to

~ is again the nominal relaxation time

T

while the term involving

"C'

"'"

,

r t

w

has a much shorter time scale . T reI Therefore, the system will first reach a quasiequilibrium distribution,

<\> contribution.

which will be further modified by the action of the

'Y "

One verifies easily that, if the

contribution conserves the kinetic

+

energy, this is not

true for the contribution . One obtains easily for

the variation of the kinetic energy per

'C>E

-1(

kin

)\d~ \

=N

';)t

"-(211' i)

N

N

• - (21fi)

-1

N

2 r:.N (mv.!2)(~f

i=1

-1 -1

(VII. 4.13)

star:

~

1

N

l'd~\

~ ti

-1~ d~\ N L.. ~

i=1

41f~2Cm-l (dl1d~

J

R._

0

lot)

2 ( izt (mvJ2),dZ e - f(Z)z-'f;:lo(O)

(mv,2 /2) 1

g-1

f

f

r

(t)

0

(v

v ,t)

0 ~1'~2'

>0

where (VII. 4.14)

It

(VII. 4.15)

w = (v

~1

are respectively the relative Therefore,

= ~ 1 - ::'2

the

t

+ ~2 v )/2

and center of mass velocities.

contribution plays the role of a source term

-

34b-

F. Henin

r

and leads to a continuous 'increase of the kinetic energy. In we have the following picture of the time evolution

of

0:

other words, first , a qua-

; the reI aged system then remains in the quasiequilibrium state but with a time siequilibrium distribution will be reached on

a time scale

t

dependent temperature.

VII. 5 -

R ole

0

f the in it i ale 0 r reI a t ion s .

In this whole discussion, we

have assumed that there were no

initial correlations. Then, we have seen that there is a continuous increase of Kinetic energy. Because of total energy conservation, we have at the same time a decrease of potential energy (the complete energy balance can be verified in detail but requires the evaluation of binary correlation Fourier coefficient and will not

be considered here) . This

continuous exchange between kinetic and potential energy of course occurs for any system when the non markovian description is retained . The particularity of the gravitational plasma is that it occurs at lowest order, which finally is due to the fact that there exists no approximation corresponding to instantaneous collisions. However, we may wonder whether this picture could not be affected if initial correlations were present. As we are dealing with

long range forces, once initial correlations are

present, there is no mechanism by which the system can loose the memory of these conditions in a short time as it happens for systems inte;,racting through short range forces. The fact that the true collision time for such a system is very long on the time scale over which we discuss the behavior of the system, has the consequence that neither can we consider the collisions as complete not· can we assume that

the

system has forgotten its initial conditions. Therefore, we have to retain f

both the non markovian character of the collisim term and the destruc-

- 347-

F. Henin

tion term in

the master equation.

We can easily see that the presence of initial correlations will indeed modify our results. Let us take as the initial condition a function of the hamiltonian: (VII. 5. 1)

f(O)

= f(H)

If one adds to (VII. 2.2),

the destruction term and computes it

for this initial situation with the same assumptions as the collision term, one verifies easily that the increase of the kinetic energy which results from

the

f

contribution is completely

cancelled by the destruction

contribution. This example clearly shows us the important role played by the initial correlations in

the description of systems interacting through

long range forces. It would therefore be of great importance to have realistic models of non equilibrium correlations.

References.

1. I. Prigogine and

G. Severne, Physica ( to appear, 1966)

2. S. Chandrasekhar. ··Principles of Stellar Dynamics n U. Of Chicago Press (1942) , Dover, New York (1960) 3.J.Brocas, Bull.Acad.Roy.Belg. cl. Sci. 50, 765 (1964) 4. K. Hauboldt, Physica 28, 834 (1962) 5. 1. Prigogine, Nature 209, (1966)

CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C. 1. M.E. )

T. KAHAN

"THEORIE DES REACTEUHS NUCLEAIRES METHODES DE RESOLUTION PERTURBATIONNELLES, ITERACTIVES ET VARIATION.NELLES" .

Corso tenuto a

Varenna dal 19 al 27 settembre 1966.

-- 351 -

THEORIE CINETIQUE DES REACTEURS NUCLEAIRES METHODES DE RESOLUTION PERTURBATIONNELLES ITERACTIVES ET VARIATIONNELLES .

par T.KAHAN

Avant d'aborder la th~orie mathematique dt sport qui

r~git

I '~volution

r~acteur nucl~aire,

des

ph~nomenes

l'equation de tran-

neutroniques dans un

il me parait indispensable de Mfinir et de preciser

un certain nombre de notions majeures en physique

nucl~aire

.

On admet en physique atomique que chaC{ue atome est par un centre ou 'loyau

positivement~:

que toute]a masse de I 'atome, autOllr

charg~,

constitu~

a lui

contenant

seul pres-

du quel gravitent un certain nombre

d'electrons n~gati!s(ou n~gatons) .

"Y&f' Q{lU

t ~D""Ue.

~s

Dans un atome neutre, la charge positive du noyau de signe contraire

a la

un multiple autre Z..

somme des charges de la charge

'nombre atomique' de l'atome . (Z Tous les me d' H ) et de

n~gativ€S

des

e.Le nombre

est &gale et n~gatons:

c 'est

Zest appel'; Ie

= 1,2,3, ... )

noyaux se composent de N n';utrons: particules

que. Le nombre A=Z+N

Z

protons (noyau de I 'atode charge electric.

- 352-

T, Kahan

est Ie nombre de masse du noyau, Ainsi les nombrffi de masse du proton et du n{!utron sont l'un et l'autre (!gal taires de masse

a peu pres

tiques

d'une seule et

diff~rents

~gale,

aA =1 .

Ces deux particules t11emen-

repr{!sentent, au fond, deux

m~meparticule

~tats

quan-

fondamentale, dite nucl{!on,

Une r~action nucl~aire est un processus qui a ·lieu lorsqu rune particule nucleaire - nucleon, noyau, photon, - entre en choc avec une autre particule . Considerons une reacteur du type a+C-+R+b qu'on ecrit aussi

qa,

b) R .

Cette notation signifi~ qu'une particule a frappe Ie noyau cible C et produit Ie noyau residuel R et

une

particule

emergente b

~ .~

~®

Ainsi par

exemple

pour donner lieu

a

un neutron (n) frappe

Ie noyau de

un noyau de lillium avec emission

iO n t 5 B -'> 3 Li 7 +

'to

fl., 1.0.

base

~B

10

d rune particule'"

B(n, '" ) Li .

Le terme fission signifie rupture dlun noyau lourd en deux (plus rarement en plus de deux) fragments un pMnomene exo energetique , c 'est

sensiblement egayx. Clest

a dire un

reaction accompagn~e d rune.

enorme liberatioh d'lmergie A E(suivant la relation d'Einstein

ld:""I\"~iol\

Am

= c 2 AE

d.tMas:sc

til tt,'Y411)

(EIII ..

,i' li~i",)

- 353-

T. Kahan

La premi~re fission decouverte fut celIe delI'isotope 236 de l'uranium obtenu en

bombardant

avec

thermiques ( crest

des neutrons

l'isotope rare (0,7

de la vitesse d'agitation thermique (E

% ) 235

de celui-ci

a dire avec des neutrons animes o

= 0,025 ev a T = 300 K) ) suivant

la reaction :

y n + U235 _ ) J236 _> F +F neutrons+ rayons P +rayor (instable) 1 2 +energ en Fl et F2 etant deux fragments nucleaires teis que 40Z297 qui sont des nuclides in'ltables jusqu 'a suivant

ce

qu 'un

qui

nuclide stable soit

se

atteint

T 137 et 52 e desintegrent en chaine seion le schema

- 354-

T. Kahan

Fait tres imputant, il apparait (en moyenne 2,5 neutron) sont que

~mis

la fission a eu lieu ; ce on

petit nombre

de neutrons

qu 'un

petit nombre de neutrons

par les fragments de fission apres les

(N" % pits

n~utrons

prompts

retard~s

; en tres

sont emis par les

fragments au bout d 'un temps relativ~ment long apres la

fission (Fraction

- 355-

T. Kahan

de seconde a

55 se.)

Ces neutrons retardes jouent un rale

maji;!ur

dans les reacteurs atomiques . Reaction en chaine . Le fait quI un neutron capture par Ie noyau

J35 provo que

l'~mission

avec une ~nergie cinetique

dlenviron 3 neutron (2,5 en moyenne)

de quelques Mev

aU total, entraine la

possibilite que ces nouveaux neutrons ament en degager par la suite

@~

'S+0~

2

3 =9 ,

~ A

puis 3 3 = 27 ,

Ia chaine continuant ainsi noyaux [issiles ,

a

flJ)

en

• principe, jusqu'a

I 'epuisement des

condition que chacun de ces neutrons induise

nouvelles fissions. En cinquante

de

generations, on aurait de la sortf

neutrons presents dans Ie syst~me . Il se produit ainsi une chaine de reactions ou e neutron initial agit comme une allumette appliquee corps combustible : la chaleur degagee met Ie feu

a une pattie

du

corps et

par la flamme

a

un

de I 'allumette

la chaleur resultante provoque Ia

combustion de proche en proche d'autres

portions

jusqu la ce que

Ie tout soit consume et son energie chimique libereE! Faisons quelques ordres de grandeur 200 Mev au cours Ceci est equivalent

. II se degage environ

d'une seule fission par un seul neutron du 0,03 e v

a

- 356-

T. Kahan

3,2.10

f\J

La fission complete

-11

d 'u

1~

de

watt. sec libere donc

l'enorme quanti-

d' energie de

t~

8,2 ,10

autour de la fission

est

temperature. ordinaire

de '" 0, 0 3 ev on voit

que I '~nergie degagee

d'u possede

d'environ: 200

7 10 9 fois

Htv -;

O,03e.v

plus

kw.sec.

a la

Comme chaque atome une energie thermique

7

.

grande que I 'energie calorifique

re par la fission

est suppose etre de

de

I 'U . Si

10 - 8 sec,

la 50 0

10 25 neutrons auront ete, serait atteinte en

oil

Se.c.o",,jL

->

des neutrons est

que la fission: et une

absorMe par des processus de capture autres du systeme

.< t·lOn genera

moins d'une micro-

bQl"lb(.atomique . En realite, une pot i un

autre partie s 'evade

Ie temps necessai-

Neanmoins

si la perte

des neutrons

n'est pas excessive la possibilite d'une reaction ell chaine et d'une explosion subsiste. Si d'autre part, on part 11 de

liberation de cette energie et

ble dans

Ie temps, on obtient

a la

plier

maitriser

a

la cadence

une utilisation contr81.a-

un reacteur nucleaire (ou pile atomique) .

Reacteurs nucleaires. Le reacteur nucleaire est un dispositif comprenant

une matiere

pouvoir entretenir

fissile en quantite suffisante, disposee une reaction en

telle machine est possible,

de

faGon

4.

chaine contralee En principe, une

car elle exige seulemeYlt

que la vitesse de

production des neutrons par fission ayant lieu dans Ie reacteur soit egale

a la

vitesse

ses de parte.

de disparition des neutrons

due

a toutes

les cau-

- 357 --

T. Kahan

Cette condition minima1e se traduit

en

disant que, pour chaque

noyau subissant la fission, il faut qu'il se produise en moyenne au moins un neutron

qui

induise la fission dans un

autre noyau. Cette conditio;,

s 'exprime par un facteur de reproduction ou de multiplication reacteur defini

C

comme

k

du

la

nombre de neutrons d'une generation nombre de neutrons de

guelco'lgue

la gem!ration immediatement prece·

dente.

a

Si C est rigoreusement egal la reaction

1 ou quelque peu

en chaine pourra

superieur

s 'amorcer, mais si

la chaine ne pourra pas s 'entretenir . Si C 6 > 1 il suffit

C

.c:::. 1,

d 'un petit

nombre de neutrons pour amorcer une chaine divergente de fissions. Pour emp~cher introll..!ire un

qu 'une

telle chaine echappe au controle, on peut

absorbeur de neutrons. Si

C < 1 , la chaine, au lieu de

se propager , finirait par se desamorcer . . Ceci etant, les neutrons liberes par fission possedent des ener100

a

200

Mev . Bien que de tels neu238 trons rap ides puiss rut induire la fission aHa fois dans I 'U (abondan235 ce normale 99, 3~, ) et dans 1'U (abondance 0,7 %) , les progies cinetiques de I 'ordre de

balites de fission (ou section efficace de fission) sont bien plus petites par les neutrons rapide::;

(energie'"

a 200 t1~II)

que par

les neutrons thermi·

ques (energie - 0,03 e v) nest donc indispensable de ramener l'energie de ces neutrons rapides . (In est dit

moderateur

ainsi

a des

energies thermiques (v. thermique -200m/

conduit

a

qui sera d'autant

rapides au prix du

placer

sec

)

dans Ie reacteur un materiel

meilleur qu'il amenera les neutrons

nombre minimal de chaos elastiques au r, i .. eau

- 358-

T. Kahan

thermique . Llhydrogene serait ideal, nletait la reaction de capture I 2 H (n, Y) H, c. a.d .

On

utilis~

avec

succes Ie

sous forme de graphite

carbon~

pur, leau lourde

etc. Dne autre difficulte tient

a ce

de neutrons , dans la region dite de

que

i.Sf

liD

resonance,

est un

fort absorant

a environ

E = 100 e V

~I;';;,:':: )f~ - "

I

....

--~--------~I----> Joe

Un te11e capture donne pas lieu

a la fission,

E

~v

Si les neutrons rapides sont

ralentis, leur energie passera obligatoirement par cette region. Cette difficulte peut Nre tournee en

pla~ant

1'uranium , par exemp1e, sous forme

de barres au sein du mocterateur, au lieu de Ie melanger intimement.

Les neutrons qui naissent pres de parois de reacteur peuvent s len echapper et Nre perdus ainsi pour 1a reaction ramifiee 'Comme cette fuite des neutrons est un phenomene de surface et que 1a production des neu-

-

359-

T. Kahan

trons est un effet de volume, on peut rendre le rapport d 'espace I'olume aussi petit qu 'on veut en augmentant la masse du aussi le r~acteur

On entoure

r~acteur.

d'une enveloppe protectrice de materiau, dit rHlecteur

ayant un grande section efficace pour la diffusion des neutrons et une faible section efficace pour la capture et qui l'interieur du en oxide de

r~acteur

. Ce reflecteur peut

~tre

les

neutrons verS

construit en graphite,

beryllium, etc. La figure suivante donne une

sch~matique de la fission d 'U 235 l'U238 dans un R~acteur nuc1~aire .

tion

rHl~chit

et de la capture des neutrons par

. Re.fle.xio 1'\

Rl ~le.)(iol'\ '\\ /

\\

repr~senta­

- 360-

T. Kahan Chapt. I L 'equation du reacteur et les ~quatiol ,s de transport cin~tiques L 'etat dynamique d 'un reacteur est determine avant tout par Son bilan ou budget neutronique. Le point de Mpart d'une analyse cin~tique est par consequent une relation de

math~matique

qUi exprime

ce bilan au COurs du temps. Cette relation

Ie nom

l'~volution

ponte,

math~matique

riu reaction.

