Advances in kinetic theory and computing : selected papers

dx

2 dx

dv

= o, i e [o,

a?
J{x,v)dv,

2

e ra, i 6 [0,1],

(2.16) (2.17)

'CO

/ ( O . v ) - g{v),

v > 0,

=0,

[Q)

V

= 0,

v<0,

(2.18)

(2.19)


(2.20)

where g(v] = F G(Vr,v). T h e assumptions (2.8), (2.12) are now easily expressed by w r i t i n g g i n terms of a given function i(w) and a scaling factor A according t o : g(v)

= A (").

(2.21)

7

Indeed, formula (2.21] insures t h a t the thermal velocity associated w i t h g is of order e. N o w , we choose the scaling factor A such that the (scaled) injected current j is of order L Since, 3

roo

h

"

/ JO

the simplest choice is A =

roo

vg{v}dv e

2

= X(

/ Jli

w~i{w)dw,

- 2

Therefore, wc shall now be concerned w i t h problem (2.16)-(2.20) w i t h

(2.22)

7

g{v)

= «•(*] -

where 7 is a given function u £ have : "a- =

V

t

=

=

9 l

/

">0,

(2-23)

—• -y(v) e 1 R . Note t h a t w i t h (2.23), we

[0,00)

ff»df

+

= -n-n

=

n

y

/

7(^)rfw,

(2.24)

(2.25)

f 9 ( t > ) d f = i-

J-, =

/ Jo

KJ7(uj)dw,

( ) V ( w ) * ' = EtTi

f-r =

/ Jo

i" 7(i") ^ ,

£

/ Jo / Jo

^7(;).

n

%

-

( J

1

)

1

3

(2.26)

= * ( - ) *

(2-27)

These are respectively the (scaled) injected density, injected current, injected kinetic energy, and the t h e r m a l velocity of the injected particles. A g a i n , we p o i n t out t h a t v * — 0 ( e ) , w h i l e j c ~ O ( l ) , w h i c h is consistent w i t h o u r requirements (2.8) and (2.12). We also note t h a t n - = O ( ^ ) , w h i c h can be connected w i t h the fact t h a t p given by (2.23) does n o t converge i n the d i s t r i b u t i o n a l sense. (However, note t h a t vg (v) —• 6(v) i n the d i s t r i b u t i o n a l sense, where 6 is the d e l t a function.) g

s

g

6

l

T h e s o l u t i o n of c


(2.16)-(2.20), is now dependent

E

on e. We shall denote by / ( i , v),

the d i s t r i b u t i o n function, electrostatic potential and electron density. c

For the sake o f completeness, we recall here t h a t f {x,

c

E

v), >p {x) and r i ( i ) satisfy

the system of equations :

»Sr + l € i r ox dx a

<

2 dx

0

= '

*ePM],*.*.R,

(2.28)

dv

2

p

= '{x) r[0,v) n

= / = g'(v)

F&iV)dv,

/'(l,w) -

/(0) -

ie[0,l], v>0,

=

0,

0,

(2.29) (2.30)

v < 0,

(2.31)


1.

(2.32)

x e [0,1].

(2.33)

A n i m p o r t a n t q u a n t i t y is also the t o t a l current :

vf*{x,v)dv, -00

I n t e g r a t i o n o f (2.28) w i t h respect t o v shows t h a t f €

are now concerned w i t h the behaviour as e —• 0 of f ,

is independent of x . We E

£

i£> , ra a n d

j

E

8

2.2.

The limit e-> 0

T h e o r e m 2.1 ([15]) There exists a sequence (r.f') (2.32) such that , as e —• 0, we have : f

e

-> /

in

M (\0,l\x]&) h

u? ->y>

in

of solutions

weak star

C'([0,1]) strong

of (2.28)-

(2.34)

(2.35)

where Mb([0,1] x IR) denotes the space of bounded measures on (0,1) x IR. The limit distribution function f is given by . f(x,v)

=

J.

6{v-

^pjx)),

(x,v)

€ |0,1] x IR.

(2.36)

ip is the solution of % « dx 2

n(x) = - T L = , ^fipjx~

^(0) = . 0 , ^ ( 1 } = 1,

(2.37)

and j = lim f where

is given by

=

titfa,^

(2-38)

(2.25).

C o m m e n t s o n t h e o r e m 2.1 : F o r m u l a (2.36) shows t h a t the l i m i t d i s t r i b u t i o n function is t h a t of a monokinetic beam of particles (i.e. a delta function i n the v variable), e m i t t e d from the cathode w i t h velocity 0 (because <Jtp(x) — 0, for £ = 0). Before giving a rigorous proof, we show how t o understand i t intuitively. I t is well k n o w n t h a t any solution / of (2.28) is constant along the characteristics. These are curves ( X ( t ) , V ( f ) ) , solutions of the differential system :

2

T h e energy invariant W(x,v) (2.39)

= v -
| ( o w

-

provides a first integral of the system

= °>

2

<-

40)

so t h a t the characteristics are contained i n the curves defined by ' 2

v

— ip(x) = constant.

(2-41)

T h e solutions of (2.28) are constructed by t r a n s p o r t i n g the b o u n d a r y d a t a a l o n g the characteristics.

9

V

t

F i g u r e 1 : Phase p o r t r a i t o f characteristics o f the stationary V l a s o v e q u a t i o n associated w i t h the p o t e n t i a l (p or (p i f x^>0 E

F r o m the assumption (2.23), as e —• 0, all the mass of the b o u n d a r y d a t a concentrates at the p o i n t ( z = Q,v = 0 ) . I t is n a t u r a l t o expect t h a t all the mass of the s o l u t i o n concentrates on the characteristics issued from t h i s p o i n t . F r o m (2.41), t h i s characteristics is i n c l u d e d i n the curve defined by the equation v = ± v / ^ ( r ) . E q u a t i o n (2.36) e x a c t l y states t h a t the l i m i t d i s t r i b u t i o n function is a measure s u p p o r t e d by the characteristics v — y/
E q u a t i o n (2.38) is one o f the most i m p o r t a n t conclusions of t h e o r e m 2.1 ; I t states t h a t t h e t o t a l current j flowing t h r o u g h the diode coincides w i t h the injected c u r r e n t j - , , o n l y i f j is below t h e value I f j - exceeds | , the t o t a l c u r r e n t j saturates t o the value g. T h i s remarkable value w i l l be referred t o as the scaled Child-Langmuir current . y

t

10

F i g u r e 2 : t o t a l current as a f u n c t i o n o f the injected current

fat

-

g.

(2-42)

B y going back to the unsealed physical variables, formula (2.42) turns i n t o (1.1). Indeed.it is an easy matter to check t h a t the current u n i t corresponding to the scaling (2.15) is

J = eNV

L

(2.43)

=

T h e plot of j as a function of j - , is given i n figure 2, and displays the saturation phenomenon which takes place at the value j ' c o T h e saturation phenomenon is due to the occurence of a potential barrier which reflects a fraction of the e m i t t e d electrons back to the cathode. These reflected electrons do not contribute t o the t o t a l current and reduce i t compared w i t h the e m i t t e d current. Indeed, we shall see i n section 3. t h a t for j , > jch and £ small, tp is negative close t o 0 (see figure 3). Since

0), but, due to the non-linearity of the problem, its effects are s t i l l present i n the saturation of the current j to the value JCL • I n the process S —• 0, an " i n f i n i t e s i m a l " potential barrier a u t o m a t i c a l l y adjusts so as to allow a transmission of the current exactly at the value j e t , whatever the injection profile 7 is. T h e analysis of the "infinitesimal" potential barrier is made possible by a boundary layer analysis that, w i l l be summarized i n section 3. c

c

E

c

c

F i g u r e 4 : Phase p o r t r a i t o f characteristics o f the s t a t i o n a r y V l a s o v e q u a t i o n associated w i t h the p o t e n t i a l E

(p w h e n x

E c

>0

12

2.3.

Sketch of proof of theorem 2.1

I t is interesting t o give a sketch o f the proof of theorem 2.1, because i t explains w h y the m a t h e m a t i c a l analysis is still incomplete in many cases of application of t h i s theory. T h e first proof of theorem 2.1 was given i n [15] by use of explicit solutions of the p r o b l e m (2.28)-(2.32). T h e n , another proof based on functional a n a l y t i c a l arguments was given i n [10], [11], and used for extending the m a t h e m a t i c a l analysis to the case of cylindrical or spherical diodes. T h i s second proof uses an uppersolution concept which first appeared i n [21], I t can be used t o extend the theory to more complex situations like magnetized flows, collisional problems, etc, as we shall see i n [5] and [6] (see also references cited i n section L ) . However, the first proof is more i n t u i t i v e , and gives a better physical understanding of the p r o b l e m . Since b o t h methods are usefull, we shall outline b o t h of t h e m . F i r s t m e t h o d : p r o o f o f t h e o r e m 2.1 v i a e x p l i c i t s o l u t i o n s o f t h e p r o b lem

(2.28)-(2.32)

Existence of solutions for (2.28)-(2.32) follows from [17] or [21], under very m i l d assumptions on the b o u n d a r y d a t a j(v). Furthermore, i t is an easy m a t t e r to show t h a t 'p is s t r i c t l y convex. Therefore, i t reaches its m i n i m u m value

0, ip < 0, i n w h i c h case >p is nonincreasing on [0. x ] and nondecreasing on [ x , l ] (see figure 3 ) . I n each case, i t is easy t o draw a phase p o r t r a i t of the characteristics. T h e characteristics issued from x = 0 at the critical velocity v = v£ = ±y/—\p\ is of particular importance. I t corresponds t o particles w h i c h reach the top of the potential barrier \p\ w i t h zero velocity, and w h i c h can either go back t o the cathode or continue t h e i r way t o the anode. Therefore, this characteristics is singular, w i t h four branches meeting at the point (x = x\,v = 0). Using t h i s critical characteristics, we can p a r t i t i o n the phase space [0,1] x IR i n three regions (see figure 4) : E

c

c

!

l

c

E

E

c

c

c

E

E

=

{ ( x , f ) , 0 < x < 1, v > v V ( z ) - & }

f ! | = { ( z , u ) , 0 < x < x%, v

2

Q | = {(x,t>), 0 < x < 1, v < sign{x

(2.44)

< < / ( x ) -
(2-45)

-x*)vV(*)

(2.46)

s

w i t h EJ^ possibly e m p t y i f X . = 0. T h e region f l , encloses characteristics corresponding t o particles leaving the cathode w i t h a velocity larger t h a n the critical velocity and thus, travelling t h r o u g h the diode up to the anode. T h e region fif, encloses trajectories of particles leaving the cathode w i t h a velocity smaller t h a n the critical velocity v\, and thus being rcflertpri by the potential barrier back towards the cathode. T h e region Q j corresponds t o particles e m i t t e d from the anode. Since no particles are e m i t t e d from the anode (see boundary c o n d i t i o n (2.31)), the d i s t r i b u t i o n function is zero along the characteristics contained i n Since, all the characteristics reach the b o u n d a r y ( 0 , 1 } x IR, the solution of (2.28) is uniquely determined from the boundary d a t a g (v) according t o the following formula : c

13

(2.47) (2.48) F r o m (2.47) a n d (2.48), i t is easy t o o b t a i n e x p l i c i t formulae for the density and the c u r r e n t j : for 0 < i < x we have : e

n

e

E

c

wg (m)dw

w h i l e for r

£

(2.49)

< x < 1, we have : CO

tug (uj)diu E

(2.50)

/ and similarly :

roo

<"9 ( tu) dtw

(2.51)

E

/ , roo

-

J

(2.52)

r ^ 7 M7(u

I t is also i n t e r e s t i n g t o evaluate the kinetic energy defined by : +oo t

of t

/ -oo

V f'(x,v)dv.

A g a i n , the e x p l i c i t formulae (2.47) and (2.48) allows to give an e x p l i c i t expression : for 0 < x < i f . , we have :

s

-

poo e

(2.53)

t (x)

=

/

c

2

c

wg (w)\Jw

+ ip (x) dw +

/"V ^ /

c

2

wg (w)\Jw

4- ^(x)

dw, (2.54)

w h i l e for i f < x < 1, we have : oo E

t (x)

-

E

2

_wg {w)y/w

/

E

+ip (x)dw.

6

(2.55)

F r o m (2.51) or (2.52), i t is readily seen t h a t j does not depend on x. T h i s fact is also verified by i n t e g r a t i n g equation (2.28) w i t h respect t o tJ. A similar t y p e of i n f o r m a t i o n can be obtained for t by m u l t i p l y i n g (2.28) by v and i n t e g r a t i n g w i t h respect t o v. T h i s leads t o : c

14

or :

E

C=(i) E

^ ( ^ r f = * independent

of x

(2.57)

e

c

Note t h a t n and t are given by different expressions i n the cases x < x , a n d x > x . T h i s is due to the presence of reflected electrons i n the region x < i f . , while no such electrons exist i n the region x > x . T h e reflected electrons do n o t c o n t r i b u t e t o the current (see (2.51)) b u t they do c o n t r i b u t e t o the density a n d t o the t o t a l kinetic energy. c

c

c

c

T h e energy estimate (2.57) is essential t o obtain the following a p r i o r i estimates: L e m m a 2 . 2 There exists a constant C independent

of s such that

s

l k | | w i - = n w " . i ( ( o , i ] ) < C,

(2.58)

£

l l f i M 0 4 ] x K ) + l|n H(..((0.lll ^ (

Proof.

C

2

-

5 9

< - >

S k e t c h o f p r o o f o f L e m m a 2.2 c

F r o m the explicit expressions (2.54) and (2.55) o f t ,

i t is fairly easy t o prove

t h a t i t is bounded i n L ° ° 11^ 111* CPU)) < C. Therefore, since k

c

(2.60) c

defined by (2.57) is independent o f x, dip /dx

w i l l be uni-

formly bounded as soon as we show t h a t i t is bounded at one p o i n t . More precisely we show t h a t there exists a constant C, independent of e, such t h a t for any £, there r

exists one p o i n t x - i n [0, l j , such t h a t . < C s

(2.61) 1

L e t us suppose first t h a t i f = 0 (i.e. t h a t f is non decreasing). T h e n , since ip is convex and reaches the value 1 at x — 1, we obviously have : | ^ ( 0 ) |

< 1,

(2.62)

w h i c h shows t h a t i ' = 0 is a convenient choice i n t h i s case. O n the other h a n d , i f i f > 0, t h e n *p reaches its m i n i m u m at i f and we have : !

w h i c h shows t h a t i

£

c

= x

c

is the convenient choice i n the second case.

F r o m (2.57) i t follows immediately t h a t

15

I ^ H t ~ < | o , i ] ) < C,

(2.64)

T h e n . Poisson equation (2.29) yields : -1 J2 I K I t a M i - l X 2 f * l - t f « - f « I «

M B

• E s t i m a t e (2.59) allows t o show t h a t f sures. Indeed, we have :

fn

£

(p

1

converges i n the weak t o p o l o g y o f mea-

-

f

in

M ([0,l\xtR)

-*

n

in

M (\0,1])

—

C°(\0,1)}

weak star,

b

(2.66)

weak star,

b

(2.67)

strong.

(2.68)

F u r t h e r m o r e , i t can be shown t h a t

0. T h i s allows t o show : 1

L e m m a 2.3 Proof.

The limit distribution function

f is given by

(2.36).

S k e t c h o f p r o o f o f L e m m a 2.3 c

I n t h e following lines we shall sketch why formula (2.36) is valid i n [ , 1] x IR for any £ > 0. T h e v a l i d i t y o f (2.36) i n [0,1] x IR involves a delicate argument t o show t h a t n o d e l t a function located at (x,v) = ( 0 , 0 ] can appear i n the l i m i t . We refer the reader t o [15] for the details of t h i s p o i n t . Let i>(x, v) be a s m o o t h test function w i t h compact support i n (0, l j x I R . According t o w h a t precedes, we have (x ,ip ) — ( 0 , 0 ) . T h u s , flf, -> 0 and for sufficiently small e, the s u p p o r t o f tp is entirely contained i n Sl\ U fif. Therefore, w i t h (2.47) and (2.48) we can w r i t e E

E

c

f f'ipdxdv [O.iJxR

=

f ~j{JJn; *

<

p£)il>(x,v)dxdv,

£

2

and w i t h the change of variables w =

flbdxdv '[0,l|xR

=

I Je Jf

E

^Jv —

f, W7_(}") ,, J. =*k Jc +
(2.69)

e

22

/

2 i i t 2

+

^foUrf-i:

e

(2.70) c

where > 0 is such t h a t suppip C ] £ , l ] x IR. N o w , up t o the e x t r a c t i o n of a subsequence, we may assume t h a t —tp /e converges t o some value 6 £ [0, ooj. T h e n , Lebesgue's d o m i n a t e d convergence theorem appliees t o (2.70) and leads, as e -> 0 t o : €

2

16

/

= js f

ftdxdv

i<(x,^fjx))-p=

(2.71)

with js =

/

wy(w)dw

< J't

I n particular, by considering test functions ip such t h a t ip(x,v) s m o o t h , one realizes t h a t

(2.72) — jjf(x), with #

js = j -

(2.73)

W i t h (2.71) a n d (2.73), we o b t a i n t h a t / is given by (2.36).

• E q u a t i o n (2.73) does not give any i n f o r m a t i o n on the value of j since the value of S is n o t k n o w n (and could possibly depend on the choice of the subsequence). Now, we show t h a t the value of j is unique, and t h a t the whole sequence f converges t o / . For t h i s purpose, we need to analyze the l i m i t Poisson equation (2.37). T h i s is carried t h r o u g h by elementary i n t e g r a t i o n techniques i n the next l e m m a : s

L e m m a 2.4 « If j > jet. Oforx £ ( 0 , 1 ] . •

such that ip(x)

>

WS: S 3CL- problem (2.37) has a unique solution suck that Oforx (0,1].

€

• Furthermore, j = jcL dtp/dx(0) = 0. Proof.

problem (2.37) has no solution

is the unique value of j such that the solution

satisfies

s k e t c h of t h e p r o o f of l e m m a 2.4

M u l t i p l y i n g equation (2.37) by dip/dx(x)

leads t o a first integral of the equation: * W - t , dx

("4)

w i t h 5 — dip/dx(0). T h e n , equation (2.74) can easily be integrated, and the value o f S is found by m a t c h i n g the boundary value ip(l) = 1. T h i s leads to the equation :

s: .

, -**

= 1.

(2.75)

I t is readily seen t h a t the m a p p i n g 5 — i j is monotonically decreasing from [0,1] onto [ O . j c i , ] - T h e conclusion follows.

•

!

T h e proof of theorem 2.1 is now almost complete. Since the l i m i t of j as e —* 0 exists, we necessarily have, by v i r t u e of l e m m a (2.4) : 0 < j < jet- We also have, from (2.73) : 0 < j < j (since obviously j ' < j - , ) . Therefore, we have : 7

4

17

0 < j < Mm(j , 7

J C L

)

(2.76)

To show t h a t j is a c t u a l l y equal to Min(j-f,j L), we need a technical l e m m a , the p r o o f of w h i c h can be found i n [15] and w h i c h w i l l be o m i t t e d i n the present paper. C

L e m m a 2.5 End

The convergence

of tp' to ip holds in Ike C ' f l O , 1]) topology.

o f t h e p r o o f o f t h e o r e m 2.1

Let us Erst assume t h a t dtp/dx(0) > 0. T h e n , by L e m m a 2.5, for e s m a l l , we have : d

0. I t follows t h a t ip\ - 0, and by the aid of the explicit f o r m u l a (2.52), t h a t j = j y , Therefore, by l e t t i n g e —• 0, we get j = j - , < JCL- L e t us now assume t h a t d

c

j -

M*n(J„jc/.).

(2.77)

T h i s concludes the first proof o f t h e o r e m 2 . 1 . R e m a r k 2.6 T h e characterization of the l i m i t current j has been o b t a i n e d from an analysis of the l i m i t p r o b l e m (2.37). Therefore, the convergence theorem cannot be used t o o b t a i n i n f o r m a t i o n a b o u t the existence of solutions of the l i m i t problem. T h i s r e m a r k w i l l be crucial i n more complex geometries, where no simple theory like L e m m a 2.4 is available.

T h i s p r o o f relies o n the a v a i l a b i l i t y of an explicit s o l u t i o n (2.47) and (2.48) for the Vlasov e q u a t i o n (2.28). T h i s e x p l i c i t s o l u t i o n provides (i) : a u n i f o r m L°° estimate of the t o t a l kinetic energy £ thanks to the expressions (2.54) a n d (2.55). T h i s L°° estimate is used, v i a the energy i d e n t i t y (2.57) t o give a u n i f o r m L ° ° b o u n d on dtp' /dx, w h i c h i n t u r n , provides a L ' b o u n d on fand j t £

e

c

( i i ) : an e x p l i c i t f o r m u l a for the l i m i t / of f v i a the weak formulation (2.69). I n more complex geometries, n o such explicit f o r m u l a is available. However, the above p r o o f c o u l d be carried t h r o u g h i f the exact solution w o u l d be replaced by a suitable estimate. T h i s can be achieved by the upper-solution concept w h i c h is developped i n the following second proof. Second method Vlasov equation

:

proof of theorem

2.1 v i a upper-solutions

of

the

(2.28)

T h e u p p e r - s o l u t i o n concept for the Vlasov equation first appeared i n [21] for the existence p r o o f of s t a t i o n a r y solutions. T h e n , i t was used i n [10] a n d [11] for the analysis of t h e C h i l d - L a n g m u i r a s y m p t o t i c s of c y l i n d r i c a l l y or spherically s y m m e t r i c diodes (see following sections). However, i t applies as well for the simpler case o f plane v a c u u m diodes, a n d leads t o a somewhat simplified proof ( a l t h o u g h less t r a n s p a r e n t from a physical v i e w p o i n t ) (see [9]). For the sake of simplicity, the p r o o f is r e s t r i c t e d t o c o m p a c t l y s u p p o r t e d b o u n d a r y d a t a 7 ( f ) , b u t i t seems to

18

be o n l y a technical matter t o extend i t t o boundary d a t a w i t h suitable decay at infinity. T h e first step is t o construct an upper-solution of the Vlasov equation (2.28), w h i c h w i l l give a first set of a p r i o r i estimates, i n particular on the s u p p o r t of / . T h e key p o i n t is t h a t such upper-solutions can be constructed as functions of the invariants of the characteristics of the Vlasov equation. We assume t h a t £

TM

-

2

C(M ),

(2.78)

a n d t h a t there exist 2 constants M > 0 and V > 0 such t h a t : 0

0 < G(u) < M,

2

SuppG

C { u , 0 < u < V }.

(2.79)

0

T h e n , the function

**(*,«)

v

= G (

2

~ £

l

x

)

)

(2.80)

is a function of the energy invariant and, as such, is an exact solution of the Vlasov equation (2.28). Furthermore, i t satisfies :

1

2

v

£

=

-^G(^)

e

=

i c ( ^ ) > / ' a ^ 0 ,

F (0,v) F (l,v)

= f(0,v),

V>0,

(2.81) D O .

(2.82)

T h u s , F'- is an upper-solution of the Vlasov equation (2.28). T h e s o l u t i o n if* of the Vlasov-Poisson problem (2.28)- (2.32) is not unique i n general, b u t i t can be shown (see (21]) t h a t there exists at least one solution / such t h a t : £

c

c

f {xj)

< F (x,v),

V(x,v)

e [0,1] x IR.