d'~quation

~~~~------------

La prototype de l'equation du

r~acteur

est l'equation integro - dif-

ferentielle de transport de Boltzmann qui formule Ie bilan du nombre de molecules dans un ~lement de volume de l'espace des phases Soit

f(

.., -..,

v ,t)

'l.,

la fonction de distribution d 'un gaz,

moleculaire.

Le nombre de molecules cOllte!IUeS dans l'element de volume de phase d7, dV' est pour definition (1)

dn = f(i'>,

v~

t) d;> dv", avec

la densite num~~ique des

=Y dv'"

.,

n( .. )

etant

molecules gazeuses.

Quelle est l'equation sous

n(;?)

qui

r~git

l'evolution

de f au cours du temps

l'action de diverses causes ? Pour l'etablir, analysons l'evolution

de f.dans l'espace des phases.(i?,v). Dans l'intervalle de temps ben l'absence de chaos entre mol~cules , les coordonnees des molecules

-,

~J -)

v'

"-

, .., ."?

F v +m

-)

, v,

ft=

.)

F m

deviennent

+ v &t

= '7+

• (2)

(? , ~)

...') v

at

= -">

F

=forme

I

exterieure

- 361-

T. Kahan

m = masse des mol~cules. Au bQut de d

1 d v;

bt,

les mol~cules initialement dans

se retrouveront dans l'el~ment de phase ('I. + -:j

-7

9 t, ";f + ~

m

8 t)

lI\ol~t.llrt!l ~~i ;. f... ~u h eI, dI.,s h,,1Iii IF"s~u

kOH.lt "it> "I;~

E ~e "',,,f

J( I'h

... e)':'''I~Q. .. rC+

VoluWtc.

de

l'101t c.ufH ~'"

d.t' d~ r:i. e't""~Q. ...1- t'

Ii_ el".a

.. f... ruit.

~o .. 1- ...... .,.,' C\ cil

,~, .l~

Par developpement en serie de Thylor

f(~ +,,~

(3)

=

..I)

-')

f( 'I.

, "3+

,

v, t) +

-?

F m

i

bt, v

i

df dt

f+-

+"t)= F

it +2 d)I,

m

ft

La var iation totale Ie long des lignes de ,lux sera ainsi

+

(4)

Admettons

maintenant que

't

-'> F.'a(

-.:::-=s. ~ Vv m

dV

(I

r

soit beaucoup

plus grand que la

dur~e de choc. Alors plusieurs chocs auront lieu dans l'intervalle

quelques uns vont faire entrer quelques mol~cules dans l'Nement dont d'autres vont

sortir

des molecules hors de

dl

gt

dP d1,

dont et

dV' (voir fig. pag.22).

- 362-

T. Kahan

Si nous designons la variation par unite de temps de f a la suite des chocs pour

(~t 6 chocs

(5 )

l'equa. (3) prend Ia forme 001)

f(~+ fr~ ~? +r v , t +& t) - f( 1, v~ I;

(6)

Soit

Sf

(- -) ~t

cSt chot,

-")

~f

(7)

-

-at

~f

.0;>

+ v. -

li'

+

~f

F-

~V>

m

Crest Ia I'equation generale de Le premier nombre de de

)

(7)

Sf

( -)

/) t

Boltzmann.

n'est autre chose que la derivee totale

f Ie long de la trajectoire (2) . Reste

Ie du second membre de

Dans

Ie

-)

a ~tablir la forme fonctionnel-

a savoir

budget neutronique d 'un reacteur les forces exterieures

F, contrairement jouent

(7)

chocs

a ce qui se passent dans les

gaz traites par Boltzmann ne

pour les neutrons seule entrerait en ligne de

aucun rMe .Car

compte la pesanteur dont I 'effet, pour des vitesses moyennes thermiques de

2200

m/

S{'C

et

des libres parcours moyens

des neutrons de quelqu~s

centimetres, est cependant negligeable. De m~me, choc

neutron-neutron est un evenement fort rare:m~me dans

2 S('C' 1'I ne se trouve que Ie cas d'un flux neutronique de 1014/ cm. 8 22 4,5.10 neutrons dans un centimetre cube compare a 10 molecules gazeuses.

I

- 363-

T. Kahan

D'autre part, Ie sort d'un n~u'.ron au cours

du choc contre les

mat~riaux du r~acteur tels que mod~rateur, matiere fissile, madriaux de structure, rMrigerant, poisons , etc. est tr~s vari~ . Les chocs I

elastiques et

laissent Ie nombre de neutrons

in~lastiques

alors

inchang~

que la fission et les processus de capture provoquent un changement du nombre de particules et dont l'~quation de Boltzmann ne tient pas compte. Depuis la

de la physique neutronique on a

cr~ation

quation dE's transport d'un

degr~

de

g~n~ralit~

des

formul~

~­

variable pour la diffusion

des neutrons. Dans notre

expos~,

nous allons,

forc~ment limit~,

~tablir

une for-

mulation matMmatiquement aussi g~n~rale que possible mise sous une forme

ind~pendante

Apr~s

avoir

du temps et tenant compte des neutrons pr~cis~

quelqUl's concepts de des

pr~senterai l'~quation g~n~rale

r~acteurs

physique

retard~s.

nucl~aire

je

sous une formulation de

transport. Avant de proceder ~ l'etablissement de l'~quation des r~acteurs. il est indispensable de pr~ciser que nous avons introduits Rappel de physique

un certain nombre de concepts atomiques

di!ja. nucl~ail'e.

Soit un faisceau

de neutrons monom~trique de vitesse . fca.lscc/lv v qui penetre darls une cOllche de substance !I.e.. I

•

3

contenant n parlicule , par cm (cf . fig. 1) Ces neutrons subiront des chot's de

lI'~~ro .. S

e------'";;>

diverse~">

sortas. Soit P()) la trons

probabilit~

dans Ie corps

a la

de si!jour des neudistance z. La

perte de neutrons par absorption dans Ie

~,~ /~. /

&-->

/ /

--

/

- 364 --

T.Kahan

faisceau (1 )

ou

inddent peut ~tre mise sous Ia forme (S =surface du corps)

- d P(z) = P( z)

6 d~signe

)C

n.6S d z S = P(z)

no

d z

Ia section efficace microscopique

du r.oyau individuel absor-

bant qui est dMinie de Ia fac;on suivante. Soit N Ie nombre de neutrons par cm

3

voyage ant dans Ia

viennent

direction

dans notre faisceau et qui

frapper une couche mince substance (cibIe). Le nombre de

processus ou de

r~actiolls

au nombre de neutrons f par unit('! d'aire de Ia

ou de chocs

observ~s

sera proportionnel

venant frapper Ia cible et au nombre de noyaux

cibl~.

Nombre de r('!actions (chocs)

(2)

z

2

cm • seQ

= G' x· nombre de noyaux cibies cm 2 neutrons

N v---2

cm. seC!

Nv , Ie

flux, est Ie nombre de neutrons venant frapper une surface unit(,!

de Ia cible par seconde. La constante de proportionalit~' est pr~cis~ment la section

efficace microscopique

pour l'('!venement donn('!.

L'~quation (2) montre que Gala dimension d'une surface. Beaucoup de sec-

tions efficases de r('!actions

nucl~aires

sont conprises entre

10 24 cm 2 . Les sections efficaces sont souvent 2 lieu de cm

1 barn = 10

n

r~sulte

de

-24

cm

indiqu~es

10 - 27 et

en nbarns"au

2

(2)

fraction de noyaux cibles reagissant par sec. Nv

,

ce .qui peut slinterpr~teren attribuant

a chaque

noyau cible une

surface

6

- 365T. Kahan

a la

perpendiculaire Ie

direction de mouvement du neutron incident

a frapper

neutron incident arrive

.

S~

cette surface, il r~agira avec Ie

noyau: cible correspondant. ()

d~pend

bien entendu de la vitesse (ou de l'energie E"

du neutron incident et du produit sera

de la r~action

(n, y ) et (n, p).

r"\V~J)

Ainsi G(E)

diff~rent

pour la

r~action

Vun des

majeurs de.1a

th~orie

nucleaire est de donner une expression pour la

objectlfs

grandeur de (). Revenons maintenant

a

(1)

dont l'int~gration conduit

e

p ( z)

(2)

compte tenude

la relation

~

(3)

P (z) dz

L lequ. (2) permet de dMinir

.~" ~

(4)

a

-noz

de normalisation

1.

un libre parcours moyen

z P (z) dz

o ce qUi

conduit

a son

tour

a l'introduction

de la section efficace ma-

croscopique

!

(5)

=

n 6

c lest la surface efficace non pas par

y = v

A

v"

~

Si Ie reacteur renferme un de noyaux , lIon

a la

r~gle

3 ' cm . Definissons

d lun neutron par seconde par :

encore Ie nombre de chocs moyens

(6)

noyau mais par

(see

-1

)

melange

de melange

homog~ne de diverses especes

- 366-

T.Kahan

y=

(7) n.

1

~tant

la

densit~

du

[y. i

i-~me

Les considerations sorption, mais les m~nes

1

= v

! n.

6. = v

ill

constituant du

prec~dentes

ri

£

i

r~acteur.

se rapportaient

a des processus dlab-

raisonnements valent pour touts les autres

m~mes

ph~no­

nucleaires. Ainsi on parle de section efficace pour la diffusion (ou chocs)

~lastique (6" ), pour la diffusion (choc) in~lastique) (G.) pour la fission (6). ea 1 f pour la capture de neutrons (6). et pour l'absorption (6 = 6+6) . Tous

c

a

c

f

les concepts introduits jusqu la present s 'appliquent aux processus precMents: il suffit de mettre l'indice approprie correspondant. En divisant grosse modo la bande d'energie E des neutrons en trois regions 1) la region de

r~sonance

comprise entre 0 et 10k e v ,

2) la region de vitesse moyenne de 10 M ev

a 0,5

Mev, et 3) la region

des neutrons rapides de E > 0,5 Mev, on obtient l'allure suivante de la section efficace de capture

c en fonction de E = m

i 12

Le reacteur au sein duquelles neutrons evoluent dont la composition et la densite varient dlun point

poss~de

a l'autre

une structure

. En passant de la

zone de fission au reflecteur, etc. En outre, il se produit au cours due temps des changements tels que : - combustion, rec;:hauffement, intervention des bazzes de regulation du fluc neutronique etc. c lest pourquoi les sections efficaces et les frequences de choc sont fonctiona ala fois de la position

(? ) et du temps.

A fin de ne

- 367T. Kahan

pas sure harger

l'~criture,

ces variables r et t seront impicitement

suppos~

dans nos formules sauf avis contraire. Les divers processus de diffusion affichent des

d~pendances

de

l'~nergie

et de la direction (angle de diffusion. C'est pourquoi il est indispensable de

pr~­

ciser avec soin la nature du processus de diffusion. Pour dresser de

sim-

pIe Ie bilan neutronique, nous allons, (neutron dit primaire)qui participe noyau

choqu~

a titre

a une

fa~on

de convention, compter tout neutron comme absorM par Ie

r~action nucl~aire

. A sa place, un certain nombre n de neutrons secondaires vont s

appaitre: neutron primaire

o

neutrons secondaires

~

Ce nombre n

est nul (n = 0) dans Ie cas de l'absorption. n = 1 pour la p p d iffus ion et n =jJ pour la fis s ion . s Ceci pos~, envisageons un neutron d'~nergie cin~tique E', se d~pla~ant dans la direction du vecte:lr unit~

n vers Ie noyau suppose d 'abord ou repos

La

Mecanique quantique des processus nucleaires nous donne la probabilite . .-t -:7 ~ ~(EI, £1lr-i E,n) dEd£1/ 4IT pour que Ie neutron secondaire entre E et E

soit emis avec une

engendr~

+ d E dans un ctme d

d aut our de ~

~nergie

comprise

(fig .... )

-i -

[L,t:

Oneutron secondaire Fig ... On peut ainsi dMinir pour Ie processus

envisag~

un nombre de chocs

differentiel =

(8)

=

Comme probabilite,

est

t;')

yx. ~ (E', v'l: (E')

norm~

a

~

..

£11 ~ E, £1) 1


1/

-t

4lTdEd £1 '

-+

7-

4n ~(E', QI..+ E,' n) dE d n

- 368-

T. Kahan

_1 iJ~d

(9)

rf = 1

E d

417'

Le nombre de chocs

total, difini

par, (6\ est

donc

(10)

Dans ce sens (11)

est la section efficace differentielle pour la diffusion d

-::::>

u

autour de

-)

0 et

dans

la bande

r~el1e

Slil slagit dlune capture diffJrent de z~ro ~mis.

(12)

=0

E

que si

dl~nergie

dans llangle solide

d E aut our de

de neutron, alors

fJ>

E .

ne peut

~tre

, clest a dire que si aucun neutron nlest

doit donc poser

On

rc

dE

dO

lci c> est la fonction Par

=VI

r

1 41T

(EI). - -

c

-)

~ (E) dE d 0

de Dirac.

int~gration

y

(13)

La diffusion

c

=

SSr

~lastique

C

est

dE d

fi') =VI LC(EI )

associ~e

aune cession

dl~nergie

par

Ie neutron incident au noyau cible· On peut alors ecrire : (14)

rd

dEd

rr

= VI

La valeur de la perte

Id

(EI) _1_ 4tr

dl~nergie AE;I

diffusion resp. de son cosinus 0=0. (

1)

(qui d~pend de l'angle de

se calcule par

la ml!canique du

choc. Pour,la fission J on peut admettre une distribution uniforme des neutrons de fission sous toutes les directions spatiales. En outre , Ie spec-

-

369-

T. Kahan

tre d I~nergie de ces dant de

(15)

l'~nergie

rf

une large mesure,

des neutrons, causes de la fission, de sorte -')

dE dO

Comme

f(E) est, dans

n~utrons

i = ):

= Vi

que

->

1

2:f

ind~pen-

(E') 41T'

f(E) dE dO

lion a pour Ie nombre de chocs total par seconde pour

tous les processus mis en jeu

Y=

( 16)

vl.

= v

(r,d

+

~ +~) c

f

et pour Ie nombre des neutrons secondaires

produits par ces processus

par seconde (17)

on v(E) est

Ie nombre d'accroissement de neutrons par fission.

L'int~gration

)) r

(18 )

de (17) donne d E d

rt

= Vi

Problemes majeurs en Nous voila tants pour

(l.'d + 'V I

r.; )

de transport neutronique

th~orie

maintenant en possession des concepts les plus impor-

l'~tablissement

de

l'~quation

Avant d'aborder l'analyse

des

r~acteurs

math~matique

du transport et de la migra-

tion des neutrons, je vais brasser un tableau rap ide des problemes majeurs dont

Ia solution incombe

a la

th~orie

du transport.

En consid~rant la variation avec Ie temps du nombre total des neutrons N dans un systeme

Sans tenir compte de la

r~at:tif

, on voit que (f)

(n,2n)

- 370-

T. Kahan

(a) Production par des Sources ind~pendantes - capture - ~vasion

() N du systeme - - - t ~ N C>

hors

~ N

- - , slil est positif

ou Ie terme

d

+ Production par fission

-

(~>

t

0 ) represente llaccroisse-

par unite de temps de la population en neutrons du reacteur.

ment

( 0 N < 0) "0 t

s lil est n~gatif

I

'

la diminution par unit~ de temps, prise

est

avec Ie signe negatif "Production par fission" signifie llexc~s du •

J

hberes par fission

sur Ie nombre

n ombre de n~utrons

de neutrons absorMs en produisant

la fission. Des sources

telles que les neutrons) dus aux rayons

ind~pendantes

cosmiques, sont toujours presentes dans un reacteur mais elles sont normalement negligeables compar~es aux sources artificielles introduites

ou

a la production par fission. Ainsi en llabsence de sources artificielles, Ie premit!r membre de (1) Supposon~ma"ntenant

soient

se reduit

'Z..

=R

chimique et la densite

demeurant les memes. La production de neutrons sera sensiblement proportionnelle au volume occupe pour la matliere fiss~le et variera ainsi par un facteur r

3

La capture va changer aussi par un 3 facteur r, et comme elle est necessairement

inf~rieure

a la

production,

elle representera une function constante de cette derniere trons hors fois

d~

production par fission.

que les dimensions lin~aires R du systeme

augmentees dans Ie rapport

la compos ition

a la

J

Llevasion de neu-

reacteur , sera toute-

sensiblement proportionnelle

a la

I /

R en devenant R I = 'l.R ,

-

371-

T. Kahan

. . par un facteur r. 2 -L' ~qu. (1) fournit alors surface frontiere, et croitra, amSl 3 3 2] NLF r - C r - E r ::

(b)

r

N, car la production par fission, la capture

ou figure aussi Ie facteur

et I '~vasionl seront toutes proportionnelles tante.

F , C, E et

A

sont des

J

L.'equ, (p ) rnontre soit

~

o

,telle

3 - E r 2]

a la

quantit~s

imm~diatement

population nelltronique exissensiblement constanteS,

qu'il existe une valeur de

'l..

que par

<

r

est

r

o

,

n~gatif

3

2

( Ar - Er o 0

en I 'absence de sources ind~pendantes,

la dens ite

neutronique diminue-

ra exponentiellement (f) N :: N e. o

Pour

r" >

'1.

o

- tiT

T )0

,toutefois, est positif

et la population neutronique va croitre exponenHellement

N = N

o

e

t / T

(f) Dans les problemes dependant du a~mps , cette decroissance peut n '~tre qu 'approximativement exponentielle ;puisque A et E peuvent dependre eux A memes d~ temps pur, exemple si des fractions de la population neutronique les diverses portions du reacteur avec Ie temps.

- 372-

T. Kahan Si r = r

o

} Ie r~acteur est dit critique} et si rest une dimension

lin~aire caract~ristique, 'to porte Ie nom de valeur critique de cette

dimention. Pour r > 1... en effet la population croU indMiniment et tl. moins que l'~tat du r~acteur varie avec Ie temps,le r~sulta risque de solder par une catastrophe. ~

La determination de la dimension critique est Ie probleme Ie plus important de la confront~,

du transport neutr-onique . Il arrive aussi qu'on soit

th~orie

soit avec des problemes fonctions.du temps, soit avec des pro-

blemes oU 1'1\xistence de sources, independante de neutrons entre en jeu . Forme

g~n~rale

de l'equation des reacteur;(Bilan neutronique)

La grandeur fondamentale de la

-")

( 19)

N( r,

th~orie

-")

t, E , rl)

c 'est la densit~ des electrons au point vitesses v

?f

='./l e E/m

bable de neutrons dans une d

(20)

sont dans la b,ande

sont dans la bande d

pour

de E Le nombre pro-

= dE . d rl . d V

-)

-)

(v=vrl)

N(r~ t, E, (2) d'1! . Le bilan

alors

cet element se met sous la forme: d1\r

(21 )

'ON

F

gains

- pertes

Commen~ons

par le_-ialcul des pertes neutroniques

de vecteurs.

par

(22)

v)

-)

d rl autour de

de volume

el~ment

de l'espace des phases 'Vest

d E autour

v autour de

-)

/If

densit~

Id/ E d/6.1 ( en cm -3 ); , al'instant , dont les

sont dans Ie cone d 'ouverture

et adont les "energies

pont les vitesses

de transport est la

r

N(

--, 't I t,

-':>-':>

. Definissons Ie flux

E, rl) = v. N .

- 373-

T. Kahan

Pour tial

cons~q uent,

la parte nette par fuite hors de l't'!lt'!ment spa-

d V se traduira pas

V· t

(23)

i?-

t

d'lfz - div

-">

-v.'l N. d'1J

d- 'IT

En
qui

n~utron

aura subi

un chor quel( onlue sera tout d'abqrd compt~ comme l,er'e. Par (16) , 1'on a pour

la

"perte par

choc"

(24)

-YNdV

Venons -en maintenant aux cessus nuclt'!aires qUi venir dans

(d

1T

gains, Les neutrons secondaires dlls ----

ont lieu dans

dV

I

-)

= d V d E' d 11'

aux pro-

peuvent par-

-'>

= dV d E d 11) . Le nombre de neutrons dans l't'!lement

d '1f ' est ~

N('t,t,

(25)

il

-"j

E',I1')

faudra comptabiliser (

(26)

) 11

Enfin, il

dE'dl1'dV. Par (17)1

un gain de -)

N (:, t, E', 11')

a lieu de tenir

5~

=-

r":

-')

11'_:> E,I1) d E' d

-'>

n'

compte de sources exterieures S(t , t, E,iL)J '!If

Le bHan (21) prendra donc

(27)

r (E',

v . \7N -

r

avec (22)

N +

r~

qui est en principe, l't'!quation des rt'!acteurs

I

a N'

(26) , la forme

r

d E' d 11' + S .

ou nous n'avons pas tenu

compte des neutrons retardt'!s . Dans la fission il se crt'!e entre autres des fragments avec exces de neutrons (/\Ui sont des t'!lt'!ments nt'!gatogenes. Le noyau

crt'!~

par

*) Pour la stabilite'd'un noyau, il ne faut pas que Ie nombre de neutrons N = A-Z exede une certaine limltl'a. Pour les noyaux It'!gers N ~ Z

- 374-

T. Kahan

fission peut

~tre

assez excite pour emettre un neutron immediatement

apres sa naissance. Le fragment de fission en question, noyau mere du n~utron,

~miE

a une vie moyenne, determinee . Clest pourquoi Ie neutron sera

en

moyenne avec un

certain retard apres la naissance du fragment

de fission au cours de la fission. Les neutrons

des neutrons dits prompts,.nes lors de Ie fis-14 '/, 10 sec. On distingue actuellement 0 groupes

farol1 se distinguent ainsi sion en

ergtmdres de cette

retar~s,

mains d'environ

d ifferent5 de neutrons retardes . Comment tenir compte de ces neutrons retardes dans llequation des reacteur.s? Revenons pour cela a llequ. (15)

rf

(15)

La fraction la fonction

~

= VI

dE d!1

(3 (E I)

Cl

-(3

~

1

-)

L f (EI) 41T f(E) dE d !1

(EID de ces neutrons sera emise ~ompteme

de maniere

retardees

. L Ion aura

designe Ie nombre des neutrons retardes du

(1

~ f3i ou ;S i

=

.

i

A.-eme groupe rapporte

a

f (E) des neutrons prompts o est representable entre de larges limites, par la formule 1 neutron prompt

(28)

. Le spectre de fission

f (E) = 0,48

o

e

-E

sh

[2E

(E en Mc.v )

avec la condition de normalisation tP

~

(29)

f

o

(E) dE = 1 .

Le nombre de secondaires nes par secondea la suite des \..

sus envisages sera des lars, au lieu de

divers proces-

(17)) ->

dE d!1

- 375-

T. Kahan J

D'apres (30) , on peut separer Ie gain par

('

(31 )

-"')

o

du gain par

~

N('t, t

)

E', 0)

-')

densit~ de probabilit~

rf

-":>

rf = ~ (E')

(33)

11"

O')v (E')

4lT

+

1

dMinie par

t0

complete

-litE')]

~;1i(EI)

d IIJ'

rd +r i) dE' d 0'

-">

E',

La

(

fission

(32)

dans

entr~e

-">

dE' dO'.

(15) a pour expression

fo(E) +

ritE)}

Pour Ie nombre des neutrons prompts,l'on tire de (32) , avec (33) ,

LQ. o

d1T

(33-1)

47i

-~

;9(E')])l(E') 'i'r(E') v' N(-:t, d, E',O') -)

X

dE' d 0'

En portant Ie second terme du second membte de (33) pour

r

dans (32), g on obtient les neutrons retardes. 11 convient, toutefois,l d'observer que les neutrons

a l'instant

retard~s i engendr~s

qui sont n~s des Envisageons,

fissions ayant eu lieu

a titre

II.1 .-;

G,('!:, t) 1

a un

dUB

instant ant6rieur

d

t' < t.

=

Leur concentration a pour expression - /) ,(t-t') G. (;;?, 1

t')

Q.

1

En admettant que les fragments de fission qui ayant lieu dans

meres

aux noyaux

d'exemple Ie i-erne groupe de noyaux meres, ayant la

constante radioactive (34)

t I sont

V, demeurent dans

de fissions

r~sultent

d V, Ie nombre d

C.(? t')d V 1

- 376-

T.Kahan

de noyaux ml!res I

est

par

determin~

tre dans

engendr~s

(32)

pendant l'intervalle de temps dt' dans d V

Ie nombre total de fissions dans d V dt' il faut met,

a la

place

de t

avec (30-1) (35)

d

CJi. t') d

=d

V

et

sur E

int~grer

-'>

et

n

. 11 vient

b

V dt' [

r1(E'h) (E') L/(E)

-) -'> -'> X v' N(r, t', E' n') dE' d n' . Dans l'intervalle dt \.

(36)

des

= Il. 1

int~grations

,

il se produit

-')

G. ( r,t") e

(t-t')

par sec dont chacune conduit

quent, Ie nombre de neutrons ment de volume dV par d v.).

1

r~tarMs)

re1al'd~.

a l'instant

nes

f r/~(E')

- ~(t-t')

t) dans

par

cons~-

l'~l~-

f

00

1

e

-00

XVi

a un neutron

seconde, sera au total t

(37)

-~1

1

0

N(

'1,

t

n

1

-)

-';)

I,

Lf(E')

E', n') dE' dn' dt '

Comme les neutrons retard~s naissent avec la distribution d~~nergie f.(E) , et que leur distribution angulaire peut passer pour isotrope J c'est 1

_)

la fraction

f.(E) d E d n / 4Ti qui p~netre dans l'~l~ment de l'espace 1

phases . On obtient ainsi comme contribution de m

des

groupes de neutrons

retardes .

-A ,(H') (38)

e

X~(E') ....

,

Vi

-)

1

J o

->:>

-.,

N(II" t, E', n') dE' dn ' dt'

Des lors II'~qu. (27) se mettra par

(31), (33-1) et (38), sous la forme.

-

377-

T. Kahan

dN

'd

=-

t

-) v

1 4i/

f (E) o

rO()(J N' 0

n

X que

8.ppelerai

dE' d

->

n'

0()

o

-0()

1

f3~ v'N'(t']

n

i f

dt'

~quation g~n~rale

lesquelles il faut

~

r elf i -/).(t-t')

f,(E)

des

r~acteurs

NJ 2,

ture, j'ai indiqu~ par des primes (') dans sur

v' N' dE'd

f

0

~A. 1= 1 n'

1

(

m 1 +8 + 4rr

n'

(~+ r.) d E' d

5 n) (l-!3'h)~) 0()

+-

(39)

'

\7N - '( N +

Pour simplifier 1'~crietc, les variables

int~grer

Si Ie reacteur faut poser

dans

contient divers mat~riaux fissiles, k=l, 2, ... r, il k k k ~ (39) "\l ,A. • It. . f. et l.. et sommes par rapport .I.... 1 1 1

k.

~

Leas particuliers : Une Supposons maintenant fissile pour entretenir

s~ule mati~re

l'

fiss He.

qu'il ya essentiellement Ulae !:eule mati~re

la chaine de neutrons par exemple

u235 ;

2~ Si , en outre. les fissions sont dues pour une part pr~pond~rante ~

des neutrons d'une petit ed lnde d '~nergie

ques respe

(r~acteur

a neutrons

r~acteur ~ neutrons rapides) • alors la contribution

thermi-

I'.

1

des

peut t:!tre supposee com me ~tant largement ind~pend!3~ 11(E')_ de la distribution ~nerg~tique des neutrons. (() E' =~ - 0)

fragments de fission dante

Pour la contribution des neutrons neutrons

contenus dans

l'~l~ment consid~r~

poser en outre d'une faeon (40)

1 4lT

retard~s

abr~g~e

dans (39) au nombre de

de l'espace des phases" on peut

-

378-

T. Kahan -")

oil. C,(? ,t) est la

des noyaux

densit~

1

correspondants . La

m~res

derivation de (37) par rapport au temps, fournit alors un syst~me de!'M. I

equations

int~gro-differentielles

qui. sont

a joindre a

(39):

dC,

(41)

~

-"-

L 'equ. (39) se met, des lors) sous la forme

~N -") Joo ~ ~ =-'II'SlN-'fN +

(42)

ot

0

+ f (E) (1o

J

-)

N(r+r,)dtdO'+b d

1

J n)' V' ~/N"JN' dE' .iE' + 1 X\,f. C. f 4-rr 41T =1 m

-'>

00

fo)

_1_

0

1 1

Elle porte aussi Ie nom d'equation du transport ou mann en theorie de transport des neutrons.

1

de Boltz-

d'~quation

,

Pour abreger, on peut introduire l'operateur de pertes et de diffusion

tI)

par

H = i. V + 'f

(43)

-

Jf

dE d& (r'd +/\) en see.--1

o 0

I

et I' operateur de production. -'>

dE' dO'

(44)

ot il convient de poser

(45)

(t-\ -,

4 (J

t

= (1 - f~ ,0)

f.

. 1

_1

'dt

-1

en sec.

t = i = 1,2,- --

Des lors) les equations(41) et (42) prennent

K0 N - HN

OC. (b)

v'

~= 0 pour les neutrons prompts et

pour les retardes. la forme

I

V' l:~

./' = 4 fi (j.

1

K.

1

N -

m

+_1 4 TI'

~ i=l

II

f C i i

+0

A.1 f.1 C,1

(If) Les op~rateurs sont indiqtes par un chapeau

.A

1\, par ex,) K

_M'\

J

- 379-

T. Kahan I

,

Les equations(39) respt. (45) determinent univoquement la densi-:>

t~ N( It), t, E, n) , compte tenu des conditions initiales et aux frontieres

donn~es j:"- .sur une frontiere d~limitant Ie vide (:f! = it f) il ne peut pas

, ,

y avoir de flux de neutrons venant du vide pour penetrer dans

l'int~rieur

-)

ni ). -")

du rE!acteur (direction (46)

N(

"I' t, E, ->nil =0

II~ En outre, dans Ie cas d~pendant du temps, il faut se donner les gran-

deurs : .:.>

d'apr~s

positif

1

nulle~

r~acteur

est stationnaire (critique) lea

dans (45)

En portant (45

dC i

b ) dans A

-"

L'operateur

"'K"

temporelles

(45-a) il \lie.,,~ alors

(K - H) N + S

=0

soit

doitJd'apr~s

,.

n

o

+

2..

i=1

f, 1

(441~tre

(I. , avec 1

/'

L N = - S avec L

f(E) avec le spectre fiE) global defini pour f(E) = (L ~t1) f

deriv~es

=0 •

j

(49)

N doit e'tre

.

~

(48)

C.(.;:, t = 0

son sens physique, all sein du reacteur partout continu, fini et

Si Ie sont

rn: Enfin,

>

-::}....

N(/I., t = 0, E, n) et

(47)

forme alors

~

1

....

......

=- K - H a la

place de

= const, par

co

f f(E) dE = t 0

II-Cas general : plusieurs matieres fissHes ~

Dans les equations qui precedent,j'ai suppose I d'une part que les neutrons retardE!s dans Ie rE!acteur ne proviennent que de la fission d'uneseule matiere fissile et qqe d'autre part, lafraction(3i avec laquelle les

-

380-

T. Kahan

neutrons

retard~s

du groupe .L apparaissent parmi les neutrons de fission

est inctependante de

des neutrons de fission . La premiere hypo-

l'~n~rgie

these tombe en defaut lorsque Ie reacteur contient plusieurs matieres fissiles en quantites comparables. Dans ce dernier cas,

t

je l'ai indique, mettre des indices de substance fission

et sommer par rapport aux

a haute

reacteur thermique

div~rses

il

dans

faut, ainsi que

l'int~grale

de

matieres fis·siles. Dans Ie

de substances l,lbreedes" ou

cO)1c~ntraction

sur r~g~neration il est dans certaines circonstances n~c~ssaire d lajouter une

im~grale

de fission aussi pour ces substances ferUles .

cas o'll' dans un r~acteur tel que celui dont il vien d 'etre que-

Au

stion, les fission!> aussi I bien thermiques que rapides, jouent un

.;g.1

valeurs des

s·::mt

sage de (39) (P.32)

(1) et

(3.1

de

a

a prendre

l~gitime

de fission. Si l'indice

k

divl!rses matie'-res , il faut remplacer dans (45) (p. Y1) ..... ..I>(1 -I)) k et (3.-k~ par o

k

alors de faire sortir sert

a d~signer les

les

op~rateurs

1

AU(50)

1 \' (I!.) =_L f (E) 4Ti h 0

'"?

dE' d[21

j. "'J(k) (E') ~

\.1 -

All)

-/~

(h) f

(E')

Vi

=

o

et 00 (

(51)

X

Les

~q\lations

-0

~)

(k)

_')

) ) dE' d [21 /( o .n.. (E')

~ (It)

1 (E')

(E')

L ~

Vi

=

k

(45) prennent alors la forme

Ji

l

(EI)J

= ~ i-~) k

les

pour l'~nergie correspC1nc\ante . Dans Ie pas-

(45) (p.36) il n'est plus

l'int~grale

r61~)

(R.)

- 381T. Kahan

.
(52)

("N f~)_iddt) ~t

=J

0

i 'dt

sous la forme (53)

1

1

1

1

1

1

I

generale des reacteurs (39) (p.30) peut alors se mettre

~N = ~r

L'~quation

~t

1

I abreg~e

C>t

/)(.~) f(£) C~) t S ,

k.{~ N - *~) f.~l c.[~)

= 4TT

,

L'~quation

m

1

1\

- H N+41i

N

f. 2::~A) 1

+

i=l

i1.

-co

1

(48) (p.32) pour Ie

(I.)

-~.(t-t') ""'(~)

e l K . N'(t') dt' 1

r~acteur

stationnaire s '~crit alors

sous cette forme plus exacte (54)

(~/QI--

II) N

+S

= 0,

ou (55)

Pour une seule mati~re fissile et pour des fl. constants, on retombe tl 1

partir de (54, avec (49), de nouveau sur (48) (p. 32)

R~duction de 1'equation des reacteurs fonction

tion independante du temps ou

du

temps

I 'equa-

equation stationnaire.

L'equation des r~acteurs (45) (p ~-1 ) peut se ramener transformation de Laplace

a

a une

forme analogue

(48) (pU ) tl condition que les sections efficaces

a celle

l:..

a l'onde

d 'une

de I 'equation

soient

ind~pendantes

du temps et pourvu ql,l'aucun changement dans Ie temps ne se produise dans Ie milieu reactionnel • Pour cela soumettons,

Ie

syst~me

une transformation de Laplace en posant (56)

e

-pt

rV

d

t:= X(p)

(45) (pill

a

- 382-

T. Kahan

Comme lIon a par ailleurs

1( ~~]

(57)

+

= -)< (0)

p~ (p)

,

il vient I\J

'"

- N +P N o

(1-/3)"

1 +411""

(58)

f. [- C, 1

10

-> N = N(I'!..,

ou

0

CiD = Cit J

En (59)

ou

N

+p C i

eliminant

OK

m

-~

i=1

'7. ,

t

N

~ i=1 4if

t = 0)

1

~gard

N

), f C i i

f3.1

./"

K

-'>

n) et

= 0 , E,

C, , eu

1

N

o m

/'

-pp+i\ ,

N

-I>

S N

N

-),

, t, C.

1 1

1

-')

N = N (It. , p, E, -':>

C, = C, (f!"

p) .

1

1

a (49)

(p 32) , lion a m

(3,

1

n)

.I"

)-.'

t. K,) - (H+p) N+(N+_1_L - - fC + 1 o 4'(1" i+1 P +).< i ","0

IU

~ )=-0

)

enl.Ot(:

en designant les grandeurs entre parenthese par

(60)

=0;

clest une

~quation

du type de (48) (p.32) avec p

comme parametre

(valeur propre) .

Chapter II 1.

Compte

La fonction d'influence ou d'importance (importance foncho tenu des conditions aux

limites et des contJiitions initial

corresp'(rldant~s , les solutions de l'~quation g~n~rale des r~acteurs (39) p. So

ou (45) (p'. 31) decrivent la distribution de la densite n~utronique

- 383-

T. Kahan N(r, t, E,

comme fonction de la postion

n.

de la direction de ,vol

1,

rI)

du temps t, de l'energie cinetique E et

Crest IS. la description differentielle par des

equations aux derivees partielles du budget neutronique. Or, en tMorie cinetique, des reacteurs dont la mission est de livrer les fondements theorique pour la commande, Ie contrtle et la regulation de reacteurs, on ne s 'interesse gu~re

a la

repartition neutronique en detail. Ce qui importe da-

vantage ce sont des relations integrales qui decrivent Ie comportement du reacteur dans son ensemble. Un grandeur de ce genre qui caracterise cette allure globale seraH par exemple Ie nombre de fission, par seconde rapporte au reacteur

enti~r.

Si Ie reacteur

est homog?me et nu

(c'est

a

dire sans rMlecteur),

il est possible d'~tablir une cin~matique simple des reacteur~ en admet.-

tant simplement que l'allure de N que Ie centre (ou coeur) llallure

en

un

endroit donne du reacteur, tel

du reacteur, est repr~sentatif pour l'ensemble de

du reacteur tout entih .

En effet l'egalisation (ou Ie retour

a IIE!quilibre) de

perturbations de

densite se fait si rapidement dans Ie sein du r6acteur que la forme de la distribution du flux dans Ie reacteur reste presque inalteree dans la plu part des cas pratiques tandis que sa grandeur on tes variations. De ce point de du reacteur comme composee facteur independant

de

I/~"-i

on peut imaginer la densite neutronique

d'un facteur

dependant du temps et d'U.l1

t. -?

E, rI) = T(t). F( ~~ E, On pellt alors prendre la comport~ment

sou intensite subit de for-

fonction

-':>

rI).

T(t) pour mesure rdative pour

du reacteur qui sevait par exemple proportionnelle

puissance du reacteur.

Ie

a Ia

- 384-

T. Kahan

d'indiquer n'est pas approprie. 11 en est plus les

particuli~rement

ainsi pour

reflecteurs dans lesquels Ie budget dans Ie rMlecteur dif-

r~acteurs ~

f~re s~nsiblement

de celui qui

te conception plus large de la

r~gne

f

dans Ie coeur du reacteur. Dans cet-

cin~tique

des

r~acteurs

la fonction d'influ-

ence (ou importante fonction des Anglo-Saxons) joue un

rtJle ~tninent

I

et decisif. 2.

DMinition de la fonction d'influence Le

ses

contient un certain nombre de

r~acteur

propri~t~s

nucleaires. Leur

r~partition

mat~riaux

dote s de diver-

est sujette entre autres

~

des

variations temporelles qui se font sentir en partie pendant des periodes breve:;(r~gulation,

d~marrage,

arrt!t ), en partie sur des intervalles de temps

plus longs (~puisement des mati~res fissiles), accumulation des poisons, breeding). A cela viennent s'ajouter Ie rtJle telres que la du

r~acteur

temp~rature

jouent

jou~

,la pression, et la

dans la

r~action ramifi~e

par les grandeurs

densit~.

d'~tat

Les diverses zones

,des r6les distincts. Des

neutrnns qUtl se trouvent dans les zones frontieres se perdent plus aisement par

~va sion

neutrons

~voluant

vers

l'ext~rieur

dans Ie coeur.

sans provoquer

de fiss ion, que

En outre, l'effet d'une zone

les neutrons varie avec l'lmergie de ceux-ci cal' nel

d~pend

des noyaux qui s 'y trouvent

En principe, on peut d~termin~

dans Ie

et les grandeurs

r~acteur

d'~tat.

d~crire

d~termin~c.sur

rendement

ainsi que de

l'importance d'un

de volume

en donnant sa position, les matieres

par exemple par Ie role d'un neutron

ve, et ayant une vitesse _)

V

nv )

=

I

=

o~U7,..,

pr~sentes

11 est,toutefois, physiquement plus raisonnable

d 'exprimer cette importance par Ie budget neutm nique. On peut ser l'endroit evisag~

r~action-

des neutrons

l'~nergie ~l~ment

les

~nergie

E) et une direction ,

J

caract~ri­

qui s y trou;..

d~termin~e _

, en consideration des reactions en chaine a ve-

- 385-

T. Kahan

nir . On parvient de la sorte il. une

fonction d~finie pour tout Ie volume

du reacteur, de la position , du temps, de fonction des

l'~nergie

et de la direction,

qui porte Ie nom de fonction d'influence (importance fonction

Anglo~Saxons).

II est

d'influence n 'a pas pour trons regnant

a

souligner des maintenant que cette fonction I

mission de caracteriser la distribution de

neu~

momentan~ment dans Ie reacteur qui depend en effet des I

cenditions initiales donnees • a mais plutot l'etat du reacteur tant geometrique que materiel (nature, etc. des mat~riaux) La fonctian d'importance a total de neutrons

;re~u

ce nom parce que Ie nombre

"filles" qu 'un neutron initialement introduit

fournira

au total au reacteur en chaine est une mesure de 1"'importance" qu 'a Ie

,

neutron initial pour entretenir la reaction ramifiee. Un neutron introduit sur la frontiere d'un reacteur n'a pas beaucoup de chance de laisser beau.coup de descendants dans Ie reacteur parce que lui et sa prog~niture risquent

de d'evader.

C'est precisement'ce que la fonction

d'influen-

ce prevoit

sur h. frontiere du reacteur Ie flux et par suite l'imp ortance

est tres petite. II est utile pour les raisonnements suivants, de raisonner sur un reacteur critique, et cependant exempt abstraite.

de neutrons . Cette conception

permet de se faire une repr~s~ntation concr~te

car 1'on

n 'est pas oblige I de la sorte, de distinguer dans Ie reacteur divers groupes de neutrons comme etant consta-ntement distincts . Cette introduction des reacteurs initialement exempt de neutrons ne restreint lite

de nos raisonnements et

pas la

peut s 'etendre aisement aussi

critique ayant un nouveau neutronique arbitraire. A vrai

genera~

a des

reacteurs

dire, cette friction

est rendlle possible par Ie fait que les neutrons presents dans Ie A

reacteur ne se genent pas les uns I

. "" de leur meme les autres en raison

faible densite et qu'on peut negliger les chors entre neutrons.

-

386-

T. Kahan

Afin de preciser Ie cocept d1influence ,

pIa~ons

par la pensee, dans Ie -)

reacteur critique excemptde neutrons au point r qui appartiennent au princeau (E , o 1I.(1)

,.

->

S(r , E , rl

l

-110 ) :

o

, S neutrons au total

-)

~ ~.,

0

~

= S -o(r-r) 6 (E-E )S(rl - rl )

0000

0

0

0

Notons que Ia fonction d1influence se rapporte Nous considerons pour llinstant un nombre ne pas avoir

a tenir

a Ia normalisation

a un

Q suffisamment

a travers Ie

au bout d1une periode assez longue seront consommes par m~me

par Ia suite

neutron)

Ces neutrons se repartissent par diffusion

absorption. En

grand pour

o

compte des fluctuations. Je procederai

a un

neutron .

reacteur et

evasion et par

temps un certain pourcentage des absorptiorE est pro-

ductif et il se produit un niveau )

zero dans Ie reacteur. Ce niveau

de puissance determine, different de va dependre de la position, de lIener-

gie et de la direction des neutrons de" demarrag~' (1) et il definit la fonction d1influence (~une.constante de normalisation pres). Suivons

, pour pr~ciser les idees, Ie destin,

des neutrons de

demarrage introduits dans notre reacteur. 1nitialement Ie reacteur ~tait exempt de neutrons et llequation du reacteur stationnaire 1-48) voir

.,..

(1-(48) etait

p.Jt , a sa-

./'

(K - H) N + S = 0

satisfaite en vertu de

N = O. Demarrons alors

(2-1) et Msignons la densite de ces electrons par

a llinstant ~

t=O avec

~>

N (F, t, E, rl) avec o

-')

-')

N ( r, t, = 0) E, rl) = S o

Abstraction faite

des neutrons retardes, cette distribution obeit , par

(1-45) (de pag. 31) (II-2)

'dN

o

~ IJ t

=-KN

0

- 387-

T. Kahan

Comme je n'envisagerai Jusqu'ici que de fission est

de

supprim~

m~me

des neutrons de ctemarrage, Ie terme

8 = 0 pour t

suffisamment long , ces neutrons seront

> 0 ). Au bout d'un temps

consomm~s

par

~vasion

et par

absorption: -">

-)

N(r, t. =00, E,n)= 0 Int~grons

(II-2) par rapport

No e S o

)

une probabilite de

pr~s),

en

des

s~jour

d~signabt

par

I

N

dt

o

neutrons de source (a la normalisation

la'vie moyenne' des neutrons. En intt!grant

(II-2) d'apres (I1-3) par rapport au leurs de

N pour t

et dMinissons par

00

.A

(II. 3)

at

= 0 et

temps, il vient compte tenu des va-

t = 00 ,

soit

"... -

(II-4)

8=H N o

Cette relation peut s 'interpr~t N

o

se trouve en quelque sorte

n

I::

de la faGon suivante. La rt!partition

"rassembh~e

II

>

par

l'operateur

A

+ K de

sorte qu'il rt!sulte une distribution finale pronctuelle S. La signification de la relation

(II-4) res sort plus

particulierement

sur Ie volume du rt!acteur ainsi que par rapport et (II. 5)

a toutes

en int~grant (IL-4)

a toutes

les t!nergies

les directions [(par (7), (23) et (IO)')

~)5 8 d E d ri d V 80

=

1m -Ion

Sf) H No IdE d n d

=

dE

V

d~dV [v.v+v-[ nf dE'rln'(r /\\) N] 0

-- 388 -

(II. 5)

e

S =

o

Cette equation de bilan montre que les neutrons introduits/en partie slevadent, 11 travers la frontiere F resulte

que

(II. 4)

du reacteur, en partie sont absorMs.

n

~

Nest une fonction des conditions de demarrage o

)11 dlignition". --)

N = N (r, E, rI, o 0

Integree sur E et

-'>

rI,

-~

-")

~

0

,E, rI ) 0

0

cette fonction permet d lindiquer Ie nombre des

absorptions ayant eu lieu en

-;

r

:

(II-6)

Une partie de cas absorptions est productive et

lion a pour

la densite des neutrons de fission produits 11 llendrc;it de ~

-)

(II. 7)

SA (r , E, rI, '1

D~signons

leur naissance :

-'?

-7'

r , E , rI ) = 0

0

0

Rv

f(E)

JJv

-')

~

N dEdrl o

la distribution de cette premiere

g~n~ration

par

-.,

E , rI ) ; o 0 elle depend aussi du temps car les fissions ont ferents et elle obeit

a ll~quatinn

II-2

eu lieu

a

des instants dif-

. L lintegration par rapport au temps

dlapres (II. 3) conduit 11 une probabilit~ de presence

N1 .

Le nombre de

fissions produits par ces neutrons peut se calculer dlapres (II. 6) et finalement on obtient de maniere analogue la deuxieme gelleration

a (II-7)

, la repartition de source de

S2.

On peut proceder de proche en proche

de cette

maniere,

f\

la distribu-

- 389-

T. Kahan

tion extreme de source (II-!) Ie

r~acteur

au cours des

se substitue une distribution interessant tu<' l-

gen~rations qui

se relayent par egalisation de dif-

fusion . Cette distribution ne peut toutefols fisant de

g~n~ratiDns

ne sont

tionnaire, compatible avec

un nombre suf-

a savoir '

....

A

(K - H) N + S = 0

(1-48)

Cette

, lorsqu

qu tune distribution de neutrons sta-

succ~d~

(1-4)

~tre

est avec

~quation

S = 0 , une

~quation

homogene et par suite, par

un th~oreme classique, la distribution de neutrons stationnaire dans Ie reacteur, n'est

d~finie

qu'a une constante pres qui caracterise leniveau

de

puissance. Le niveau de puissance d 'un reacteur critique est fonctionl de l' histoire anterieure, dans notre cas du processus de mise a feu de

ou

d~marrage.

n resulte,

de ce que je viens dire, la distribution asymptotique N

00

doit admettre la

forme

m~me

stationnaire N(r~ E,

que les fonction.s propres du reacteur

flf . L'amplitude,

par contre

, est une fonction des

conditions de source seule : -")

A = A( I\,

-")

E , !1 ) . o 0 a ,

On peut donc poser -

(II-8)

N

et 1'on obtient mens ion,

-"?

00

0

a

un

conditions

s'etablit (r

a

-")

-'>

N(r, E, !1), par S, o

une

fonction sans di-

neutron

+

Cette fonction

-)

0

par integration et division

normee

critique qui

-/

= A( It. , E , !1 )

N

represente la teneur en neutrons du reacteur

apres la mise

-)

a fe·u

, E , !1 ) . Par definition,

000

avec un neutron

+

N

peut donc

se

dans les concevoir

-

390-

T.Kahan

comme fonction

stante de normalisation. Nous allons ,dans un instant fonction

d'influence est bien l'adjointe

.. Jusqu'a present ,-

nous n'avons

de

la

possede une puissance

J determin~e,

d~montrer

densit~

une conque cette

neutronique

l'influence que pour Ie

analys~

diff~rente

N. r~a-

Ie r~acteur critique

cteur exactement critique, exempt de nedrons . Si ,

de zero ,les raisonne-

se font d 'une maniere completement analogues aux pr~c~dents. L'in-

ments

troduction de neutrons que

a part

d'influence (on d'importance), mise

r~gnante

dans

jusqu'a ce finisse

~tat d'~quilibre

l'argumentation

r~acteur, perturbe la distribution neutroni-

qu. apres une

suffisamment longue, un

p~riode

de nouveau par

de puissance, les neutrons furent

Ie

s'~tablir.

11 va de soi que Ie niveau

en fonction des conditions,

introduits,

va se trouver

de puissance

s'~tablissant

augment~'

sous lesquelles Dans ce cas donc,

asymptotiquement qui sert

a

definir la fonction. d'influence .

2. L~quation d'influence fonction du temps Une fois

d~finie

la fonction d'influence et

gnification phys ique, reste Pour

cela Ion peut

a

rechercher

s 'appuyer sur

un

mise en

l'~quation th~oreme

~vidence

a laouelle

sa si-

elle satisfait

de conservation pour

l'influence. L'influence de toute ce des neutrons de Ce

th~oreme

dans lesquels la mise par la fixe

,

individuelle doit

de

d~marrage r~sulte

jusqu'a

l'~tat

Ie processus

a

la distl1ibution de cette

Ie reacteur vide

~tre ~gale

a l'influen-

d~marrage.

,nous avons poursuivi les

a feu

pens~e

g~n~ration

exactement

des raisonnements g~n~rations

d'~quilibre.

une

Ou

gen~ration

g~n~ration

prec~dents

de neutrons depuis

peut en effet interromprf quelconque . Si done on

et qu'on mette

avec cette distribution, Ie

de nouveau m~me

a

r~sultat

feu fi-

-

391-

T. Kahan

nal asymptotique doH sf~tablir qu'en Iaissant se Mrouler

Ie processus

initial. Ins~rons

dans Ie

vide, au point -1, au total So neu-"> (E. 12) . L'influence de chaque neutron est

r~acteur

trons appartenant au pinceau d'apr~s

a

(II.9) proportionnelle

+

-'>

N (II. , E, 12) Parmi

S neutrons, o

un nombre

0

atteignent Ie point

-)

-')

r + dr, puisque,

P (r) = n€". e

a

Id 1 (

-~

S) = S (1 o 0

(II. -10)

~gal

SI

par (I. 2) de p.

-nl! r

~

= c'e

-~

r

et par suite (1- ~(dr))

Ces (II. 11)

8 0 neutrons I

+ -)

portent avec eux une

_)

-":>

S N (r + dr, E, 12 = o

influence I!!gale '5"

8

o

(l-~dr)N

+ -'I

-">

a

(fig. -)

(r+dr, E, 12)

1)

- 392-

T. Kahan

d'~nergie E qui partent du point ~Q

Les neutrons (E,

0) ,

diffus~s hors dece 'pinceau sur le chemin qui mtme au

sont

-:>

-"-;>

point voisin r + dr

,

diffus~

dans le pinceau

. Dne partie d'entre eux se retrouve dans le pinceau

-")

(E, 0) --,.

Sur Ie parcours

S

dr

o

d ~ dr

+ -")

-) + £dr,

-":0

E', fl' )

-')

puis que Ia collis ion a eu lieu sur Ie parcours Q-)

S (E, !1_>

.I

nous faut encore la probaiJilite neutrons aboutisse

, cette

-">

r et r + dr (fig] ) . II -> E', fl') pour que l'un de ces c~

-)

dans Ie pince-au

de p. 20.

une colli~

avec lui une influence

sian. Chacun d'lnt.re eux transporte N (r

neutrons ant subi

(E', fl') . D'apres(.r.f;.

probabilit~

(1. 8)

Au total, Ie pinceau initial perdra par collision; sur Ie parcours question,

l'influence

((

+ -')

))So~dr

( 1C(12)

v

00

-':>

a l'influence

argument~e

+ -?:>

-

fl

que 1'influence

de l'influence parvenue au

-"?

point r

+ -)

+->;,

-.,

+.y

-":>

+ d r soit

S N (r, E, fI) o

c

emport~e

--;>

totale des neutrons de source -:<>

->

(E, fl.->E', fl') N ( r + E.dr, E', fl') dt'dfl'

D 'apres Ie th~oreme de conservation, il faut

des cho cs ,

~

0)-")

+ tdr, E', fl') )<)(E, fl._>E'fl')

)! r 0

-">

_'>

N (r

S o =dr

(II. 13)

en

-')

S N( r, E, Il) =S (l-c.dr)N (r +dr , E, fI) a . 0

S

+...£ dr ooJ v

o

S'"I (E, fI-., E', fl') N+ (r + [ -'>

fl

-)

-"?

-':>

.-"?

di, E', fI') dE'dfl'

par

~gale

- 393-

T. Kahan

D~veloppons

s~rie

+

alors

de Taylolj suivant les puissances

So d r et passons tion integro

de dr,

pag.U

+

'>

'1'

+

o : : V. VN -'YN + l

-)

-)

au pomt (r + dr) en

divisons ensuite

dr -) O. On obtient

a. la limite

differentielle de la fonction

(II. 14)

.

la fonction d'influence N

I

de

ceUe

fa~on I

[cr. (I-2j

d'influence

par

1'~qua­ de

+ ., I -') N I dE'dn'

qui

.....+

(II. 15)

H

et A

(II. 16)

K+:::

~

V(t) . v

~E)

00

f )' o

et compte tenu de

(de pag. 27)

(I-3D)

tf dt

dE'd6' f(E)

n

(1-15) (pag. 2~, peut se met-

tre sous la forme

A+

(II. 17)

(K

+

"A

- H) N

::: 0

Comme l'indique la notation, ces (1-43)

c 'est-a.-dire

op~rateurs

(de pag. 51 )

(1-44)

sont

resp (1-48)

pour toute founctions ~

et



qUl

adjoints aux de p. 31) sont

I

definis a.

l'int~rieur du r~acteur. l'on a la relation dMinition des operateurs adjoints I

f l' i f#I(i - H) cp . : S j S~K+ -'H+) r.

(II-18)

R~acteur

0

R~acteur

La fonction r~acteur

d

V dE d n

-"

d'influence

critique. Elleest

0

d V dE d n

n

+

Nest donc adjointe univoquementd~termin~e

i'l. la

par

d 'une condition aux limites. Cette condition aux limites d~rat1on

que

des neutrons sur les frontieres du

pas d'influence sur la

r~action ramifi~e

densit~

N dans

Ie

(II-I8). compte tenu r~sillte

-')

de la cons i..::>

(r ::: r ) n'ont f.. •.. ~· si leur direction de vol pointe vers r~acteur

- 394-

T. Kahan

l'~xterieur

-)

(direction

rlext ) . Cette partie

du flux d'influence doit done

s 'annuler

+

N (rl

(II-19) En outre,

-~

,E, rl

f~D.!-.

+

doit

N

~tre

) = 0

ext

dans tout Ie volume du reacteur, continu, fini et

posit if. L'~qu. (lUg) repr~sente la condition aux limites pour la fonction d'influen-

ceo Son sens

est justement l'oppose de la condition aux limites (1-46)

(p. 32) pour la du

r~acteur,

densit~

neutronique

l'influence entre

N: tandis que des neutrons

dans

hors

s'~vadent

Ie reacteur. Cette inversion trouve

+

son expression dans signe contraire du terme flux de H ,

, La fonction d 'infl uence d~pendant du temps Reste

a 'proceder a une

jusqu 'a present ,la les operateur

ft+. H

generalisatiGm. D'apres la dMinition donnee

fonction d' influence est determinee pas (II-17) ou

A+

et K

correspondent

a l'etat

critique du reacteur. Modi-

fions maintenant l'etat du reacteur par exemple , en une barre de reglage v~nt

renfermer Ie

retirant lentement

(qui absorbe des neutrons) . Dans ce cas

r+

... +

H et K

temps en tant que parametre et il sera necessaire d 'in_ I

troduire la fonctio'1 d'influence dependant du temps. Si l'on fait abstraction

des

neutrons retardes et des sources

ext~rieures, l'equation des reacteurs dependant du temps (1-39) (p.30)

pour expression (II-20)

dN t) t

a

........

= (K - H) N,

Formellement parlant, l'equation adjointe a pour forme ~ '" + =(K-H)N

(II. 21)

Or la founction

d'influence a une signification physique claire

elle

-

395-

T. Kahan doit caract~riser l'etat ramifiees dont il initiales.

du reacteur, eu egard aux diverses reactions

peut ~tre Ie siege lors du choix

Or l'etat

de

diverses conditions

J

du reacteur ne depend pas, de conditions initiales

de quelque genre qu'·elles soient, pour la densite neutronique, mais peut

...

etre modifie

par des interventions exteriElUres

On peut ainsi , par me

vide

exemple, modifier

en retirant une barre de

son ~tat ant~rieur en renfor~ant

ou

~

a nouveau retabli.

l'~tat du reacteur consid~re com-

reglage et ramener Ie reacteur dans

la

barre. Une fonction qui caracterise

+

Ie reacteur ne doit donc pas dependre explicitement du temps comme Ie N defini

par

(11.21)

11 n'a va pas de

m~me

exemple, un reacteur vide

pour la densUe neutronique

Si~

par

surcritique qui est caracte'rise par une fonction

d'influence constante dans Ie temps, est demazre d'une J

densite neutronique qui change tres rapidement (rite par

N.

(II-20). Les conditions de mise

a

fa~on

determinee, la

dans Ie temps

sera. de-

feu servent de condition ini-

tiale. 11 resulte des consideratioiE pas a une definition

precedentes que (II-21)

ne se pr@te

ph,\siquement admissible d'une fonctiont d'influence

variable dans Ie temps. 11

est plus raisonnable de continuer

fonction d'influence par l'equation

(II. 17)

m~me

,

definir la

lorsque l'etat du

reacteur vade avec Ie temps ou n 'est pas critique. Chapter III .

Les equations cinetiques des reacteurs

Les equations des reacteurs (I (39) (de p. 30) 31)

decrivent Ie comportement de la densite neutronique

et caracterisent par de la position /

rapt. (1.

de

consequent l'etat

differentiel

l'energie et de Ia direction

etablir maintenant des relations qui

-?

,-

)) (de p. -'>

N(r, t, E, !2)

du r~acteur qui depend

a I'instant

t. Nous allons

decrivent l'allure integrale du reacteur

- 396-

T. Kahan comme fonction du temps dans lesquelles les

positions,

div~rses

~nergies

et directions dans Ie reacteur ne figurent plus. Pour les dJ.stinguer des I

equations des n~ti

ques des

a ~tablir ant

ces relations

r~acteurs

recu Ie nom d '~quation~ci-

r~acteurs.

n faut tout d'abord faire Ie choix d'une grandeur qui prenne la place de

et

N

choisir

a

qui puisse exprimer

l'~tat int~~ral

du reacteur.On pourrai

la puissance du r~acteur qui est proportionnelle au

cet effet

nombre de fissions par seconde ayant bien dans tout Ie reacteur , Pour obtenir la puissance du r~acteur

a int~grer

sur Ie volume du

les directions

;1

R.

a multiplier

N par

v l. f et

sur toutes les

~nergies

et toutes

Vona

r~acteur,

frl Nv 2i dVdEdrl

~

0

En tant que grandeur int~grale qui ne dJpend d~sormais que du temps, la puissance

se

fort bien

pr~te

que3 dont Ie reacteur est Ie

a

caract~riser

les processus dynami-

si~ge.

Il est toutefois plus utile de choisir une autre grandeur, '-

a

la puissance,

n(t) tion

apparent~e

a savoir Ia teneur en neutrons, ponderee au symbolique . Pour parvenir a cette grandeur il faut multiplier N par la fonc. d'influence N+ et int~grer ensuite sur Ie volume, les ~ne_'gies et les ,

directions : (III. I)

r foc )rN+

n(t) = )

R

I

-') -") -') -"") -:> + E, rl) N( r t, E, rl) X d V d E d rl

(TL

0 {l.. ,.d

!

(N+, N), avec

(;, t')-:; (

'V

0

fonction

+

Quel est Ie sens physique de

n (t)

se l'influence d'un neutron en

(r~ t, E, rl) . Le produit N . N+ d V d E d rl

I

? Comme Ia

f ~4f6 V d E d r? rl

~

N caracteri-

mesure donc I'influence de tous les neutrons (E; Q) dans l'element de volu-

- 397-

T. Kahan me d

V. L 'int~grale

(Ill. 1) mesure enfin

a l'instant

les neutrons se trouvant de la

ut~rieure

(III-I) est

en

de r~acteur, ainsi que

les 'energies

cette fonction

sous certaines conditions, n(t) admet la

des

simplifi~e

a

~quations

en une partie

cinHiques J

+ --.

-'>

N (r , t, E, n) et integrons

par

+ dN (N , - \ - ) = ( \ r)

et

+

(IIl-3)

t

dfi C.1

(N ' F ) = Les operateurs

si l'etat du

r~acteur

sur V, t.

."

H et

('»

+< (N , K

0

+

N)

+

+ ".

4'" 13,(N , K. N} - '>I.(N f. C.) 1

A

K

change au

1

1

contiennent Ie temps

1 1

atitre

de param~tre

cours du temps. Comme alors dans (II.

2t

non seulement la densite rna is aussi la 'fonction d'influence de-

, en vertu de

pliguees

s~parer

il vient

(III. 2)

et (B-3)

peut se

N

une equation differentielle valable pour n(t) , multi-·

plions l'equation (1- 45) (p. 31)

n .

r~acteur.

spatiale et une autre fonction du temps

Pour parvenir

p~nd

joue un rMe decisif en theorie

du temps et que

ind~pendante

Forme

et

n

en est ainsi lorsque nous n'avons a faire appel quIa la fonction

d'influence fonctio~1

des diverses zones

et les directions neutroniques . 2)

variation avec Ie temps que la puissance du

n

sur l'evolution

une serie de raisons

1a signification

~vidence

des perturbations. 3) Enfin, m~me

r~acteur,

fonM sur toute

1) Tout d'abord il met

En deuxieme lieu,

dans Ie

en cha!ne .

r~action

Le choix de

t

l'influence totale de tous

de peu

(II. 1 'f) p. 46) dlinter~t

, du temps, ces nHations sont fort com-

pratique, On peut parvenir cependant

a

des

equation:; eSBentiellement plus simples si llont tient compte du fait que dans

- 398-

T. Kahan

les cas pratiquement

int~ressants

du temps ne

que

diff~re

tr~s

, la

foncton d'influence

peu de sa valeur

d~p~ndant

stationnaire

+

N o

Si l'on pose donc

+ -,

._")

+ -")

.."

+

'>

N (r, t, E. n) = N (r, E. n) +~N (i\ t. E, n) o

(Ill 5)

+

l'on aura 4N

« N+ n

· + + dans (II. 2) et (III-3) . Comme on do it anaet on pourra sub shtuer NaN o lyser l'etat critique d'un reacteur avant de passer aux problemes cin~tiques, -">

il faut supposer connues.1a distribution de densite critique tJ ('i-:: E. n) et par suite

auss~

la fonction d'influence stationnaire.

~

Des lors, (III. 2) prend la forme, en utilisant Ie spectre global fIE)

(1. 49) (de p. 32) et en rassemblant les neutrons produit aU

d'apr~s

total dans l'expression K N : (III. 6)

= - (N

-

~ fl

I". (N

. 1

1

+,;. +-" • H N) + (N • K N) o 0

+,;. 1 • K. N) + 40

~\ 1\.

IT .

1

,

1

1

+ (No/t, C.) 1

+ + (N 0 • S)

Ici, l'influence des neutrons exterieurs est donneepar

+

sIt) = (N

(III.t )

o

• S) ;

la contribution des neutrons retardes est decrite par C. (t) = 4- (N , f. C.) 1

):l:n posant

encore (N

~

(III-9) il vient

i

a l'aide d n( ~ t

(N

+ ~ • K.N) o 1

+""

"0

0'

K N)

'+,.

(No' H N)

1

1

avec (J = i.. 'i..f3. . 1

(N • K N)

de la production totale

+ ,;. (III. 10)

+

1

(III-B)

,,;.

KN

+ ....

~ (N ,H N) (1-~J3)-1 ---,;..,.0+_ _ (No IN)

1 1

-- 399 -

T. Kahan +.1' (N ,K N) (N ,if N) represente ici Ia production d'influence par seconde divise par

Ia °perte d'influence ayant lieu dans Ie meme temps. D'apres (1-48) (p.32) (K - H) N = 0 (K N = HN , ce rapport est egal a reacteur critique et

1 pour Ie

> 1 pour Ie reacteur hypercritique. Cette grandeur

peut t!tre comme comme un . (III. 11) ( t)

~

+ (N N) o La quantite + ,A.

facteur de multiplication effectif generalise (N+ , KN)

~

" (N , HN) o

represente Ia teneur en neutrons, divisee par Ie

(N ,HN\

perte-influence pour seconder . L'inverse de cette grandeur ayant la dimension d'un temps,

combien de temps il prendait jusqu'a

indique donc

,

ce que la teneur en neutrons soit epuisee par les pertes de Ia production g{meralisee

a parti:r

de 1'arr!€

elle revet donc la signification d'une duree de

vie

let) des neutrons dans Ie reacteur fini . Nous poserons donc

(III. 12)

l(t)

(N+, N) o

+ ./'-

(N ,HN)' o

Avec ces notions nouvelles,

(III, 10) et (III-3) prennent leur forme

definitive (III. 13)

oil.

N est

fonction

suppose "factorise' en un produit d'une fonction de t

de r. I

Les equatio!1'.> (II-13)

portent Ie nom d'equations cinetiques

du reacteur et expriment son allure dans Ie temps en fonction

I (t)

et d'une

de

It (t)

£,1:

. La discussion de ces equations constitue'le probleme central!!! de

- 400T. Kahan

la

cin~tique

des

r~acteurs.

Les conditions de

validit~

de ces

~quations

sont 1) Une seule

I

2) les

~.

fissile;

mati~re

independants

1

"'"

3) les operateurs

de

l'~nergie

...

H et K sont voisins de leur valeurs stationnai-

res. Chapter IV.

des

Cin~tique

r~acteurs

a 1'approximation

de Ia

th~orie

de

la diffusion ,

Nous avonS

d~vp.lonne

du transport rigoureus. Or,

I..

dans ce qui precede (ch. I d~jli.

la solution de

a III) Ia theorie

probl~mes

stationnaires

est, dans Ie cadre de la theorie du tra.nsport , fort delicate, liee ~ des calculs

fastidieux

et

ne conduit Ii.

des resultats

que dans des cas exceptionel!; . Pour les ficulds ne font que

SO'lS

forme close

probl~mes cin~tiques

, ces dU-

croitre. Force est done de faire appel, pour l'ana-

lyse de ces problemes connue sous Ie nom

a une m~thode

d'approximation utilisable qui est

de theorie de la diffusion . Cette theorie de la diffu-

sion, comme approximation il. Ia tMorie du transport , analyse la densite neutronique sans egard

a Ia

distribution angulaire. Lea rMuliats s 'accor-

deront done d'autant mieux avec ceux de la theorie exacte que Ie mouve1

ment neutronique dependra mains d'une direction tion de la

th~orie

privil~giee.

L'approxima-

de la diffusion fournirA done des resultats utiles pour I..

des milieux homog~nes a condition de se restrindre

ades domaines qui sont

Elloignes d'an mains quelques longueurs de diffusion des fronti~res, des borrls ou

~es

sources

TMorie de la diffusion 1) Rallentissement de neutrons de fission jusqu'aux energies thermiques 2) Ralentissement des neutrons de fission en tenant compte de Ia possibilite de leur capture dansla region de resonance 3) Diffusion des neutrons thermiques

- 401-

T. Kahan

Passage

a la theo.rie

la diffusion

d~

a la

Le passage de la tMorie du transport peut s'effectuer

simplement en supposant que la distribution de densite

tr~s

N possede la symm~trie spherique , dans ce cas

.a..

n

direction ne

d~pend

)" = cos

N(r, t, E, n) des neutrons mobiles dans la

(fig. IV-I) on voit que

pas de

t

N n 'est fonction du rayon

->

Si l'on enviSage la densit~ -)

tMorie de la diffusion

,mais

f)-

de

Aft

N 01)

de

9.

I

Comme l'element d'angle solide s'ecrit

lIon a (IV -1)

2iT

J

,-7

N('!., t, E, n)

=0

-/

d;--df=21fd)-"xN{r, t, E, n)

= N(r, t, E,)", )

ctr

Partons de la premiere des equation<,(I-45) (p) 1 seule matiere fissile. Le terme pertes -":>

v.9'N,

-,-? (v = n

v)

-=I

_'>

N = \f(

( d

(J-14) (p. ! -1) conduit (I~3)

;::;::;- = - v (}t

a

+ ~ 'Q N 1'1-

';)~

)

'd N ) ,P ". Or par rapport a o.f ' eu liN

L'integration de (I-45)

N

2-pr

= v(/'-

~N

HN contient Ie terme

or d'apres la fig. (IV.I)

fV

(IV-2)

) en ne consider,ant qu'une

.10-

1_",,2

+....:...L...- -

-

dN l~/~ "\(J N (j- + -) 'Qr r 0;-"

-v N

egard

a

(IV-2) et

a

- 402-

T. Kahan

ou J)

est Ia contribution de diffusion (sans diffusion

lJ

(IV-4)

00

f

= _1

41f

+1

t1T

-1

o

S 7' )

dE'

0

A

X ~LE- (E' -}1 et Fest

in~lastique)

2 ii

Lf' ( d 'f v'~

d

)0

N'

d

E'~ ;

la source de fission F =

(IV-5)

J II

+1

00

i£. 2

')\I'2.. f

(l-fi

v'N( r, t,

E~')

d E'

~'

1 -

+"2 L).J.C. . 1 1 1 1

)

Afin de que

v)(

s~parer

Ia variation angulaire developpons Ie flux neutron i-

une s~rie de

N en

fonctions

00

2n+l -2-

v N ( r, t, E,j-) = ~ n=O

(IV-B)

ou

I':)

rh

nY-)'\f' n ( r, t, E)

1

(IV-7)

¢n(r, t,E) =

Comme

£

P o(~) = 1 , Ie premier

P n~)

de Legendre

E~) P n y..)

v N (r,t,

terme du

d?

d~veloppement est inde-

pendant de la direction. Si nous poussons cette s~rie jusqu 'au

[~t;-

terme

a la

)=;:J '

}

-"'.>

dependance de r.l

avons dit en guise de plicationJ de

la

nous n 'obtenons qu1une approximation grossiere quant • Cette approximation ne vaut,d'apres ce que nous

pr~ambule

th~orie

, que si les hypotheses consernant l'ap-

de Ia diffusion sont

donn~esl On posera

Itapproximation de la tMorie de la diffusion (IV-B)

-

/\:) N =

n=l

2n+l p¢ L. -2 n n n=o

-

1

2

P(> o

=~~+~~~ 2 0 2 1

it "r +"",..

donc (IV-B)

deuxi~me

dans JIV-3)

0

+~ 2

peD 1

1

=

donc ;;

- 403-

T.Kahan

(IV-g)

1 3+(- S +-M.-S) 2 0 2F 1

Le flux de somme a

que

ete developpe dans (IV-g) de la

vN

On peut multiplier (IV-g) successivement

m~me

fas;on

par les P (...... ) et obtenir.:, tn

grrtce aux relations d 'orthogonalite's des P (}') par integration sur toutes n les dirtctions, m systerre d 'equation~diff~rentielles pour ~ i . Multiplions d'abord

(IV-g) par

P oy...) = 1 et

integrons sur

de

d)'"

- 1 a' + 1, il

vient 1

(IV-IO)

-

d4;o -

v ~ t

~ + - t )"tI. - -1 '( rj., + 5+1

= - (-"

'&r

Multiplions en~""~ ( (IV-II)

7.

1

par P 2

v

0

-1

y..) =~ et

l'Ot 1 ct 1..1. r+ .7.1 --=-----y" +_IJA.,If/+F) v~t; 3c1r v I / ' 1

Par

(IV-4) et

+1

S}

(IV-12)

-1

~=

(IV-B)

o

5 (1-/3')v '~ o

I

-

d",,+SI /

lion a

00.

f

integrons:

.

I

f

Le calcul des integrales de :iiffusion

q>'0

dE'

+ ~(). f, C,

,ill

)+1r ",F drJ.... = 0

1_1

(IV-12) necessite l'introduc-

tion d'hypothesesconcernant les processus de choc. Nous supposerons iol qu 'n s 'agit d 'une diffusion purement c1assique des neutrons sur les noyaux de la matiere du reacteur et que cette diffusion

est isotrope

sterne du barycentre neutron-noyau. Si A est la

masse

perte

I

d'energie lors du choc

neutron- A est

dans Ie sy-

du noyau, la

- 404-

T. Kahan ..

(IV-I3) ou E'

uE' = E' I'~n~rgie

1-.,1.

A-I 2 , avec rj... = ~+1 )

-

(1-)")

~

du neutron avant et

E

cette

apres la colli-

~nergie

sion. Je donne

Ie

final

r~sultat

WJ..

+1

(' ~ dt<.=.!..) ~'

(IV-14)

)-1

/

2 E

d

I

+1

0

!I

E/rJ..

d'¢> dE'; (~)'-clt.=.!. .f"'dl~'dEI 0

2 E d

F

1 1

d ' = d (E' ) et d' = d (E') sont des: fonctions alg~briques de E '. 00011 En portant enfin (IV-I2) et (IV-14) dans (IV-10) et (IV-l1), on obtient ou

l'approximation

dPo

1

(IV-15)

PI pour Ie

v ~t

Po(r,

flux

(.1. +..!..)
= _

@'I,.

'Z.

t, E) : J

I

-r ~ +r~~d ~ ~ + 0

1

E

0

1- c}.

E'

00

+ et son

~ ( 1{i 'lv ' z: ~ ~ 0 dE' +~A i

f0

premier moment

¢I (r, t, E)

(Les quantit~s prim~e () ) dependant (de p. 31rpeut effectuer

de

-'_i = -

Ii

DCl.ns

S....

(IV -15)

ainsi comme condition 00

f '"A'i ",It~ fTo

i

i

et

(IV -16)

cm I

3

it partir

compl~mentaire

dE'

o

I

figurent des integrales de freinage

dont la signification physique est

Ie nombre par port~s

. La seconde ~quation (1-45)

a,;-v ,

~ c +

t

E

~t~

E')

elle aussi d'apres (IV-I) . On peut en outre .Ade Ki N par rapport apres introduction

l'int~gration

(IV-17)

de

s'int~grer

(IV-B), et obtenir

E!J,.

+ So

J.. It= _.!. '04'0 _tnh + (j~, ~ dE' +v,H g ch 0/ 1 -k).O.;{-J. E' SI

(IV-16)

de

Pie i

la suivante : elles indiquent

et par seconde des neutrons qui par freinage ont des bandes

d'~n~rgie sup~rieures

d

~'

, au cours

- 405-

T. Kahan

d 'un seul

chot

a

I '~n~rgie E. Ces

de freinage ne permettent pas en

gen~ral

tion simple du systeme (IV-15, 16) integro-diff~rentielles coupl~es.

int~grales

L \

la solu-

01 E

T

d'~quations

En effet,

comme les deux limites d'integration

d~pen-

dent de E, il est imposs ible de les transformtlen tiation

~quations diff~rentielles

et

d~cov

par

diff~ren­

v

pI age . Font exeption Ie frei-

nage en hydrogene pur (J.. = 0) ou dans un diffuseur

de grand poids atomique (A»l)

oil on peut posero ~ 1 .

Bien entendu, peut ne

~tre

,

'

,Ie systeme

g~neral

pr~c~dent

ne

qu 'approximativement, selon la situation existante . Je

r~solu

puis abQrder; c..: que

cinetique, en

J

dans Ie cas

les

cas les plus importants au point de vue

du plus simple vers Ie plus

proc~dant

compliqu~.

2. Cin~tique des r~acteurs monoener g~tique.

a) Zone proche. Pour les caicuis d'orientation, on peut, en Ie

r~acteur

que en

thermique en approximation grossiere ' comme

supposant

,

que

tous les neutrons sont

se. Dans ce modele fort prirnitif des neutrons et

a la

se font