(2.83)

We shall consider one of these. F r o m (2.79) and (2.83), we immediately o b t a i n bounds on / 2

Suppf

c {(x,u), 0 < v

s

2

E

and on its support:

2

-

(2.84)

M If]

<

(»,i.)£[0,l]xIR,

(2.85)

and from (2.84), we deduce : 2


2 0

E

n ( x ) = 0.

(2.86)

T h e n , by a m a x i m u m principle argument, we show t h a t necessarily, s

-e V

s 0

< i / ( i ) < 1, M

ie[0,l).

(2.87) c

T h i s estimate already gives an L ( 0 , 1 ) weak star convergence of ip f>0.

towards

19

W e n o w again decompose the proof into the same Lemmas 2.2 t o 2.5. Proof.

S k e t c h of p r o o f of L e m m a 2.59 : 2 n d v e r s i o n l

E

We are l o o k i n g for an L estimate o f / , i n order t o get weak star convergence of f t o w a r d s / in M {\0,1] x IR), However, the estimates (2.84), (2.85) are n o t sufficient. For convenience, let us introduce e

b

s

V,

2

2

2

= {veWl/0
(2.88)

0

so t h a t we can w r i t e , w i t h (2.84) : Suppf

= {(z, ;)£|0,l]xIR, ;eV 1

1

E t ( l )

}.

(2.89)

Easy c o m p u t a t i o n s show t h a t there exists a constant C, independent of e and ip, such t h a t £

M e a s ( V ) < Ce.

(2.90)

I t follows from (2.85) and (2.90) t h a t f™ M C / f(x,v)dv < -= Mea (V , ) < - , (2.91) J-oo * % w h i c h does n o t give any useful bound. However, the same t y p e o f c o m p u t a t i o n shows t h a t re ix.) =

S

k

v dv

2

< Ce ,

v (x)

Vfc>l.

(2.92)

K 6

I n p a r t i c u l a r , (2.92) shows t h a t the t o t a l kinetic energy t (2.53) is bounded. We recover a feature w h i c h i n the first proof was deduced from the analytic expression of the s o l u t i o n . T h e n , the use of the energy i d e n t i t y (2.57) provides the conclusion of L e m m a 2.2 i n the same way as i n the first proof. Therefore, we can e x t r a c t a subsequence such t h a t (2.66), (2.67), (2.68) holds t r u e . F u r t h e r m o r e , from (2.87), we have t h a t

0.

•

Proof.

S k e t c h o f p r o o f o f L e m m a 2.3 : 2 n d v e r s i o n

T h e s u p p o r t estimate (2,84) i m m e d i a t e l y gives t h a t Suppf

C {(x,vj,v

= v ^ W h

(2-93)

a n d therefore, / can be w r i t t e n : f{x,v)

= n(x)6(v-

w i t h n £ A4t([0, l j ) .

yRz)),

(x,v)e

[0,1] x IR,

(2.94)

B y the fact t h a t the current j is independent o f x, we

deduce t h a t : y-CO

/

J —oo

vf{x,v)dv

= JifslvVx)

(2.95)

20 is independent of x. To go further, we need the conclusion of L e m m a 2.5 which can be o b t a i n e d at this level of the proof by a careful use of the energy identity. Again, we shall not discuss this p o i n t and refer t o |10] for details. T h e n , i t can be deduced t h a t

> 0,

V z e [0,1).

(2.96)

Therefore, from (2.95) we have nf>) = —===, vV(x)

on (0,11.

(2.97)

A last technical p o i n t is t o show t h a t formula (2.97) holds t r u e on [0,1], t h a t is n does n o t c a r r y any delta function located at x — 0. A g a i n , we shall skip this proof. F i n a l l y , we can w r i t e / according to (2.36), w h i c h concludes the proof of L e m m a 2.3

•

The proof of L e m m a 2.4 and the completion of the proof of theorem 2.1 are done in a similar way as i n the first proof. T h i s ends the second proof of theorem 2.1 T h i s second proof does not require the knowledge of an explicit formula for the solution of the Vlasov-Poisson problem (2.28)-(2.32), b u t o n l y t h a t o f an uppersolution. Therefore, i t w i l l be applicable i n all the situations where such an uppersolution is available and allows fine enough estimates. Examples of such situations w i l l be given i n the next section, concerning c y l i n d r i c a l l y or spherically s y m m e t r i c diodes, or i n [6], for semiconductors. B u t before passing to the analysis of these cases, i t is interesting to have some i n f o r m a t i o n on the b o u n d a r y layer (or spacecharge layer), which is responsible for the c u r r e n t - l i m i t a t i o n phenomenon.

3.

A n a l y s i s of the boundary layer

T h e analysis of the boundary layer is interesting for several reasons : first, i t provides a precise knowledge of the shape of the "infinitesimal" potential barrier w h i c h is responsible for the current l i m i t a t i o n . B u t also, because this analysis can be done by explicit calculations, i t gives precise estimates of the convergence rate of ip and j , towards
e

T h e boundary layer equation is obtained by rescaling the system (2.28)-(2.29). We introduce the following change of variables . 2

x = i»"jf, v = e6, f = i f c , ifi m c ^ , n • 1 *

(3.1)

T h e resulting equations for the d i s t r i b u t i o n functions fi(£, 6), and for the potenc

t i a l V ( ) are :

21 ^cW

1 dip* dhf

»„

e

3

h {W)dB,

a

£;E[0,e- / ],

(3-3)

-co £

/i (0,(5) -

f

fi (e-

3 / 2

,f)

7

(t?),

(3.4)

= 0, fl < 0, E

1&*(0) = 0,

8>0,

V (£

_ 3 / 2

) = e

(3.5) -

2

(3.6)

T h e scaling (3.1) can be shown t o be the o n l y one such t h a t e does not appear in the equations (3.2) and (3.3) and such t h a t the cathode b o u n d a r y c o n d i t i o n (3.4) is independent o f e, W h e n e —> 0, we formally end u p w i t h the following p r o b l e m :

^

= u(@=j

h(i,8)d9,

A(0,fl) = 7 ( 9 ) ,

V(0) -

fJ>0,

g — oo,

hfi,9) ->0,

0.

- £

?e[0,co],

4 / 3

(3.8)

(3.9)

6 < 0,

(3.10)

as£—c<>.

(3.11)

T h i s last p r o b l e m is a m o d e l for the cathode boundary layer. We show t h a t i t a d m i t s a unique s o l u t i o n under the c o n d i t i o n t h a t j > J o t , w h i c h is the regime w h i c h displays a current l i m i t a t i o n and a p o t e n t i a l barrier. We also show t h a t , i n t h i s case the rescaled p o t e n t i a l tp converges to the b o u n d a r y layer profile ip, and t h a t the difference is of the order of e. B y going back t o the " unrescaled" p o t e n t i a l •p , we o b t a i n an a s y m p t o t i c behaviour of

e

l

E

E

I n the case j - , < jcL-. ^ converges t o + 0 0 , w h i c h shows t h a t the rescaling (3.1) is n o t adequate for the description of the p o t e n t i a l close t o the cathode w h e n no c u r r e n t l i m i t a t i o n is present. However, since no p o t e n t i a l barrier appears i n t h i s case, finding the correct scaling w o u l d not present much interest. T h e existence of solutions of p r o b l e m (3.2)-(3.6) (or to some extent, of (3.7)(3,11)) can be o b t a i n e d d i r e c t l y from the theory of [21], However, i f the injection profile 7 is n o t increasing, a simple calculation yields the result and provides more i n f o r m a t i o n t h a n the abstract t h e o r e m of [21]. T h i s m e t h o d first appeared i n [17], a n d is intensively used i n [16], a n d i n [8], L e t us summarize i t now. S o l u t i o n o f t h e r e s c a l e d V l a s o v - P o i s s o n p r o b l e m for finite e ( 3 . 2 ) - ( 3 . 6 )

22 T h e rescaled energy T*(x) is defined by : +oo

/

(3.12)

f?VM)
L e t us introduce H and Tpl according to : g ~ « r * ^ 4

t

f £ » r % t

T h e n , ( £ , V £ ) is the p o i n t of m i n i m u m of V

£

(3.13) C

T h e rescaled energy T {X)

is

therefore given by the explicit formula (which is deduced from the rescaling of (2.54) and (2.55)): for 0 < f < £ , we have : £

oo

/ /^s and for £

£

f/—$i _ e (9)^e

2

+ ^lOdd

1

+ 2

9y(6)y/W+Wf)d9,

(3.14)

=

*V ¥tQ 3

< £ < e~ ^,

we have :

s

r (0

=

/

O-rW-yW+WIOde.

(3.15)

T h e n , the energy i d e n t i t y (2.57) is recast i n the new variables according to :

Let us first assume t h a t £

E

> 0 : T h e n , we integrate the energy i d e n t i t y (3.16) E

on

E

and use the fact t h a t ( d t / f / d i ) ( £ ) = 0, we get : (*£)"
= 4 / "

0 7 ( ^ ( ^ + ^ ( 0

-

y/PT*i)d8.

(3.17)

T h i s first integral allows t o integrate the equation :

f**tt)

dip

T h i s equation can be p u t i n i m p l i c i t form. We let Jt*«) = V - ' ( 0 - < ,

X

(

3

-

1

9

)

and we rewrite (3.18): ftfttaW] with

- € - 6 ,

(3.20)

23 and

GiWc.x)

=

\ ei(e){^8 •'v *J

2

+ ip} +

x

-

Je

2

(3.22)

+ i>%)de.

C

I n the same way, the i n t e g r a t i o n of the energy identity (3.16) on [ 0 , ] leads t o the following i m p l i c i t r e l a t i o n , defining V on the interval ( 0 , £ | ££

£

£

(3.23) with

(3.24)

Ja< and

Now, V f E = e~ ' and * equations for : 3 2

£

m

E

u

s

t

= e~

he determined from the b o u n d a r y conditions (3.6). L e t T h e b o u n d a r y conditions (3.6) lead t o the following

2

K < f t - 0

-

S*-&

(3.26)

-

g

(3.27)

E

W e can e l i m i n a t e (; from these two equations and get : E

F, ( , #

E

- #*) + F tfg, - O

= H

2

£

(3.28)

T h e left hand side o f (3.28) can be shown t o be a m o n o t o n i c a l l y decreasing f u n c t i o n of ipf as soon as 7 is non increasing. Therefore, the existence o f i/> is guaranteed as soon as 5 is i n the range of the left hand side of (3.28). T h i s c o n d i t i o n can be e x p l i c i t e l y found and coincides w i t h the c o n d i t i o n j > jch£

E

y

L e t us now assume t h a t = #g = 0. We can the previous case and integrate the energy i d e n t i t y m u s t be careful t h a t (di/> /d£J(0) is n o t necessarily 0. (dil> /d^)(0) , we o b t a i n t / i ( 0 for £ > 0 i m p l i c i t l y by £

£

E

ftfsv^tf))

= &

use the same m e t h o d as i n (3.16) on [ 0 , ] , Simply, we Therefore, i n t r o d u c i n g 5 = the formulae : c

E

eafti

(3.29)

(3.30) i-OO

Gi(S\ip)

= 5

e 2

+ 4 /

Jo

t?7C) ( ^

+ V< - 9).<0

(3.31)

24 T h e n , again, S' is i m p l i c i t l y determined by the equation £

£

F (S ,4 ) -

E

3

£

(3.32) f

F is a m o n o t o n i c a l l y decreasing function of S , and the existence of 5 t h a t (3.32) holds is guaranteed as soon as E is i n the range o f fg(..,4f?J. condition can be proved t o coincide w i t h j < jcL3

£

£

such This

7

S o l u t i o n of t h e r e s c a l e d V l a s o v - P o i s s o n p r o b l e m for c = 0 ( 3 . 7 ) - ( 3 . 1 1 ) We proceed i n the same way as i n the previous proof : either the m i n i m u m p o i n t (£c> $c) o f the p o t e n t i a l ip is such t h a t § > 0, and ip Is necessarily given by the formula ( w i t h ip — ip + x) '• e

c

or we have

c c

FiWoX(O)

=

FA^.X(O)

=

£

>t" .

(3-33)

5
(3.34)

c

6»~{.

= ip = 0 and i i is given by ( w i t h S — (dip/d£)(0))

:

c

ft(*,»XflJ

=4.

(3-35)

T h e problem is now t o find under which condition the behaviour at infinity (3.11) is satisfied. I t can be verified t h a t no solution of the type (3.35) satisfies such a behaviour. O n the other hand, a solution defined by (3.33), (3.34) satisfies the correct behaviour i f and only i f j > jcL and i f ip is (uniquely) defined by : y

c

e~,(B)d9

£

= jcL-

(3.36)

s

A s y m p t o t i c b e h a v i o u r o f i/> , tp a n d j'~ £

€

£

We are now ready t o study the asymptotic behaviour of ^ , ip , and j " as s —• 0, in the case j > jcLy

P r o p o s i t i o n 3.7

We suppose that j - , > jcL0(f),

Then, we have :

C - 6 +

O'e),

(3.37)

and = 0(f) + 0(e), uniformly for § belonging to any compact subset of

(3.38) fft . +

Proof. S k e t c h of p r o o f of p r o p o s i t i o n 3.7 I t can be shown t h a t the equations (3.28) for ipl and (3.36) for ip can be w r i t t e n in the form : c

H W £ , e ) = p,

(3.39)

25

£

such t h a t , ip appears as the regular branch of solutions of an i m p l i c i t equation passing t h r o u g h t h e p o i n t (e~,ip) = (0, T/J ). T h e n , (3.37) for ib follows from the i m p l i c i t function t h e o r e m and the estimate of £ is an easy consequence. T h e same a r g u m e n t s can be applied for V> (£) by recasting (3.20), (3.23), (3.33) and (3.34), i n the framework of the i m p l i c i t function t h e o r e m i n a similar way. c

c

£

£

•

2

2

e

R e m a r k 3.8 G o i n g back t o the original variables x% = £^ (.l and ifi' = £ ip , we o b t a i n t h a t the b o u n d a r y layer length i f of order 0(e ' ), and the p o t e n t i a l barrier height, of order 0(e ). A n a p p r o x i m a t i o n of the derivative at the origin is p r o v i d e d by the q u o t i e n t 'p\fx\ — 0 ( e ) , which shows t h a t t h i s derivative tends t o 0 as e ^ 0. T h i s is consistent w i t h the fact t h a t the C h i l d - L a n g m u i r c u r r e n t is associated w i t h a zero cathode electric field : (d-p/dx)(x — 0) ~ 0. e

c

3 2

2

1 / 2

T h e constant i n front of the O(e) t e r m i n the estimate (3.7) can be made explicit. e

I t is more usefully reformulated i n terms of the " unresealed" p o t e n t i a l
inhere tcL

We have :


( ^ V ]CL

3

( l

- z

s / 3

) e + o(e),

V x £ [0,1],

(3.40)

is defined by : CO

/

(3.41)

1

f^

We now t u r n t o the current. F o l l o w i n g the same arguments as i n the proof of p r o p o s i t i o n 3.7, we can prove: P r o p o s i t i o n 3.10

We have f

4. 4.1.

= 3CL + 3 f c t £ + o ( e ) .

V i e [0,1],

(3.42)

T h e c y l i n d r i c a l or spherical diode Introduction

A l t h o u g h s t i l l one-dimensional i n space, the p r o b l e m of the c y l i n d r i c a l or spherical d i o d e is n o t a t r i v i a l extension of the cartesian case for the following reasons: (i) : N o a n a l y t i c a l f o r m u l a like (2.47) and (2.48) is available as we shall see later o n . Therefore, o n l y the second p r o o f developed i n section 2.3. w i l l succeed. We can consider t h i s second m e t h o d as a p r o t o t y p e for possible extensions o f the theory

26

direction of election fiow

Anode

Cathode Case p > I

Case p < I

F i g u r e 5 : D i o d e g e o m e t r y i n the c y l i n d r i c a l or spherical case

t o the general d-dimensional case (d > 2). However, many open problems remain t o be solved as we shall see i n section 5. (ii) : We shall also see from these examples t h a t i t is too much t o expect t h a t j = Min(j ,jcL)Indeed, there are situations where a current j larger t h a n the C h i l d L a n g m u i r current jcL can be found (the C h i l d - L a n g m u i r current jcL w i l l be defined as the current w h i c h produces a vanishing normal derivative of the p o t e n t i a l at the b o u n d a r y ) . T h i s example shows t h a t conjectures for the general d-dimensional p r o b l e m are not easy t o formulate, and even less easy t o prove. y

4.2.

The cylindrical diode

We first consider a cylindrical diode which consists of two coaxial metallic cylinders of circular section and of infinite length. We denote by R, and R- the cathode and anode r a d i i respectively. T h u s , we have Ri < R$ i f the cathode is surrounded by the anode and Ri > Ri i n the converse s i t u a t i o n . T h e scaling is chosen similarly as i n the plane diode case except for the length scale t h a t wc choose equal to the radius of the cathode H j , and for the density and d i s t r i b u t i o n function scales N and F t h a t are defined by formulae (2.13) and (2.14) w i t h L replaced by R . T h e scaled c d i s t r i b u t i o n function S (t,vt,v$,v:) now depends on the radial distance r, and on the r a d i a l , azimuthal and l o n g i t u d i n a l velocities, (v , v$, t i ) . T h e scaled cathode radius is (by definition) r = 1, while the scaled anode radius is r = p, where p is the aspect r a t i o of the diode p = Rq/Ri. Therefore, the p r o b l e m is set u p i n the interval N o t e t h a t p > 1 i f the cathode is surrounded by the anode and p < 1 2

x

T

z

c

if the cathode is outside the anode (see figure 5). T h e dimensionless density n electrostatic p o t e n t i a l tp* now depend on r. T h e diineiisiouless Vlasov-Poisson problem is w r i t t e n :

and

27

1

df

c

v*

1 d
9

VM

df

,

,

, (4.1)

T o express the b o u n d a r y conditions on the d i s t r i b u t i o n function, we make the same hypotheses as i n the plane case (see section 2.1.): we assume t h a t no particle is injected at the anode, and t h a t the injection profile at the cathode has a t h e r m a l v e l o c i t y o f order e, and a finite i n c o m i n g current n o r m a l t o the cathode boundary. T h e b o u n d a r y c o n d i t i o n s are thus w r i t t e n :

e

f (p,v)

3

= 0,

v e I R , sign(p-\)v

< 0.

r

(4.4)

where the expressions sign(p — \)v > 0 or < 0 express the conditions for the velocities t o p o i n t inwards the d o m a i n , whatever the configuration of the cathode and the anode are. As i n the plane case, the cathode p o t e n t i a l is set equal t o zero, and the anode p o t e n t i a l is chosen as reference p o t e n t i a l . T h u s , the scaled anode p o t e n t i a l is equal t o 1. T h i s gives : r

e

* (l)

-

0,

y'ip)

-

I-

(4-5) c

N o w , the conservation of the current implies t h a t the current intensity

i (r)

flowing t h r o u g h any cylinder of radius r between the t w o electrodes does not depend on r : E

i (r)

s

= r

v f (r,v)dv

independent

r

E

A g a i n , we are interested i n the l i m i t of f , T h e o r e m 4.11 (4.1)-(4.5)

([10], [11])

E

of r, £

rS[l,p],

(4.6)

E

tp , n , i , w h e n e —• 0 !

f

There exists a sequence (/ ,^ )

of solutions

of

such that , as e —> 0, vie have : c

f

3

— finM ([\,p\

x I R ) weak star

b

c

(4.7)

l

i' -» i € IR,

where I R

3

0 < i < t, =

denotes the set of incoming

3

I R , s i o n ( p — l)v

r

/ J**

(4.8)

V "fM dv

velocities at the cathode : I R

> 0 } - The limit distribution

(4.9)

T

function

f is given by :

3

=

{v €

28

f{r,v)

= r v w )

rtnrf

(t,v) £ [i,p] x I R , 3

6(Vr - \Mr))6(v )6(v ), e

1

(4.10)

is the solution of :,

= 0,

= 1,

(4.11)

Comments on theorem 4.11 : T h e value o f the l i m i t current 7 w i l l be made precise later on. Indeed, i t requires more eare t h a n i n the plane case as we shall see further. Formula (2.1) espresses t h a t the l i m i t d i s t r i b u t i o n function is t h a t o f a m o n o k i netic beam o f particles (i.e. a delta function i n v), e m i t t e d from the cathode w i t h velocity 0 (because \ / v ( r ) = 0, for r = 1) and flowing r a d i a l l y from t h e cathode t o t h e anode (because v — ve = 0 ) . T h i s last feature is new compared w i t h t h e plane case. Indeed, the geometry o f the l i m i t i n g flow is independent o f the details of t h e emission profile 7 even i f 7 is extremelly n o n isotropic, a n d sends much more particles i n one p a r t i c u l a r d i r e c t i o n . r

I t can now be explained w h y no simple analytical t r e a t m e n t o f t h e p r o b l e m like in the first proof o f section 2.3. is possible. Indeed, the characteristics are now defined by t h e set o f o r d i n a r y differential equations :

r = IV,

v

T

K = y

I d
vv ~ • T

V e

=

9

0 v

*

=

°'

'

4

1

2

'

We notice t h a t , i n a d d i t i o n to the energy invariant = \v\ -v {T) 2

W(r,v)

e

(4.13)

there are t w o other invariants ; t h e angular m o m e n t u m m tv

C(r v) t

(4.14)

6

and the projection o f t h e m o m e n t u m i n t h e z direction, w h i c h in t h i s scaling coincides w i t h v . :

Therefore, the m o t i o n o f the particles can be reduced t o a one

dimensional m o t i o n associated w i t h the phase space ( r , f ) , t h e equations o f w h i c h r

are :

where the "effective" potential fp%ft particle and is defined b y :

c

depends o n the angular m o m e n t u m o f the

C~

(4.16)

29

N o w , i n t h e e x p l i c i t formulae used in the first proof of section 2.3., the p o t e n t i a l

$

t

:

a t l

r

s

E

e

e

c

(Hi¬ Proof.

S k e t c h of proof of T h e o r e m 4 . 1 1

T h e key p o i n t is t o use the three invariants

the energy i n v a r i a n t (4.13), the

angular m o m e n t u m (4.14) and v, t o construct an upper-solution of the form :

F < ( r , u

M

)

= l F

l

(

^

+

^-^V,(^)F (g) 3

(4.17)

Indeed, i t is easily shown t h a t a function of the t y p e (4.17) is an exact s o l u t i o n of the Vlasov e q u a t i o n (4.1). I t is thus possible t o chose the functions F,, F , F n o n negative such t h a t the r e s t r i c t i o n of F to the b o u n d a r y ) r = 1 [ dominates the b o u n d a r y profile 7. T h e n , we have : 2

3

l

/>.*,(*,«,) Now, when £

< F*(r,tV,W,B,).

(4.18)

0, all the mass o f the upper-solution P* concentrates on the

characteristics defined by : v

2

=
v

g

-

% -

0.

(4.19)

so t h a t , i f f* converges t o a measure / , t h i s measure must be o f the form : f(r,v)

= n(r)6(v

r

- y/^?))5(v )6(v ). B

!