~~~1 E' . IAns

on peut 1'~quation

th~orie cin~tique, traiter

effectuer les ainsi

(IV-15

~nergie

a

17)

int~grations.

obtenue

1th ( r, t ) = ~(r, t)

meme

Ia production,

que

. 11 vient

de

. Avec

sont

anim~s

mono~nerg~ti­

de la

m~me

vites-

la diffusion et 1'absorption

hypotheses~

I

d

~ J E' 0

ind~pendant, de l'en~rgie et

Le flux ne depend plus alors, dans r

et

I

que nous designerons par

-

406-

T. Kahan

:i4, (IV-1S)

- -

v

Gl

o = -( ) - + -) 2 ~ - ~ tf.. 111 --, 'I' +(1-(/J,.J).' > 'f' V +~fI.C.+ S , t 'Or·"to a 0 vI 0 1 1 0

1

(IV-19)

1~-1

1

dl/> 0

-

'&t 3 @'ldC. _ _ I =~.vl. ~ -7\. '0(; 1 f 0 1 v

(IV-20)

- ~~ ~ + Sl

'

C

Dans ces equations figurent la section

11 D

Ie parcours libre moyen de diffusion

d1absorption

I

et

defini par

_1_ =~ = ~ _;; ~ AD D Fod'

(IV-21)

q].li , pour une absorption negligeable, se transforme en moyen de (IV-22)

Ie parcours libre

transport _

~

~r

.fa =;cos ~

ou


= [

-:- tr

,Et

=L

~tant

d

(1 -

j)

.

0

l'angle de diffusion dans Ie barycentre.

Pour simplifier les calculs, nous passons de la symNrie spheri-

a

que x

et

alors de

plan ou ~

de t. en

des reacteurs homogenes • Multiplions

nous

(IV -] 9) par (-v) et derivons ensuite

m~me

Po

a un restreignant a

la synletrie par rapport

, (IV-1S) par rapport

de second ordre en

par elimination de


a t.

ne dependra que de

par rapport

a

x, dJrivons

On obtient alors une equation

t. De m~me

pour

on obtient une equation pour ~ ,

0

(IV-20)

+a

2

¢ 1 + Q1 =0

Ici, les contributions des neutrons retardes et des sources

exterieure~

sont

- 407-

T. Kahan

designees de

fa~on

ab1"egee par

Q =. -v~)

(IV-21)

dC

dS

~ 2 v c:. C.) + v (_ _1 -[ S ) ,D 1 ~x D 0

(_ _i+ . i ~t

o

1

dS at

_ 'IJ _ _o

t dC i 3.\~ 2

y

v

2

dS0

a

'Ox

3

1

~

(IV-22)

v(~ a+~

a 2 :; v 2 ::

I

- (1

-(3 ))) :>- f J


1)

.[f

Le systeme (IV-20) n'est pas encore

J \

I

completement decouple

dependent encore de par (IV-17) 1 0 Pour etudier la distribution neutronique

diat,

puisque

¢

Q et Q

o

1 v~)

,!l

En outre

a1 .:.

'ds

2

+ - - - ( - S,+

dit

dans Ie voisinage imme-

"zone proche" d'une source variable dans Ie temps; il faut faire

appel au systeme (IV-20) . Zone lointaine - Pour l'analyse des phenomenes dont les reacteurs sont

a une distance' '!. grande h.» ~tr ) ( dite zone lointaine)

Ie siege,

transport:

a la longueur du falre appel a l'equa-

(comparee on peut

tion de diffusion ordinaire I.es equations differentielle5 (IV - 20) se di>,tinguent des equations de la diffusion normalement utilisee par temporelles faisant

(i /dt 2)La suppression de ces d~rivees

croltre la vitesse

indefinimert

v/'f. :\ et 1/'2:.0.-

n ant constant Posons etudi.ons la

v

t'"fj donc dt t

fa~on

,,0

I

l'apparition de

secondes derivees secondes s 'obtient en

(v_) 0)

tout en mainte-

V.

dans la premiere equations

dont la seconde equation

(IV-20) et

(IV -20) pour Ie premier mo-

- 408-

T. Kahan

ment ~~ du flux

se trouve modifie . Pour

me (IV-IS) et (IV-I9) (p. 59 ) . On voit

.~

dans

deriv~e

cela gu'on

~liminahon

(IV-IS) pour queJ apres

seconde ne figure pas. De

l'~quation

, retournons au syste-

de

doit

annuler Ie terme

~ 1 dans

(IV-19) ainsi

(IV-20) la

abr~g~e,on

tire

pour Ie premier moment du flux 1

"& avec

a~o

~I = - 3I D '""\"

(IV-23) !o (J v.

-.)\7~pour

sph~rique

n~utronique

d'introduire Ie vecteur de la

indiqu~

-> -'>

J(r,

~->

(IV-24)

3.P 51

la symetrie

II est alors tout

courant

+

0..,

/ (r, t) =- D'V

t)

par

¢o (1,

t)

8i I 'on supprime dans

~limine avec ce qui reste,

- D grad

rb 0

lie

(;? ,t)

¢1

(IV -11) (p. 56) Ie terme ~

¢1 de

densit~

I dt

et qu 'on

l'~quation (IV-10), il apparait I'op~ra-

teur

,

qui se confond, dans Ie cas de symetrie spherique, avec Ie Iaplacien 2 <::;7 : D.. . 8i I 'on s 'affranchit de la sym~trie spherique que nous avons utilis~e

pour

(IV-I0) et du temps

(IV-25)

~tablir

de maniere simple I 'approximation. PI' on obtient avec

8 1 = 0 , I '~quation de la diffusion

~ = ¢o

pour

.!.vOlt ~

=

-v. (DtJ
Dans Ie terme source sultait de

mono~nergetique ind~pendante

1

1

C +8 i

0

8 nous avons supprim~ Ie. barre qui r~o

1 'int~gration de (IV -1) P . 54 ). A cette

dre l'~quation des noyaux meres

(IV-20) (de p.59

~quation

il faut adjoin-

- 409-

T, Kahan

~c

~t

(IV-26) En partant cin~tiques

=I/3-i J~f 111_}, ~ F\

i

~

de

(IV-25) et (IV-26},il est aisl! de formuler les equations

pour Ie r~acteur

f = V ~f

(IV-27) il vient

1~

~ C. ~ (1 t

.h

A.

(I-~) K 'f+ ~~. .

1 A

Q

monoenerg~tique, en posant ,

et

I - ':\ = v II1t

(IV-28)

C i

A.

A K 'f - /),

/ ~i

1

1

c.1

Comme I '~quation de la diffusion est autoadjointe I L'~quation

de la diffusion (IV-25) est une equation fort simplifiee

non seulement par rapport rapport


I 'on a

a l'approximation

~

I 'equation

PI sous

du transport

mais aussi

par

1a forme (IV-I8) et (IV-I9) (de p.59

La distribution de ~lmergie dans Ie reacteurt j 'ai montrl!, dans ce qui precede que sur l'exemp1e du rl!acteur monol!nergl!tique, qu'on peut travailler dans 1a p1upart des cas, avec une l!quation

de diffusions qui

est de premih ordre dans Ie temps. L'Hypothese essentielle pour la

a la

validite de ce procedl! est de se limiter ment

zone lointaine,

respective-

a des instants dans 1esquels la majorite I!crasante des neutrons de

source ont deja subi des chocs Le passage de 1a forme de diffusion ordinaire temporelle de

a

de 1'approximation PI

gen~ra1e

~te effectu~

par 1a suppression de 1a derivl!e

~l'

Le traitement monoenergetiQue du approximation

grossi~re

et n'est

gu~re

reacteur ne represente qu'une

satisfaisante pour 1a p1upart des

buts en vue. Obst pourquoi on fait usage des l'energie

a 1'equation ~

~quation!>

dependantes de

(IV-15) et (IV-16) (de p. 57) ou Pon aura pose ~4>

/'Q t = 0 .

- 410-

T. Kahan

Mais ~e sous ne peuvent tout

cette forme simplifieej ces ~quations (IV-15)

Mred~coopl~eB

a fait

et

int~gr~es

particuliers. lei aussi

gueur, telles que par exemple la m~thode

etant

sous forme close que dans des cas

des

m~thodes

m~thode

approcMes sont de ri-

de diffusion des

group~.

Cette

des groupes rassemble les neutrons ou divers groupes en fonc-

tion de leur J

(IV-16)

mono~nerg~tiques

type (IV-25)

a l'interieur I

~nergie,

desquels ils sont

. On obtient ainSl

un

cons'id~r~s

systeme

comme

d'~quations

du

(IV-26) (de p. 61 ) .

M~thode

de diffusion

Appliquons cette pour cela toute la bande

a

deux groupes

m~thode

des groupes

a

deux groupes. Nous divisons

des neutrons en Ie I1groupe rapide l1 qui

d'~nergie

de fission maximale E _ ( - 2 Mev) a l'~n~rgie, o mettons, E et en Ie ngroupe thermique n qui va de E a E = O. Affeca a 1 au groupe rapide, l'indice 2, an groupe thermique . On tons I' indice s'~tend

de

l'~n~rgie

obtient ainsi pour Ie groupe rap ide

l'~quation

de diffusion.

C, 1

+S

1

avec (IV-30)

? (i.

(avec duise

a um!

t, Eo) = ~n~rgie

0) G (E'-~ E)

inMrieure

a

~tant

la

E. En outre

neutrons de fission appartiennent au

probabilit~

que la collision con-

, on a admis que tous les

groupe rapide. Le facteur

(J

est

suppos~ constant dans Ie groupe rapide, respt. dans Ie groupe lent Le flux nrapide n est (IV-31)

A'I

."')

If,. (r, t) =

donn~ -)

par

(r, t, E) d E

-

411-

T. Kahan

(ou

~1 est bien entendu

dans cette section le flux rap ide et non pas

le prermer moment du flux!

Les constantes du groupe sont definies

par

Pour le calcul J on se sert de I'hypothese d 'un spectre d 'energie independant (au moins

par domaine)


(IV-33)

E) =

~ (t:

de

la position:

t) . Z(E)

De meme, I'equation de diffusion pour Ie groupe thermique est (IV-34)

1 d!\>2 - = v2 )t

2

2

) - L a cp 2 + q (r: 2

q(~ t, 0) = 0 et il n'apparait pas

lei

fission dotes Mre

V. (D 'il tP

a

2

pratiquement, de neutrons de

d'energie thermique . Les constantes de groupe. doivent

formes de ~c. _1

dt

a (IV-32) compl~mentaire prend ici analogue

manH~re

La condition (IV -35)

t, E) + S

=_~

i

la forme suivante

C +11 )) i

(. i-1 I

Formulation matricielle Eu egard

a

J

(IV-30l/on peut introduire dans (IV-29) et (IV-34) la de-

finition (IV-36l avee (IV-37)

5'\

El.... E

a

,

4> G(E'- E) dE'

- 412-

T.Kahan

qui definit une section efficace de freinage . Pour limite ELI il faut mettre la plus petite valeur

A l'aide de (IV - 36) et

29)

II

~

~

'I' (r, t) =

(IV-39)

Eo d Eo / J..

, il est possible de mettre les

(IV-39) sous une

(IV-38)

de

sup~rieure

<1>1

11 Ci (

C.1 VI r, t) = (0); (~)

~

(IV-

;J.

forme matricielle en posant : 1

(tP 2 );

~quation·s

t~t.

0

v

2

'

Matrice de perte et de diffusion

avec

Matrices de production

(IV-40)

Ceci permet de rassembler, avec une matrice de source maniE~re

analogue

des r~acteurs

•

(1) ~ v (It "\

ClC,

~qu'

= ~0

S constante de

(IV-29). (IV-34) et (IV-35) en les equations

d'apres la m~thode de diffusion II

(IV-41)

les

v

a deux

~ _ H~ + ~~. ~. + ~ .

1

1

groupes

,

1

..

v

-ri- =Ki q> -/)

Pour passer aux

"

i Ci

~quations cin~tiques

seIon (I1I-l3) (p. 52 ) • il faut

connattre la fonction d'influence. Pour cela. formons les op~rateurs adjoints

.'" =['J -r,) 1

(IV-42)

[en

Ht

o

J2

'

~

supposant pour simplifier

J<.

K

13

+

(VI ~. =\.V2 'l. f2

I ii 12i-r =

00)

/J - A ce qui entraine

- 413-

T, Kahan

(IV-43)

oJ

a

lesquels appliqu~s I

la fonction

d 'influence


liurent les ~quations dif-

.

ferentlelles pour la fonction d 'influence stationnaire .A

A

(IV-44)

(K - H) cl>

Une fois

r~solu

+

cette

=

0

~quation.

on peut former les grandeurs

n. s. c .• k et 1 de maniere analogue aux grandeurs du

ch III et

1

de ce chef les pes, Nous avons (IV-45)

~quations cin~tiques

a dMinir net)

)'5 -~ +~ 1

Au lieu des ~. ~, 1

1

Sf

dans

',It): f~\ dV ; ( + "'-

~ (t) = ,L~+ .... ~ d V Itt> H ~ dV

sit) = (fSd V; v

d'apres la mElthode des deux grou-

en particuller

J~ "'·4>d Vi

lit) =

~tablir

....

dV

H~d

V

(mI3) (p, 52),1'on a l'expression',

f~; (f3il\)l~f/~l +A2V2'E.f/~2) dV (IV-46)

)~ + (Vl~ 1

qui se confond avec (r~acteur

f

lorsque

rapide) •

~

~

fl

fl

cPI

+~2"£f

c1J

2

2) d V

0 (r~acteur thermique) ou ~~ 0 f2

Faisons encore une remarque sur la relation entre que et flux, Le flux rapide et le flux lent sont dantes (IV-47)

par

les

relations "

reli~s

aux

densit~ densit~s

neutronicorrespon-

- 414-

T,Kahan

""L cp

(2)

extr~males

rendent (3)

(~quation homogenel.

fonctionnell~ suivantes(L ~ =

les

S~ LVI

J =

En effet en

= 0

d't,

~J

0,

posant

(4)

4>

~1 et

ou sant

la

deux solutions distinctes de l'~qu, (2) et en ut ili-

2 sont

propri~t~

(1)1 lIon a en effet

b J = J[~i +5~] - J[~] =s( Q>2 +b~i'L(~1 +~~) -S~2 L ~'~ d'C =)4>2 L 2 L~1

d-c

dT +

+5~2 '1 f~ 1d-c"+y' 4>2-'"L ~~1 dt = a[O~I~~J 2 : 8'[ ( f¢)t] C.q.f. .i), ·sr~,'i41.J 1:

En dlautres termes , lorsqu'on commet une erreur "sur les solutions de (21

, la

fonctionnelle

ne suhit qulune variation du second orde

J

e((b cp)Z} De

envisageons

m~me,

l'~quation

avec

second nombre ou

~quation

inr..omogene) .I>-

L J..

(5)

t

't'

1

=

f.,

(equation inhomogene)

1

~tant toujours un op~rateur lineaire sym~trique,

et ou

les

f. sont des fonctions 1

poser I

(6)

et

cp.1 les

n..' 'fI i

,1>._1 = L

v~rifient

f. 1

tI>.

1

1t~quation

ou

sources

possMant un inverse

donn~es,

' On peut alors

-- 415 -

T.Kahan

n n I en va pas de

m~me

,pour la fonction d linfluence

~

lion a

au contraire

+

(IV-48)

ce qui

" N r~sulte

de ce qui suit. Par (IV-39) et

mettre Ie flux et la

densit~

(oJ 1 !. ~ ~quations

(IV-50)

stationnaires sous la forme

1

v

J-,

-J)v N +\J ~ 1 1 1 '- f 1 ./-

- J v N2 2 2 1 Nl i.