(4.20)

T h e n , the conservation o f the c u r r e n t intensity i leads t o f o r m u l a (4.10). A l l t h e remainder o f the p r o o f of t h e o r e m 4.11 (like o b t a i n i n g the a p r i o r i est i m a t e s w h i c h guarantee the existence o f t h e l i m i t measure / ) follow exactly the same lines as the second p r o o f of theorem 2.1 (see section 2.3.).

•

We n o w t u r n t o the d e t e r m i n a t i o n o f the l i m i t current i . For this purpose, we investigate t h e existence and uniqueness of solutions o f the semilinear elliptic p r o b l e m (4.11). We first define the C h i l d - L a n g m u i r current as the value of t w h i c h produces a s o l u t i o n o f the p r o b l e m (4.11) w i t h a vanishing derivative {df>/dr)(r = 1) = 0, i.e. a vanishing electric field at the cathode. M o r e precisely, we define a function xi )' as the s o l u t i o n of the Cauchy p r o b l e m (for r > 1, or r < 1) : r

30

Existence and uniqueness for the singular Cauchy p r o b l e m (4.21] is shown i n [11], T h e n , the C h i l d - L a n g m u i r current ici.{p), which now depends on the geometry of the diode via the aspect r a t i o p, is defined by :
3 2

-

Xipy ' *

P > 0, P * I .

(4.22)

2

Indeed, icL.

problem

(4-11)

has no solution such that 0 forr

£ (p, 1].

• / / i < i c / , , problem (4-il) has a unique solution such that Oforr

£

(PM • Furthermore, i = icL dtp/dr{l) = 0.

is the unique value of i such that the solution

satisfies

T h e surprising result is i n the case p > 1 : L e m m a 4 . 1 3 Let p > I (cathode inside the anode), •

•

There exists i {p),

0 < i {p)

max

0

mal

— ifi

satisfies


— ifi

satisfies i > t

< i ax(p). m

m 0

then ;

< + o o , such that: problem

(4-11)

i ( p ) , problem (4-H)

has at least one

has no

solution.

solution.

There exists Po > I such that W * { / > ) = icdp).

Vp e [1,

P o

\,

(4.23)

and W •

> tc/.- in(po.oo).

There exist values of p € [ l . o o ) and i E [0, i solution exist.

m n l

(4.24)

( p ) ) such that more than one

These Lemmas are proved i n [ l l j . T h e i r proof is much more technical t h a n t h a t of the corresponding lemma for the plane case ( L e m m a 2.4], Indeed, due t o the weight r i n the Laplace operator, no first integral is available any longer. T h e existence proof relies o n the use of the fixed p o i n t theorem. T h e complicated p a r t is t o o b t a i n a p r i o r i i n f o r m a t i o n on the behaviour of the solution near the singular p o i n t r = 1, so as t o fix the correct functional s e t t i n g for the fixed p o i n t t h e o r e m .

L e m m a 4.13 has an i m p o r t a n t physical consequence. I t states t h a t , i n the l i m i t e —• 0, the C h i l d - L a n g m u i r c u r r e n t (i.e. the current associated w i t h the s o l u t i o n w i t h a vanishing d e r i v a t i v e at r = 1) is n o t necessarily the m a x i m a l possible curr e n t i n the diode. I n [11], n u m e r i c a l s i m u l a t i o n s are presented, w h i c h s u p p o r t t h i s conclusion, b u t w h i c h also show t h a t the relative difference between ici(p) and i m a i ( c ) is v e r y s m a l l . T h e e x p e r i m e n t a l verification of t h i s difference must be difficult. However, from a t h e o r e t i c a l v i e w p o i n t , i t is i m p o r t a n t t o know t h a t the regime o f m a x i m a ! current is n o t necessarily associated w i t h a vanishing cathode electric field. I n d e e d , i n the physical l i t t e r a t u r e on space-charge l i m i t e d flows, t h i s fact is usually taken for g r a n t e d . T h e m e r i t of the present analysis is t o e x h i b i t a counter-example. T h e t w o L e m m a s 4.12 and 4.13 allow t o precise the value of i i n t h e o r e m 4 . 1 1 . We have : L e m m a 4.14 • Let p < 1 (cathode outside the anode), orp > 1 (cathode inside the anode) with p < po where p is defined in Lemma 4-13 (small aspect ratio). Then we have: 0

i = Afin^JctG*))•

Let p > po (cathode inside

(4-25)

the anode with large aspect ratio),

i

-

Mm(i„i

i

e

{ i - , , i c 7 i ( p ) } , ifiy

c t

then !

( p ) ) , if i, f [ i i , ( r t , * « « ( / > ) ] , C

€ [ict(p),imax(p)]-

(4.26) (4-27)

We notice t h a t L e m m a 4.14 o n l y gives a p a r t i a l conclusion concerning the value of i i n the case p > po (cathode inside the anode w i t h large aspect r a t i o ) a n d h £ [ i c t ( p ) , i m a r ( p ) ] (injected current i n the i n t e r v a l comprised between the C h i l d L a n g m u i r c u r r e n t and the m a x i m a l allowed c u r r e n t ) . I n t h i s case, we do not k n o w , up t o now, how t o d i s c r i m i n a t e between the values i and I ' C L ( P ) - Indeed, since z < i {p), both and ICL{P) are allowed values of the current i. I f i = z , the associated p o t e n t i a l ip satisfies (dip/dr)(r = 1) > 0, and i f i = icL(p), i t satisfies [dtp/dr)(r = 1) = 0. I t m a y be possible t h a t these two solutions are b o t h l i m i t p o i n t s of the sequence ip 7

7

max

7

c

4.3.

T h e spherical

diode

I t is i n t e r e s t i n g t o investigate i f the same results extend t o the spherical case. We o b t a i n the s u r p r i s i n g result t h a t the p a t h o l o g y w h i c h appears for p > po (cathode inside the anode w i t h large aspect r a t i o ) i n the c y l i n d r i c a l case does not show u p in the spherical case, and e v e r y t h i n g behaves p r o p e r l y like i n the plane case. L e t us investigate the spherical case i n more details. W e consider a spherical diode w h i c h consists of t w o concentric m e t a l l i c spheres. We denote by Ri and R? the cathode and anode r a d i i respectively, and by p the

32 aspect ratio p = KijR\. A g a i n , we have p < 1 i f the cathode is surrounded by the anode and p > 1 i n the converse situation {see figure 5). T h e scaling is chosen s i m i l a r l y as for the cylindrical diode. T h e scaled d i s t r i b u t i o n function f (r, v ,a) now depends on the radial distance r , on the radial velocity r y and on the squared n o r m of the angular m o m e n t u m a = \x x v\ 6 [0, co). c

r

2

T h e dimensionless Vlasov-Poisson problem is w r i t t e n

+

* £ ^ d r ar {

r

2

+

£

5£>J£-*

d f ar

_

}

«

n

(

= ^TCJ-^)-

= g (v ,a)

r

= _L / r(nW.,,a)
)

r e | l , 4

(4.29)

+

l

f{l,v ,a)

r

* « *

r

"f e K,

a s 1R+, sign(p-lfa

> Q, (4.30)

/ * ( p , v a ) = 0,

tveIR,

r i

^(1)

a e ! R , sign(p-l)v +

= 0,

V

< 0,

r

(4.31)

' ( p ) P 1.

(4.32) e

Now, the conservation of the current implies t h a t the current intensity i (r) flowing through any sphere of radius I* between the t w o electrodes does not depend on r

s

i {r)

=

E

/ u / (r,i; ,a)dt' tia JRiR,. r

r

independent

r

e

A g a i n , we are interested i n the l i m i t of f .

£

of r,

re

!

Iff, n , i , when £ —• 0 : e

T h e o r e m 4.15 ([10], [11]) There exists a sequence (f',f ) (4.28)-(4.32) such that , as e — 0, we have : f

E

— f

in

M ([l,p]

5

V) —
t

f

— i

in

TR,

xJRxlR )

b

in

(4.33)

of solutions of

weak star

+

(4.34)

1

C {{hp)) strong

0 < I< i

y

= /

(4.35)

u^iv.ajdtydo,

(4.36)

where D denotes the set of incoming velocities at the anode i D+. — {{v ,a) fR x I R + . sign{p - l)v > 0 } . T/ie l i m i t dfe^Piiut&m function f is given by . +

r

£

T

f(r,v)

=

'

i(ur - y^H)

5(0),

(r, v ,a) T

€ [ 1 , p] x TR x JR+,

(4.37)

33

and ip is the solution of *-

*%)

{ r

-

„(r) = - ^ = ,

„ ( ! ) = 0, * , ) - ! ,

(4-38)

W e again define the C h i l d - L a n g m u i r current as the value of i w h i c h produces a solution of the problem (4.38) w i t h a vanishing derivative (dip/dr)(r = 1) = 0. L e t X(r) be t h e s o l u t i o n of the Cauchy p r o b l e m (for r > 1, or r < 1) :

Existence a n d uniqueness for the singular Cauchy problem (4.39) is shown i n [11]. T h e n , the C h i l d - L a n g m u i r current ici(.p)- is defined by : 3 2

icUp)

= X[p)~ ' .

p>0,p*l.

(4.40)

T h e n , for the p r o b l e m (4.38), we have the following : L e m m a 4.16 o fort • Ifi

• If i > ich-

problem (4-38) has no solution such that

e f>,i).

< icL,

problem (4-38) kas a unique solution suck that ip(r) > 0 forr

e

(PM • Furthermore, dip/dr{\)

i = icL(p)

is the unique value ofi such that the solution

satisfies

= 0.

C o n t r a r y t o the c y l i n d r i c a l case, there is no difference between the cases p < 1 and p > 1. N o w for the current i i n theorem 4.15, we have the ; L e m m a 4.17

We have: i = Min(i ,i i(p))y

C

(4-41)

T h u s , the nice behaviour of p r o b l e m (4.38) allows us t o completely determine the value of 1. T h i s analysis o f the c y l i n d r i c a l a n d spherical diode shows t h a t extensions of the convergence result f r o m the plane diode case t o fully m u l t i d i m e n s i o n a l cases can be very delicate. However, some formal results can be obtained by i n t u i t i v e considerations. We shall detail t h e m in the next section.

34

Insulating. boundary F

Cathode

Figure 6 : D i o d e geometry i n the general m u l t i d i m e n s i o n a l ease

5. 5.1.

T h e fully multidimensional case Setting of the problem

We consider a diode i n a d-d i mens ion al space (d = 2 or 3), w h i c h consists of 2 disjoint electrodes modelled by 2 s m o o t h (d — l)-dimensional manifolds To and TV We suppose t h a t the electrons Bow i n the d o m a i n Q l i m i t e d by To, F j and an artificial (or insulating) boundary r , (see figure 6). We assume t h a t the length scale has been chosen such t h a t the diameter of the d o m a i n Si is of order 1. The natural extension of the C h i l d - L a n g m u i r problem (2.28)-(2.32) to this d-dimensional case is: l n

(5.1)

(5.2)

r(x,v)

1

c

=

g (x,v)

!'{x,v)

f(x,v)

=0,

= r(x.v-),

7(1,7),

(*,») € E J ,

(5.3)

feM0eET,

(5.4)

(T,t.)eE

(5.5)

(5.6)

35


| £ t »

= Q,

mmTu

xeT

i n s

(5-7)

.

(5.8)

We have denoted by E T , the set EQ

= { ( * , « ) , z e
(5.9)

where is the outwards u n i t normal t o F at the p o i n t x and ( . , . ) denotes t h e inner p r o d u c t o f vectors of R T,^ is the subset of the b o u n d a r y of the phase space n x I R corresponding t o i n c o m i n g velocities along Tg. Definitions of Ej" and S,~ are the same m o d u l o some obvious changes. 0

d

d

3

c

T h e f u n c t i o n g (x, v) describes the emission profile of the particles at the cathode To- F o l l o w i n g the C h i l d - L a n g m u i r hypotheses detailed i n section 2 . 1 . , its dependence on e is such t h a t w h e n e —• 0, all the mass concentrates at the zero velocity, w i t h a n o r m a l i z a t i o n constant l / e ' ' ' w h i c h insures t h a t the t o t a l injection c u r r e n t remains finite. N o particles are supposed t o be injected at the anode F j . A l o n g the insulating boundary r , specular reflexion is assumed : i n f o r m u l a £5.5). v' is the s y m m e t r i c of v w i t h respect t o the n o r m a l f[x), namely v' = v — 2{v, v(x))v(x). T h e cathode p o t e n t i a l is assumed constant and taken as reference p o t e n t i a l , w h i l e the anode p o t e n t i a l tp\ is assumed positive : + , 1

i n 3

0.

(5.10)

F i n a l l y , along the i n s u l a t i n g b o u n d a r y F i „ , the normal derivative o f ifi is assumed t o vanish, w h i c h is a usual artificial b o u n d a r y c o n d i t i o n for the electrostatic p o t e n t i a l . A l l the remainder of t h i s section w i l l not essentially depend on the part i c u l a r choice of the b o u n d a r y c o n d i t i o n along the artificial b o u n d a r y T . 5

i n s

5.2.

The limit e - > 0

We shall n o w t r y t o guess f o r m a l l y w h a t is the l i m i t of p r o b l e m (5.1)-(5.8). Details of w h a t follows ean be found i n [12], A s i m i l a r ( a l t h o u g h simpler) problem relative t o the m o d e l l i n g of a m u l t i d i m e n s i o n a l S c h o t t k y diode i n semiconductor physics has been analyzed i n [13] (see also |6]) and has led t o a complete m a t h e m a t i c a l proof. T h e s t a r t i n g p o i n t o f the analysis is the use of a change of variables based on the characteristics i n the weak f o r m u l a t i o n o f the Vlasov equation (5.1). We recall t h a t the characteristics are the solutions of the o r d i n a r y differential system : X

= V.V

= \v^{X{t)).

(5.11)

I f t h e caracteristies h i t s the b o u n d a r y r , at a p o i n t X, at t i m e t w i t h a v e l o c i t y V, we e x t e n d i t by the characteristics s t a r t i n g at t from X w i t h the reverse velocity V We shall denote by (t;y,io),K {t;(y,w) £ UE[, 1 e [0, T 4 ( t / , w)}, the p o s i t i o n ( i n phase space) at t i m e t o f the characteristics s t a r t i n g n s

e

a l

36 at t i m e 0 from the p o i n t (y.vj). Note t h a t , since ( ; / , w ) belongs t o Eg U E J " , y is a p o i n t of the b o u n d a r y and w is an i n c o m i n g velocity. [0, T°vaiy, to)] is the m a x i m a l interval of definition of the characteristics. We shall assume t h a t the field is regular enough so t h a t (X (T {y, to); y, w), V CE!nax(Vi w); y.ui) is a p o i n t where the characteristics leaves the d o m a i n (i.e. belongs t o U E j " where c

!

s

XOI

£

+

= {(z,i>), i e r

0

l

v € IR'', (v,v(x))

> 0},

(5.12)

and analogously for Ej~). N o w , i t is easy t o prove [13] t h a t the m a p p i n g (x,v)

£

x = X (t;y,w), s

w i t h V = {(t,y, iv), {y,w) € EQ UEJ, a local diffeomorphism. Its jacobian is : dxdv

=

(5.13)

£

v =

V (t;y,w),

t £ [0,T^ {y,

£

vj)}} is one t o one and is

al

(w,v(y))dS(y)dvjdt,

(5.14)

where dS(y) is the surface element on Oil. I n all the remainder of this section, we shall assume t h a t the mapping (5.13) is a global diffeomorphism, i.e., is onto. T h i s excludes the possibility of orbits not meeting the boundary like closed o r b i t s , dense orbits, or unbounded orbits. O f course, t h i s assumption has t o be verified i n any rigorous proof and m a y create a serious technical difficulty. Now, we shall consider the weak l i m i t of / and Ig(f ) be the the following integral .

£

Let 8(x,v)

be a s m o o t h test function

s

(5.15) B y use o f the change of variables (x,v)

le(f')

£

=

—t (t, y,w),

we have :

E

J f (X (t;y,w),V<(t,y,vj)) vc

(5.16) £

£

0 ( X ( t ; y, w), V ( t ; y,w))

(w, f ( y ) )

dS(y)dwdt.

B u t since f- is constant along the characteristics, we can also w r i t e :

J V

°

(5.18)

c

1

w i t h V , given by the same definition as P , b u t associated w i t h E Q alone. T h e change of variables u = w/s leads to :

le-in

=

[

c

i{y,u)B{X {t\y,eu),V'{t\y,sv.))(u,t'(y))dS(y)dudt.

(5.19)

37

E

where Q

0

= {(t,y,u),

(y,u) £ E ", t 6 [ 0 , T ^ ( y , 0

a l

e l l

)]}. £

I t is n o w possible t o guess w h a t the weak l i m i t o f / w h e n e —• 0 is. W e shall suppose t h a t f —• ip i n a s m o o t h way. W e shall denote by X(t; y, w), V ( f ; y, tu)) t h e characteristics associated w i t h the l i m i t p o t e n t i a l , and (0, T (y,w)\ t h e i r interval of d e f i n i t i o n . N o w , there are t w o cases, whether the t r a j e c t o r y ( A ' ( f ; y,£u), V (t; y.eu)) r e t u r n s back t o t h e cathode or not (see the one-dimensional case, section 2.3.}. For y £ r , we define : E

max

e

c

0

U+(y)

= { « € I R " , {U,u{y))

< 0. l i m ^ f j , , ^ )

U°(y)

= {«eR ,(M[»))<0,

> 0},

(5.20)

l i m l £ „ ' y , e u ) = 0}.

(5.21)

and i

I n s p e c t i o n o f the one-dimensional case (section 2.3.), shows t h a t we may very w e l l have Meas(U°{yj) > 0. T h e convergence o f (X (t;y,eu),V'{t;y,eu)} is different o n U°(y) and U (y). I n t h e first case, we obviously have E

+

E

l i m ( X ' ( t ; y , £ w ) , V ( t ; y , e u ) ) = (y, 0 ) , u e U°(y), £--0

(5.22)

whereas i n t h e second one, we have :

e

lim(X'{t;y,eu),V (t;y,Eu))

+

= {X(t;y,

0), K ( t ; y , 0 ) ) , u e W ( y ) ,

(5.23)

£—•0 and Um7* (]r,C») = r—ztv.D), u € « + ( » ) . £—•0 a B

N o w we r e w r i t e t h e integral defining hi!')

= fc dS(y] B

/

{

u

e

R

M

(5.24)

according t o :

u

.

u

f

v

l

l

<

0

1

^

i ^ - ^ > * (5.25)

7(»,u)ff(Jf«(t; !/,£«), V " { t ; y , e n ) ) (u.i'fo)), and w i t h (5.22), (5.23) and (5.24), we i m m e d i a t e l y get

with kffi

=

IrJS(y)(f l(y,n)(u,u{y))du} unv)

(5.27) W

0(A(t;y,O),V(t;y,O))dt).

N o w , we define t h e n o r m a l injected current j ( y ) at the p o i n t y 6 To b y : 7

j,(y) -

/

7{».«)(«."(l'))*'.

(

5

2

S

)

38

m i d the current j(y)

carried by the characteristics s t a r t i n g at the point y by ! Hjf)-f

+

-r{y,tt){u,v(y))d\i.

(5-29)

d

Since U (y)

C {u € JR , (it, u{y)) < 0 } , we have : My)

Note t h a t we may have j-,(y)

< i(y)-

< j{y).

(5.30)

T h e n , the integral I (f) a

can be r e w r i t t e n

according t o :

T

W f l -

AV0)

/ j(y)(f ~" Jr„ Jo

8{X(t,y,0),V{t,y,0))
(5.31)

Therefore, f* Formally converges to / i n the weak topology of measures, where / is a measure such t h a t there exists a function j : y e r

0

—> j(y)

g IR

> = h(}), for all smooth test function

+

and (5.32)

8(x,v).

We now t u r n to the Poisson equation.

—< n weakly, we can pass t o the

c

If n

l i m i t i n the Poisson equation and get Aip -

n(x)

i e £2.

(5.33)

To express rt, i t is more convenient to w r i t e (5.33) i n a weak form. L e t ip{x) be a s m o o t h test function which vanishes at the boundary. We have : -

/ VtpVipdx Jn

=

/ f(x,v)ip(x)dx. JnrTR

(5.34)

d

B u t the integral at the right hand side is n o t h i n g else than h(f)

for a test

function 9(x, v) = ip(x). T h u s the Poisson equation is w r i t t e n :

-

5.3.

[ V V^dx= Jn

T

[ Jr

v

,

l l

J^y)(/ " " '' Jo

0

0 ,

tb{X(t;y,0))dt).

(5.35)

The multidimensional Child-Langmuir problem

We summarize now the multidimensional C h i l d - L a n g m u i r problem for the potential. Let j : y € r —• j(y) e I R be given. T h e problem is t o find a potential function

+

-

[ V^Vipdx Jn

m

f Jr

m

j{y)([ 0

"

l

V

' 0(X(i;»))
(5.36)

Jo

for any s m o o t h test function ip defined on fi w h i c h vanishes at the boundary, where X(t, y) is the solution of the o r d i n a r y differential system:

39

X

= V
X{0,y)

= y,

X

= 0,

a n d T (y) is the m a x i m a l interval of definition of X ( . , y). subject t o the b o u n d a r y conditions : raal

tp(x) = 0,


z € r , 0

x e T,

(5.37) F u r t h e r m o r e , tp is

(5.38)

(5.39)

(5.40) A s far as we k n o w , the m u l t i d i m e n s i o n a l C h i l d - L a n g m u i r problem (5.36)-(5.40) has never been stated i n t h i s way i n the l i t t e r a t u r e . We notice t h a t the use o f t h e characteristics X(t,y) is very close t o the numerical procedures used i n the E T P - E G U N code [18]. I n t h i s code, the particle trajectories are launched from the c a t h o d e and the electric p o t e n t i a l is c o m p u t e d inside an i t e r a t i v e procedure w h i c h is r u n u n t i l l a self-consistent stationary solution is reached. Indeed, particle t y p e m e t h o d s are p a r t i c u l a r l y well suited t o the resolution of this problem, due t o its use o f " L a g r a n g i a n coordinates". However, no m a t h e m a t i c a l l y rigorous setting of this p r o b l e m was available u p t o now i n the l i t t e r a t u r e . N o existence t h e o r y for p r o b l e m (5.36)-(5.40) has been developped yet. We may conjecture t h a t existence should hold for small (in a sense t o be defined) c u r r e n t j , a n d t h a t no s o l u t i o n should exist for large j . A n o t h e r feature of t h i s p r o b l e m is its n o n l o c a l i t y : the right hand side of Poisson equation (5.36] depends on the p o t e n t i a l v i a the resolution of the equation of the characteristics. T h i s non-locality is a consequence of the fact t h a t several trajectories X(t, y), s t a r t i n g from the p o i n t y on the cathode w i t h velocity 0 can meet inside the d o m a i n . I n the plane, c y l i n d r i c a l or spherical cases (sections 2. and 4.), the very restrictive assumptions o n the geometry a p r i o r i insure t h a t these trajectories do not meet. T h i s yields semilinear elliptic problems w i t h a local non-linearity (problems (2.37), (4.11) a n d (4.38)). Below, we show t h a t the m u l t i d i m e n s i o n a l C h i l d - L a n g m u i r problem reduces t o a semilinear e l l i p t i c problems w i t h a local non-linearity i f we make the a d d i t i o n a l a s s u m p t i o n t h a t the trajectories X(t,y) do n o t meet. T h i s corresponds t o the so-called l a m i n a r regime i n physics t e x t books [20] -

5.4.