2

=0 .

adjo)fntes correspondantes ont pour expression

soit h)lZf - ""J l ) v N4 +~ .J 1 1 1 "ll2Z'f

Dans les deux v,

v - J v N 2 2 2 2 2 N;

demi~res ~quations

+

differentielles pour

N1 et

~quations

et

~~


,

N+ 2

-1

+

.0-

0

=0

(IV-50) • on peut

la seconde par v2 et on trouve pour

0

v 2, N2

1

pa'r

(IV-40), Pon peut en effet

soit

(IV-49)

Les

2

diviser la premiere

effectivement que les l!quations

+

N2 se aonfondent avec les deux premieres

Dans Ie cadre des hypotMses restrictives que nous avons introduites, il est

possible d 'appliquer la

cin~tique

des reacteurs len principe

a

des

-

416-

T. Kahan

substances moderatriceSquelconques . La difficulte principale qui se presente alors est la determination des constantes de groupes, en particulier pour

les

moderateurs si

ve red uite

la description

contenant de llhydrogene. Cette difficulte se trou-

lIon fait appel

a la

du processus de freinage. Cette theorie dont va nous en-

tretenir Ie Prof. Pignedoli conduit simplifiee

il

theorie de ll§ge de Fermi pour

est

vrai

a une

seule eq uation de diffusion, fort

, qui represente aSSez bien Ie comportement

de gros reacteurs thermiques pour certains buts.

-

417-

T. Kahan II Partie Methodes variationnelles en cinetique des reacteurs I

1. Introduction . - L'impossibilite ou l'on se trouve habituellement rt!soudre

a des

de maniere rigoureuse les equations

problemes de diffusion,

de

differentielles attacMes

atomiques nucleaires au autres, ont amene

divers auteurs, (Schurniger, Hulthen , Kohn, etc)

a proposer

)

des me-

thodes de resolution approchees cl l'aide de principes variationnels, .

t

.)

d 'a-

,

spect et de mamemen tres dlfferents. Ces principes s appliquant tous des problemes de

nature, nous avons

m~me

tent~

de voir,

des

a

1951

T. K. et G. Rideau .C. R. Acad. Sc. 233 1951, (1849), J. de Phys.13 (1952) 326, T. K. G. R. et p. Roussoupon pos des methodes d'approdimations variationneIEs dans la tMorie des collisions atomiques et dans la physique des pilES

nucl~ai~

res, memorial des Sciences Mathematiques, 1956»'il n' existerait pas un principe plus

gen~ral

liers. C'est ainsi que nous avons

et~

fait l'objet des presents expos~s et qui principalement en

a titre

de cas particu-

a proposer

Ie principe qui

englobant tous les autres amen~s

a

re~u

de nombreuses applications I

de la diffusion neutronique et en theorie des

th~orie

reacteurs nucleaires 2.

Formulation du principe I

g~n~ral.

I

Considerons un operateur lineaire

v~rifiant (1)

la relation

r'fl

~

,/0.

L

s~metriqueJc'est-a-dire

suivante (fJ

edd,;

=5e 1'1'd~

T

quelles que soient les fonctions

If et e.

Les solutions satisfaisantes a ~

L ='j} (I =bttll~ loc~l)

- 418-

T. Kahan /'

L ~

(7)

Elle

J-

....

= L (L

i

rendrons

-1

=0

f, - rh) 1 'f 1

extremales

une

fonctionnelle du type

(3)

(8')

=J(L

-1 f

=f(i t -1 cont

2

Y2 ) (f 1 -

-

L0/1) d T

=

f·f -IjJ f _i-If LljJ +\jJ~O/} dl: 2121 2121 f 1J- f"-

car

f

sont des fonctions derivees

=jt~2L

- 0/

=ft&2 (L ~ - f 1) - ~1

f.

FL

1

d2J

Les

¢' rendent 1

(8)

J

Quand on

~t'f2

substitue

-1

fonctionnelle

(8)

1" 1

se reduit

f~. ~.J

f, et

2

suivante

"0

et 1fl2 des solutions exactes de (5) , la

a (ou bien avec

et I 'on peut

constant puis que

done extremale la fonctionnelle

(1'1'1 - f 1)-'l'1 a

dt

f2} d"t

f d -c qui est 1 2 sont des fonctions derivees conues .

en omettant Ie terme

If par (1)

2 1

ainsi utiliser (8)

0/ 2->f1 a

-54' 2 f1d

1: )

pour Ie calcul approche de cette fonction-

nelle. Generalement , dans la conduite des calculs, il sera commode de transformer (5) en une equation rep res enter

~

L

integrale, ce qui

va no us amener /

a

par un operateur integral dont Ie noyau sera symetrique

- 419-

T.Kahan ;.

quand

Lest symMrique.

3.

Application aux L'~quation

qui

problemes de diffusion. r~git

canique quantique peut se

un probleme de diffusion (scattering) en

r~duire a

M~­

Ia forme suivante (d1:"!d r ::. d 3r)

ob Ie noyau K( 1,{') est g~n~ralement un noyau sym~tri.que[K(.(',Ii') ~ .:;>,-'\1 A-. 0 . . , ) ::>.1 = K( r, r~ et ob '1'1 (!l.) represente l'onde incidente VIr, etant Ie potenti.el

De

du

centre diffusion

m~m~ de nombreux problemes en th~orie du transport des

neutrons ,se posent sous Ia forme d'une

~quation int~grale

du type de

-1', est

sym~trique

Fredholm (9 bis)

ou les limites

sont

ou sym~trisable et Fa.isons, dans

tt

(9)

-(1)-

PO,ur

donn~es

et Ie no:ra.u K (rt',

t')

Ie terme s.olU'ce. est

s(

r~el,

une fonction donn~e.

(9), Ie changement de fonction inconnue 4>1 :

p!'end alors la forme (1) ,~et

qUl slgmfl€

,

operateur, j'utilise la notation

5

L~= F (-;, 11)

....

S

-7

-?

L = F( r, r ')

{q,( 1! r)) d1 r

tr

- 420-

T. Kahan

( 11)

.->

dr ' En appliquant en revenant au

~

~~

a.

notre principe variationnnel relatif a. (8) et

1 initial , nous oQt enons l'expression suivante, sta-

tiona ire vis-a.-vis des solutions de (9) .

iTfr] ~ -50/2 ~;

V

-)~(t)

( 12)

d~ -f~1 '1': V d;?

V K

(f, r1 )

V(r?I) 0/-1

~ et VI 2 sont remplac~es

Quand

de (9) , l'expression

+)

'1\ V ~ 2 di'

(~~) d1 dr1

.

par des solutions


tp 2

(12) a pour valeur

( 13)

Seule

qui

quantit~

lis ions ou elle

L'op~rateur

L

-1

pour

vl-

un interet physique en

L, qui

J

theorie

l'amplitude de diffus ion (cf.

repr~sente A.

.A

ait

~qu.

des col-

(17))

figurro dans (11), doit avoir un inverse J

que les conclusions prec~dentes soient valables. Demontrons

qu lil est bien ainsi . Posons pour simplifier 11 ~criture (11-1 )

On cherche un noyau G(r::

1')

51/(i', -f" 15Q?"- r'l

(11-21

=

car [(f-r I)

~(r' - -;.'1)

tel que soit v~rifi~e 1'equation

-'Ib (i", )

est Ie noyau d1l'ophateur

unit~

1, I}~'

I (I ¢(f)

= 6' (~--f')¢(r') df' = ~ (r) . (11-2) fournit l'equation lntegrale suivante pour

~(7J r"').

-

421 -

T. Kahan

La resolution en

est

Ie noyau resolvant

relatif

(U-4)--y

(?,f')

avec la (11.5)

immediate

U(t, ~') ,

an noyau =

~ (i--f') +r (i'?,r"')

forme bien connue du noyau resolvant

r (r,

aussi un im1rse

H~(i"

a droite

pi.lisque

-14<:

+ -;")

= ~ (r - r') -

a

Diffusion en

(r, r')

il vient

, I

~ ~ ... eh ~

.

+ -7 --)-

Mais

G(r, r')

est

verifi~

(t(t, 1') r')

+

r(~ ;"', )

df /I

=

S';t - r:")

m~meru noyau r~solvant

D'apres la definition

4.

lion

":>-)

appelant I

t.)] i (;.". ;') d~"

-5'iftCl, t") r (~\ =

.,

en

~') = U(f, htJ1i(?, f" d-' (r'i ill

Ceci montre l'existence d'un inverse

(II. 6)

car

M~canique

r Ct, ;, ) .

quanti que .

Nous allor~ appliquer,

d 'abord ; titre

d' illustration,

la

theorlE

generale qui precede aux problemes de diffusion de la mecanique ondulatoire ( 14)

. Dans ce

cas

[ V 2

+

nous avons k

2

a

eonsiderer l'equation

-')lnh-)

- V(r)

0/ (r) = 0 ,

et la solution de celle-ci qui se reduit

I 1-':> 1\ 1 = k, en l'absence te

du potentiel diffuseur

solution verifie l'equation integrale

que ondulatoire

a

a l'onde

-~

plane

-",

=

e

r

1

V (r) . On sait que eet-

(ef. L . .L.. Broglie

la theorie du noyau , Hermann

~ ~

~

i K

,de la Meeani-

, t. III p. 35(55) et

- 422-

T. Kahan

and Massey , Theory of Atomic Collisions 2

Mott

Kahan, Revue de Physique Th~orique

(15)

=

ik1 .i

e

1 --41f'

t.

Moderne,

j'

ed. p. 116 (3) - T.

II etc.

et admet Ia forme asymptotique "'.>

~

ik r

( 16) ou

¢1(-::)

->

est

l'onde incidente donnee

par f(

00

ji:"

,dont Ia

A.

1 Mott et Massey

(d. -')

1

~ 1,- &2 ) ;: - 4'iT

J ik~. r>

~..,

Cette expression valeur stationnaire

(18')

e

p)

f( 't 1, -11,2) est

(V-S)

- 41T J [,} - ""(

. de

e

V

k\ I"-r'l \1--[-')

comme il resulte de nos formules (9)

v~lriationnel

,Je vais

defini

dans

valeur cif

-)

CPl

Vir) , au

lk. r ( 1 V(f)

e

sur Ia formule

dans la directlvn

t

de

la direction

p.114 (30)

-)

-

£.

J

2

-)-> (r) dr

facteur

- 41, pres,

Ia

S",

-)~

bosons

r

~~ lVi'" -; V(tl
(

ci,pe

ikr

Ia quantite

ll- ~I +)

fondee

e

-)

+ f(kl ' k)

I' amplitude de diffus ion

suivant

~

(17)

r--;>

-..,

e 1

-)

-)

f( k ,k) 1

est

-)

'~ 2(1)

r

2 dI<

V(t')

a

di>

.

liJ /(')

Tl

d? dr'

(13) . La methode variationnelle

(18) est une des formes que peut prendre Ie prinpar ( 8) .

indiquer maintenant

Ia generalisation de notre methode aux

et aux fermions. Generalement, I'equation de diffusion (14) est ecrite dans Ie barycen-

- 423-

T. Kahan

tre de deux particu1es semblab1es. On sait qu 'aIors l!indiscernabilite particules est responsable de phenomenes stique particuliere suive pour effet

par les deux

d'echange regis par la stati-

particules,

phenom~nes

de modifier la valeur des sections efficaces

obten~es

ne tient pas compte de cette indiscernabilite . Si Ie vecteur donn,e la direction

e

designe l'angle entre

de connaitre est

iI'

des ondes incidentes, est pris pour axe des

...,

9

~

kl et

f( 11'"

f( B) ~

qui ant quand on

_?

qui

t~

z et

S1~' -:1 t.

I

- II. 2 ' 1a quantite qulil est interessant

- /1)

Ie signe + on Ie s 19ne -~ui -

avec

vant que lion considere des bosons ~-,

des

oU

des fermions,

;~

IU,i.

OU lion a pose

-'>

L2) = f(11"-/1) noterons .p (;) (r)

f(t, + 1

A partir de maintenant, nous

a

integrale suivante relative .....

( 18;.1)

£Pi (+)

-<)

1

411"

¢ ~+)

X

tamlis que la solution relative

4>~-) (~) (18-2)

a

et satisfera

<1>(;) (f)

une onde incidente suivant

ik.r

(t) = e

e

-i~

1

Quand Ie potentiel est

Ia solution de 1'equation

a

) (t')

e kli'--i"l

Ir\"¥' I

r .

'l. 1 .

vch

dt, -")

une onde incidente suivant - k

1

sera notee

l'equation integrale

t

-")

-)

(r') dr' uniquement foJinction de la distance I

rI=r

, on

verifie aisement la relation ( 18-3)

lI'I(+)

't'1

-">

(-r)

8i 1 'on tient compte de la statistique du probleme de diffus.ion s 'ecrit :

(effet

d'echange), la solution eXClc,e

- 424-

T. Kahan

+ q,(-) (?)

_1 (¢(+) (1)

'f2

.-t ( sta t"lstlque -1 de B ose- E'mste}n )

(statistique de

Fermi- Dirac) ,

et il· s 'agit de tenir compte, dans le principe ge. Reprenons donc (18) terme que nous notons

a partir

reli~e a V~+)

par

( 18-5)

Y~ (-) (r) -I

a

approch€~s

de maintenant 'VI

des solutions

On fait la meme convention necessaire

(18), de cet effet

terme. Ala

fonction d'essai

(+\f) ,

(+) ;-') (r)

'fA

En d'autres termes, les f0l1nctions des valeurs

,

exactes

pour la fojdnction

~~+)

9'2(~») ,

y} -) (f)

associons

Y (-)

et

1

(1) et

d'essai

d 'echan-

?

[r) sont

(F7 ) ,

Yz(rl ' 11

est donc

d'introduire les combinaisons

'If: (+) .A

+

4' (-)

-

1

1 ), -

V2

(+)

(CVt

(-)

~\f 2

(suivant la statistique adoptee) dans une formulation variationnelle devant tenir compte de l'echange. Puisque j'ai

a calculer

ajouter on retrancher les valeurs Soit

V~·:

+)e

V( It )

V(r)

1 fl'J"(+) ""9. 4 TT T 2 (r) V(r)

f(e)

'+

f(Ti' -e), j'aurai respectivement

de (18) relatives

a

-':>

+ k2 et

a -

--,

k2

a

- 425-

T. Kahan -'>

\ e

-")

(+)

-ik2' r _?

(

V~~) Y 1 (i~) di?

ik r

+)

e 1

v(r)y~-)(f)

+ d(-

( 18-6)

-jv:+)

v v;(-)

) -4f~2 1

Etant

donn~

- -)

d~

-

,,_")

e iktr-r'l

_)

(r)V(r)

\1-P'1

I

la definition des foJinctions ,

(18-7)

( 18-8)

(18-9)

=

+\0/(-) (~)

-) 1

V(r) liii·(+) (r) +Ijr(-) (r)) 'T 2 - 2

=.! ((y(+) (r) +,i}(-) (r») 2)

1

_Tl

V(r)

(~+) 2

d? =

(i) + \f(+) 2

f;)) d-;?,

-

426-

T, Kahan

repr~sente

l'onde incidente

~ quand on tient compte de la statistique, De 1 .llt~ f? -& m~me pour la quantit(! (e 1 + Q 1 ) , Donc, finalement, on

suivant la direction

t

obtient les deux princiPet2variationnels bles

, soit aux

bosons , soit aux fermions

f( e) + f( 'If - e)

=

.:':>

v, S,

11{2 -r [( e

+ e

-)

.-7

+c

(+) V(r)

-1"1 Y.

+e

~~ (+) (f)

- -.!...( r-"

-?

-lk 2 , r

.;:.J2

.t?::::> 1"1'!' (18-11)

ci-dessou3, directement applica-

411") ...... t

(+)

V(r)

8

F":"I

2

~

'3 ~+) (i'I) df

(tl

V(r)

+

dr? -

(+) (t) 2

V(r)

(t) df

-

f(6) - f(1r-e) =

(18-12)

~ V. s. tr·;~'·? -. -;);,.f 'k

+ ) .'

~

Vir) :::: (-) It) di'

-')

,.~- • ' ,.r

1

V(r):;:

I-~

rr)

d.t-

- 427-

-

'f

T. Kahan

(-)

-::::. 2

,......._ (-)

;;t,

_

(r) V(r)

(~)

dr , -

2

__ 1 (~(-) 4iT)'-' 2 Dans les forrnules

If)

Vir)

e

ik rf-1'\

t)

V( ')""-' (-) d-> dr" r'-="'-1 r r

\1-t'l

"-:' ....... (+) et '7' ........ (-)

pr~cedentes,

remplacent les quanti-

tes

essai

et representent les fonctionsi tera qu'elles sont assujetties de

la statistique de

dans Ie cas

qui doivent

utilis~es

. On no-

a la condition d'etre paires dans Ie cas

Bose- Einstein et

a la condition

d'~re

irrPaires

de la statistique de Fermi - Dirac

NOlls allons montrer maintenant comment

(18) permet d'arriver aux

(Phys Rev. ~ (1948) 1763 - Hulth(m , XO

methodes proposes par Kohn Congres

etre

des Mathematiciens scandinaves _ Copenhague

1946).

~

5

Methode de Kohn - A l'aide de la fonction de Dirac, on peut ./' 2 2 mettre l'op~rateur L =9' + k sous forme d 'op~rateur integral dont Ie

noyau

G(;'

:r1)

est

~videmment symEitrique . Ceci ~tant, l'~quation (14)

s'ecrit

)

V(;~ cp (r)

(19)

lavec

=) G(l, j?,) ¢ (i~')

L=(I?+

k 2)

di?'

=~G(;,-f'{--]dr'),

En posant

+

(20)

on obtient, pour les

, 'PI (r)

" VCr) J

,

l'equation integrale suivante

- 428-

~\f!

(21 )

au

j'ai

I

I

(0 0

-'>

(r) = -'f'

J

-)

1

T. Kahan

-1-

+

(r)

v;0

pos~

(22) I

En ~crivant notre principe variationnel relatif aux ~ 1 ' on obtient une m~thode de calcul approcM pour la i(l? -) e

(22)

quantit~

+k' ) Ii' 1

2

si bien qu 'en revenant aux

Vet)

t

dt -

/,11 f(

-'>

initiaux, l'amplitude de diffusion

est, au facteur 411 pres, valeur

stationnaire de la

-)

-:?

f('{ l' --{2)

quantit~

~

j" '1.[

X

V1(r) dr oJ( :') 1

- e 2

oil. j'ai

~ f~ 2) ,

\l L + k

2J 'K (f) d?

utilis~

(24)

Pour Ie ca1cul de la Green:

"&

de

dernH~re int~grale

df

r:

2

2

S(f~ -g~t\.)dS=/fY'g-g;Yf)

(23), on utilise la formule de

dl:,

~ ~tant une fonction d'essai que lion supposera avoir la forme asymptotique

e

(25)

ikr r

On conduira Ie calcul de la

fa~on

indiqu~t.

dans Ie memoire de Kohn

- 429

-~

T. Kahan

(cor. cit. p. 1766) . pour retrouver exactement Ie principe variationnel qu'il

a propose. 11 a lieu de remarquer que Ie principe (18) que nous proposons,

bien que d~coulant cision

sup~rieure

de la m~me methode, fournit n~anmoins a

celle obtenue

une pre-

par la meth,)de de Kohn . En effet,

en prenant pour fonction d'~ssai simplement l'approximation d'orde z~ro (onde plane) , Ie principe de Kohn fournit la premiere approximation de Born, ta:ndis que notre principe (18) donne directement jusqu'a la seconde approximation.

Methode relative a l'equation radical..!de la diffusion.

6.

Nous allons maintenant deduire les methodes relatives

a l'~quation

radicale de l'~quation de la diffusion directement de notre principe variationnel g~neral. A cet effet, prenoDs pour les fonctions

VlIet ~ 2

qui figurent dans l'expression

la forme

't'i (r) -~

(26)

(18) un

00

=~

(2n+1) i

n

(27)

--,

n

\¥2 (i") = ~ (2n+1) o

de

P (cos 8)!\ ( /rl) ,

n=O 00

-) au 1\ (r)

I\.

dev~loppement

t

'"

A P n (cos 12

";? ) '\ dt\),

a la forme asymptotique in'n. e . A 0 - - s m (kr

-) 1\) r-)

n

00

kr

-)

la direction ayant pour

{I ~tant cor.doonn~s

Nous aurons egalement pement suivant

prise comme polaires

a utiliser

axe des ( @, 0)

Ie deveIop-

rJ)

~) Mott-Sneddon, Waye mechanics and its application Oxford, 1948 p. 386

- 430-

T. Kahan

-41'

e

~k

_)"'>/ 1r-rl

I r-rl 1- -

0

n

k

r,

~

t

(kr) Co ___ P cos ...

.11."

/\

I •.,,;;YI II'.';J

3~(~ r):~ <\

(28)

:R. (t ) =V~ ainsi que l'expression

P

n

(cos

t

J

i

_-:,A

r.

~f~

)

~tant

les

e ) Pn

n

I (n+m) (

polaires de

coordonn~es

a I'origine

r)

,

(cos ( 1) +

I) celles

,(9!

entendu Ie potentiel V(r) , qui va intervenir) ne de Ia distance rz.

(k

-n-

p"" (cos 9 ) p'" (cos 9' ) cos"" (

2: (n-m).

"'~~

(k r) + (-D J

r~) = p(cos

n

+ 2 (9 ,

~ (kr)

00

"'- 2 .. +1

411'"

(symHriie

d~pendl

J

de rl . Bien

par hypothese" que

spherique).

Ceci ~tant, (18) se met sous la forme 00

41T~ (2n+1) (-It P (cos o

I..

n

00

[~·f a-2 k

J

n+Tt

o

(30)

-J (r

(k' r)r (r) V(r) drn

00

o

n (r))2 V(r) dr -

--k1frn(r) ou j'ai

6)

V(r)f (k r )
pos~

r

rn(rl) dr dr'lJ ,

(r)

/\ I\. (r)= - "r ' -

Quand

y;. et V2 sont des solutions exactes

d'onde , il est clair

que

"I\. devient

d iff~rentielle radicale / relative obtiendra d e s f 1 et

'II 2

a

L (r) n

la valeur n

diff~rant

2 de 11~quation

peu les

,solution de l'~quatiol1 du


moment cinHique . On et

- 431-

T, Kahan

dans

V2

des

(26)

ainsi

L (r) . La variation des 't'- et n 2 nlest pas la variation la plus g~n~rale, mais clest

obtenue

la seule qui

diff~rant peu des

An

importe dans Ie cas de la symNrie

envisageons ici.

La premiere variation de

sph~rique

que no us

(30) qui resulte de eette varia-

suite de llorthogonalite des poly-

tion radicale, doit t!tre nulle et, par

n(j'mes de Legendre, Ia premiere variation de chacune des quantites entre crochets cette

de

est nullc, Quand

30

entre corchets

quantit~

r

t\.

devient

(r) se confond avee ~gale

a

G (r) = L (r), n n

2i"1,.", -1

e

Uk 2 ) hasa te habl'tueI (-1-) Done, 11 expreSSlOn ' es t 1e dep en t re eroc he t S

OU'1 .. provenant

de

e

(30):

lit. .

-1 = V.S.[ 2 - ti k

~GdU: k2 t~ 2'" In+1~ (kr)l: (r)

co

-Jo (rn (r))

(30)

2 V(r) dr-

dr ' {.

)( V(rl) rhh') dr fournit une m~me

m~thode

V(r) dr-

variationnel1e de calcl approchee des

d~phasagfS.En

temps, (31) donne l'equation integrale que doit verifier G (r), n

a

llJRemarquons que la fonction G (r) introduite ici differe de celIe que lIon trouve dans Mott et Masse~, cit, ch. II et ch VII, en ce qu1e:De a· la forme asymptothique :

i"l

c n -~

,~

p

('111. - n

fI/2

+II'!) ~n

•

- 432-

T. Kahan

~

savoir

l.\~

G (r) = n

(32)

k~2

J

r \fJ (

- l.

n+1/2 (k r) -

kr )

»

k)oJ n

~ (k r) <

V(r') G (r') dr' . n

Nous allons maintenant etabLc' en Mtail les calculs conduisant /

la formule (30) . Procedons terme

co

n'=o

1 2k

Ce qui donne

~

(t n' + 1) (2n + 1) i Pt\.' (cos 9 )

J 1 ,( ),1 kr

I

(kr()

n)

o ...,..

o

\f I .

n=o

co

f

-')

d'abord l'integration sur dr' en tenant compte du

ne depend pas de

L.

terme

,

Nous effectuons fait que

~

r;> r (

p l(cOS 9') P n

n

V (r') 1\ (r') r" dr>' I\.

(cos 0) sin 6' de' .

En utilisant les relations

d'orthogonalite des polynomes de Legende ii')

il vient

~)

a

Valiron : TMorie des fOt1'nctions (Masson) p. 207

- 433-

T. Kahan

1 (e ik 1-r-r?11 _ -:) _) -4,,) Ir~ _rl/ V(rt) 'V-;(r) dr 00

r-

(30-2)

t·p

2n+1 {

n=o

(cos 0) n

(OOt.e! 'I.~)llt(l~v(rl) A (rI2) drl

Jo

'I.

I!.

11\

<1

Nous devons maintenant effectuer llint~gration sur df. ble

t~gration

If}

'f

(Yl"\

1-' -

ferons disparaltre les termes

e

-f

2-

00

'-Y 1(r-,l) dr'"")-~ dr l

00

A

)( ('T" P

2

n

T

~

12

0/ 2(r)

--)

qui con-

(2n+1) (_l)n Pn(cos

a

00

),

[ 1\ ~ (r) V(r) n o

1\ (rl) r

n+n l

=

p (cos n

dr dr 1

n

00

V(r)

r

(cos 0) p 1 (cos 0) sin

) o

)(

V(rl)

ko;:;o (2n+1) (2nl+1) i

''a,(r l ) r

4rr

ik\1-r1 /

If'-r l !

4~i2(r-1 V(r)

=-

de

et il reste

:7..

X V(rl)

ainsi en effectuant d1abord llin-

A

sur T nous

tiennent cos

6

V 2( -t) et

n1apparaitra que dans

. La varia-

Ode

=

®)

(kr ) ~ h~ .., n r

2

n

r

>r<

/2

dr dr l .

Cal cuI du terme

if?; (t)-, V(r)

f1f'ii"J

Vir)

ne contenant

Y,lf) df pas

If',

. en effectuant

d labord llint~gration

-

434-

T. Kahan

'P en

sur et il

~limine les termes

de

V It)

qui contiennent

cos

m If,

reste

-f~ (r)

(30-4)

%(1)

V(r)

00

d? =

00

L

=-2JT!

(2n+1 )(2n '+1) i

n+n'

n=o n'=o

00

x

)

CA o

Jo(IT'

)(

1\ ,

(r) V(r)

n

h.

r

2

P , (cos n

®)

dr

P (cos 0) P , (cos 0) sin 0 d 0 = n

n

00

= - 4fT

I..

(2n+1) (-It P (cos n

n=o

I 00

("n(r))

2

V(r) r

2

@)

dr,

Toujours en utilisant les relations d 'orthogonalit~

A

des polynomes

de Legendre. Pour Ie calcul

des deux termes restants, on doit utiliser Ie

veloppement bien connu (30- 5)

e

l*)

/\ (2n+l)t P (cos 1{. t n n=o

f~,t

00

L

J et en

employant

quantit~

c)

t*l L.

Ie

~

n+

1/

2

)

(kr) ,

d~veloppement

.I'-

(29) des polynomes de Legendre

.,

.

e

d~-

ik.t

V(r)

'V;

(f)

de Broglie, loco cit p. 20 T. Kahan, Precis.. de Physique

-:>

dr

,

Theorique

t.I1 (P. UF.)

, la

- 435-

T-. Kahan

devient, apres

Lf:

j -) .~

1

(30~6)

~limination

2' r

e

des termes contenant

_00

~

'\"00

nt-" '

1/ n+ 2

J

n

int~gration

sur

(r) dr =

= 2Ti?.. '- (2n+l) (2n'+1) i n=o n'=o

P

par

_'>

__';>

V(r)

'f

(cos B) P

(~'t)

n

I

P (cos @ ) rt

V(r) A (r) r2 dr n

(cos 8) Sin 8 d 8 =

00

471

~ (2n+l) (-It P (cos ®) n

o

Le calcul du terme

d)

s 'effectue de la meme fa(,:on . L~ encore,les termes contenant

Y'

et pro-

venant du developpement de \V2(~ s'elimine It par integration sur ~ 1'utilisation des relations

d 'orthogonalite des polynomes de Legendre don-

ne

00

'" 4 7T [

t

(2n+ 1) (- 1

n=o ;

00,

(]I

o V2kr

et

I n+ 1/ 2 (kr)

P (cos

n

Vir)

®) 1\

)r) r

2

dr,

- 436-

T, Kahan

Si bien que (30-8)

l'expression (18) devient .finalement

_00

L

411'

n

[1 faf:" 0

-.!.

rr

A-tn

7,

I n+I ,

Ikr)~lr)

Vir) dr -

VIr) dr -

(rll, (1'»

o

n

(cos \!Y )

2

00

- [

~

P

(2~1)(-1)

(1') V(r) j(kr )

~(kr~ ) V(r') rn(1")

n.,J

dr dr' ]

Methode de Schwinger deduite de notre principe general (3) Nous allons montrer maintenant que la methode de Schwinger~ se

deduit de notre principe general (3) en donnant une forme convenable

a

requation integrale (32) , A cet effet, introduisons une nouvelle fonction

fu

(r) reliee

a G(r) par

_i'l..

1 ,

n

n

(r) -== k G (1') _e_

n

cos"!..

n

Cette nouvelle fOJlnction

verifi~

1vn =[fk"2

ntI 2

(34)

(r)

(_I)n

-r

';( V(f kr,( 2

ii") J ,SChWinger·· 76 (1949) 21 ,

00

o

-

r J

/v kr V- 2 J

/

alors l'equation integrale (kr)-

7

J ntI/

2

({ I' 7 )

/ (kr ) V(r') -n-l 2 <;

~qn(1") dr' ,

Conf&tences in{dites, 1947. Blattand Jackson, Phys, Rev,

- 437-

T.Kahan

On peut alors

1?n(r)

(36) apr~s

~crire

avoir

(34) sous la forme

=ft1'l (r, r') V(r') 1n(r') dr' ,

fkr

pos~

I1J r'):- - 1 !J'Z(r J (kr)f~- kr' J 1/ (kr') + , tgT[ n 2 n+ 1/ 2 2 n+ 2

(37)

(_I)' \1pTKr( + __ __ k

On

sym~trise

J

n+l/ 2

l'~quation

avec

( k r~r7 )-<. 2 J -n- 1/ 2 (k r ., ) .

finalement Ie noyau

1tn(r) = (V[r)

(38)

et

'2

pr~c~dent

~In (r),

prend la forme

i~ n

=0

i.) ~'-"I d VI'ldl l" "I ~H

(39)

en posant

dr' ,

En appliquant alors Ie principe (3), on conclut que l'expression cot g IIJ.. n = V's{J'1! (r) VIr) dr -

(40)

-1-~lnfS 1'1 VI'I'f"; n+V I

X

,~fiikr 2-2- J- n+.(1 ...(fiffi": ~

1.'

1/ 2 (kr»)

V(~) 1(r') dr

2

k'

dr'

+

l2}

I n+1/ 2(kr) VIr) tn(r) drJ

a une valeur stationnaire pour les solutions exactes du (40) fournit la m~thode de calcul approcM

de cotg

"Z n

probl~me

propos~e

de diffusion, par

Schwinger (lit) ~') cf".

cit.

par exemple, Blatt-Jackson, loc. cit . p. 21 , J, Schwinger Loc.

- 438-

T. Kahan

Gl!n~ralisation(1) a des op~rateurs

8

non

sym~triques

du principe va-

riationnel general I

1

Considerons un op~rateur lineaire

L

et l'~quation homog~ne (sans

second membre /'0

Lq! = 0

(41) avec sa

transpos~e

hermetique

(2)

(42) Nous allons montrer en appliquant notre technique generale que la quantite (43) nulle quand

~', Ij)'

respectivement, ne

viennent coincider avec Ie souhons et

diff~re

du second ardre quand

de zero

'4'

if

et

I

\j7 de (41 et (42)

que par des termes infiniment petits

diff~rent

de

¢ et ~

par

des infi-

niment petits du premier ordre . Posans en dfet ,

11 vient alors

('

(

(lJ=)~'

+~

-

L ~'d~

=)(~+r~)+~L =

t Lr

s(r~

La premiere integrale est

(\iJ+bo/)dt: d"t

nulle par

=

5~

+

+ L ~ tf d"C.

(41). Ql.iant

a la

seconde, elle peut

s'ecrire (I) P. ROllSSOpoulos. C. R. Accad. Sc. Paris t. 236 (1953)1859) Kahan - Rideau ....

Roussopoulos, loc. cit. 22 (2)

'"

l'operateur adjoint

(43-1)

S0/+

L

L

+

adjoint

$dt

a ",.Lest

,

defini par

=JH +0/ ~

=<~it 1jJ)

-

439To Kahan

s~+[, ~!id< {lrft+~dTr ~O en appliquant la dMinition

du transpos(CI hermitique (adjoint) d 'un opera-

teur et I' (CIquation (42) . Le cas plus complique de 1'equation inhomogene ou avec second mem-

ijJ

ore ou

et

tp

sont solutions

de

A

Ly

(44)

./'0+ L

(45) ....

= f,


=g

I

se raml:'!ne au precedent en posant _

-

-"-I

Y= 'V - (L

q; et 1>

ou

) f,

th ./0-1+ 't' = ~ -(L) g,

I verifient maintenant des equation> du type (41) et (42)

0

Il vient alors

(46)

J

1¢

'-"LIP' dT =) ('f'-(L- 1)+g(L(o/'-(L

-It f) d"t

=j~Lifl-4"L 6~-ltf- (i-It g(L =

tV')

_(I.-It gi,rL- 1t

~

)[q,'LIti' -<\>' f - 4>10/' +¢~

=S[

- g Ij) ,

dC(I) = d1:

q

+ cp

Finalement , nous pouvons ecrire

(1- 1)+ = (L +)- 1 .

(1 )

+ Jo -1 + A -1 +A + +1 -1 -1 + En effet de 1 = 1 ~ (L L ) =(L ) L , on h l< (L ) =(L ) (A +)-1 A -l mltlque L =L 0

o

•

A.

,A

0

Si L est her-

- 440-

T. Kahan I

La notation V. S. sigilifiant comme precMement , que la mier nombre de

(47) est la valeur stationnaire (ou extr~male) de la fOI

ctionnelle du second

membre.

Les formules hermitique

au pre-

quantit~

1'-+ ...

pr~c~dentes

(L = L) car no us

se simplifient quand

a

aurons

.A

L est un

consid~rer

op~rateur

alors comme equa-

tions

(47) s'ecrivant dans ce cas

(48)

Remarquons que Ie principe variationnel (47) contient a titre de cas particulier

notre principe variationnel pour

op~rateur

symetrique

(8).

Ce principe s' ~crirait

::;;; ~ V, s'rry-; ',d Hf~}2 dt -J~;

(49)

=V.S·t lJ T

~ et

If' ~

I

etant des

('f',f l )

+(o/~f2) 0/ l' 'Y2

/

formes approchees de

~o/;

dt }

-(Y'2,Lo/ 1)},

solutions des ~quationll

diff~rentielles

'"L

(50) A

L

~tant

un

montrerons

op~rateur

ais~ment

sous forme d 'un (51 )

UJ

T2

= f 2'

SYtmMrique tel que)' \f1

L Lf2

que (49) est identique

a

op~rateur int~gral

=) ~ Llf1

d

't •

(47) en envisageant

L

d'{

Nous

- 441-

T.Kahan Si "Lest la combinaison de les remplacer par des

d~riv~es

de divers ordres il surfit alors de

d~riv~es de divers ordres de la fonction ci

dont les noyaux sont des

op~rateurs int~graux

~

I

(cf. p. 7 J 11 ). eeJ

on Mduira aisement que Ie noyau prGpre au transpose hermiti-

~tant,

.A

que de

L sera

et que Ie noyau d'un sym~trique

pourra

sera

m~trique

op~rateur

de T et "'( I, Des lors, la seconde

se mettre

sous

+ .... +

) d'~ 10/ lrl) L (1;I,.1:)

ce qui permet de conclure que, les deux

~quation

une fonction

( 50) qui s I~crit

=f{r)

dans Ie cas des

+

,

op~rateurs

symetriques,

~quations A

L

~quivalentes

n

lui-m~me

la forme

(

sont

de Dirac

I

ill

, 2

::

f

2'

L

+ +

0/ 2

= f2

+

.

suffit maintenant d'ecrire Ie principe variationnel (47) en partant

du compte d ' 6quation

pour retrouver Ie principe variationnel (49). On peut gen6raliser notre prinCipie d'~quation.