The "laminar beam" hypothesis

L e t ip be a s o l u t i o n of the m u l t i d i m e n s i o n a l C h i l d - L a n g m u i r p r o b l e m (5.36)-(5.40) w h i c h satisfies the l a m i n a r hypothesis. T h e n , the m a p p i n g ( [ , y) e O —• x € f i such that. X(t,y)

= .r,

(5.41)

40

where O - {{t,y),y € V ,t e \0,T (y)]}, is a one t o one diffeomorphism from O i n t o Q. We denote by ft the range of the mapping (5.41). We may define a function u: a! £ R o - * u{x) E R by : 0

max

0

d

Bpf(*,»M -

V{t,y).

(5.42)

We notice t h a t u{x) satisfies the boundary condition i £ T .

(5.43)

Now, we show t h a t the measure / asociated w i t h is of the form

« { x ) = 0,

by formulae (5.31] and (5,32]

}(x,v)

0

d

= n(x)S{v-u(x)),

i e f t , v£lR

(5,44)

F i r s t , 7i is chosen identically 0 where u is not defined i.e. i n n \ f t . T h e function n is determined i n fto from the fact t h a t / is a measure solution of the Vlasov equation (5.1). Indeed, inserting (5.44) i n the Vlasov equation (5.1) leads t o the set of equations : 0

V . ( n u ) = 0,

fu.V)(u) -

xefto,

^Vtp = 0,

x E £V

(5.45)

(5.46)

E q u a t i o n (5.46) is a direct consequence of the definition of u by equation (5.42), b u t the latter also yields the boundary condition (5.43). N o w , let 8{x,v) be a smooth test function, and consider the d u a l i t y bracket < f,8 > w i t h / given by (5.44]. We show t h a t this d u a l i t y bracket coincides w i t h L)(f) given by (5.31). We have : =

/

n{x)6{x
(5.47)

and, by the change of variables (5.41), we have =

f n(X(t,y))J{t,y)6(X{t,y),V(t ))dtdS(y), Jo iy

(5.48)

where J{t, y) is the jacobian of the change of variables ( ( , y) —» x. T h e n formula (5.43) coincides w i t h (5.31) i f n(X{t,y))J(t,y) does not depend on t. L e t Y.((,J/} be a smooth function on O, which vanishes on a neighbourhood of the boundary dO a n d let %X~ We have :

J ftMX(t y))J(t,y)} (t,y)dtdS(y). o

7

X

(5,49)

41

B u t , according t o t h e "laminar'' hypothesis, there exists a s m o o t h function ip(x) w i t h c o m p a c t s u p p o r t i n i i such t h a t i>(X(t.y))

-

(t,y).

(t,y)

X

E O,

(5.51)

(t.y)eO.

(5.52)

A n elementary c o m p u t a t i o n shows t h a t :

| | { * , j , ) = {iiWMZ(t,m,

Therefore, the change of variables (5.41) i n (5.50) gives, w i t h (5.45) :

l

x

=

-

=

-

[

n(*)(».V)*(Jt)d*

I V.{m)(x)$lx}dx Jflo

w h i c h shows t h a t n(Xit,y))J(t,y) t o (5.31) leads t o :

(5.53) = 0,

(5.54)

does not depend on (. Identification of (5.48)

n(X[t,y))J(,t )

(5.55)

mj( )

iy

V

T h e same c o m p u t a t i o n allows t o determine the boundary c o n d i t i o n on n, w h i c h w i l l be shown t o be : x E r ,

( { n u ) ( z ) , i / ( i ) ) = -j(x),

{5.56}

0

Indeed, i f i n (5.50), we assume t h a t x is o n l y 0 i n a neighbourhood of the set yer },

{{T (y).y).

(5.57)

0

mai

we have, by the Green formula:

J^n{X(t,y))J{t )^(t,y)dtdS(y) >y

= - j

n(X(0,»))

J ( 0 , y ) v ( 0 , y)

dS(y). (5.58)

T h e same change of variables and formula (5.55) lead t o : / n( )( .V)yb(x)dx Jsio X

U

= -

f j(y) {Q,y)dS(y), Jr

(5.59)

X

0

or, again by the use of the Green formula and of (5.45)

f JTO

(M(&,Ks))#*)<&Ce)

= " f n(X(Q,y))J(0,y) (0,y)dS(y). Jfo

B u t , by definitions (5.51) and (5.41), we have ${x) b o u n d a r y c o n d i t i o n (5.56) follows from (5.60).

(5.60)

X

= x ( 0 , y), x 6 T , a n d the 0

42

5.5.

The multidimensional laminar Child-Langmuir problem

We now summarize the m u l t i d i m e n s i o n a l C h i l d - L a n g m u i r problem under the ass u m p t i o n of laminar beam. L e t j : y £ To —> j(y) £ I R be given. T h e p r o b l e m is to find 7l(x), l i ( i ) , f{x), satisfying the equations : +

V . ( n u ) = 0,

(u.V}(u} •

» € « ,

i v i p = 0, n(x),

(5.61)

x E 0.

(5.62)

i Gfl,

(5.63)

subject t o the b o u n d a r y conditions : u L r ) = 0,

x e T ,

((n )(x),!<(*)) -

i e r

U


1 6 To,

=

((u)(x),v(x))

(5.64)

0

= 0,

0,

r e r ,

rer

r

a

l

,

(5.65)

(5.66)

i e r „

-

0

.

(5.67)

n

„

(5.68)

(5.69)

T h e b o u n d a r y c o n d i t i o n (5.68) is made t o insure t h a t i f a characteristics h i t s the b o u n d a r y r , „ , i t docs i t tangentially, so t h a t no particle flow crosses t h i s boundary. I t is a consequence of the laminar beam assumption. s

A g a i n , no existence theory has been developped yet for system (5.61)-(5.69). We notice t h a t this problem is set i n the framework of " E u l e r i a n coordinates'', i n stead of " Langrangian coordinates" as used for the general m u l t i d i m e n s i o n a l C h i l d L a n g m u i r problem (5.36)-(5.40). Consequently, the numerical resolution of the laminar C h i l d - L a n g m u i r system (5.61)-(5.69) may be easier and cheeper, because more standard fluid dynamical tools can be used. O n the other hand, i t cannot yield non-laminar solutions. I t is well established t h a t , in particular i n the case of large space-charge beams, non-laminar flows are produced. Therefore, the l a m inar C h i l d - L a n g m u i r model (5.61)-(5.69) should be used w i t h care, o n l y when the laminar hypothesis is guaranteed t o be fulfilled.

43

6.

Conclusion

T h e analysis of v a c u u m diodes o p e r a t i n g i n the C h i l d - L a n g m u i r regime (i.e. when t h e applied p o t e n t i a l between the electrodes is much larger t h a n the t h e r m a l energy associated w i t h the d i s t r i b u t i o n f u n c t i o n of the e m i t t e d particles) leads t o a very unclassieal p e r t u r b a t i o n p r o b l e m for the non-linear Vlasov-Poisson system i n a bounded d o m a i n . T h e analysis of t h i s p r o b l e m is completely solved from a m a t h e m a t i c a l v i e w p o i n t i n the one-dimensional cartesian case, and m o s t l y solved i n the c y l i n d r i c a l l y or spherically s y m m e t r i c case. I n the fully m u l t i d i m e n s i o n a l case, a formal a s y m p t o t i c technique yields new descriptions of the C h i l d - L a n g m u i r regime. T h e first one is suited for general flows ((5.36)-(5.40)). T h e second one ((5.61)-(5.69)) consists i n a restatement of the first one i n a more conventional Quid-dynamical framework, b u t is o n l y valid for laminar beams ( t h a t is when the trajectories of the electrons do not cross). M a n y open problems remain unsolved, like existence theory for the general C h i l d - L a n g m u i r p r o b l e m ((5.36)-(5.40)) or the l a m i n a r one ((5.61)-(5.69)). Also, no convergence proof o f the Vlasov-Poisson system (5.1)-(5.8) t o these t w o C h i l d L a n g m u i r systems has been obtained yet. O t h e r open problems arise i n the various extensions of t h i s t h e o r y t o more complex situations. Magnetized flows a n d beams i n a n e u t r a l i z i n g background w i l l be reviewed i n the second p a r t of the present w o r k | 5 ] , together w i t h a constructive a l g o r i t h m for the d e t e r m i n a t i o n of the C h i l d L a n g m u i r s o l u t i o n . T h e specific extensions of t h i s theory t o semiconductors w i l l be reviewed i n the t h i r d p a r t [5],

References [1] N . Ben Abdallah The Child-Langmuir regime for electron transport in a plasma including a background of positive ions, preprint, to appear in Math. Methods and Models in the Appl. Sciences. |2] N . Ben Abdallah, Convergence of the Child-Langmuir asymptotics of the Boltzmann eqation of semiconductors, preprint, sub. to SIAM J. on Math. Anal. [3] N . Ben Abdallah and P. Degond, On the Child-Langmuir taw for semiconductors. Proceedings of the Institute of Mathematics and its Applications, Minneapolis, to appear. (4) N . Ben Abdallah and P. Degond, The Child-Langmuir law for the Boltzmann Equation of Semiconductors, preprint, to appear in SIAM J. on Math. Anal. [5] N . Ben Abdallah and P. Degond, The Child-Langmuir law in the kinetic tkeory of charged-particles; Part 2, magnetised flows, plasmas and numerical algorithms, (in preparation). [6] N . Ben Abdallah and P. Degond, The Child-Langmuir law in the kinetic theory of charged-particles; Part 3, semiconductor models, (in preparation).

44 [7] N . Ben Abdallah and P. Degond, A mathematical model of magnetic insulation, (in preparation). [8] N . Ben Abdallah, P. Degond and C. Schmeiser, On a mathematical model for hot carrier injection in semiconductors, Math. Meth. in the Appl. Sci. (to appear). |9] P. Degond Kinetic Equations for plasmas and semiconductors, Lecture notes, Siena, October 1992 (manuscript). [10] P. Degond, S. Jaffard, F Poupaud and P.A. Raviart The Child-Langmmr asymptotics of the Vlasov-Poisson Equation for Cylindrically or Spherically Symmetric diodes: part I statement of the problem and basic estimates, internal report 9313 CMLA ENS de Cachan submitted to SIAM Appl. Math. [11] P. Degond, S. Jaffard, F. Poupaud and P.A. Raviart The Child-Langmuir asymptotics of the Vlasov-Poisson Equation for Cylindrically or Spherically Symmetric diodes: part II, Analysis of the reduced problem and determination of the Child-Langmuir current, internal report 9313 CMLA, ENS de Cachan submitted to SIAM Appl. Math. [12] P. Degond and F Poupaud, The multidimensional Child-Langmuir problem in the kinetic theory of charged-particles, (in preparation). [13| P. Degond, F. Poupaud and C. Schmeiser, A mathematical analysis of a multidimensional Schottky diode problem, (in preparation). |14| P. Degond, F. Poupaud and A. Yamnahakld Simulation particulate et analyse asymptotique de I'quation de Boltzmann pour la modlisation d'une diode Schottky, contract report, (English translation in preparation). (15] P. Degond and P.A. Raviart, An Asymptotic Analysis of the One-Dimensional Vlasov-Poisson System: the Child-Langmuir law, Asymptotic Analysis 4 (1991), pp. 187-214. [16] P. Degond and P.A. Raviart, On a penalization of the Child-Langmuir emission condition for the one-dimensional Vlasov-Poisson equation, Asymptotic Analysis 6 (1992), pp. 1-27. ]17] C. Greengard and P.A. Raviart, A boundary Value Problem for the Stationary Vlasov-Poisson Equations : the plane diode, Comm. Pure Appl. Math 43 (1990) pp. 473-507. [18] W. Herrmansfeldt, Electron trajectory program, SLAC technical report 166, September 1973. [19] I . Langmuir and K . T . Compton, Electrical Discharges in Gases . Part II. Fundamental Phenomena in Electrical Discharges, Rev. Mod. Phys. 3 (1931). pp 191-257. [20) J. D. Lawson, The physics of charged-particle beams. International Series of monographs on physics n75, Oxford University Press, Oxford, 1983[21] F. Poupaud, Boundary Value Problems for the stationary Vlasov-Maxwell System, Forum Mathematicum, 777. |22| T . Weisterman, A particle-in-cetl method as a tool for diode simulations, Nuclear Instruments and Methods in Physics Research A 260 (1988), pp. 271-279.