(52) les

Envisageons

par exemple Ie

variationnel pour des

syst~me

syst~mes

homo gene

~

L. (w.) :: 0 , J ..,. J

L. etant des op6rateurs diff6rentielles oli lin6aires. E notation vectoJ rielle , (52) peut s ' 6crire

- 442-

T. Kahan

(53)

ou

·L

) n

qUi est du type (41) . On peut alors montrer que l'expression

J=f(f,l)~,

(53)

est

extr~male

ou

verifie l'~quation adjointe

Consid~rons,8. titre

d

1-

(55)

L aij

et

une fo nction

c

..

d

(56)

C 'est le cas simple ou

'

(i, j = 1,2,3, ... , n)

coordonn~es

x. et a ij = a ji I

."...

Llf = 0 avec les conditions aux limites

J = -

0/= 0

5

dans

(41) et (43)

x. I

J

i'

d

~

- a .. alJ

ij

dl& (a1J..::..;:r)+c!f]dr= dx. J

d ,/I

..2.:f dx. J d ID -d .:::.....I

n. dS + a 1

dx. ~x. .1 J

prend la forme

f1f,"

A I.{JL\Odt= -) I

~ i' d J) =- - ( a J .::::...I...)d"t+ a J _ T dx. dx. dx. I J J

conpte tenu

.

dl./ ) =0 dX j i $

(57)

(58)

+c

sont des fonctions des

telle que

a (53)

d'application, 1'op~rateur sym~trique

(a lJ - ) dx j

dx. I

oi:! les

d"t

_ c

ij

rJ

d

"\. a I

d If d -.p d'T d dx. X. I J d~ '

de la condition aux limites (57).

.

2

~ dt-(('f d't.= )

S \f c

2

d 1:

-

443-

T. Kahan

Soit maintenant d dx. J

(59)

Ie

syst~me

d-t

( ij

dx:") 9J.(3 !-'"

~!3

0

1

ou et J..j (1 = 1, 2, 3, ... , m ,

i, j = 1, 2, 3, ... , n

et ij

(60)

aJ(3

j 0 ij I = at-{2 = i:1~

, c.;..p =

13 ot.

avec les conditions aux limites

-

(61)

d

'PrJ..

dx.

1

n)

o

j 5

Dans ce cas , Ie principe vabationnel

a

appliquer

Contentons -nous d I~noncer Ie principe sans Ie

Jl-

(62)

J =

(58) et (62) jauent

sque la

9

m~trique

ij' aJ(J

~ dl/J~ dx.

un rtle n lest pas

Principe variationel

1

r J

cd-j1

f

tA

I{'

du type

d~montrer

(1]

important dans la

d~finie

est

(54).

:

d~ de l'hypercycle Ior-

m~thode

positive (j)

g~n~ral, c )mme extension

du

principe de

Ritz-Rayleigh I

L'inconvenient des principes variationels est de fcnrnir une approximat on

d~pendant

la forme

ala

mllcanique

diff~re

pas essen-

appliqu~e,

fournisse .1t une mllthode de calcul qui ne

tiellement du calcul des perturbations

(47) ou (48)

de la normalisation particulie-

re adoptlle pour les fonctions dlessai. De plus, quantique, ils

de

dont on sait, par

ailleurs, les dif-

*)cf. J. L. Synge, The Hypercycle in Mathematical Physics, p. 292 et (Cambridge Unniversity Press, 1957) L. Cairo et T. Kahan, Variational Techniques in Electromagnetique, p. 58 / Bmacjoe Mp, dp,; 1965

- 444-

T.Kahan

ficult~s

. Aussi donnerons-nous maintenant un principe variationnel

ral; extension du principe Ritz-Rayleigh qui dans I

d~ja ~te.

kson (loc .cit.) et champ

de

de

~tudie

a la

fa~on dl~taill~e

par ScllVinger, Blatt et Jac-

utilis~ pour la d~termination des dephasages dans un

forces centrales V(n) .

Rappelons dlabord Nous

application

quantique, ~chappe aux objections pr~c~dentes, Un cas parti-

mecanique culier a

Son

g~n~­

allors

bri~vement ce

qu1est Ie principe de Ritz-Rayleigh (1)

Ie dMuire au demeurant par la suite (65) (83) de

notre principe variationnel g~neral Le probl~me pose est celui de la determination de

.A

ou

Ia plus

.....

A et

B

petite

valeur

sont des op~rateurs

lisant cette derni~re hypoth~se,

de sorte

que nous sommes

propre de

hermitiques d~finis positif . En uti-

on peut poser

ramen~s

a

1

B- 1/2 A B -"2 ~ _~ Ie

"ili J

probl~me

de valeur propre la plus simple possible. On verifie aisement

que A

AI

est comme que ( 63)

= ""B

2 A "'"B

I

1 2

un operateur hermitique dMini positif. On peut alors

,,=

d~montrer

V. S.

(l)Pour les d~tail~ cf. par exemple comant -Hilbert, M~thoden der mathematischen Physik ,ou Schwinger Lect. Nots Haward iJ. • 1947) .

- 445-

T. Kahan

En effet

est

en tenant compte de ce que

A cp I

=~

4l

donne

A=(~,A'~)

(¢' I~) pour les solutions exactes


OHI\. trouve i apres l'utilisation de l'hermiti-

cUi, une premiere variation nulle. Notons que aucun usage n la Ceci

~te'

fait du caractere

dans cette

d lop~rateur dMini

Id~monstration,

positif de

AI

•

n1entre en jeu que dans la suite dEB d~montrations permettant de conduire a la conver-

gence du

proc~d~

Ceci

vers la plus basse valeur propre d lop~rateur.

rappel~,

envisageons de

nouveau

des

~quations

de la forme

(64)

la

quantit~

que nous voulons calculer

~tant

(65)

on peut alors (66)

""+ L les

op~ratcurs int~graux

du second membre

~tant

bien

conjugu~s

hermitiques

- 446-

T. Kahan

d'apres ce qd

il

ete dit plus I aut.

Sous cette forme (66) , 1a determination de

1/ Q est ramenee

a 1a

resolution d 'un prob1eme de valeurs propres generalisees ,

L~ =AM~ Lt =A M+¢

(68)

J\

avec l'operateur (69)

AM.= ~

prob1eme au quel on

peut lcvidemment appliquer Ie principe de

igh , tout

au

moins

CL}

d 'Ul. ,point

l( L't ..} d't' Ritz-Rayle-

de vue formel , car Ie fait que

le~ ope-

rateurs integraux dans (68) ne sont plus obligatoirement definis positifS nous obligent arenoncer aux developpements classiques sur la <.:onvergence de 1a

methode. Nous obtiendrons donc

(69)

qui pour Ie cas particulier OU

~

(70)

Q

car dans ce cas dre


=

v'S'

Lest hermitfque s 'ecrit

~)~'+LQ'd"( ~ lllji'+ fl d-rf2 J ' on doit pren-

la seconde equation de (64) Mait

!.pl. crest un principe de la forme (70) qu'ont utilise Schwringer

Blatt et Jackson dans Ie cal cuI

.1\

des dephasages, l'operateur L etant alors

un operateur integral symetrique reel , donc hermitique.

10.

Unicite ou multiplicite des principes variationnels. N,.ous avons

variatinnnels (47)

r'"

associer a' meme type d'equations

et (70) , sens,hlement differents, par

deux principes

Ie calcul de la

mt!me quantite. Aussi peut-on se demander si les principes (47) et (70)

- 447-

T. Kahan

a

sont les deux seuls que l'on puisse associer (45)

,ou si , au

des equations du type (44)

contraire, il serait possible d'en trouver un plus

grand nombre. Crest cette seconde partie de l'alternative qui se trollve realisee. En effet, posons J-

(71)

+

=S~

f dT

J1+ L~\f'

,f3 =

d -c,

'( =) ~' /

d (' ,

et considerons une fonction analytique

a laquelle et

9'

a vis

nous imposons d '~tre stationnaire vis -

autour des solutions exactes

la valeur correcte de 0< quand

~'

de et

et

(44)

r'

des variations I(> ,

(45) et de se reduire

co'incident

a

avec ces solu-

tions exactes , Il suffit pour cela d'imposer la condition F(,f.., J.. , rf.)

quel que soit

.;..,

mais

ceci entraine

-oFI~; : •• ) et en tenant compte de donne les variations

f~

f.J.

d'ou resulte

~~(f'

+ =

,fi

~ F(v-,~, .1-)

1(j.

+

do.,(#.)

~rj,

= J..,

m.;..~,,) ,(M;.+ :lFI 3~~) ~~ I,

S.f. + f'(' , la

condition de stationnarite

etant arbitraires,

JF(c1\,~,d-) }F("',~,,J..)

d(J

d(1

+, dF(d--,:;l,o-)

'.

~F(do., fl., rJ. )

JF(d-.,,,, ,d-)

df1

dJ

-;)i

=0

= 1

Mais F etant suppose analysique , il existe au mains une valeur

d- o

de

-- 448-

T. Kahan

r:J..., (JA

r autour de laquelle elle est d~veloppable en s~rie

En portant les tions ecrites, serie telle

d~veloppements

de Taylor .

obtenues dans les dernieres equa-

on obtient des relations entre les divers c6efficients de la

qu 'un arbitraire illimite subsiste dans Ie choixe de ces coef-

ficients. Le nel F("

r~sultat

,(I, '() independant

17'

c/t!ssai celle de

est tout autre si nous cherchons un principe variation-

~'. Si en effet l'a norme de ~' est multipliee par I" '

et

rp'

par ;-',

~

L' independance Me

quels

des normes particulieres des fo,lfnctions

que

~ d., "("""y- ct~

est change en

vis-A-vis

en

veri-

)'" et ,?:

soient

= F(q.,/S,Y').

r

11., ,}I-"" i.. ' il vie)!t

1" =

particulifir

F(eJ. ,A ,'r) = F(I,

'/'

et

_.P- Y3 .

de la nor me ,entralne que soit

F(.rol,,;"-Fj1,I"Y) choisissons

en

t- , 1)

fA r

F (i- (l ,y) doit Mre une fonction de la seule quantite ~ '<',11 et la

condition

que F(d-.!,' Y ) F( d-

c 'est

a dire

Ie principe

= rA

quand '"

f3 . ( ) =

=(1 =Y conduit

A prendre

d-; .

(70),f· jj . On voit donc que

ce principe se distin-

gue de l'~nsemble de to us les autres possibles ce qui sans doute. a fait son int~r~t

et

son succes.

11.

Principes variationnels generaux de la theorie des collisions en

m~canique quantique. En

~crivant

sous

diff~rentes

formes

I

convenablement adaptees les

- 449-

T. Kahan

equations pur

fondamendales de

la theorie de la diffusion (1) , nous avons

deduire tout un ensemble de principes variationnels, appblicables

sp~cifiquement Ces

a la theorie des collisions (cf. aussi plus haut 14, p. 74

principes ont deja

ete

enonces par

Schwinger (2) , Schwinger et

Lippmann(3) , Goldberger(4) , mais aussi sans autre justification qu'une simple verification: (pa 'colle ")on trouvera dans notre M~morial des Scien-

,

,

ces Mathematique5la theorie detaillee

et la justification rigoureuse de

ces diverses methodes. 12.

Operateur

lineaires

particuli~rs

en physique des reacteursnucleaires

Les equations lineaires qui decrivent des systemes interessants en physique des reacteurs nucleaires peuvent toujours Nre mises sous la forme A

L ~

(72)

ou

<;

=

.I-

f

represente la distribution

lineaire et

f

I

est

./\.

, (L = K - H)

d'etat inconnue , "L

est un operateur

vecteur donne, (decrivant d'ordinaire

unt source ext~-

A

rieure

S si Lest

relie a la conservation des neutrons ) .

Le cas homog~ne

f =0

est inclus,

implique un parametre indeterminee solution

; ce

cas peut

tion in. . llOmogene lorsque

~tre

A

pour vu que l'operateur

pour assurer

.A-

H

l'existence d 'une

connu comme limite (singuliere) de 1'equa-

f tend vers

zero

et

Ie parametre s'approche

d rune valeur propre.

(1) T.Kahan G. Rideau, P. Roussoupoulos loco cit. p.14 (2) loco cit. (3) Phys. Rev. (4) Phys. Rev.

~ (1950) 469)

~

(1951) , 929

a 22,

p. 28 a 49 , etc,

- 450-

T. Kahan

¢

Bien que la d~termination de Ie systeme

d'~crit

par (72) , lIon

fournira toute I Iinformation sur d'ordinaire ala moyenne

s'int~resse

pond~ree

(73)

(g

+

ou

est un certain vecteur

g

ction

d1'

d'habitude, une

donn~,

quantit~

du genre se·

plut~t qulun vecteur complet

efficace,

possible de deviner ant

+ (+ '

grossitrement

la forme de la solutbn en s lapP)lIY-

sur des raisons physiques, Ie probleme de calcul revient a convertir )

cette information de mediocre pr~cise en ce

qUi

concerne

qualit~

la valeur de

Notre princip~ variationnelle permet vecteur adjoint

a 4>

t/> + correspondant

,

~

en une seule piece de information

+

(g

,~).

de Ie

fQ.i.,.<.

comme solution

DMinissons Ie de

l'~quation

adjodnte ~+

(74) et

L

consid~rons

I de

..,+ 'I'

+

= g

nouveau, notre

fonctionnelle

r,t

~ + + +'" Jl~' ~Ij = (g ,~I) + (~I ,f). (~I , L)~ I)

(75)

ou les arguments ¢I,

cI' +

sont maintenant

a

I

reacteurs arbitraire'i independants . Si, soit l'autre

v~rifient

se r~duit

(73) et (74) Ia fonctionnelle

+

(~ ,f) Pour des

(77)

1+

C\>, soit

J[~I

comme des

IP :, soit

l'un et

= +J

a

(76)

Hons

regarder

exactes

=(g + ,$)

J+ variations petites des arguments ~ I , cf> aut our des solu·

cp,

Q> + , lIon a de

nouveau:

- 451 T. Kahan

I

II en resulte June fois de plus, que des erreurs du premier ordre dans les fonctions n '~ntrainent de Itordre une classe

4> '+

d 'essai

de

l'ordre de

10

+

de fonctions

stationnarite de

d'essai

peut se fonder sur la

J . Remarquons

que si la classe (t) -

diff~rentielle

est en effet l'equation

inhomogene

propri~t~

de

(t)

./---

(a)

L'r' (x) = fIx) ,

ci

li~e

a

certain<,conditions aux limites. Par inversion

(b

(c)

(d)

I-

'V

(x)

L

fIx)

=)

f(x')

ou

,

-1

'fix)

'"

'i-I fix) i

fIx)

~(x -

puis que L n 'agit que sur les

= ) f(x')

= AL

x) dx'

coordonn~es

-1 Sf(X')

non primees (x)

~ (x'-x)

dx' =

-1 S"(X'-X) dx' ,

;S

d;f

f(x') G(x', x) dx'

(e)

G(x, x') ~tant la fonction G(x', x )

(f)

dif

de

Green de

(a), (.. "Cb

.... -1 (' L Q (x' - x )

d'ou '" -1 r L G(x', x) = d(x' - x)

(g)

Si par

-1

que les erreurs du second ordre dans( g , .') (donc -1 2 -2 de (10 ) = 10 ); de plus, Ie meilleur choix parmi toute

I

d lOU

~' (mettons

et

exemple

une integrale de

L =~x 1... , on peut

mettre

-S (x'-x) sous la forme d'

Fourier

q- Carir-Kahan loco cit.

D

(Paris et Blanche (London)

)

- 452T. Kahan

~(X'-x) ~ 2/1

alors

-" -1 L

(en effet

n

c

(e

ik(x-x')

)

e

ikx

0

d",

.

ikx ik

A_I ikx L L e

A

e

ikx

ikx

ie k

A

= L

=

eikx)

vient alars pour la fonction de Green ,11._1(" 1 ( .... -1 ik(x-x') _.!. .Jeik(X-X') G (x, x') = L () (x'-x) =27T)L e dk -211 ik dk

J

permise de fonction, d'l!ssai n'inclut pas mation

noptimale" choisie par

la fonction

des

~carts a la

side dans Ie fait yenne

pond~r~e

Ie prindpe variationnel sera relative

aux diff~rents pointtide 1'espace des phaselO (~

solution qu~lle

de

<:p

de

et, par

vl

exact (1) .L'utilite de la

fonctionnelle (75) re-

fournit

variationnelle d'une mo-

une

estimation

quelconque de la solution.

+

En particulier, si on choisit de vecteur

a

w(x) dans le. produit scalaire, puisque w(x) dMinit

de poids

1'importance relative

la solution exacte, l'approxi-

base, ou peut

g

successivement comme un(serie

evaluer les

composante~

consequany Ie vecteur solution

En plus, notre principe comprend

correspondante<j

complet.

la plupart

des principes variation-

( 1) Tout op~rateur lin~aire nous llavons vu, est repr~sentable (moyennant des fonctions sous forme d 'un op~rateur int~gral (note sur p Ii 1 - DiS)

f)

(a)

11+=

j

w(r') H( 1',1") q>{r') dr'

oil w(r') est la fanction de La fonction de poinds peut (b) (d)

~(r)

"w(r)

H(r, 1") =

+

avec (b) H (1',1") = H(r', '"I.) eliminee , en posant

rc>"cI~, ~tre

cp (1')

Vw(r)

H(r, r')~ w(r"ij

sans chan~er les syrretrie du noyau H. Au deme"rant, la libhte dans Ie choix d; Ie fonctian d'essai peut ~tre utilis~e Ie cas echeant pour symetriser les equations du syst~me.

- 453-

T. Kahan nels

utilis~s

pour les

l'operateur est

q;

+

=¢et

titre

probl~mes lin~aires.

de

cas particuliers. Si

""Lz + L et g+=f, alors

auto -adjoint (hermitique)

A

a:

(75)se reduit

(78) fonctionnelle utilisee par pour la solution A'/\

-

'..

ou

(79)

J

La condition .A ( fb' T; H ~) soit

f = 0 , s i l' on remplace

homog~ne I AA,

L

ind~pendament

par Huang(2)

du probleme de Hllnt

Pour Ie cas

AI

Kourganoff (1) et

t

par

l'op~rateur unit~ • alors

I .. 1 est

[~IJ =

2

que

J 2 est

( ~, ~ ) soit maintenu

constant,

justement Ia condition que

a Ia condition

assuJettie •

extr~male

est

extr~male

(la valeur propre

supplementaire que

A apparaissant

comme

multiplicateur de Lagrange ) et elle est equivaIen"k. au principe de Ritz-Rayleigh que

...

( "", L

.r

J 3 [IP] I =

(80)

A.I )

~ (~ ,q)

(dit quotient de Rayleigh)

soit stationnaire. On peut aussi admettre

f> I independantes ~

forme

c

+ +

et c I choisir

Enportant dans

des

,ou c

de iacon

fonctions et

a

c

T

rendre

d'essai de la

sont des amplitudes scalaires

T

stationnaire (3)

(75), il vient

(1) Kourganoff, V. Basic Methods in Transfer Problem, Oxford U. P.(1952) p.141. (2) Huang, S. Phys. Rev. 88(1952) 50 (3)

Davison

B, Neutron

Transport

Theory Oxford U. P. (1952) p.141.

- 454 -

T. Kahan

En portant dans (75), il vient

- +

~galant a z~ro

En

a

+

,1

JLq' ,¢'J

(81)

ce qui

suit c

(82)

=c(g,
d~riv~es

les

comme

(f +,f)-cc+ (
de J

choDe

par

rapport

a

c

et c + conduit

optimal pour les amplitudes

,+

c

et

/(1)

+

=~----..J} /( g

¢')

'+ (¢ , H~')

.... 1·' ¢ '+,Hill'

En portant

+

A

(82) dans (81) conduit

a

une fonctionnell e independan-

te de la normalisation

[r/,

J, '+, fh .. 'I' r

(83) (83)

a ete

utilise

Franc is , et

+

+ ,IP ') I (quotient de Rayleigh generalis~) (¢ , H 4»

J1 -_it',+... ,f)(g

par Schwin ger (2) dans Ie calcul de diffusion, et par

al. ( 3 ' dans des problemes neutroniques; Goertzel (4 ) a

donne une autre d~monstration de (83) . Il d~finit Ie scalaire

et il

+

1/(gt , ¢

~ =

(84)

multipli~

(72) cO: d.

t

<ef> =

-AL rf" 't'

(85)

II

par

)

f, pour

1=

~ (/, ¢»

pour obtenir

+

=!) fIg , ~ ) .

dont 1'~quation adjointe s '~crit (1)

1'"" ",', JJ

Jc

"G}J

=

+

(g ,

+


+

'-I-

A

(¢, H ') = 0,

+ '"

~ = (~' , f) - c(~' , H ~') =

~~)thYS' R~V;

°

78 (1950) 135, H. Levine et J. ,schwi.mger, Phys Rev. 75 (1949)1423 Francis, N, Stewart .J. Bohl, L. et Krieger T. , Seconde conference de 4 Geneve ,Pager A conf. 151 62. ,1947). ( )Goertzel, G. Discussion at the Gathriburg Conference on Neutron Thermalisation, may 1958.

pi

- 455 --

T. Kahan + +

..A

(86) Comme

ces

propre,

Ie quotient

(¢'

f)

sont mises sous la forme avec ~ comme

equations

,

de Rayleigh g(meralise

tement pour fournir voyons dont:

\++

cP = I'lg

L

(83) peut

~tre

utilise imlidia-

+

1/~ =(g ,~) . Nous

une estimation variationnelle de

que la demonstration indiquee par Goertzel

en 1958, n'est

a (70)

autre chose , formule par formule, que notre demonstration 68)

d~

parue

(p. 99) f'.

Si

r

'956 (1)

.....

L:; I - K , de sorte que

A.+..

valeur

.A....

les equations du

syst~me

deviennent

.A-

.A.

H",= f=(J-K)¢>=,
+

g = f et

+

K = K, alors

(83) peut s'ecrire

(87)

(puisqu'une fonctionnelle

differente

se l'est) ; c'est Ia Ie principe

de zero sera stationnaire

utilise par Marshak (2)

pour Ie

de Milne Pour Ie cas par

......

AI

.A.

-

L , f

.~

si son inver-

probl~me

,

d'une equation aux valeurs avec H remplace ;. .... + + g = 0, et H =H, et prenant c ~' comme fon-

ction d'essai, lion obtient

(88)

Comme I 'amplitude doit triviale , il en

, ~tre

dift'~rente

de

zero pour une solution non:

resulte

(1) Kahan _ Rideau _ Roussopoulos, loco cit (1956) p. 24. 25 .

(2) Cairo - Kahan Blackie, loco cit. p. 53 Phys. Rev. 71 (1947) 688 .

- 456-

T. Kahan

" 4»

(4)1' L

(~, ~ ce qui est Ie quotient de

13.

Lagrangien et

)

Rayleigh obtenu prec~demmentcf. (80) ) .

d'Euler.

~quations

L'analyse pr~c~dente 12) Dr~suppose que les ~quations lin~aires "A If. + + + (L'f ::f)p.l02 et (74) (L ~ =g )(p.l03) sont donnees et on en conclut Itallure

(0

sta!Jonnaire de la fonctionnelle (75) (p.lo ~ )

L []=(g

a l'interpr~-

'" +.1L 41') AI La reciproque est aUSSl. vrale " e t con d"t +( rt I.ft+,f) t¥' Ul tation de

comme

J

Ie lagrangien du

a partir

syst~me

dtLquel

obtenir des ~quations de definition (~quation d'Euler) ui appendice Supposons en faibles

,~')+

on peut

(-A..,t
effet que la fonctionnelle (75) soit stationnaire pour de

ind~pendantes ~ ~+

variations

~~

et

autour des

solutions

~+,

et ~ . Par l'equation (77)Lr Ie ~ ).

+ .,A.+ +

(g - L

(90)

~f

Comme

~$

et

stants

s'annulent dans.

re d«.

J

entrame

b ~)

(+.,A.

+ (~~ ,

sont arbitraires, toutes

dans

Par consequent,

la quelle estimation

ecrire

les equations pour

toute

~

)=0

les composantes des facteurs

....

car~cU!re

re~

stationnai~

les equations

~tant donn~

me physique, on peut

f - L

les directions; Ie

.,A. L
(91)

9.

!p,

les

A+

L4>

+

=g

~quations lin~aires

imm~diatement

peuvent

+ syst~~

untforme lagrangienne J de

t!tre dMuites et

fonctionnelle

decrivant Ie

lin1!aire

qui

desir~e

fournir~

une

de la solution.

Signification et importance de Ja fO.r1nction adjointe De nombreux

travaux

ont

et~

consacres

ces

derni~res

ann~es

- 457-

T.Kahan

pour de

interpr~ter

et utiliser la notion

de fonction adjointe

• en theorie

la diffusion neutronique , en particulier en physique des

r~acteurs

nucleaires. ce

Nous allons I dans tion de la fonction

adjointe

pour un

sorte un point de vue plus Pour Ie systeme

H.

H

Le paramMre

nous avons vu que) dans

prend

.

source

de

continuit~

pour

sera done pas

la

une constante dHerminant

conditions

sources

Oll.

repr~sente

Ie

de

op~rateur

fUite).

la multiplication

(neu-

)

suppos~

AC'l. "\CA.l,a (t

criticalit~

inferieure a la valeur

sour ce exterieur.(. ne

nulle partout.

LI~quation inhomog~ne

les

....

K un

ext~rieure,

tronique) du systeme et sera pour l'instant n~cessaire

ll~tat

la forme

op~rateur de pertes (absorption et

un

est

arbitraire. On obtient de la

-'l

J

S une

la significa-

-}.K~+ S

comme une equahon

flux neutronique, et

lin~ai re,

r~actif

0=

production

syst~me

relief

g~n~ral.

.A.

(92)

et s 'interpr~te

suit mettre en

t /"iJ t :: 0) ,l'~quation

~

stationnaire

qui

aux limites notre

singuli~res,

(92) a un

non

terme

de somme (S:f). Comme

nulles peuvent

~quation

(92)

se traiter comme des

se mettra

sous la forme

g~­

n~rale A

(93)

avec genes (94)

J..

L't'=f

.... L

A

:: AK -

.A.

H

"j:

La fonctionnelle J

f :: S, avec des conditions aux

et

duale


W(L~-f)-g+"'J

limites homo-

- 458-

T. Kahan

donne , par

variation

de



+

+

l'equation

(93) et l'equation ad-

jointe L

avec

.I'

L

+

I

defini

fanctian

~

"h+

=g

A

<''t' '

par

L'equation 1a

+

.A-

(95)

f

L

(93) est dite

>= <

parfois

L

+ +

¢ , f>

l'equation du systeme, et ~ ,

du systeme

Remarquons (96)

J

Lorsque nelle. Jest

que

(b

=)a

(93)

donnee

peut se mettre sous

(94)

dx

et

forme

f,; [L.... +¢ + - g+)-f
(95\ sont satisfaites

par llune

des

deux

+

dx g (x)!P(x)

(97)

-1

la valeur de la fonction expressions

+

= - (g

,~)

~=J

b

(98)

la

~ fIX) = -(f ,4>+)/

dx fIx)

a

Cbmme en general

Ie choix

hon adjointe est

dans une mesure

des reacteurs,

a un coup

du

on prend pour

certain de liberte

processus pour

+

g

1a section

une

obtenir

inhomogene

+

g

dans l'equa-

tres large, arbitraire (en theorie

physique

choisir

au systeme , qulon d~sire

terme

efficace) correspondant

donne) Nous disposons grandeur interessante

avec une precision

de beauJ,

re1i~~

du second ordre.

- 459-

T. Kahan

Pour mettre en E!vidence Ia signification de Ia [onction adjointe gE!nE!ral,

dcms Ie cas

~ (x

fix) =

(99)

I'Mfet d June source unitE!

consid~rons

- x ) o

En portant dans (98) il vient (100)

En rapprochant fonction

t~ressante

adjointe est

de l'E!quation (98) , on voit que Ia

la contribution d June source unitE!

a la

quantitE! in-

J. Dans ce sens, la fonction adjointe est une fonction de Green

pour une certaine

quantit~ reli~e

t~ ~tant d~termin~e Lorsque alors

(100)

l'~quation

l'~quation

au

l'inhomog~rleitE!

par Ie choix de

l'~quation

(en l'occurence J) , la quanti-

syst~me

adjointe g+ .

"'t~ :: f est un probleme de valeur initiale, ... + + + ""adjointe (95): L 4> = g doit etre un probleme de (93)

valeur finale. Soit par exemple

.A

L (x, t)
(100-1)

et la fonctionnelle (100- 2)

l'~quation

J

=

(T

J

dt

b ~

dX[q>

+

(L~ - f)

- /(X, t)¢ (x, t)

J

o

La valeur de la fonctionnelle est chacune des deux suivantes

-)0 T

(100-3)

J ::

t

T

(100-4)

J

=-

Soit f la source

+

b

dt

~a

dx

b

dt

unit~

Sa

dx

t

g (x, t) ~ (x, t) :: - ~ dt <

fix,

+

t)~

T (x, t) = - ) o

dt

g~~ >

+ < fl ~ >.

- 460-

T. Kahan

f(x, t)

(100~5)

=~ (t-t ) ~ (x-x o

L'equation (100-4) devient

= - ~+

J

Dans un

0

)

alors

(x , t ) o 0

physique, une source telle que (100-5) peut affec-

syst~me

ter I 'avenir , mais non pas Ie passe. Avec cette source (100-5) doit

~tre

nul pour des temps precedents to (pour t < to' '1> (x, t)

,
(x, t)

=0 ) ,

et l'equation (100-3) devient T

J = -~.

(100-7)

b

dt

5a dx

+

g (x, t)

P (x,

t)

=.0

a

La fonction adjointe est done une contribution integree interessante J pour l'intervalle de temps

t

adjointe est une condition de valeur finale

+ g (x, t)

(100~8)

=f

+

la quantite

< t ~ T. Si 1'inhomogeneit~ o alors

~(t ~ T)

T

et l'equation (100-7) devient b

(100-9)

et

J,

J

=-

5a dx; +T

(x) ; (x, T)

la fonction adjointe est la contribution

a

1'instant final

cp+

une description plus fondamentale du

syst~me

une certai~ quantite telle

que

T.

Dans un sens, la fonction adjointe offrant

a

. La fonction

des sources en general .

syst~me

comme

que la fonction du

nous dit comment Ie systeme (reacteur)

va se comporter pour une source exterieure tion adjointe, d'autre part

peut ~tre concue

particuli~re

, nous dit comment Ie

donnee. La fonc-

r~acteur

repond

a

- 461-

T. Kahan I

On peut presenter ce rl!sultat sous une forme differente.

(A <.:\ ) ,

Tant que Ie systllme est sous critique melle de (92) peut s'l!crire

=

'"

(101)

T

Dans la

.... -1

L

"

-'I.

(H - ~ K)-

f

1

f

reprJs~ntation coordonn~e ,on utilis~ les fonctions propres de

l'op~rateur

x. dont chacune peut

coordonn~

leur correspondante entier

la solution for-

"'-

n (c'est

dl:

a dire

~tre l!tiquet~e

par la va-

x (c. fl.. d rJ.. ) plutdt que pour un indice

~ n)

(1)

4>

(102)

l' x

=~ (x-x') arbitraire 0/:

(x')

x

On a alors pour vecteur

:

(103) ./"

(104)

L

L(x, x') tat ion que

~tant ici

Ie noyau de

l'op~rateurL.

crest

dire que la

".

abstraits

d'op~rateurs

'V

xx'

tels que L,et de vecteur

J

.

repres~n-

abstraits

tels

I

sous la forme d'opl!rateurs intl!graux et de fonctions integrales

peuvent Mre considl!rl!e laquelle des fonctions

comme une representation delta sont

utilis~es

de composantes dans

comme vecteur

de base

(cf. 102) • Si la sommation dans ce cas intl!gration formelle sur des indi-

ces rl!pl!tl!s est sous entendue • les l!quationS(93) et (95) peuvent s'l!crire ~

(105) puisque

L

L

+ xx'

( 1)Alors pour

IV

xx'

~

x'

= L XiX quelconque

f

+A x

4> x'

.

Lx' x

+

= gx

- 462-

T.Kahan

~

Des lors

(101),

s'~crit

repr~s~ntation coordonn~e

dans cette

comme (106) Cette repr6sentation en termes de source (f) du flux '1> x au point

d~ a une source externe au point source exte~ieure pr~s~nte au point x' x

I

5

est jusH\'ment la fonction de =

x' (c. a.. d

~(X-x'))

somm~

x' ;

sur

fois Ja

iH _}. K) xx-I,

Green G(x, x') du syst~me : (e ,'a. d ~x =

fIx') G(x, x') dx') . La solution adjointe de (95)

s'~crit

de la mt!me

fapon sous la forme +

(107)

¢x

= (H

+

-

"-1

I~ K)

xx'

+

~,

r.A. J ....K) -1J+ + xx' g = (H _AK- 1\-1

= L(H -

Xl

'X'x

..... -1+ puis que (L) = (L ) .l'AA_l+ .... -1+)'+ En effet I de I = ( LL ) = (L ) L on

/

Xl

... +-1

En faisant appel A la mt!me

interpr~tation

~videmment donn~ par Ie flux en Xl diZ +

s

multipli~

~:

=

g

par la section efficace

(x', x)

crest

~I

'

g

dire que

toute

quantit~

unit~

espace en x.

du genre section

a une x'

la fonest

x

source unit~ en

x,

a. d)

et somm~ sur

Xl (c.

I

la vitesse

repr~sent~

pr~cMente

pr~cisement

par la section efficace

de phase du systllme lorsqu¢ I on

n convient

de faire observer

que

efficace peut t!tre utilism pour la sour-

ce adjointe et que chacune donnera lieu La dMuction

Ie

Ie flux adjoint en x est

se produit dans I 'entier

introduit une source

a l'heure, pour flux adjoint ~ +

dx' ) .

avec laquelle Ie processus physique,

+

en

"+-1 ,,-1+ (L) = (L ) .

que tout

permut~s,

ction de Green avec les arguments

l

r~"'~

a une

solution adjointe distincte.

demeure valable seulement pour. un

syst~-

- 463-

T. Kahan

me sous critique . Pour discuter Ie cas Ie consid~rer comme un cas s'annuje tandis

que

limite

A.

H

th

'rn ./-

Ii - XK inverse

solution

de

=An

a

la limite, introduisons les vecteurs pro-

A

K,j.,., 't" n

.

++ H ~ In

...... ++ "" '\" n

=11n K

2:: (A n - A) K~n (Q> +n , )

_AK)-l

(92) (dep.110)

(~+, n

m~me

K-1 f)

=[ ~ (tl I\n - ~

c.a.

+ ,

n

(

;n -,\

..A.

~-1

f)

.....

H~=AK~+f, peut

d (92)

la source externe

qui engendrerait des

"-"-1

K

Rtf., 'fn

="n ~n ,

n~utrons a

f, fois son importance par

, avec'\' .A

s'~crire

~n

est 1'int~grale du flux

vitesse que

(1)

Si

Iorsque la source exterieure

s '~cr~ (1)

~ =~

(93)

la

nous allons

Ie d~veloppement de Schmidt usuel pour I'op~rateur resol vant. La

comme

ou

=

{H

(110)

,

R - ~ K a pour r~solution spectrale

L'op~rateur

et son

=0

homog~ne

(108)

(109)

avec f

Ie paramHre .>. tend vers la plus basse valeur pro-

pre. Pour effectuer ce passage pres du systeme

homog~ne

= 'E n

a £P.

n n

- 464-

T. Kahan

a

rapport

la source adjointe .

r,

On peut maintenant envigaset le cas ou

A

propre la plus basse

les termes impliquant ront et

sup~rieurs

la solution pourra s '{!crire ,.h+

l' ~

(94)

....

('fo ,K

1\ o

-1

f)

-~

Ao- A.A'

alors

taus

au premier mode s 'annulle( 1)

comme

.t-

'Yo

Physiquement, Ie r{!acteur syst~me

de P~tat

la solution

meme vitesse que

des modes

vers la valeur

. Si nous faisons maintenant d~cror-

critique

a la

tre la source ext~rieure

a

correspondant

o permanent pour Ie syst~me

tend

se comporte de plus en plu

~omme

sans absorption pour les neutrons dans Ie mode Ie plus bas

s i bien qu 'un neutron donn{! et ses mode {!chappent aux neutrons dans

descendants

pertes pendant une

les

modes

(au sa pr~geniture

rfto dans ce

longue p{!riode. alors que des

superieurs sont pendus'sensiblement

m~me vitesse ind~pendanmment de la valeur de il n I y a pas de perte

un

nette de neutrons

dans

A.

a la

A la criticabilit~,

Ie mode Ie plus bas, et

son amplitude croitrait lineairement avec Ie temps si la source ext~rieliJ

re ne s 'Hait

annul{!e elle aussi.

L'~quation

che

de ~ . Il

(94)

en

devient exacte

dans 1a mesure du flux

resu1te que la forme

sera celIe du mode

Ie plus bas,

de

ind~pendant

pr~s

ou'\

s 'approa de la criticabilite

la distribution de source

qUi ne dNerminera que l'ampllttude relative . Par cons{!quent,

a

tion d'influence (importance neutronique relative processus physique fonction

Ii. 'f' +0

(x)

a un

facteur d'tkhelle

pr~s,

d'influence correspondant

a

dans Ie systeme sous-critique tendent tous

(1) Eneffet

\ -"4 -)

110

n importe quel

qui peut se produire dans Ie r{!acteur sera la

diff~rentes fonctions sique

la fonc-

0

'

f

-J

0 avec ~ -). o

"

m~me

.en d'autres termes, les chuque processus phy-

a se

F 0,

si

confondre (aux

iF o.

- 465-

T. Kahan

amplitudes relatives pres) lorsque Ie

r~acteur

raisonnable pour des raisons physiques car un entretenir Ie mode Ie plus bas daps steme

l'~tat

devient critique. Ceci est r~acteur

personnel;

critique ne peut

a mesure

s'approche de la criticabilite,es neutrons avec leur

que Ie sy-

prog~niture

persistent assez long temps pour oublier la distribution de source. L' portance ft d 'un neutron relativement aux des phases d'un systeme critique peut

diff~rentes positions

dans l'espace

donc se mesurer par

n'importe

quel processus physique et pas seulement en termes de

densit~

de fission

de puissance .

ou; de niveau

I A. Application des

a

'~m­

m~thods

variationnelles

a l' ~quation

de diffusion

deux dimensions

,

Le formalisme qui vient d't!tre pr~sent~ peut etre utilis~

, par exemple.

pour ramener de maniere syst~matique l'~quation de Boltzmann aux d iverses formes simplifi~es g~n~ralement r~acteurs I en

de

l'~nergie

mises en ouvre

en physique des

partant -avec la d~pendance complete vis -

a vis

de la position

et de l'angle de diffusion.

Au lieu de proc~der ainsi, nous allons, avec Salengul, une application

choisie

qu'on peut utiliser pour

de faeon r~duire

a illustrer

un lagrangien

l~ genres donn~

consid~rer

d 'approximations

a une

forme plus sim-

ple, bien que les rMultats finaux pr~sentent un certain int~rt!t Le motif et tivariant pos~s/

est

l'int~r~t

d'~tablir

principal qu'il y a

,

a reduire

intrinseque.

Ie lagrangien mul-

un certain type de compromis entre les buts op-

d'une part d'une. simple description qui diminuera la

mation mise en jeu

et par consElquent

et d'autre part, d'une

pr~cision

siques essentiels du probleme .

Ie temps

suffisante pour

quantit~ d'infor~

consacr~

repr~senter

aux calculs)

les traits phy-

- 466-

T. Kahan

A titre de cette application, qui conduit la plus simple, consid~rons un probleme

au type d 'approximation

a deux

composition est une fonction arbitraire de deux

dimensions dans lequel la coordonn~es.

Cette situa-

tion peut se pr~senter dans Ie probleme d'une r~partition plus uniforme

,

de la puissance dans un

r~acteur

changeant les

du

propri~t~s

la position. L'effet sur la

en redistribuant Ie combustible ou en

coeur

(partie centrale) comme fonction de

r~activitE!

(1) et la distribution de puissance (en

particulier si Ie reflecteur entoure completement Ie coeur) se trouve compliqu~

du

fait

de la non

de diffusion

L'~quation

a

de I '~quation de diffusion.

un groupe peut

s'~crire

- 'V. D~ ¢ + t a ~ =.!K »~f q> ,

(95)

ob

s~parabilit~

~

est

la constante de

multiplication

comme un parametre commode

5:

du systeme (2) qui est introduit

valeur propre. Si Ie systeme est ther-

mique et sensiblement homogene pour les neutrons rapides, consid~r~

a condition

comme Ie flux thermique,

"\) l.. -) !

(96)

p

e

- B

B2 (3)

r~sonance

1:

V "2-

g~om~trique

et l, p

et

~ sont

r~acteur

r~activit~

res-

dit de fission rapide, la probabilit~ de (-\.Ioih .\c

et I ':fge neutronique . Comme ( 95) est auto -adjointe,

(1) Pour un (2) La

de faire Ie remplacement

f

est Ie facteur de forme

pectivement Ie facteur local

peut ~tre

2

f

ob


son

infiniment grand R = C ( C = criticalit~)

est la quantitE! (C = C-1 on

~crit

aussiJC

J~/£ , h =v~/Z:a

+ +A 1 -1 (k=(N R N) (N ,HN). K =op de production H = op de diffusion et de pertes o 0 Ie "Bakling" de la distribution fn (3) Les Anglo-Saxons appellent B2 De

'4' (,-)\ 2 II! I ~. Ijin\'1. 1+ Bn r.,.{'l'), ot il resulte B2 = _ A~ /~

est une me sure de la

Pf

2

convexit~

de la surface

'¥

n

a Ie second membre(flamvage)

n (x, y, z)

- 467-

T.Kahan

lagrangien peut s '~crire J

(97)

[~J :: (dlt. [D (V~)2+! 4>2 _ .!. ~I. <J>2] )

a

k

f

On a om is les termes provenant de l'int~gration par parties qui fournit Ie terme de fuite tions qui

; l'espace fonctionnel ne comporte que les fonc-

I

verifient des conditions aux limites, de sorte que des variations

de flux aux limites ne fournissent aucune contribution. Pour fixer les les

id~es,

des

z

supposons

que Ie

r~acteur

soit uniforme suivant la direction

et que sa section droite soit dans Ie plan (x, y) soit rectangulaire

! ~+-+---1

l

t ).

Pour approcher

i-

d~tail

en

produit de founctions d';pendant


(98)

Comme

l'~quation

::

la distribution de flux, choisissons un

separ~ment

~ (x).

de

de diffusion est

SdX ~.f2

ou

l'int~gration

Le Iagrangien (100)

(x):: 1,

s '~tend sur Ie r~duit

de y

4t(y). homog~ne,

normer les flux moyens dans les directions x (99)

x et

et

)dytpi (y)::

il sera commode de

y de la fa\!on suivante 1,

r~acteur ~ntier

est alors

d~ 2 D ( dy 2) +

- 468-

T. Kahan

et les

d'Euler correspondante,> sont

~quations

-i4 (r d2

(101)

-D

(102)

-D

+

dx

1

d2~

ou

2 -2

2

+ (L

dy

a1

+D

+D a2

2

1

B2lcp 1 1

=l..v~ 4> k1

f1

1

B2)th - 1 ~~ J.. 2 i2 - k f2 l' 2 2

les coefficients sont donnes par D1(x) =

(103)

L a1 (x) =

(104)

(105)

\J

dy

~~

j tP: dy

(y) D(x, y), D 2(y) (y) La (x, y),

~ fl (x) = jdY ~~ (y)v~ (x,

y)

2 5dA~2 D(x,y)

B (x) 1 \

Y 2 lY~2 (y) D(x, y)

(106)

~ a2(y)

,vtf2

=)dX~: (x) D(x,y), = ) dx

dx4>~ (x'lt, y)

(y) =)

2

B (y) 2

4>~(X)!.a (x, y)

5 (::1)2 dX

D(X,y)

= -Ih2- - - -

dX't'1 (x) D(x, y)

,

Com me les deux t\quations sont coupMes par les moyennes des 2 sections efficaces, on peut admettre une valeur de ~ (y), calculer B (x), 2 2 2 . 1 puis admettre une valeur de ~ 1 et calculer B 2 , recalculer 42 et ainsi de

suite. Pourvo\.l que ce procMt\

ensemble de flux self-consistants leure approximation forme

2

t

B1 et Bt

produit/ q> donnent

1

it~ratif

I

converge, il definira un

et ~ 2(y) qui fournira la meil-

~ 2 pour Ie flux r~el . Les facteuI""de

les fuites

appropriees

transverses correspondantes. Le procede est

dans les directions

l'analogue exact de la me-

thode de Hartree-Fock pour Ie calcul du champ self-constant en physique atomique.

- 469-

T.Kahan Ayant et (102) systeme

I

obtenu

des solutions self-consistantes des Equations (101)

on determine une valeur de k par

la constante de multiplication au

,

chacune d'elles. Pour etablir llidentite des deux valeurs

propres calculees de cette tivement par CP1

fa~on

J

multiplions les deux equations respec-

et 4>2 et integrons-les;

il vient

5dx~~ 1) ~ fl

(107)

(108)

,:;

~a2

En faisant usage des dMinitions les deux

sont

egaux

a

I

zero

a la

(103)

valeur de

+

a (106),

D

2

B 2)

2

il vient

k obtenue

kl = k2 et

que

en posant

: J = 0 il slensuit que lE5 valeurs calculees pour tout

ces equations sont coherentes comme il se doit. La derivation variationnelle ass que la distribution de flux rEsultante est la meilleure solution (comme estimee p l'effet des erreurs sur la valeur propre)a l'interieur des limitations imposees par la condition de separabilite.

II. Applications Nous avons vu

du principe

variationnel au probleme de Milne .

que maint probleme de la th~orie du tr.ansport

des neutrons se pose sous la forme

dlune equation integrale

de Fredholm (1)

4> (x)

~5L

(x, Xl)

~ (Xl) dx l +

S(x) ,

du type

- 470-

T. Kahan

ou les limites sont que ou

, et ou Ie noyau

donn~es

L(x, x')est

r~el

symMri-

sym~trisable,

Si la solutien exacte de (1) est Ia plupart du temps inconnue, il est relativement ais(! de trouver des solutions Si l'on pose alors

a trouver

Ie probleme revient ~(x) (3)

Iy'(x) =

.

,

des valeurs raisonnablement correctes pour

dans[~)Jil vient

(2)

Portant

~

~(X)=~a(X)+'f>(x)

(2)

approch~es

AJL(X, x')o/(x') dx' +f(x)

ou f(x) = )

(4)

J

L(x,x')

q,a. (x')

dx ' - ~ a(x) + S(x)

Si d'autre part, G (x -) x) est Ia fonction de o c.a.d. fq(X) =)G(x, xo)
J

G(x ,x) = ~J L(x, x') G(x ,x') dx ' +

(4)

A I'

"'!l L(x, x )

0 0 0

de notre L

les

Green de

~quations

(x-x')

(3) et

(4)

-

L(x, x')

...

dx,

prennent Ia forme

(3')

L

(4')

L G(x , x) = o

Notre

int(!grale

op~rateur

(x) = f(x ) , o 0

th~or~me

montre que les deux

(x )L(x, x )dx o

L{x,x ) o

0

0

et

quantit~s

f(x )G(x , x) dx o 0 0

(1)

- 471T.Kahan

sont

~gales

~ (Ylx )

(5)

la

~

0

m~me

valeur stationnaire (S. V)

) L(x, x ) dx ~ 0

Sf(X o )G(x ,x )dx

SGI(X~X

0, V

S ./'"

0

....

oJ 1

0

GI(X,- xo ) et 'YI(XO) sont des valeurs

savons par ailleurs

)f(x )dx (tyI(X )L(x, x )dx 0

0

00

pour Get

approch~s

'P . Nous

que la fonctionnelle (5) garde sa propri~t~

tionnaire: inMpendamment de la signification particulihe de G(x -J x ) et

0

5G' (x-;x )L WI(X ) dx

00

OU

0

sta-

\f(x), fIx ), o

0

L(x, x ) •

o

0

Un cas

particulier important est Illstimation de

Si L(x, x ) = 0 et o

lim

x -')

00

lim

Sex) = 0 ,

Pon aura lim

(5 bis )

x _)

ou

J'"

est

un facteur constant

lier par des

(6)

lim x~oo

G(x , xl = ·o.x) o j7\ 0

00

physiques.

consid~rations

Y'Y(Xo) L(x,

~ d~terminer

x0 ) dx 0

= r(f(x ) ~

J

f

=r

0

a

variationnel mis

(7)

S

f(X )

o

0

0

fIx orr ""(x0 ) dx 0

~tre ~valu~

par exemple par notre

sous la forme particulit!re de Schwinger

= V. S.

) dx

!p (x

Dans ces conditions

(x )dx + 11- f(x )IJJ(x ) dx 00 I oT 0 0

Le dernier terme de (6) peut principe

dans chaque cas particu-

{

(rx (J.l )f(x )dx

~

I

I.f (x

o

J )L hll(X )dx 0

2}

Too

- 472-

T.Kahan

n

ne nous a pas

dans (5)

(par

, . l'equatlOn (7),

on

res

a

~t~

possible de

exc~B

ou par

d~terminer

la direction de 1'erreur

dMaut). Toutefois dans Ie cas de

a montr~ (1,2,3,4) que,sous des conditions particuli~­

"""L

f(x), la fonctionnelle (7) est un maximum. o Nous notons aussi que notre fonctionnelle (5) nous fournit une

impos~es

estimation

de

et

Ij/(x) pour chaque point dans l'intervalle d'integration, en

contraste aux mNhodes

expos~es

dans

(2), (3) et (4) .

Le probleme de Milne. Nous allons appliquer avec J. Devonght, la

d~termination

du flux

semi-infini diffusant

de

de neutrons mani~re

la fonctionnelle (5)

(ou de photons) dans un milieu

isotrope et sans capture , qui

mite

par la vide et qui entretient un courant

l.'infini

: c 'est Ie

probl~me

est li-

constant venant de

de Milne

La situation physique de ce probleme produite par la surface

a

de Milne est assez bien re-

ext~rieure d'un r~acteur nucleaire sans

ecran

Il se pose aussi en astrophysique sous la forme du transfert du rayonnement de l'interieur du soleil vers sa surface. Ce phenome-ne est essentiellement

r~gi par

qui gouvernent Ie transport des neutrons dans les En

r~alit~,

radiatif au entre prises

sein

c 'est des

a

les m~mes equations

r~acteurs

propos des problemes souleves

~toiles

que la plupart

sur ce probleme majeur

des

de Milne.

(1)

J. Devooght,

(2)

Phys. Rev.lII (1958) 665

J. Le Caine, Phys Rev. 72 (1947) 564

(3)

R. E. Marschak,

(4)

B.Davison,

Phys. Rev. (71) (1947) 694

Phys. Rev. (71) (1947) 694

par Ie transfert

recherches ont

et~

- 473-

T.Kahan 1

lci no us avons ~ '" l'• SIx) = 0, L(x,x') = "2 ~(I x-x, et 1'approximation de diffusion nous donne I/f> a (x) = x • c'est ~ dire 1

fIx) ="2 E 3(l x l)

• ou co

En{jXp= Comme lim

L(x. x')

e -t

~

t

Ix! n

dt

= 0 • on pourrait appUquer aussi bien (7) avec

(cf. (3)f..;)co

'" (co) =

(8)

lim

puisque

lim

S

L(x, x')

x_) co

't' (x')

dx'

fIx) = 0

x -., co

et obtenir (9)

+yv.S I

Des considerations physiques (ef (3) et B. Davison, Neutron Transport Theory (Oxford U. Pr, New York, 1957, p. 210) mont rent que 1'-= 3. Notons que la solution (9) nlest autre chose qu'une estimation variationnelle bien connue de la longueur dite d'extrapolation en th60rie du transport. Si d1autre part, nous voulons d6terminer drons (2')

'f' '(x) =

Zo et en

0/ (co)

rempla~ant G(x. xo) par

m (x ) =

par (5). nous pren-

[ef. equ. (2)J

• nous obtenons

- 474-

T. Kahan

dloll il

par

r~sulte

int~gration

3+z

o 4/(00) = 4+6z

o

Le r~sultat

travers ma

d~pend de la normalisation choisie pour prendre

de moins de

11 reste a

z

1 =-

"1'( 00) qui differe

[~-

mais si lIon prend une valeur compatible avec un sch~-

il faut

it~ratif,

11/ (x)

= 0/(00) ce qui donne immMiatement

0, 7071

0,5

d~terminer

o

~

'/0

de la valeur exacte 0,7104 .

(x) pour tout point du milieu

~

llaide

de

la fonctionnelle (5) . Comme la fonction de Green pour llapproximation de diffusion dans un demi-espace sans capture 2 IT nOus choisirons

I x + xo I -

la fonction de Green

est

217 Ix - x

0

approch~e

I, suivante

(11)

qui jouit

de la

propri~t~

lim X~

c'est-~~ire

}A-= 4".

00

suivante :

GI(X, xo) = 4f1' Ixo + ~ (oo)J ' cf (5 bis ) p.124) . Llassociation de l'~quation

- 475-

T. Kahan

(5) avec

(3)

conduit

ti

Des integrations

directes mais longues conduisent au resultat

final (13)

qui admet la forme asymptotique correcte

La valeur sur la frontiere

(interface libre

4-'(0) = 7/ 12 = 0,584

differe

de

1 ()

/a

de la valeur exacte

L'erreur n 'excede jamais presumer

1,5

0,577

.

#10 en d'autres points. On peut

que l'erreur plus grande pres de l'interface libre est

ti 1a forme assez

due

incorrecte de la fonction de Green ti 1a frontiere x=O

Conclusions Pour terminer, cipe variationne1

je voudraise

souligner

que Ie nouveau prin-

(5) dont la precision parait suffisante pour 1a plu part

des objectif!)ti atteindre,

joilit

de deux proprietes

int~ressantes

majeures .

1.

Nous n 'avons fait aucune

hypothese

sur

Ie

I de~loppement

de la

- 476-

T. Kahan fonction d'essai

Ip'(x) en

tenons avec Devooght sai

grossi~rement

de fonctions, et

s~rie

une

satisfaisante avec la fonction d'es-

pr~cision

'fI '(x)

approch~e

nous ob-

n~anmoins,

=

Z0

•

On pourrait obtenir une pr~cision bien plus grande si I 'on avait

fait appel

a un

E (x) avec n

des coefficients

2

n 'avons pas

Nous

En

a

If '(x)

en une

calculer un

la

de

bien s'appliquer

a des

re, contrairement aux

expos~e

pourrait aussi

I

problemes en theorie du transport

tion exacte est inconnue, tel

dont la solu"'-

que Ie probleme de la basse

m~thode.s

classiques

sommes

E (x), ce qui est long et n

qui vient d'l!tre

m~thode

et nous

extr~mum

. g~n~ral,

de fonctions

s~rie

inconnus •

de calculer les coefficients

dispens~s fastidi~ux

de

d~veloppement

(cf. (2)

0)\

a (4) et

de la

sph~-

Weinberg et

Wigner , The Physical Theory of Neutron Chain Reactor) . The University of Chicago 1958) valables pour des totiques. Nous avons et non pas Ie cas nies.

par

0/(00) qui,

ob tous les points Les fonctions de

t!tre prises

une

~tabli

d~termination

densit~

directe de

'¥ (x)

asymr-

par

(5)

aU demeurant , serait impossible dans

du milieu

r~actionnel

sont

a des

distances fi-

Green approch~es G'(x; xo) pourraint toujours

sous forme de fonction de Green appartenant

mation de diffusion qui sont tout

a fait

faciles

a

a I 'approxi-

construire pour un

grand nombre de problemes. Signalons, pour terminer, un travail

int~ressant

Brooks, Journal for Mathematical Physics, sur la

m~thodes et ses applications

a la

U{eorie

de Koshin et

g~n~ralisation

de nos

du transport des neutrons

- 477-

T.Kahan

Signalons encore: M. D. Kostin et M. Brooks" generalization of the variational method of Kahan, Rideau and Roussopoulos . II . A variactional Principle for linear operators and its application to Neutron - Transport theory"

Journal of Mathematical Physics. Vol. 8 . n. 1. January 1967.

-

478-

T. Kahan

APPENDICE 1 1 Rappel des notions fonda.mentales en calcul des variations.

L' objet du cal cuI des variations est de trouver des fonctions y(x)

qui

rendent stationnaires ou

pe

J&(XU J (y)

des fonctionnelles du ti-

extr~males

. Vne fonctionnelle

JkY] est

llne variable qui rev~t

une valeur numerique particuliere pour chaque fonction portee

y(x) qui

y est

Example simple b

(A. 1)

J [y]

Chaque fonction fonctionnelle appelE\e

J

~ y(x) dx .

=

y(x) fournit

[Y] . Pour

une seule valeur

num~rique

de la

cette raison , une fonctionnelle est parfois

IIfonction d 'une fonction" . Soit

donn~e

la fonctionnelle

(A.2)

Ie probleme fonda mental

du calcul des variations est de trouver donc une

fonction y(x) telle que des accroissements du premier ordre dans cette fonction

(y _") Y + by) induisent

'y(x)

seulement des accroisse-

ments du second ordre dans la fonctionnelle J (J -)

.u t;?

J ); en

d'autres termes lorsque y(x) -)

(A.3)

J

(A.4)

En port ant (A.3) une

~quation

&] -) dans

y(x) +

J + 0

f

y(x) , /!nax 'Y(X)] 2

(A.2) et en imposant

determinant y(x). Cette

~quation

(A.4), on obtient

porte Ie nom d '~quation

d 'Euler du probleme,. Dans la section suivante, nousetablirons les ~quations

d'Euler pour plusieurs problemes.

- 479T.Kahan

2.

Probl~me

simple . Le

J

cas

ou la fonctionnelle est

de la forme

b

J =

(A.5)

est un des

les

probl~mes

sant varier y(x)

ry :

plus simples du calcul des variations. Fai-

comme dans b

fJ Sa

_, 10

(A.6)

(car

=

~x =

0

(y' = dy/ dx)

F(x, y, y') dx

A.3, on obtient, au premier ordre en

ClF ( jF \ +-. Oy') dx, 'dY Y ()y'

(- 0

"...,

I

I

A

t

ou (A.7)

En integrant par parties, il vient

(1)

~J

b = (

'OF d (""""" - Oy dx

'OF

1o

Si la fonction y(x)

est

assujettie

~

(A.8)

~~ dx flY

r

tJ Y

I

verifier

1

dF r ~ + -OY' "y

les conditions

(aux limites (A.9)

alors

~y (a)

et

~y(\,)

(1)

b (

)a

dF ("

7"\"':: fI y'

d y' dx

d¥ent etre nuls et les termes aux limites de

Sa .()-.y'

=b

., F d(fy) d F~b dx = dx '0 y'

r

- -

d

~F

(

L..::....-~dy dx

dx ~ y'

- 480-

T.Kahan

(A.8) sont nuls. Si non, y(x) doit satisfaire aux conditions aux limites naturelles

~F

(A.IO)

-

,(h' Comme ~y(x)

~

(a)

~F

=-

aY'

b

( g) = 0

est arbitraire (sauf ~ satisfaire, Ie cas ~cheant

certaines conditions aux limites ), y(x)

do it

v~rifier

par

(A.8)

l'~quation

(A. 11)

pouifque la variation du premier ordre en (A. II) , ~quation

J

soit

nulle. L'~quation

difrJrentielle ordinaire, est dite "~quation d'Euler du

probleme. Pour

d~terminer

si cette valeur stationnaire ou

minimum, un maximum ou un ncol n, on peut ~tudier

vement. 3. des

la valeur

extr~male

n~gative,

ou de signe

ind~

est un minimum, maximum ou un col respecti-

L'effet des termes aux limites. 11 est souvent ~quations

est un

,2 la variation second b J=E; (b J )

Si cette variation seconde est dMinie positive, dMinie t~rmin~,alors

extr~male

int~ressant

avec des conditions aux limites autres que les

(A. 9) ou (A. 10) . La fonctionneUe peut mes de limite, conduisant ainsi

~

~tre modifi~e

une

~quation

d'obtenir

~quations

par l'addition de ter-

d'Euler avec des con-

ditions aux limites diff~rentes. Supposons par exemple que la fonctionneUe (A. 5) . Soit

modifi~e

ainsi:

b

J =

(A.12)

Par Ie

1 a

F(x,y, y') dx - gI[ y( aU + g2[ y (

procM~ pr~c~dent,

on aboutit de nouveau

~

b)] 1t~quation

d'Euler (A. /I ) , mais les conditions aux limites naturelles sont ainsi :

modifi~es

- 481 -

T. Kahan

(A.13)

"0 g1 dF (a) + (a) d y' ()Y

(A.14)

() F (b) + - - (b) gY ~Y

0,

og2

4.

Probl~mes

=

o.

avec plusieurs variables dependantes .

Si la fonctionnelle depend

de plusieurs fonctions

y (x) , y 2' •.. y n

sous la forme b

=.1.

(A. 15)

on obtient

un

syst~me

dx F(x, Y1 ,y{

d'equations

,y 2'

d'Euler

Y2' '" , Yn'

yIn) ,

simultane (i = 1, 2,1/ .. , n)

(A.16)

avec les termes de limite. ( i = 1, 2, 3, ... , n)

(A.17)

5.

Probl~mes avec plusieurs variables ind~pendantes.

Si la fonctionnelle la forme (A.18)

Jest definie

sur

plusieurs dimensions sous

b dX 2 ..• { n a

dx

n

n

ou (A.19)

, (x = 1,2, ... , n) ,

- 482-

T. Kahan

alors

l'~quation

d'Euler

est-

l'~quation

aux

d~riv~es

partielles

(A.20)

6.

Fonctionnelle. avec des d~riv~s d'ardre s_u~~!!eur Soit

l

(A.21)

'

(1)

(2)

(n)

J = ~ dx

F (x, y, y

)ar parties

seront necessaires pour obtenir

,y

,"" y

)

ou (A.22)

alors

n

quat ion

int~grations

d'Euler

~F

ay

--

7

en

g~n~ral,

Prob1~mes

dF

n d - - + + (-1) dx (1) '"

ay

d n ~f n ay(n) dx

=0

dtordre 2n.

une equation

avec contraintes;

multiplicateurs de Lagrange. b

Supposons que dre stationnaire

It~quatian

assujettie

Cette contrainte nous arbitraire9

en

dx F(x, y, yt) sait

(A.5) ; J =

a la contrainte dx

(A.24)

tement

l'~-

qui sera

(A.23)

c 'est la,

I

E

ren-

(x, y, y') " 0

interdit

de prendre des

y(x) , de So.ft

1 t~quation dtEuler pour ce probleme.

a

a

variations compl~

que (A.l1)

ntest pas

- 483-

T.Kahan

Le proc~d~ du multiplicateur de Lagrange revient ~ multiplier 1'~quation terme

b

~

dx

[F(X,

En faisant varier

y

,1 'on a

r.ab

dx

J =

(A.25)

(A.26)

)

de

fa~on

~

(F + \

~

pour

est

appel~

r~soudre pour

d~

G) -

~

x.

A -.

V(x,

proc~d~

Le

pour

y(x,

paramN~e

la fonctionnelle

(A.28)

est similaire,',

L'~quation

d'Euler

.

il -peut

d:' (F+~G)

Nre

y, y')

=

(A.5)

= O.

t.) et faire appel ~ (A. >. choisi de cette fa~on

"multiplicateur de Lagrange". Un

te souvent appliquee

Le

AE(x, y, y')]

l'~quation

On peut r~soudre 1 '~quation (A.27) 24)

Ie

~ y [L (F+ A E) _ ~ ~(F+ -1E) 0y dx;3 y'

satisfaire

(jdy

(A.27)

y, y') +

~ est un paramMre arbitraire ,

Puisque

et ~ ajouter

J; 1'on obtient alors

la fonctionnelle

r~sultant ~

choisi

~

(A. 24) par une constante arbitraire

autre type de contrainest

0

sauf que Ie multiplicateur est une fonction de

devient

(A.29)

Les et

~quations

A (x)

.

(A.28) et (A.29) sont

r~solues simultan~ment

pour

y(x)

CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C.I.M.E.)

C. CATTANEO

n

SULLA CONDUZIONE DEL CALORE"

In questa conferenza sono state svolte considerazioni contenute nei seguenti lavori : " Sulla conduzione del calore" , Attt Semin. Mat. Fis. della Universita di Modena,

1'

3 (1948)

"Sur la conduction de la chaleur" , C. R. de l'Acad. des SC. de Paris, 247 , p. 431 (1958) .

Corso tenuto a Varenna (Como) dal 19 al 27 settembre 1966

CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C.I.M.E.)

Cataldo Agostinelli

"FORMULE DI GREEN

PER LA DIFFUSIONE DEL CAMPO MAGNE-

TICO IN UN FLUIDO ELETTRICAMENTE CONDUTTORE"

Corso tenuto a Varenna dal 17-al 19 settembre 1966

FORMULE DI GREEN PER LA DIFFUSIONE DEL CAMPO MAGNETICO IN UN FLUIDO ELETTRICAMENTE CONDUTTORE . di Cataldo Agostinelli (Universita - Torino) 1.

E' noto come in un mezzo

cibilita. finita, e s:>ggetto a

elettricamente conduttore, di condu-

un campo magnetico , questa , col tempo, si

diffonde nel mezzo e il coefficiente di diffusivita magnetica logo

'7

, ana-

a I coefficiente di condicibilita termica e alla viscosita cinematica

nei fluidi viscosi

in movimento, e\ inversamente proporzionale

alla con-

ducibilita elettrica. Ora in

questa lezione

ne del campo magnetico in un di un

mi

propongo di stabilite per

la tliffusio-

fluido elettricamente conduttore, dotato

assegnato movimento, delle formule integrali che gnneralizzano

queUe ben note di Green. Queste formule, che sono

cosi feconde nei

Fisica matematica, sono gill state come

diversi rami della

si sa impiegate da tempo nella

idrodinamica pura. In effetti H. A. Lorents aveva ottenuto dei risultati interessanti nel caso dei moti permanenti e poi dlcato

un'ampia memoria nella !Viluppo di

G. W. Oseen aveva de-

una teoria

generale del mo-

vimento dei fluidi viscosi mediante l'applicazione delle formule generalizzate di Green (1). Recentemente , in una memoria in corso di stampa nella Rivista della Associazione Italiana di

Meccanica Teorica e Applicata (A. I . M. E . T. A)

queste for mule sono state

da me estese

al

caso del movimento di

un; fluido viscoso incomprensibile, elettricamente conduttore, in cui si ge-

(l)G. W. Oseen, Sur les formules de Green generalis~es qui se pres~ntent dans l',hydrodynamique et sur quelques-unes de leurs applications. "Acta Mathematica n, B.34, 1911, pp. 205-284, B. 35,1912, pp. 97-192.

- 490-

C.. Agostineili

nera un campo magnetico (2), mentre la Prof. ssa Maria Teresa Vacca ne ha fatto l'applicazione

al caso piano (3) .

Qui, come ho gia detto, mi limitero a considerare soltanto Ie formule generalizzate di Green per la diffusione del solo campo magnetico.

2.

Nel caso in

cui

siano

trascurabili la corrente di spostamento

e la corrente di convezione, Ie

equazioni di Maxwell e l'equazione che

espriD;le la legge generalizzata di Ohm, risultano

...

..

rot H

I , rot

r

E =-

...

r:JH-

""t

• \ div H = 0).

.. -+ I = C"'{E + ". v 1\ H)

l'

Prendendo il rotore di

-

ambo

i

membri della prima di queste equazio-

ni, eliminando quindi la densitii. di corrente

E, servendosi

delle altre

si ha

H = ...div

..

e

I

il

-

campo elettrico

due, e osservando che

.... ,~......

rot rot

-+

H-

r

~

II

H=

2

...

-.a 2 H ... 'JH

-1I2 H = orot (vAH) - r·t'.l')t Dividendo per

t!'V'

1

rG"

con 11 coefficiente di diffusivi-

(2)C. Agostinelli, Sulle formule integrali

di Green in Magnetoidrodinamica.

tii. magnetic a, si

e ponendo " - -

ottiene

I

-

I

(3)M. T. Varra, Sui moti magnetoidrodinamici piani di un fluido viscoso incomp,rensibile elettricamente conduttore. nRendiconti del Seminario Matematico dell'Universita e del Politecnico di Torino", Vol. 25 0 , 1965-66.

- 491-

C. Agostinelli

....

~

= rot (H 1\ v)

(1)

che l! llequazione della diffusione del

....

div H

(2)

Alle

equazioni (1) e

campo magnetico, con

0.

(2), dove il secondo membra della (1) 10

considereremo come noto, associeremo Ie equazioni

lJii' , J. 2 ~I H + "J t -

(3)

+

div HI

(4)

con la condizione

U'

=0

= 0,

,

(5)

A

u

essendo il vettore

grad

II = 0 ' 2

......

HI e 10 scalare

Le equazioni

(3), (4) ,(5)

VI

delle incognite da determinare

ammettono Ie solutioni:

~ ~ "?'#' .. til = rot rot ('f'i) = grad ;;;:- - U2 1(. j ..~."

_

dove

,

...Ha :: rot rot ('f' j) ::: grad ')y -..1 '(, j.. , 2 ...H; ::: rot rot (1"k).. :: grad IJZ ')", .... - ~2'f'. k,

(6)

...

aggiunte (4)

. 'J

?",

V' :: - ( - + 114.1..,.) 1 nx? t I 2 VI ::

L/~::t:. + tltJ 'f"')

2 'Jyy)t

'2

'J )'1'

U' :: - ( - + 17J:y) 3 /,)z?t I 2 '

"9

i, j, k

sono

i

versori

di una terna di

assi

cartesiani orto-

gonali O( x y z) di riferimento, con la condizione che la funzione

r

(4) Si osservi che se si assumano come equazioni aggiunte, associate alle (1)e(2)le

... 'JH* , A2 H* + /)t :: 0, ~...

4.

div H :: 0 • ~

I

basta porre

H· = HI - gra
per avere le equazioni

(3) e (4) • con la condizione (5) •

--- 492 ---

C. Agostinelli

verifichi l'equazione (7)

Considereremo di questa e uazione una soluzione che dipende soltanto 222 dalla distanza r = (x - x ) + (y-y) + (z-z) di un punta P(x y z) o 0 0 da un punto p (x y z ), e dal tempo t 000

Se poniamo A 2'f' =

0

f ' la

(7) diventa

'. '1+

'1 J 2f +0t

(7')

Essendo ~

funzione

=0,

soltanto di

!. '

r e di

si pub scirvere

(8)

Ora

se

I

F(r, t) e

cilmente che

e

Dalla teoria del

soluzione

anche

di

questa

soluzione

calore si

equazione

si

la

e

ha che r

soluzione della

2

e pertanto avremo ~

da cui si deduce

r= -1~J2 -

2 - r /)r 2

IJ

(rr)

f

1~

= =-r I) -r

~r (rf)

e quindi

riconosce fa-

=r

F

(r F)

(8)

la funzione

- 493-

C. Agostinelli

(9)

jr 1'rc:t

t

(r, t) , , - 1- o

che t:

la

soluzione

arbitraria

,

rl

errata

della

(7), dove

che assumeremo come raggio

nel punta

In base HI

Ix' tano:

(10)

HI

HI

ly'

al

1z

valore (9)

del vettore

di

1

per t > t

-12 W "- 1-; +~:._1_[1_3 Ix !\" 1\ a 2 'I/y liz r

r

2

(t-E)

lz

=--- =

0

{)z

per semplicita

(11 )

E(r, t)

(12)

3(x-x )(z-z ) o 0

('(

-E)

r

si

1

=

27(t o -t)

E

(x-x )(z-z ) o 0 2

E

r

e posto r2

la funzione scalare UI si ha 1 UI

E -J---

r

rlt

dove

2

o +[l--_ 2

"

'J2y1 HI

0

prima delle (') , risul-

'12'1' 3(x-x )(y-y ) (x-x )(y-y ) ~" . 0 0 (f-E) _ 0 0 1y I") x')y 4 2

HI

< t ,ed

componenti

(x-x )

r

e per

Ie

2

--0-1 (x-x )

0 ~t

con

o

'f

definito dall

:J2

So

una sferetta

della funzione HI,

t: una costante

rl

t: definita per

P (x y z ) . Essa o 0 0 0 t: per definizione identicamente nUlla

centro

.11'

411 (to-t)

e

il valore

rl(x-x ) 0

E{rl, t) .

2t3(t -t) o

Le componenti degli

altri due

le funzioni scalari corrispondenti

vettori

U2, U3 I

(12) con opportuno scambio delle coordinate.

,

H~,

3, e

H

i

valori

si ottengono dalle

del-

(10) e

- 494-

C. Agostinelli

S ,ci~e per r " r' ,si ha 't = 0, -t-Ie quantita o H' ,H' ,U' assumono i valori ly lz 1 2 (x-x )2 (x-x )) E , H' = _1_[1 _ 3 -_0-1E (r' t) + [1-~ (r, t) Ix ,2 ,2' ,2 211 (t -t) r r r 1 0 Sulla sfera

3(x~x

H':: .. lz

(x-x )(y-y )

)(y-y )

0 r,4

HI:ly

( 13)

0

E(r',t)-

r'

3(x-x )(z-z ) o 0 E(r', t) 4 r' x - x U' = 1

(14)

3.

O

2 r'

2

0

(x-x J(z-z 0)

-

r

,2

E(r' , t) 21(t o -t)

I

E(r ,t) t - t

0

2

0

Cib premesso consideriamo, nel campo in cui si muove il fluido,

un dominic

D (t) limitato da

col tempo. Sia

un superficie

S(t), in generale variabile

P

(x, Y , z ) un punta interno a questa dominio, e o 0 0 0 una sferetta con centro in P e raggio r' sufflcientemente

sia S o 0 piccolo in modo che quest a sfera nelPintervallo di tempo 0 sia sempre tutta contenuta nel Formiamo ora

~

t ~ t

o

dominic J) •

mediante Ie

equazioni

(1)

e (3), la seguente

combinazione

..... X HI

ottenuta te per

..

1( ~ 2 H

(15)

moltiplicando ~

H,

... ;T (H... x .....HI)+ grad U'X H =rot (Ht\v)JC .......... HI,

- A2HI X H) la

~

(1) scalarmente per

e sottraendo quindi membro

Integrando ambo i limitato dalla

superficie

~

la

(3) scalarmen-

a membro.

membri della (15) S(t) e dalla

it' ,

sfera

rispetto al dominio DI(t) S , abbiamo o

- 495-

C. Agostinelli

[

(16)

D'(t)

-to

,(4

!

1';) .....

........

HXH'-~H'XH)dt,,-(HxH')df+ gradU'XHdl'= 2 2 D'(t)~t '(t)

.. . .

J

=

"9

rot (HI\ v) " H' dr.

D'(t)

Applichiamo

ora Ie note formule di trasformazione degli

integrali di

volume in integrali di superficie , osservando che grad e ricordando ta

inlOltre

che

in una regione della

una superficie

...

U' X H = div (U' H) , F(x, y, z, t) e una funzione derivabile, defini·

se

spaz40 contenente il dominio D(t) limitato da

8(t) , variabile col tempo, sussiste la relazione (5)

~F

J

J

~h

dt'= F d'Y+ F V d8 D(t) ") t dt D(t) 8(t) n '

dove

V

n

con cui

e

la componente, secondo la normale interna, della velocita

si spostano

i punti

Dalla (16) si ha aHora

-L

8(t) + 8 o

,

di

8(t) .

.....1

( dH X H dn

-

...

~

d! .....

dR'"

- X H) d 8 - H X H' d 1: dn dt D'(t) ..

L(

.. +

..

-/, (H x.H'.V + U!H xt)d 8 = rot(HAv)" H' dt'. 8(t)+8 n D'(t) o

..

Integrando ancora che per

t .... t

o

e

rispetto

al tempo

da

0 a t , e osservando o

H' = 0 , si ottiene

(5)

efr. E. Goursat, Course d'Analyse, t. I. p. 666 (Paris,Gauthier- Villars, 1933) .

- 496-

C, Agostinelli

...

J.H JS [~(-d. ~

dt

n

..

... X

HI -

d H'

H)+ VnH.JC H' + V' H X n d S-

~dX H)+V

(H X H')

D'(O) In questa scalare

l'

dt= t=O

J to o

V', succesivamente i vettori

2

3

.

......

~

HX HI+V!H.xtJdS+ n

1

.....-1> d t r o t (H,'\V) X H' D'(t)

. . . .

equazione occorrera sostituire al

V', V', definiti dalle

ri V'

1

n

1 ......

o +

..-+]

~...

d H''''

- ~ X

(17)

vettore

..

H' e alla

H' H' e Ie funzinni scalal' 2' 3' (6) e passare al limite per r ' ... 0 •

...H'

H'

= Hie U' = V', dove 1 1 H' e della funzione V' sono dati dalle 1 1

Riferiamoci al Ie componenti di

caso

funzione

di

valori

del-

relazioni

(10) e (12) ,

Si riconosce

intanto

f1

(18)

che

dH

S ~x o Si osservi per questo che

.....

~ dn

...,

H' d S = 0 , 1

.... l(

d. H d H d Hz H' = X H' + _.2. HI + HI 1 "Qil Ix dn ly dn Iz'

So

e che sulla sfera

risulta

'JH

d H ~H x - x y-y -")H x-x __ x H' j __x _ _0 + _ _x ~ +_x_ _ _o)HI dn Ix \~x r' /'Jy r' ~z r' f A ' ece, dove H'

,H' H' harolD i valori espressi dalle (13). Ix
dia sugli integrali, si vede Analogamente,

sub ito

e

ehe quelliintegrale

sfera e S 0 Y-Y _ _0 +V __ 0+ V Z-z { r' 1 r' r'

poich€: sulla

x-x

V = V n x

°

nullo.

-

497-

C. Agostinelli

si trova che (19)

Per

1 ....

V. H)(H' .,J.,s = 0 • S n 1 o quanto riguarda il limite dell'integrale

J t

(20)

1=

0

d'Is

d 'ft'

(1

d

~ - u'l ;) "it,d's,

o 0 intanto che si ha

osserviamo

d~'

__ 1

dn

XH =H

... ...

d H' d H' d H' Ix + H --..!l. + H ~ x dn y dn ~ dn

U'l' H"n = U'1 IH ~ \ X cos n x + Hy cos

e sulla sfera d H'

~' ~

=..2.-[; r,3

x

x-x JH Ix I __ o+

ot'

'V

y

y-y __ 0 r'

')H'

+~

'J z

rt-

2 (x-x o ) -11 E (r' t)+_I_ r' ~ ' 2 1 r ,L

2

XO }

~/.t"

dn

z)

~[1-

_1]E(r',tL to -t 412 . (X-X o

r' d H'

n

z-z __ 0 = r'

_

--....!L =

H~ cos

S risulta o

~=~

d n

.y +

9(x-x )(y:'y ) 0

r'S"

0

E(r', t) +

3(x-x )(y-y ) 0 30 r'

21

E(r', t)

)2]

2

E(r', t) 2 ' (t -t) o

(x-x )(y-y ) ~ r ,t

o 0 - - - +--::---to -t

4 ,2 r'

dHi 9(x;.x )(z-z ) 3(x-x )(z-z ) (x-x )(z-z ) __z_= 0 0 E(r't)t 0 0 E(r', t) + 0 0 E(r', t) 2 ' dx 1"5 . , 3 t -t 4 " 2r , (t -t) 21J. r' 0 " o , ... -+ x-x y-y U1 nXH = U\ (Hx_,_o + H _,_0+ H z-z,o). r y r z r Si trova che

(to _t)2

- 498-

Jto

lim

r!..O

dt

C, Agostinelli

I

II

d H'

J"'(H ----.!.Y S 11 Y dx

y-y z-z d H' + H ~)- U'(H _0 +H _0),1d S = 0

J.

1 Y rl

dn

~

• rl

o

e pertanto il limite dell'integrale

=I'1m

, I I 1m

rl~o

rl..,.o

jt

dt

0

j,

f

- lim -

r

I

"t 0

dn

1 rl

1t

x

2

2

E..I1-

_ 1] t -t o

4,

2

(X-X o )]

r'

2

2

E(rl,t) _ (t -t)2

, E(r , t)

,3

2r (t -t)

0

= - r I ...,

I

r

H (x ,Y ,z ,t) 3x 0 0 0

7

J 0

3E (

I

L t 2' (t -t)

)

21C + -3

o

y

(x

z

x 0' 0' 0'

It 01 ~

t )lim 0 rl~o 0

+ 27t rl E(r', t) 3

1' -H (x,, y ,z ,t ) 1m x 0 0 0 0 rl... 0

o

,3 r

I

E(r', t -t o

t)} dt

27C r' 3E(r', t) ,;. 2 (t -t) o

t -t

J

E(r l , t) (t _t)2

o

rl 3E (r ' , t) +

37

(t _t)2

o

=

dt

o

}\127f 37 0

r ,3 E(r ' , t) (t _t)2 0

Ma risulta

Jto r' ..... o

~

o

H (x ,Y ,z ,t)-H (x ,y ,z ,t x 0 0 0 x 0 0 0 o~

[

to -t

lim

_ S-

)1137

0

+ 21(, r,E(r l, t)Jdt-H 3

Jd

o

12TC'

0

(x-x o)

t

lim

1 [ (x-x o) 3 2r r'

+ 2'

I

H (x ,y ,z ,t) dt - 2 - -1 E r ,t) x 0 0 0 S r' rl o

E(rl,tL

t

1

~~X-Xo) ~

0

0

j

a

0

t

' _ - l 1m -, r ? 0 0

riduce

dH' x-x_ (- _ _1_x _ UI _ 0) H, Js

S

o

(20) si

l' 1m

+ .!?£rl E(r', t)jdt 3

t -t 0

[to ~

r'+o 0

tI

I

t -t o

dt

= 2 I'X..Y ~i'

- 499-

C. Agostinelli si ha percio infine

1

J

to dt S o

lim rl-.o

(21 )

...

d HI

(7-d1_ U11.~))C iLls" n

-41f{,;H (x,y,z ,t ).

Ixoooo

o

Consideriamo ora

f

r

-t 0

~

-+

.

U',HXn dS"hm rl-+ 0 Sit) 1

\im Jto dt

(22)

il

o

Posto f(t) "

i

....

i

t0

J

dt

r'x-x )

Sit)

0

I

HJCti'. __o_E(r,t)dS. to -t

2 r3

x-x 0

~

HX n - - d S 3

Sit)

r

possiamo scrivere

r Jto dt )S(t) o

;){r7. UI d s" .!, 1Jt o

+Jo [f(t) e passando

al

r' fIt ) E(r',t) dt +

21 0 t-t t O O 1

limite

- fit )] 0

r'E(t''', t) dt to-t

per

si

1 lim ) H'K n . U I dS " - f( t ) 1 2 0 riot 0

...

ottiene t

-'0

0

(23)

r

I limiti

l ,..

j dtl to

lim 0

o

degli

j ....... - r

r-:*'. HlC'n,U ' dS=v7C'Y Hxn 1 Sit ) Sit) o

integrali

-to ...

che figurano

DI(t).

tendo no agli stessi

pure

i limiti dei rimanenti

agU stessi integrali, dove

integrali integrali

nella

=v!C7 f (t o)

o

o cio~

c::

r'E(t", t) t-t dt

x-x

o

,

dS,

r

(17) estesi al

dominio

estesi all'intero dominio D(t); cosi estesi all

pero Ie componenti

superficie S(t), tendono

HI HI HI del vettore Ix' ly' lz

- 500C, Agostinelli

.

HI vanno calcolate per rl =(),

cioe mediante Ie

I

(10), ove si ponga

I

(24)

r~ o

L'equazione

...

'*

HI =H' 1

per

(17) ,

, al

~UI=UI

1

limite

per

rl~o

porge quindi

47f.f;; H(x ,y ,z,t) =1

(25)

x

0

0

D(o)

00

(HXH')

I t=o

-JtodtJ

[j(dH XHI - -dn dRl l x .. H) d S(t) n 1

o

-

to

J

dt

l

..........

rot (HA v) )( HI

D(t)

o

r

dS -

.-+J dS -

+ Vn, H XHI'

d T' ,

(25) si hanno per Ie

componenti

H, H del y z

magnetico, Osserviamo .......

che

= e pertanto

il

trasforma

nel

J

risulta ~

-+

H x Hl1 = HICrot

(26)

f

X-X

S(t o )

I

Due formule analoghe aHa campo

d-r-rn;! H
div [

rot (t 1)

=

ii X grad'r

'

primo

integrale

modo

seguente

.........

(H X HI') _ dt = D(o) t-o

-1

del

+

(rot

r

rot

r -H

..... i) H

ti'

A

x grad

secondo

t'1 +

rot

membro

...

~

+ rott+- i)Xrot H= HAgrad della

t

X

7,

(15) si

(H x gradt ,cos 'tx - H d'l') d S+ x d n t=o

.. to) S(o)

div

....

H Agrad'Y- Xi)

- 501-

dove figura

la

corrente di

conduzione

e

ed

grad'(

'J", = ~r

=LrE(rr' t)

grad· r

C. Agostinelli

. .. I = rot

H alllistante iniziale,

J

r

-"2I

E(~, f: )cJ~

]

grad

r.

0

r

Cost pure, essendo

=div

.."..,....

rot (H/\ v) )( H~ si ha

(27~ J'

.......

-to

rot (HAV) x HI d"t'

dove

4.

10

~

J+ rot

l'

(H/\ v) A H\

-00

I ....

(HA V)I\HI J( n d S +

S(t)

...

1

_+

rot H;lI'Hh v d 1";

D(t)-

e

Se il campo

condizioni gli

si

integrali

estende di

a tutto

superficie

10

spazio, sotto determinate

tendono

di volume risultano convergenti. In tal caso to

~/

HI x HI\ v

J .. +......

=-

1

D(t)

[

delle (26) e

a zero

e gli

integrali

la formula (25), tenendo con-

(27), porge

(28)

-J

to

o

dti~[iA E(ri t) 2 4" (t -t) I

.1

~

...

r grad rJ X HA v d l'"

0

e analogamente 47f.fi;

Hy(Xo'Yo'~6to) = lJJ(iA grad.,.

ICj)t"O

_)'0 d' 11p, Ejr, ') 2 r o

Jj

47 (to-t)

d l'"-

grad r

l~ii~;

d l'

- 502-

C. Agostinelli

..

(28 1)

::}I((

41Cm H (x ,y ,z ,t)

z

I

0

0

0

(IA grad'1' )(

JJ

0

4

k)

t=o

dt'-

_Jt dtjffJr k1\ E(;, t) 2 o

~t

o dove gli

conduzione

...I

e

i primi integrali dei secondi al Le

assegnata in memhri

un

spazio. Se inizialmente la dominio limitato

D , o

delle formule precedenti saranno

dominio D • o formule (28) e (28 ') si possono ovviamente conglobare nell 'u_

nica formula

4'1fM

7 (to -t)

estesi a tutto 10

integrali tripli sono

corrente di

estesi

4

vettoriale

H(x ,y ,oJ; It )= o 0 0 0

JIIJ.

(iAgradt)

Jl" -

t=o

jo ~.j~'\to~t) . .. to

E r, t 2

dt

2 r gra4"

A (H" v ) d~,

che

e

un'equazione

ogni punta

P (x ,y ,~ ,t ) ,e in 00000

della corrente di "'dei valori

integrale che definisce il

conduzione

della velocita del

ogni

istante t ,per menD dei 0

~

I all 'istante mezzo

campo magnetico

fluido

iniziale

in

un

.....

H in

valori

dato dominio

che occupa tutto 10 spazio.

CENTRO INTERNAZIONALE MA TEMA TICO ESTIVO

(C. I. M. E.)

A. PIGNEDOLI

" TRANSFORMATIONAL METHODS APPLIED TO SOME ONE-DIMENSIONAL PROBLEMS CONCERNING THE EQUATIONS OF THE NEUTRON TRANSPORT THEORyn.

Corso tenuto a Varenna dal 17 al 19 settembre 1966

TRANSFORMATIONAL METHODS APPLIED TO SOME ONE-DIMENSIONAL PROBLEMS CONCERNING THE EQUATIONS OF THE NEUTRON TRANSPORT THEORY by Antonio PIGNEDOLI (Universita - Bologna)

1. Introductory remarks. There is a very great bibliography concerning the mathematical theory of slowing down and diffusion of the neutrons within the moderating medium of an atomic fission pile. The methods used for such study must allow for the differences in the reactivity that take place in the reactors chiefly during the starting and because of the sinking and the prising of the control bars, so doing a quantitative scheme of the behaviour of the fast neutrons produced in the nuclear fission, to reach, by slowing in the moderating medium, the thermal energy. The neutrons, endowed with such energy, are "trapped" :h turn, by the uranium nuclei producing new nuclear fissions, and so on, by the well known chain reaction. The methods used for the study of the neutron diffusion in a moderator recall or upon the research, in opportune conditions , of the solution of the Maxwell-Boltzmann integro-differential equation of the transport theory, or on the research of opportune supplementary conditions of solutions of the partial differential equations, of diffusion, derived from the application of these so called .. phenomenological theory". This lecture is concerning the transport theory point of view and, in particular, the application of transformational methods to some one-dimensional problems. As well known, a neutron is a heavy uncharged elementary particle. Of the forces which act upon it the nuclear forces are by far the most important and they are only one that need be taken into a account under the conditions in which one is interested in the diffusion of neutrons.

- 506-

A. Pignedoli

Since these forces have an extremely short range, it follows that: 1) the motion of a neutron can be described in terms of it collisions with atomic nuclei and with other freely moving neutrons; 2) these collisions are well-defined events; 3) between such collisions a neutron moves with a constant that is in a straight line with a constant

velocity,

speed;

4) the mutual collisions of freely moving neutrons may safely be neglected and only the collision of neutron with atomic ding medium need be taken

nuclei with a surroun-

into account;

5) for a neutron travelling at a

given speed through a given medium the

probability of neutron collision per unit path

lenght is a constant;

6) the neutron or neutrons emerging from a collision do

so

at the point

of space where the collision took place. We will indicate with N the neutron density N in vector

a medium

is

density in a

a function

medium (the neutron

of the position, denoted by the

...o ,

1, the direction of the neutron

its velocity v and

the

time t). Let

!: (v)

be the total

macroscopic cross section for all

processes;

r (v)

is the inverse mean free path. Let c(v) be the mean number of secondary neutrons produced per collision. The quantity number of secondary of unit path. Let

c(v) E (v) is the mean

C(V') f (v',

nL. v,

the mean number of neutrons produced in the velocity range dO when a neutron of velocity

v'

and

a collision with a stationary nucleus. Let

....

a

direction S(it, v,

n,

0)

dv dO be dv

and cone

undergoes

t) be the source

strenght of neutrons in the particular volume element. The rate of change

-

507-

A, Pignedoli

n,

of N(r ,v, red to

t) with time is equal to the number of

n from

the velocity v and direction

neutrons scatte-

other directions and ve-

n

locity less the loss due to leakage and scattering out of v and The transport equation

..

...

~N(r ,v, n, t)

at

1)

is therefore : =_v

n• grad

N _ v E(v) N +

..... . v, n) N (r , v',

((

+ ) JV'c(v') E(v') f (v',

12' -

...

...

0', t) dv' dO'

+

...

+ 8 (r ,v, 0, t) , The physical meaning of this equation is the following. The rate of

a~N

change

is equal to the neutrons scattering to v and

sources minus the leakage and the scattering out of v and In the case

r

of

nplus

....n .

the isotropic scattering, the density of the neutrons at

is

1

and the following integral equation is obtained v N (r , v)

(2)

0

=

eKpl- ,EI

41T

+ ~ v'

f2

Ls

[8

(~,

, v, t - -)Pv +

(v') f (v'_ v) . N/!', v', t-

~)dV'] dV

(dV = volume element) , If the neutron distribution does not vary with time, then

N(; ,v, (3)

n) satisfies the equation:

v~ , grad N + v

=~~Vl(C(V')

r (v)

N

=

L(v') f(v', n'_v, n) N(? ,v',

n') dv'

dn'

+ 8(r, !i,V) .

-- 508 -

A. Pignedoli

This equation does not have, generally,. a solution and the following modified equation is considered : (4)

vn • grad =

N + (ex + v

~ ~ v'c(v') E(v')

E )N=

f(v',

n' -v, 0) N (r

,v',

n') dv' dO',

(0( is a constant) .

This

equation has

functions

solutions for certain eigenvalues

C\I .• 1

The eigen-

of the equation satisfy to the following conditions:

a) Continuity at the boundary between two media; b) N. = 0 at the free surface for all incoming directions. 1

A gene ral solution of (1) is :

...

-..

N(r ,v, 0, t) =

(5)

L a. .

..

1

...

N.(r, v, 0)

exp

1

( at. t) 1

1

(We assume For large

that t, it

...n,

(6)

where (

Cl

o

0(

o

ex

<0

o )

is

the N. are a complete

set).

1

is

:

v, t) =

N

o

(r.

-+

0) exp

v,

the maximum of the

(01 t) , o

I

01. S • 1

subcritical

system,

01,

0

=0 critical system,

> 0 supercritical system) .

The coefficients conditions which

a. of the series 1

are determined from

the boundary

may be specified at some particular time.

- 509-

A. Pignedoli

2. The one group theory. Consider the time-dependent equation (1) . If all neutrons have the same velocity v

0,

,

-a,

N(i'"

then it v, t)

is:

= ~(v-v 0) N (r,

ff,

t) , ...... ( f(v', 0' --.. v,O) = 0 (v'-v ) f (0' _ O) , o .... --t .... -.. S(r, 0, v, t) = o(v-v ) S(r ,0, t). o r

~

(7 )

Let:

n, t) = f N(r, n, v, t) dv , { f(n'_n) = ~ f(v', n' -v, n) dv , N(r,

(8)

....

~

S(r, 0, t)

= Jr S(r. . . , --0, v, t) dv .

Integrating the transport equation over

v,

results in the following one

velocity group equation:

(9)

[VON

= 't' (r , n, t) = angular

distribution of the neutron flux.

If we assume :

A} time independence of N ; B} No C} f

-a N Tt,

variation

of

E

and

(n' --- n) independent we

0

b

. tam

:

of

c

with

neutron velocity;

v; and integrate

(1) without the

term

- 510-

A. Pignedoli

n. grad

(10)

where

'I" (r, S1) +

.. - r .. ... { 'r(r,.. fl) : F~ (r, v,_fl) dv , S(r, n) - S(r • v, fl) dv .

r

(11 )

3. Solution of the one group transport

equation in an

infinite uniform

medium in the case of the time-independence and with at

a plane source

x =0 .

We consider now the one group time indipendent transport infinite uniform medium with That

equation for an

a plane source at x • 0 .

is the equation:

d(x)

(1 )

41T'

J(x)

where

We apply to indicate

with

is

the Dirac

this 'IT

'

"Delta distribution "

equation

the method

of

the Fourier transform. We

(t, y.) the Fourier transformation

of

'¥ (x'.J-)

, that

is we put: +00

y(x'.r) exp (-i t' x) dx .

) -00

We have

(2) i 'r

Y.

11 ('t' •Y' ) +

~

L..

11 ( t

cL

,yl = -2-

+1

f

1'l' (t'• .)I-) dy. + 41'1'

-1

Now we indicate member

of

with

Tro( 't') the

(2); and we

have

:

y.

indipendent

quantity at the second

- 511-

A. Pignedoli

(3)

Substituting back

(4)

f(

o

From

(t')

=

in

the equation (2) we have:

[_ ~

4 1T 1

the Fourier

1:'

T

arctg

't

inversion fourmula we obtain :

r

c 1: +i t' ]-1. [ 1 - -2.-1g (~. ) d't.

(5)

1 t'

L,-l't'

The total flux is obtainErl by integration over )A+1

~

2rr

!(x) =

dr'

"'f(x,,;.}

-1

that is

r

1

(6)

=-41'1'

Tx [1 _~ 2it

ei

'('

log

-co

.

(1 t)

The integral The integrand

-1

L..-

it']

-1

(~ L.+lt

r

+it log (-~-.- ) d~ " -It

can be evaluated by the method of residues. has

a simple pole where the term in the denominator

vanishes , i. e. where: (7) If c

C

("

< 1 the poles are at

(S)

(The principal

log

l..

t' =

KIf.. = tgh rooth of this

E+i't.

,,-It = 21 t

.

K, where

K is

-~-.-

±i

given

by

K (7t ).

equation

is

real). An examination of the

- 512-

A. Pignedoli

integrand reveals a singularity log

z for

z

-4'

0 • To

avoid this

the deformed contour is taken back again

. We

T =i

t' = +i

I. .

This is of the form

singularity the plane is cut and

along the imaginary axis to

i

r.

and

have:

I) an asymptotic part

at

at

of the solution

that

is arising from the residue

1<

II) a transient part which is only important near to the source; that

arises from

the contributions of the integral around the cut

(fig. 1) .

Deformed path

original path

Fig. 1 .

- 513-

A. pignedoli

For

x < 0, the integral can be e.lIaluated by taking a path

gration arises

in the lower half of the complex plane. A contribution then from the pole

We have therefore

1

c

part

This

in

T

of the solution

(10)

of the solution, that is :

t2 -K 2 K 2 2 . 2(1-c)r K - E (I-c)

dominates

A second part

at

x is

corresponds

"l

log (2"l

(-(,+1) IxlI: ) d, 2(1.+1)

o

is important near the source (-

4. Solution

-1

r. x)

c

+1)

> 1 , that is

]2

2 2 2 +'1l'c E

the integral comes from

i. e. the integral is given approximately by :

exp

(11)

exp

to

small, the major contribution to

large values of

This

exp (-K Ixl ) .

large values of x.

r.l2L('1.+1)-C!:

When

as

't' = - it( •

a first part

~ = 2(1-c) •

(9)

of inte-

when x tends to

=.!.

2

x = 0 and

Fl (LjXI ) . decreases rapidly

co

of the time-dependent transport equation

without sources in

a semi-infinite medium. Now we consider equation for

the time-dependent Boltzmann

the case of the neutron

integro-differential

transport in a semi-infinite me-

dium without sources and for the one-group theory with non-isotropic collisions . That is we consider the integra-differential equation: (1)

v

oN(x,},-, t) + ". iN(x,,,..., t) =

'dt

= 1'l0's

L +1

h N(x'l' t)

1+3 p

I

2 .}J:.JL d.r I

- ( 0' S

+ O'/lt N(x,.f' t)

,

- 514A. Pignedoli

where

'tl. = number per unit

of volume of collision c.enters

for the

neutron;

o-e

= cross section of capture of the neutron;

0'.

= cross section of scattering.

P

= constant

and (1+3P~r')/2

.

Putting

11. ()'o=!\..-t ,1IllOC:'t-1 , we obtain the equation: (2)

aN(x,}\-. t) +VJA dN(x,).I., t) = (_:::.. + l) N(x t) + J A t 'JA'

ot

r

ax

,1+3P~;i N(x", t) 2

1

v

+A

J

dj, (A, v, t

-1

We have the following boundary condit ion: (I)

for

N(x, J-' t) = 0

x =0,

"u.
We put now: (3)

Writ ing (4)

N(x'.f,t)

= exp (-wt)

F(x'JA,t) I

we obtain: (5)

8F(x,.l':' t)

at

cF(x,)L, t)

+V)J-

ax +1

=w

0

t

F(x,i,t)

= 1+3Pe~ df'l 2

,

P constants)

- 51S-

A. Pignedoli

with F(x'.fL,t) =

°

for x = 0,

J: t, p £ x, q

We apply a

(double

!

/,<0. -transformation) and we put:

+0()


=}

exp (-pt) F(O,.!.t) dt,

-0()

o

'I'(q,y.)

=~

exp (qx) N(x,y., 0) dx,

-0()

o

4>

(p, q, 1oL) = ./

oLt.L [N(X~, t)] = ,p x, q

+0()

~ e
dx ) e -ptN(X

-0()

l' t)dt}

o

We obtain the integral equation: (6)

-V(q,y.) + pq>(p, q'jl) +

vy-

+1

with (7)

c.p(P,j.l) =

°

One considers F(O)'-, t)

for

F(x, )A. ,t) and

will

as a given function

(11'11

Cf

is

and

dependent from

be determined

cf (p,y.)

=

L

m=l

~

Ifmff)' m p

=[ 1

(9)

~mn(JL) mn

n n p q

and we observe that we shall have:

is a known

the initial values of

when the function

solve the integral equation (6) we write: 0()

(8)

2

.J-< 0.

function. The function 'l"(q, y.)

In order to

1+3P)oLJt'

~-1 ~ (p,qj)

= Wo

If

is given.

- 516-

A. Pignedoli

for

(11)

~mn(y.)

L

m-l n p q

m, n=l

-

0

'I'n(y.) -- + n q

0()

L n-1 +1

0()

w

Lm,n=1

mn p q

• (m=l. 2. 3....... ) .

equation (6) • we obtain:

Substituting in the integral

(12)

.r < 0

J 1


vr [~1 1+3P.l~ 2

tf)

'fro!)'}

L Pmnr,.}]

---m-~-1 p m. n=l p q ~.

Therefore we have necessarily:

(13)

cFm1 = 'fm (!) ,

(m = 1,2,3, ...... )

= 'Y)J)'

(n = 1,2,3, ...... )

(14)
.

(15)
=W

)

o


'!)

-1

(r) =

1+3PJA:JA:

2

dJA-'.

The equations (13) give the coefficients
(m.n=1.2,3, ..... ).

of the double series

<.f m . The equation (15) is recurrent ••• ,

terms of the


we have

the functions

p.m,n ,we

have also


and the


and gives
't': in terms of the functions If . n

Applying the theorems on the

m

i, -transformation

in

of the series. we

obtain , after the calculations: (-X

t -1 ]

(n-l)!

'

- 517-

A. Pignedoli

that is (unicity of the L-transformation):

L mn 00

F(x, Y. . t) =

(17)

t

,h

'±'mn (r)

1

m-I

n-I

~

(m-I)l' (n-I)!

One demonstrates immediately that it is F (x.Jl.t) = r: for

y.<

x = O.

0 •

F(x, y. , t) exists an

It is easily possible to demostrate also that for

exponential majorant series

x w(t-- )

w H (1

+ 3P -

w

o

)e

v

Finally we obtain the density function: For

y.>

( 18)

N(x,.f' t) = e

0:

-wt

00

[

m=l

m-l { oc [ m=O

Vm+n(f)

(~-l)!

(v)'-)

00

+ 3Pwo For

y.<

L m=l

n

nl

<xm+2n-2,2n-2 2n-l v

0:

(19) N = 3P

Wo

e

-wt

C:Xm+2n-2,2n-2 2n-l mn=l v

L 00

And it is N(O, Y ,t) = 0 So our problem is solved .

for

y <0

t

n

J.::.:L

m-I

(m-I)!

x

2n-1

(2n-I)~

+ 2n-l

~2n-l)'

I

- 518-

A. Pignedoli BIBLIOGRAPHY Hopf, E Mathematical problems of radiative equilibrium, Cambridge Tracts, No 31. Davison, B, Neutron transport theory, Oxford University Press, 1957. Marshak, R, Rev. of modern Physics, 19, No 3, 1947. Lecaine, J, Phys. Review, 72, 564, 72, 1947. Lecaine, J, Canad. J. Res, A, 28, 242, 1950. Case, K., Annals of Physics, 91, No 1, 1960. Pignedoli, A. Atti del Seminario mat. e fis. della Universita di Modena, 1947. Tait, J. H., Neutron transport theory, Longmans, London, 1964.

CENTRO INTERNAZIONALE MATE MATICO ESTIVO

(C.l. M .E. )

A. PIGNEDOLI

ON THE RIGOROUS ANALYSIS OF THE PROBLEM OF THE NEUTRON TRANSPORT IN A SLAB GEOMETRY AND ON SOME OTHER RESULTS.

Cor sot e nut 0

a

Va r e n-n a

d all 7 all 9 set t e m b reI 9 {; G

ON THE RIGOROUS ANALYSIS OF THE PROBLEM OF THE NEUTRON TRANSPORT IN A SLAB GEOMETRY AND ON SOME OTHER RESULTS. by Antonio Pignedoli

1. Introduction. This lecture is concerning the problem of the neutrons distribution in a slab geometry (1) . The statement of the problem is the following. The neutron density is satisfying the integro-differential Boltzmann equation: (1)

1.v dN(x,y., t) + I.A. ()t J

,

oN(x,,)4, t) + ~N cZ: (}x f.. (x'r ,t) = -2-

J

+1

N(x, Y. I, t)d X. I

-1

where the symbols are usual (see my first lecture in this Course) . The physical conditions are the following: a) the cross section b) the production

~ is constant;

of particles is isotropic;

c) the slab is surrounded by a vacuum, so that no particle may enter the slab from outside; d) at the time t

.~

0 a particle distribution f(x,

y.. ) exists inside

the slab. The supplementary conditions of the problem are: N(a, )A-, t) (2)

=0

,

.JA < 0

•

{ N(-a ,y...t) = 0, )A->O. N(x. J-'-. 0)

= f(x.r)

t

>0

;

t >0;

; -a ~ x ~ a. -1 ~r 51.

We ask for the resulting time-dependent particle distribution. The equation (1) may be simplified somewhat by writing .

(I)Important works in this direction have been done by : J. Lehner and C. M. Wing. Communications on pure and applied mathematics. vol. 8. 1955. idem Duke Mathern. Journal, 1956. R. L. Bowden and C. D. Williams. Journal of mathematical PhysiCs. vol. 11. 1964.

- 522-

A. Pignedoli

1

N(x, y-, t) " exp (-c ~ t)

(x,

oJl). ,

t)

and choosing, for convenience, v = 1, ~ =1. We have: Ol(X,P.,t)

at

(3)

+~

=_Ll.a}(X,Y.,t)

)

ox

2

+1

r 1(x, y!, t)d JA'

)1

with the supplementary.conditions :

ita"~, t) = 0 , .f-< 0, t > 0 ; { ](-a,p, t)=O. y.>o. t > 0 ;

(4)

] {x,;;:, 0) '"

f{X~Iol),

-a::: x::: a, -1<1.\.<1. -) -

In order to obtain a formal solution, we can write the equation (3) in the

form:

d1 (x,'ilt)A • t) = A J (x' ))..l' t) ,

(5)

where A

is the time-independent operator: +1

~ .df'

(6)

-1

A formal application of the Laplace transformation with respect to t

gives:

"1 - f{Xt') =A J.

(7 ) where

}(x.y,t) 1 = j(X'f' A ) = +fJ e - At-

(8)

dt =

o

=! t, .\ fIx, .J- ' t)

•

Obviously the equation (7) can be written:

(j.. - A)

(9)

If

i

-A were

J = f.

now just a complex-valued function, the solution to (9)

would be simply:

- 523-

A. Pignedoli

}= (,( _A)-l

(10)

f.

But A

is actually an operator, and it remains the problem of -1 finding (/... -A) when it exists and determining its properties. Let us for inverse

a moment press the formalism still further. Applying the

transform operator to (10) yields:

l(x, f'

(11)

t) = (2 1l i)

r+

b ioo

-1

[(I,._A)-1

fJ

e ht

dA.

lb-ioo

The evaluation or

estimation of

(11) depends on

a knowledge of the

behavior of (A_A)-1 f . The singularities of this function are of particu'\

-1

lar importance. Thus we must study the operator

(/I-A)

we must investigate the spect rum of the operator

A. Let us -suppose

that ( J.. -A) values

-1

is

f

a

"well-behaved" function of

A

; this implies

except for certain

A.

where the operator A fails to exist. This latter event J will happen when there are functions y.(x, u.) such that: J J

( A. - A) '!'; = /.... 't': - A 'f. '"

(12)

J

J

that is when

'f.

J

is an

J

J

J

eigenfunction

of

A

0,

belonging to an eigenvalue

..( j .

If

there were infinitely many

pansion

as the following: 00

(13)

l

(x'j'-,t) = [ ,

But there is not an strates that (14)

J.. . , J

then we should have a formal ex-

.<.t e J

j=1

infinite number of

g.(x, )\.A.) • J ~ eigenvalues

the soiution (13) must be replaced by:

Aj'

and one demon-

- 524-

A. Pignedoli

~ (x, )"-, t) is, in

where

a sense,

small compared to

the other terms

of (14) . There is, of course, always at least an eigenvalue ging to the operator A. Therefore the sum

'=1

contributes the dominan~

empty and the sum Thus one can

J=

ming

L.. 1\..

belon-

J term for large

determinate the asymptotic time behavior of e At g(x,},-) and then finding

J..

kt 1 e J g .(x, y-) is never

the value of

A

J

t.

by assu-

of largest

real part for which this expression satisfies the equation c + 2

(15)

But we shall is an

+1

1,(X'j'-,,.ty dy-' . -1

expose a rigorous analysis of the problem , on which there

important literature .

2. The rigorous analysis of the problem. The action of the operator

J.L 'at

(1)

Now we g(x,

r ),

A is

=A-,.J

rectangle

Ix

I~ a

by the equation

•

choose the Hilbert space defined and

defined

H of complex valued

functions

square integrable in the Lebesgue sense over the

,1 f'1 ~ 1:

(2)

We define the usual (3)

(g, h)

=

+1

1

-1

and the usual norm :

inner product

~

of two functions

+a

t

g(x.J- ) h(x.,. ) d x ,

g and

h in

H:

-

525-

A. Pignedoli

(4)

Write: A

(5 )

Let gE

= -D+c J •

absolutely continuous in

and such the set and

() .JJ-a;' ,

J.

the domain of the operator

~

H

D.=

Dg E

that of all

gE

H

such

D be the set

x

H . Call

for

dJ

that

2

each

of functions inly.l.::: 1

}J-

the domain of the operator J g

g,

J.

1

exists for each 1: in x I.::: a

J g EH

Finally define

ciA

d J and such

and

both d])

g(a,

{

(6)

as the linear manifold of all fUnctions

Hence

A is

an

that

p) = o.

g( -a.y- )

-1,:::p<

=0 •

operator

g belonging to

O.

O<jA-'::: 1. from

the Hilbert space

H

into

itself

with domain Concerning the spectrum of the operator

A

, one can first demonstrate

that there are no eigenvalues in the left half of the lambda plane. We ha ve the following:

Theorem 1. There

is no function

I..

for some In order I",

(I..

'\IT

with

d.II

satisfying the equation

we consider the function:

CP(x)

=~

-1

=

J.. 'f

if there exist eigenvalues of A with Re(" )~O,

+1

(7)

Ay

Re (..{ ) < 0 .

to establish

rr 0,

L

1"

o/(x,}'-')dIL-'

J

in

L 2(-a,a) ,

-

526-

A. Pignedoli

and

we demonstrate the following theQrems :

Theorem 2.

-y(x,y.) ~dA satisfies

Suppose

Then the function =~

(8)

1

A'f=A.'f

with Re(A)~

0,1..10.

satisfies:

E(A Ix-x'i ) ¢ (x') dx' ,

-a where

i 00

E( u ) =

exp (-ut)

1

t

dt,

~

Re(u)

0,

uFO.

Theorem 3, There is tm( J...)

I

no

function

d A satisfying

Alf =.(rwith

Re (~ ) ~O,

0

It remains to

write J.. =

'fE:

r

examine the poss-ibility of

when

A is

real

. The case

f... real non-negative. We shall ~ = 0 is easy, The equation

A'i" = ,( 't' becomes

which yields : ~

't"(x,y.) - 'f'(-a,y. ) =~ ) d x', Ix I:: a , .J-A- _)(
(9)

Choose :x such

the right side of (9) is not zero, x = xl ' 1 -a < xl < a, Because of the .)A. factor, the right side is then not in:egrable over fl

~

since

that

f ,If-}::

1. But the left side of (9) is integrable over

'f E- d. 4 . Hence

~ =0 is not an eigenvalue. Suppose now

> 0 , By theorem 2, every solution of

satisfying (8) ,

A'f = A.'f yields

a function

4

- 52i-

A. Pignedoli

cp & L2

Conversely if solution

'f

satisfies (8). then the corresponding

is

a

A'I' = f... y.

of

We have the following: Theorem 4 A necessary and sufficient

A"I' = AY solution


condition that

~ = 0 is that there be

corresponding to to

CP(x)

c

=2

1:

\f (x •.J-)

be a solution of an

L

2

-

E(" Ix-x'i )P(X') dx' ,

for that ~ value. We have also the following: Theorem V. The point spectrum set

of points

'PO"

A of

fil > 132 > ...

A consists of a finite but

nonempty

> ftm all lying on the positive A. -axis.

Theorem VI. The eigenvalues

of the operator

we consider the adjoint

Aa of

A are of finite multiplicity. Now A (2)

It is a matter of relatively easy computation to prove that the operator

•

A

adjotnt

to

A is

given

by:

d, - + -c A•• =J U-ax' 2

. dJ'" .

(2)cfr. F. Riesz-B. Sz. Nagy. Functional Analysis, F. Nugar., New York, 1954 M. H. Stone, Linear transformation in Hilbert Space and their applications to Analysis, Amer. Mathern. Society Colloquium pubblication n. 15, New York, 1932 A.E. Taylor, An introduction York, 1958

to functional analYSiS, J. Wiley, New

- 528A. Pignedoli

The domain to both

'f

dA•

C.

H

consists of those functions Y(x,y.) belonging

db and d.J and such that jA-< O.

(a.'r) = 0.,.> 0 ;'l'(-a~) = 0,

Clearly the operator A is not

self-adjoint (this fact

com~cates

consi-

derably the problem). If 'V(x. y.} is

'tl( x,y.)

an

eigenfunction

='1'( -x'f') is

of

A,

an eigenfunction

with

eigenvalue

of A· with

A. ,

then

K.

eigenvalue

The converse is also true. Theorem 1 The residual spectrum 'RItAof

A

is empty.

Theorem 2. The continuous spectrum Let

f'

be the set

of the points of trum) or domain

I1t A d A is

of

A

contains the half-plane

of points in the right half of the

PO' A.

The points of (

dense

in

H

A-plane

are either in

(restr. Spectrum) and hence (" -

A)

One demonstrates

"Re (")

-1

~ 0.

exclusive

Co- A(const.

spec-

exists and its

the following:

Theorem.

A

The resolvent set of contains ( deleted by the point spectrum, Summary theorem, The linear operator

A of

. The resolvent

set

is: Re ( A )>0

A'f=A'f

a) is non self-adjoint; h) decomposes the spectral plane as follows: Point

spectrum: a finite nonempty point set lying on

Residual spectrum: empty ; Continuous spectrum : Re( A. )~ 0; Resolvent set

Re( A) > 0 deleted by the point spectrum

I..

> 0;

- 529-

A. Pignedoli

3. An integral representation of the solution of the transport equation. Considering the integro-differential equation: (1 )

C)JC(x,JI:., t)=

at

-IA.

+1

d)(X,JL, t)

J

S J(x,y.',t)~'

+{

ox

-1

whit the supplementary conditions : }(a'jJ-,t) = 0 ,jA< 0,

t

1(-a,j4' t) = 0 ,jJ-> 0,

t

{

(2)

> 0; >0 ;

1(X, jJ-' 0)= f(x,,.) , -a ~ x ~ a; -1 and having determined how the operator plane , we

are in condition to

A

2..r 2.

1,

decomposes the spectral

return to our original problem, that is

the solution of the aforesaid problem. We write: l(x'jA-,t) = T(t) f(x'JA) .

(3)

Symbolically we have: T(t) = exp (At) ; more properly

T(t) ,

t:>;. 0 is

a semigroup

by A. Such a semigroup exists provided: A

H; I R.( ~

< (I.. -K)

-1

Effectively the operator operators for It follows from

t

~

of operators generated is closed; ciA is dense in

,f... > 'K. for some K > 0 . A

generates

a semigroup T(t) of bounded

0 •

known theorems (3) that the solution

1 is

unique in

H,

(3) E. Hille, Functional Analysis and Semi-groups, ArneI' Math.Soc. colI. publ, VI. 31, New York, 1948

R. F. Phillips, Perturbation Theory for semi-groups of linear operators, trans. of the Amer. Mathem. Soc., Vol. 74,1953.

-

530-

A. Pignedoli

that lim

(4) t

-+

0

II px,y, t) - f(x'Jl) I

=0

,

and that (5 )

fIx, l"

b+iw t) =

lim

2TTi

w..... co

S.b-iw

The integral is to be considered the strong limit of Riemann sums.

< to


for fixed (x,y,) . If we

fix

It converges uniformly for

ous in

valued continuous

0

A

function of

on

tl ~

~

00

J

so that

(x, fA) then

is continu-

R A f is a complex

the integration path so that the in-

tegral may be considered as an ordinary Riemann integral. We now wii;lh to move the

line

of integration

in (5) to the left so as to pick up

the contributions from the singularities at

A (j = 1,2, ... , m).

accomplish this we must study the behavior of We restrict

f

to

analytic function

d.A of /..

• Then, for fixed

for

j3.,1i = 1,2, ... , m). J Let ft. be an eigenvalue

J eigenfunctions

R e (/..)

x

R.(

To

f.

and)A , R,{

f

is an

> 0 except at the points

of multiplicity

s, with

linearly independent

'if.J, 1 ''fJ, 2 ' ... , '1"J, s .

Let the adjoint functions

be 1J./ k ( x..r) =0/ k (- x I J, J,

,y)

j= 1,2, ... ,m;

k = 1,2, ... , s .

One demonstrates

that

it

is

e

(6)

JH e J

.(t

s. )J

L

k=1

R,c. f d/.. +

(f,'1/

J,

k)'f· k (x'jol) , 0 J,

<$
(one shifts the line of integration to the left of )3m but not as far as the

-

531 -

A. Pignedoli

imaginary axis, picking up the residues at And estimating the integral, one obtains: s. J m fo.t (7) } (x'P.' t) = e J k=1 j=1

L

j = 1,2,3, ... , m.

[

where for almost all (x, J-') , \

>I

is dominated as follows (setting

IS (x,y., t)! where D depends on f

p.,J

<

D(x~jA)

2

cS=~:

2

t log t,

and is finite for almost all (x, }A-) :

4. Some other results on the transport theory. From the somewhat practical viewpoint, it is desirable to obtain more information about the eigenvalues of the operator A. Results in this direction have been obtained

by S . Schlesinger (4). But

we will not consider this direction of research in this lecture . We consider some other results in the transport theory. G. Pimbley (5\as studied the multienergy state time -dependent slab problem . Thi s is especially important from a physicist's viewpoint because practical calculations of particle behavior (e. g. of neutron behavior in rea· ctors)

generally rely on replacing the continuous energy dependence by

(4)S.Schlesinger, Approximating eigenvalues and eigenfunctions of symmetric kernels, Juurnal of the Society of industrial and applied mathematics, vol. 5,1957. ideql., Some eigenvalue problems in the theory of neutrons, Los Alamos scientific laboratory reports, LA 1908, 1955. (5)G. Pimbley , Solution of an initial value problem for the multi-velocity neutron transport equation with a slab geometry, Journal of Mathern. and Mech., vol. 8, 1959.

- 532-

A. Pignedoli

a discrete dependence. Pimbley considered the equations: dN.(x'j4,t)

(8)

1

v.

(IN. \' +1.0\. __1 +'-- N

)"0 x

'a t

1

i

+1

n

2

[ j=l

C ..

1J

~ N/X',jA', t) djA' -1

i=l, 2, 3, •.. ,n

with (9)

It was discovered

that, while some results carried over quite comple-

tely, it was required in other cases to considerably restrict the matrix c. . in order to preserve the analogy . 1J For the time-dependent transport problem for bounded geometries, some very general results have been obtained by K. Jorgeps (6). Let

c;b and 1J be bounded measurable sets in position space (\' x 2' x 3 )

and velocity space (v!' v 2' v 3) respectively. Let

V

from zero (the particle velocity cannot become

arbitrarily small, and

be bounded away

"trapping" is avoided) . The transport equation can be written in such a general case in the form:

(10)

dN elt

Let N~ H,

3 = _ [,

.

1

v J'

ax

J= 1 the space of square integrable functions on-p)\r and let

I: (x, v) be

non-negative, bounded and measurable on"bN. In Jorgens analysis K is a quite general bounded linear operator on H. In must cases K in an integral operator: 6) K.Jorgens , An asymptotic expanSion in the theory of neutron transport, Comm. in pure and applied mathem., vol. 11, 1958.

- 533-

A. Pignedoli

(10')

f

K N '"

k(x, V, Vi) N (x, v', t) dv l

V·

whlch describes the scattering of particles of velocity (v~, v 2' , v

3) into

velocity (v I' v 2' v 3) . It is further assumed that no particles can enter ~

from outside.

Under the conditions stated, the problem

dN cH

=AN,

N(x, v,O) = f(x, '\T)

has a unique solution N(x, v, t) = e

At

f(x, v) = T(t) f(x, v) .

(L(~,v) =[(x 1,x 2,x 3' v 1,v t v 3 )etc. ).

With certain additional restrictions, the operator T(t) is completely continuous for

t

t where to is the maximum length of time that a para tic1e takes in crossing :l> . This io true, for example, in the n velocity

>

~

group theory. It is also true when

( *) and

k is

tV" is

bounded

and

a three-dimensional set K

integrable over ;]) >< V X V' .

When T(t) is completely continuous it has

a discrete spectrum

(plus possibly the point zero) where the /.... are the J The spectrum of A may be empty . The spectrum is certain

is the form ,0 e J

eigenvalues of A.

non empty for the sphere problem (7)

The formal series expansion of the solution function

(7) R. Van Norton, New York

N is complicated.

Univ. Report, 1960

J. Lehner, Comm.on pure and app!. mathem

,

1962.

-

534 -

A. Pignedoli

Jroblem of the convergence of this series, in the case that the specis infinite, remains insolved. Asymptotic results holding for large 'e been obtained by J8rgens (8)

5.

Researchs using the method of distributions in

sense of L. Schwartz. Consider the simple time-indep. transport operator:

Y.

a: d

+N

=

~

J +1

N(X'jA')

d.}I-"

c = const

-1

(L =1

for convenience).

ssume : N(x'r) = g(x) h(r) .

ituting in (1) we have readily: N(x,

fA'

x

J = exp (-~ J

jJ. ~ (u.) =c (1 --) y .., I 2

\l

+1

r

)-1

-

CP.., (}4)

cp.., (}Jo')

,

djA'

is a yet arbitrary. Let us require that:

J

+l

=1 .

-1

J8rgens

, An asymptotic expansion in the theory of neutron transport , Cornrnunic. in pure and appl. Mathern. vol. 11, 1958 .

Grosswald,

Neutron transport in spherically sirnrnetric systems, Journal of Mathern. and Mechanics, 1961 , vol. 10 .

- 535-

A. Pignedoli

Then manipulating we have:

.t1

(5)

f

1 =.

-1


c'\>

-]A

y'" -1

=

-

0 < c < 1,

+1

ciJ"'.

-1

v

2

11

The equation (5) has two roots + ry ; for

,

I)

0

.

-.r

For

= ~ c t gh

-1 1

(:;). v

c

> 1,

I)

0

is pure imagina-

"

> 1. K. M. Case(9) has observed that other solutions o may be obtained assuming that is a distribution in the sense of


L. Schwartz (10) Using this idea we can write :

4>

(6)

where

P

integral For

~

However

~

(r) = ~ P

2

_v_ + F(V)d(r- V)

v

-~

indo that the Cauchy princ. value is to be taken whenever an invol ving ~,,(1') occurs.

not

in ( - 1, +1) the procedure leads again

for

-1 < y <1

1

='c2

y P

to two roots

+

vo .

) gives

1+ 1

+ F (v)

-1

from which F( v ) may be found Thus for any V

in (-1,+1) there is a solution of the type (6).

Case has also examined some time-dipendent cases in

(9)

K. M. Case, Elementary solutions applications Annals

this way.

of the transport equation and their of Physics, vol. 9. 1960 .

(,lOlL. Schwartz, Theorie des distributions, vol. I, Hermann, Paris, 1950

- 536-

A, Pignedoli

p,p, Abbati Marescotti(ll) has studied the problem of the slab in

the

multigroup theory and in the case of non isotropic scattering, that is the non-stationary problem

df.

aT-

(8)

= A fi

0':::

t

' in the rectangle R:

< + co

fi(x,.f'O) =

Ixl,:::

a,ljll~ 1, (i =1,2"", n) ,

;

'I'lic'r);

fi(a,J'A' t) = 0, /A< 0 ;

fi(-a 'f' t) = O,]A> 0 . It is solved the problem

of determining

fi I: L 2(R)

One gives here a summary of the results : a) Using the theory of the semi-groups, one demonstrates a theorem of existence and uniqueness; b) The determination of the resolvent R(,{, A) is reduced to a solution of a system of integral equations of Fredholm of the second kind; c) One finds the solution in the form of a Neumann series: (9)

f.(x, 1

1Jt.,

J

_ CIO (r) t) - exp (-I.v.t) f.(x-v.rt.r' 0) + [ F. (x'r' t) , 11 1 1 0 r 1

where the first term at the second member represents the neutrons of the i th

group without interactions in the time interval

cond term the number of neutrons with

0 f---I t and the se-

r+l interactions in

the same ti-

me interval. Abbati Marescotti(12) has also solved the problem of the transport in a homogeneous sphere in stationary conditions, with spherical simmetry, isotropic scattering and with unknown source term S(r) : that is the problem:

(11) see p.P. Abbati-Marescotti,Atti Ace. Scienze di Bologna, 1962. ( 12) P. P. Abbati-Marescotti j Atti Scienze di Torino, 1962 - 63.

-

537-

A. Pignedoli

(10)

df

1-) df

~

)'\1; + - r - Or + O'f"

t

}-1

o < r < r , (r "radius of the sphere) , -

The function

o .- 1.

0

0

g()A-) is assumed

absolutely continuous in

S(r)

The problem of determining f(r'J") and

the interval

is reduced with a

unique unknown function

(11 )

'V(r)

c

"2

~+1

f(r

-1

This ra of

function

,r)

dr'+ S(r) .

is determined by solving an integral equation of

the first kind. One studies also the dependence between

Volterf

and

S.

Finally we shall say that the author (13) has studied the approximate solution of the problem in a spherical medium

!

with

c" c(r) (: B v ,

= L(r)EB v (the symbols are usual) without regular
ning g. The problem

is reduced to determiling

f(r,.I") and S(r) satisfying the

conditions (12)

f+l (0

)-1 Jo

2

r If(r:t")\ dr

ctr

r10 r


0

r

2

IS(r)\dr<+cc;

(the quantity of matter or radiation is finite). One determines necessary and sufficient conditions for g; in the case of

L." const in

every part in

which the interval is divided, one solves a chain of integral equations; in general one constructs an approximate solution, demonstrating the convergence of the method.

(13)p. P. Abbati-Marescotti , Atti della Acc. delle Scienze di Bologna, 1964.

- 538-

A. Pignedoli

BIBLIOGRAPHY

B. Davison,

Neutron Transport theory, Oxford at the Clarendon Press, 1957.

G. Milton Wing, An Introduction to transport theory. J. Miley and Sons, New York-London 1962 K. M. Case, , F. de Hoffman and G. Placzak, Introduction to the theory of neutron diffusion, Government Printing Office, 1954.

CENTRO INTERNAZIONALE MATEMATICO ESTIIlO (C.I.M.E.)

Giorgio Sestini

nPRINCIPI DI MASSIMO

PER LE SOLUZIONI DI EQUAZIONI PARABOLICHEII

Corso tenuto a Varenna (Como) dal 19 al 1966

27 settembl

PRINCIPI DI MASSIMO PER LE SOLUZIONI DI EQUAZIONI P ARABOLICHE di Giorgio Sestini (Universita.- Firenze) Introduzione. Questo seminario ha I 'unico scopo di richiamare 1'attenzione sull'impor

M

tanza dei cosi detti "principi di massimo" nella risoluzione di problemi al contorno per I 'equazioni parabolic he , ad es. della diffusione, qualunque sia la questione fisica che ha condotto

a quel problema e il tipo di gran-

dezza caratteristica del processo considerato. E' ben noto che, dopo aver superato Ie notevoli difficolta di tradurre in equazioni un problema fisico , troppo spes so anche a spese del suo stesso significato fisico, si presenta la questione della risoluzione del problema matematico cui si

e

giunti. Questa risoluzione, specialmente se

si tratta di problema non lineare, si rivela spesso cosi difficile da preferire una valutazione approssimata della soluzione, in modo da potere controllare, attraverso al confronto con dati sperimentali, fino ache punta

il

problema matematico resti aderente al problema fisico considerato. In ogni caso, almena in un primo tempo, si cerca di assicurare I 'esistenza e

l'unicita della soluzione del problema analitico in esame,

seguendo di regola una via che permette anche l'istituzione di un algoritmo costruttivo della ricercata soluzione, appartenente ad una fissata classe di funzioni. Due metodi assai usati sono quello delle approssimazioni successive, generalmente applicato dopo aver trasformato il problema da differenziale in integrale

0

integro-differenziale, oppure quello proprio

dell 'analisi funzionale che conduce a considerare una conveniente trasformazione di un opportuno spazio funzionale in

s~,

alla quale si possa appli-

care il teorema del punto unito. In tali metodi, sia per dimostrare la convergenza del processo iterativo, sia per assicurare l'esistenza del punto unito per la trasformazione

-

542 --

G. Sestini

funzionale individuata,

e di

fondamentale interesse il conoscere "a priori"

valut
danno un fonliamenta1e aiuto a questo ordine di

problemi. Di rilevante interesse

e poi il

loro impiego nella dimostrazione

eli tr,uremi di unicita. Sulla scorta degli studiatissimi problemi di tipo ellittico e, in partico1are delle proprieta di massimo delle funzioni armoniche, molti studi sono st.ati fatti per giungere a stabilire anche per le soluzioni di equazioni cti. tipo parabolico analoghi teoremi, atti a caratterizzare Ie proprieta di

speciali classi di

soluzioni di problemi al contorno di questo tipo.

Conviene subito osservare che se anche tali classi di soluzioni POS80110

sembrare assai ristrette dal

punto di vista analitico, esse tutta-

via sono del tutto soddisfac.enti a descrivere vaste categorie di fenomeni fisici del

tipo di quello classico della diffusione del primi risultati ottenuti riguardavano, come

calore.

e ben

naturale, Ie

so:uzioni di particolari problemi lineari. Successivamente sono state rilevale analoghe proprieta anche per classi pili generali di soluzioni di problem, parabolici anche quasi lineari. Per i problemi al contorno non linea-

ri, a quanto ne so, moltissimo resta ancora da fare. In cia sta il carattere di "seminario" di questa lezione. Per una sistematica, ampia e moderna trattazione delle equazioni pa caboliche, accompagnata da una ricca bibliografia,

e

ressante la ]ettura del recente libro di A. FRIEDMAN I " principi di massimo II Seguendo L. NIRENBERG guono

;.11

molto utile ed inte-

[1] (') .

•

[2J,

i "principi di massimo" si distin-

"deboli" e in "forti" .

('l I nume:'i in parentesi quadra si riferiscono alla bibliografia posta 1& termine della relazione.

- 543-

G. Sestini

I principi deboli affermano che la soluzione di

un problema pa-

rabolico , in un conveniente dominic e appartenente ad una certa classe di funzioni , ha i suoi massimi suI contorno del campo di definizione. Quelli forti affermano che se una soluz lone di un problema parabolico, in un certo dominic e appartenente ad una certa classe, ha un massimo interno al campo di definizione, essa

costante.

~

EI evidente che questa secondo tipo include i principi deboli come imme-

diata conseguenza. EI quasi superfluo accennare che principi analoghi valgono per i minimi .

In generale Ie dimostrazioni di tali principi vengono date nel caso di due variabili, ed io, nell 'esporne alcuni, mi atterro alla regola, essendo immediatamente estendibili ad eqL,azioni parabolic he in un numero qualsivoglia di variabili. Del res to dal punta di vista fisico-matematico Ie equazioni in due variabili coprono una vasta classe di problemi non stazionari (quando una delle due variabili sia il tempo) nei quali la dipendenza dal posto della grandezza caratteristica ricercata

e affidata

ad una sola variabile, cosa

che si verifica tutte Ie volte che si ha a che fare con campi dotati di simmetria (problemi piani, cilindrici

0

sferici) , nei quali la coordinata spazia-

Ie puo interpretarsi come distanza del generico punto, in cui si cerca il valore della grandezza ad un certo istante, da un piano

0

da una retta

0

da un punto. II "principio di massimo" di M. GEVREY (1913) M. GEVREY

[3J ,

che con E. E. LEVI

[4J

puo considerarsi il fondatore

dello studio sistematico dei problemi di tipo parabolico, ha stabilito che, assegnata I 'equazione parabolica lineare : (1 )

Z

xx

+ a(x, y) Z + b(x, y) Z+c(x, y) Z x

y

= 0

Z = a

d

Z

"0 a '

- 54-1--

G. Sestini

con a(x, y) , b(x, y) , c(x, y) funzioni continue di finita R

x

ed y in una regione

del piano x, y, ogni soluzione regolare di (1) (cioe continua con

Ie derivate

che compaiono in (1 ), non puo avere in R

tivi ,ne minimi negativi, quando sia c

n~

massimi posi-

<0 .

Infatti Ie necessarie condizioni per Pesistenza ad es. di un massimo positivo in un punto P

rendono insoddisfatta la

= (x 0 ,y 0 ) GR :

0-

(1).

Nel caso in cui sia b

< 0 (che e il caso dell 'equazione della dif-

fusione), qualunque sia il segno di cambiamento di funzione incognita modG che risulti

kb + c

c (se

e c.:::

0 basta operare

in (1) il

Z = U exp ky e scegliere la costante in

< 0 ) resta facilmente provato che per ogni campo

R' interno ad un contorno regolare

't'

(nel senso di essere continuo, sem-

plice e tale da formare con una caratteristica che 10 incontri uno contorni privi di punti doppi) il valore assunto

0

piu

da Z in un punto PE R' ri-

sulta compreso tra il massimo positivo e il minimo negativo dei valori che la Z assume sulla parte di

r

al di sotto della caratteristica per P.

Da questa proprieta discende subito, nelle condizioni specificate, il teorema di unicita per la soluzione di (1) con assegnate condizioni al con-

torno. Il "principio di massimo" di M. PICONE (1929) . Un altro principio deb ole si deve a M. PICONE 1'equaz ione di

( 2)

[5] .

5i consideri

tipo parabolico : a

E(Z) =

h,k

hk

(P) Z

xh x k

+

L:

b (P) Z - z h h xh t

+ c(P)

= f(P) ,

-

545-

G. Sestini

~ ahk Ah Ak

con

forma quadratica definita pos itiva, per

essendo ahk = a kh funzioni continue di

e un ad

A. i reali,

P:: (xl' x 2' ..• ,X n' t) £ R, dove R

qualunque dominio limitato e connesso di uno spazio euclideo En+l n+1 dimensioni. Si indichi con FR la frontiera di

R e con F R la parte di -t

FR,

se esiste, per i cui punti esiste la normale interna ad R avente verso opposto a quello dell'asse della variabile t

e tale che un intorno circolare,

convenientemente piccolo, di un suo punto P

appartiene a

F _tR (un domi-

nio sferico non ha evidentemente F R; un dominio parallellepipedo, con

-t

gli spigoli lat~rali paralleli all'asse della variabile t, ha come F _tR la base superiore) . Nell'ipotesi che anche bh , c, f siano funzioni continue di pER e Z (P) soluzione regolare (nel solito senso) di (2) , si ha il seguente teorema; se f (P) ~ C (~ 0) ed allora

e

e

ZIP) ~ 0 (~ 0) per

P (& FR-F _tR,

ZIP) ~ 0 (~ 0 ) per P E. R.

Discendono immediatamente , come corollari del teorema, la unicita della soluzione di (2) , neUe ipotesi dichiarate, che prende assegnati valori su FR-F _tR e, nel caso dell'equazione omogenea (f=O), se

c

~

0 il seguen-

te principio di massimo debole : nelle ipotesi dichiarate, ogni soluzione regolare di E(Z)

= 0 raggiung~)l

massimo dei valori del suo modulo su

FR-F_t· Il criterio di massimo viene poi esteso al caso di campi illimitati,

facendo alcune ipotesi suI campo e suI comportamento della ricercata soluzione per P-+oo. L 'elegante dimostrazione del teorema s i basa essenzialmente su di un Lemma di algebra Siano

L::hk

di Moutard) che ci limitiamo a ricordare:

2«..~. ~.

a hk A).k'

ij

11

o semidefinite (positive

~eor.

0

1J

11J

due forme quadratiche definite

negative) . La somma

I!hk


- 546-

G. Sestini

e non negativa hanno

10

non positiva secondo che Ie due forme, se non son nuHe,

0

stesso segno

segno opposto.

0

11 "principioforte di massimo" di L. NIRENBERG (1953)

[2] .

Sia Z(x ,x 2' ••• , x ; t 1, t , ... , t ) una funzione di n+m variabili 1 n 2 m in un dominio D ad n+m dimensioni e consideriamo 1'operatore differenziaIe : (3)

ellittico neUe variabili

xh e parabolico neUe variabili \' cioe tale che

:J...

la forma quadratica rna

qUadratiCa~ ~. ~.f~ IJ

reali.

IJ

Z(P) C.

e soltanto semidefinita positiva

per Ah e

1 J

a k ' ~ .. ,b ,

Supposti i coefficienti .

a hk '\h Ak e definita positiva, mentre la for- .

R.

funzioni continue di

2 h IJ h /'1 C , si ha il seguente teorema : Se Z e tale che rislulti

~i

P~ D e

L( Z)

2: D

e in un punto

PET la Z raggiunge il suo valore massimo, aHora e o Z(P) = Z(P ) in tutta la parte di iperpiano t. = costante, passante per P o

e appartenente

1

0

aT.

Accenneremo aIle linee

della dimostrazione nel caso di due va-

riabili e cioe per 1'operatore : L'(Z)

=

AZ

xx

+ B Z + aZ + bZ , tt x t

A>D, B> D.

Si dimostra da prima che, se in

P (x, t ) e-T la Z alSSume il suo valo-

re massimo M • aHora e ancora

Z

o

=

0

0

M in ogni punto della caratteristica

t = t , appartenente aT. o

Questo

teorema e una quasi immediata conseguenza del seguente

Lemma: se Z assume il valore massimo in punto P' della circonferenza di

-

547-

G, Sestini

un cerchio appartenente aT, llascissa Xl di pi coincide con quella del centro del cerchio, La proprieta vale anche per un contorno ellittico, avente gli assi paralleli agli assi

x

e

t.

ei riJeriremo ora al caso

B=O e b = -1 , c ioe al caso che interes-

sa i problemi di diffusione. Quanto ora ricordato implica che se in T

(4)

L"(Z)

AZ

xx

+aZ

x

-Z

t

e:

>0

e in P E. T la Z(P) raggiunge il suo val ore massimo, allora si ha Z(P)=: o 0 =: Z(P ) , per ogni P appartenente alIa caratteristica t =: t passante per o 0 P e appartenente aT, o

Premesso questo si indichi, per ogni dei punti

t

PET, con

Q € T,

S(P) llinsieme

che possono

essere

collegati con P mediante curve semplici, appartenenti a T, sulle quali, nel senso da Q a P, la tenon decrescente , Ebbene se per P E: T e -----

Z(P) massima ,

0

0

aHora si ha Z(P) =: Z(P ) per P E S(P ) ; 0-

0

in questa affermazione, valida naturalmente anche per l'operatore

o

,~

L( Z) definito

in (3), sta il principio forte di massimo, Ne segue in particolare che Ia soluzione

regolare della equazione

L"(Z) = 0 pub raggiungere i suoi valori mas simi

(0 minimi) suI contorno del campo di definizione e da cia la possibilita

di limitazioni "a priori" della cercata soluzione per un particolare problema al contorno associato all'equazione L" (Z) =: 0 Ie dei tipi considerati

(L( Z) ::: 0 , L I (Z) =: 0 ) ,

0

ad altra piu genera-

- S48-

G. Sestini

Generalizzazione di T. KUSANO (1953). Oi questo molto importante "principio forte di massimo II si hanno generalizzazioni di A~ FRIEDMAN

(6)

e di T. KUSANO

[7] . Accennere-

mo soltanto a quella di KUSANO , perche riguarda una equazione quasili ~ neare e cioe del tipo :

~

(5)

con a hk ed f

II grad

zII<

\k (P, t, Z, grad Z) Z \

x - Zt" f(P, t, Z, grad Z), k

funzioni definite in un dominio

co]

e limitate in ciascun

D'::[(p, t)t:D,

Iz I < co,

sottoinsieme compatto di D'.

II principio forte di massimo per la (5) , la cui dimostrazione ricalca passo

a passo

quella del criterio

di NIREMBERG, e il seguente :

se esiste una funzione semicontinua, positiva H(P, t, Z , grad Z) tale che :

Z f..K ~

Q

(P, t, Z, grad

rispetto a punto P

--0

Z

Z)~

a

hk

A.h A k ->

Hllll12, per ogni Ah reale

D' ; se inoltre la f(P, t, Z, grad Z)

e gradZ e si ha

f(P, t, Z, 0)

~

e lipschitziana'

0 per Z ~ 0 , allora, se in un

€: D - la ZIP0 ) e inassima , si- ha Z(P) " ZIP ) per Pc SIP ), 00

essendo SIP) I 'insieme appartenente a 0, gia definito. Conclusione. Questa una rapida e molto sommaria scorsa sui cosidetti "principi di massimo" per Ie equazioni

di tipo parabolico e un rapidissimo cen-

no suI lora utile impiego nella risoluzione di problemi al contorno originati da questioni

fisico-matematiche, ad es. del tipo di queUe che hanno

formato oggetto di studio il questo

Corso.

n fatto

che tra il principio di

GEVREY e quello di KUSANO intercorrano cinquanta anni giusti, avvalora l'importanza e I 'attualita della questione, e, a mio avviso, giustifica I 'aver-

- 549-

G. Sestini ne fatto oggetto di un seminario.

BIBLIOGRAFIA .

1

A .. FRJ.EDMAN. Partial differential equations of parabolic type, Prentice-Hall In c. , Englewood Cliffs N. J. 1964 .

2.

L. NIRENBERG , A strong maximum principle for parabolic equations , Comm. Pure Appl. Math. 6, 167-177 (1953).

3

M. GEVREY , Equations aux derivees partielles dUo tipe parabolique, J. Math. Pures Appl. , (6) 9,306 - 471 , 1913 .

4.

E. E. LEVI, Sull1equazione del calore, Ann. Mat. Pura Appl. (3), 14, 187-264, 1907 .

5,

M. PICONE, SuI problema della propagazione del c3.lore in un mezzo privo di frontiera , conduttore, isotropo omogeneo, Math. Ann. 101,701-712, 1929.

6.

A. FRIEDMAN. Remarks on the maximum principle for parabolic equations and its applications, Pacific J. Math. 8, 201-211,1958.

7.

T. KUSANO, On the maximum principle for quasi-linear parabolic eqvations, Proc. Japan Acad., 39, 211-216, 1963.

Stampa; Editorial. Gra/lea . Aoma . reI. S800154