45

EULERIAN CODES FOR THE VLASOV EQUATION

2

M R. F E L X ' . P . B E R T R A N D . A. GHlZZC-2

1. P M M S / C N R S , Universite d'Orleans, Orleans, France

2. L P M I , Universite de Nancy-I, BP239 S4506 Vandoeuvre les Nancy, France

Abstract: Eulerian Vlasov codes, using Splitting scheme for the resolution, are presented for the study o f electron collisionless plasma in electrostatic and electromagnetic regimes. Many concepts have been investigated as the problem o f fomentation in phase space, with the appearance o f fine micro structures, followed by their "cleaning" Moreover the mathematical topology o f the Vlasov equation, and phase space representation o f the distribution function are studied. T w o examples indicating the important role o f a minority population o f particles, concerning the nonlinear Landau damping and the particle acceleration in the laser-plasma interaction are presented.

1. I N T R O D U C T I O N 1.1. Particles and Eulerian Vlasov Codes. Plasma codes have now a long history since the first calculations have been performed in the late 50. This history provides an interesting illustration o f the subtle influences and connexions between [he state o f the theory, the "fashionable'' problems and the power o f the available computers. The first simulations [1-3] used a few hundreds of macro pa nicies Very quickly it appeared that the choice o f the code was depending first o f the nature o f the problem with t w o main class. Problems o f statistical physics (drag and diffusion on test particles, evolution o f velocities

distribution, spatio

temporal

structure

of

the

fields

fluctuations,

particle

correlations .). Purely collective motions (corresponding to a description through the Vlasov-Poisson (or Maxwell) model) The next important aspect was the number of dimensions which can be handled Here, obviously, the power o f the available computer plays a crucial role and in the first simulations only one dimensional problems could be handled But, then, a difference appears accordingly the two types of problem mentioned above.

46 In statistical physics we should really handle charged points in a 3D space Since the computers o f the sixties were unable to do that, theoreticians, consequently invented one dimensional particle (sheets) i.e. planes perpendicular to a straight line and experiencing a translation parallel to this line In the same spirit two dimensional particles were introduced (lines perpendicular to a plane) The statistical treatment o f I D plasma was initiated by Lenard [4] from an analytical point of view and from a computer experiences aspect, by Dawson, Eldridge and Feix [5-6] and is still going on [7] Although from a statistical point o f view, I D and 2D plasma do not exist, their study has been quite useful in the shaping o f statistical theory o f weakly correlated, real (3D) plasma. Especially the existence o f a I D code where the only error is the round off error due to the finite number o f digits in the computer words has allowed a complete separation between the phenomena of numerical and physical irreversible processes. The first one corresponds to a lost o f information while the second involves a smoothing o f informations which, on the other hand, are still conserved for the computation o f the evolution o f the system. The question can be raised o f the possible use o f these N body problem codes to treat all the problems

As a matter of fact the purely collective approach - as Vlasov model- are

approximation o f this N body problem A simple calculation delineates clearly the problem A plasma o f length L allows {L/X Y

collective modes (X.;, is the Debye length) while the

D

importance of the individual effects (for example the time at which they will have deeply influenced the plasma evolution in unit Co~' (0J being the plasma frequency) is given by nX p

(n is the density of particles). In fusion and space plasma n\' LfX

D

3

~ 10 — ! 0 ' Obviously we cannot treat 10

l!

0

5 D

is o f the order oflO* - 1 0 ° while !

particles (since N = nX (h/ D

X)' this number

!

corresponds to a L / X = 10 and n V = 10*) It was consequently realized very quickly that a D

D

numerical solution o f the Vlasov equation -which is based on the limit ( i f t i l —* ™ was a more appropriate approach and in 1962 G. Knorr [8] gave the first solution through a spectral method (a double Fourier transform) with respect both to space and velocity but the problems dealt with, were very close to the linear ones and the full problem was not brought in Indeed the Vlasov approach, i f it deals with the correct treatment of the purely collective plasma- as those met in the real world, does it by introducing a phase space i e a product o f the configuration space by the velocity space A sampling o f this phase space for a plasma o f dimension d (d=l,2,3) and of length L requires {L/X ) D

samples needed for each velocity component distribution f ( r , v , t ) Typically N

v

= 10" - 1 0

!

N ° , where N

v

is the number of

for a precise description o f the

density

Let us compare the numerical effort in the two d

methods for a 100X. plasma for which we will consider that a g - (nX. ) ' = 1 0 " ' is enough t)

0

The configuration space mesh size is taken equal to X 12 while N , = 500 For the Vlasov n

treatment we get 10 !

d

f

mesh points per dimension

For the many body problem we need

10 x ( 1 0 0 ) particles It turns out that the numerical effort to treat one particle or one mesh point is similar

4/

1

Consequently this effort is

d=l

d=2

d=3

many body

10'

10'

10°

Vlasov

irV

10

10

10

15

Although these figures must be considered with cautions they give the general ideas for deciding the use o f particle or Vlasov Eulerian code For

I D plasma the efforts are equivalent and the results given by the Vlasov code are

much better - we will see why precisely in one moment and the Vlasov eulerian code must be prefered For 2D plasma interesting results are obtained with 10' particles while Eulerian (i.e. Vlasov) codes need, at least, 10' mesh points (10'" being better)

Since these numbers (1

Gigaword 10 Gigaflops) will be available in a near future 2 D Eulerian Vlasov codes will be used more and more frequently especially in situations where one configuration or velocity dimension does not need a too frequent sampling. On the other hand fully 3D plasma will remained, for a long time, described by particles codes Now, before introducing the topic o f Eulerian Vlasov code - i.e. the topic o f this article - t w o remarks are in order.

Particle codes have been designed allowing a better approach o f the purely collective (Vlasov) limit. Particles are considered as volume elements o f the phase space gathered at the same point rj.v,. In that respect an individuality parameter will show up The improvement is on the way the fields are deduced from the positions o f the superpa nicies

An efficient

smoothing is introduced either by considering extended in space overlapping particles or by filtering the shorts wave lengths o f the fields or by redistributing the charge o f a superparticle among the neighboring points o f the configuration mesh (Particle-in-Cell) Although these treatments decrease efficiently the unwanted - because exagerated - individual effects they do not

change the scaling o f these effects and dividing these effects by two still imposes a

doubling o f the number of particles The reason is that, to kill these effects, we must smooth the fields not only on the interparticle distance but on the Debye length, but then we beggin to a

s

modify the possible collective behaviour finally nX o f order o f 10 must be used u

The possibility o f using a small number o f particles for 2D plasma with the particles codes (we indicated 10') must not hide the fact that our sampling o f the velocity space is very rough. I f we lake a 100X , 2 D plasma with the currently used X I3 D

D

s

for the size o f the

configuration mesh we have close to 10 configuration mesh points with, at each o f these

48

points, 100 particles to define a t w o variables (the two velocities) function i.e. 10 points per velocity dimension (and many 2D particle simulations have been run with much less than 10' particles) I f violent oscillations occur in the phase space (and we will see that this is the case in the velocity space) they are simply wiped out, which is a good thing i f they do not play any role, but a catastrophe if they can reappear on longer wavelength, as in the echo problems This bad sampling of the velocity space is worst in the region o f high velocity where there are very few particles which play an important role in the evolution o f the plasma (stopping of the Landau damping by non linear effects, appearance o f holes in the high velocity part o f the phase space, echo problems).

Our second, at last, remark before we concentrate on Eulerian Vlasov codes is that the particle codes present strong analogies with the Monte Carlo method

As in Monte Carlo

codes they are noisy, ignoring the too subtle correlations but efficient to describe the most important phenomena For the moment they have no rival in 3D problems, in 2D plasma these results should be compared with some test runs obtained with Eulerian codes, while waiting a factor 100 in the power o f supercomputers.

1.2. The Eulerian Vlasov Codes.

The reader whom we have convinced - hopefully- o f the virtues o f the Vlasov Eulerian Code may be surprised that no more effort has been spent on them

We see two explanations:

The first is that, in the last 20 years, the emphasis has been more on fusion devices that in basic, fundamental, plasma physics Such devices imply at least 3D phase space (for example two velocities and one position) and up to very recently eulenan codes could only treat 2D phase space Previous attempts on the numerical solution of the Vlasov equation were not able to solve, in a practical way, the big problem of filamentation in velocity space. This problem appears already on the treatment of the motion of free particles described by the equation

dl

dr

a Fourier transform on x gives

^--^•/(k.v.ij^O

(1.3)

49

The solution o f (1.3) is: f(k,v,t)=

f(k.v,0)exp

(1.4)

ikvi

We see on (1 4) that, as a function o f v, t\k.\>,\) oscillates at the frequency kt, When this frequency reach the inverse ofAv (where Avis the size o f the velocity sampling) we cannot any more follow the exavt evolution o f f This is the unavoidable fomentation phenomena

N o w in the treatment o f fljr.v.r) by eulerian codes we may heve to interpolate and this operation may lead to numerical instability The code must avoid these unstabilities and provides a smoothing o f the details o f the velocity space, hoping that these details will not influence the subsequent evolution o f the plasma

After an attempt to solve directly by

difference method the Vlasov equation, which was unsucessfull [9] peoples turn to spectral methods in the spirit o f the method introduced earlier by Knorr [ 8 j .

The choice o f the transforms was rather obvious for the configuration space. The Fourier transform on x has the advantage that the linear theory (perturbation around an homogeneous neutral plasma) make all i components independant one from each other. Slightly non linear plasma could consequently be expected to be treated by a small numbers o f k The Fourier transforms on v used by Knorr was less evident (especially Fast Fourier Transforms are fast but not very efficient in the ireatment of velocity function- see remark on paragraph 2)

The next treatment was introduced by Feix and Grant [10], Armstrong[ 1 1 ] and Sadowsky [ 1 2 ] Fourier transform were kept on x but Hermite transforms were introduced to treat the v space The Hermite series has advantages (it does not introduce spurious periodicity in v ) but has the great disadvantage o f a slow convergence due to the fact that, for large v =

n,H„( )

cosjnv

or sin-Jnv

Consequently large number o f Hermite polynomials - up to 10

4

in some calculations- was needed F H . expansion was, nevertheless, efficient in the treatment o f small nonlinearities, initially located on long wavelengths {k\

D

< 1) see [14],

The solution for the correct treatment o f the Vlasov equation was finally given by Cheng and Knorr [13] The method is based on the splitting o f the equation in two steps In the first we treat the free motion-

(15)

50 In the second we treat:

%L Oi

+

E

(x.t&-0 dv

(16)

Both (1 5) and (1.6) have analytical solution and in part 2 we will give the detailed algorithm Although the solution was given in 1976 - the method was not systematically used before 1982 where the present authors and theirs collaborators started to look at different problems both in the electrostatic regime (Non linear Landau damping, B G K holes stability, two stream instability, Wigner equation describing quantum plasma (see in this volume the paper by G. Manfredi and M.R.. Feix)) and also in electromagnetic regimes (especially laser plasma interaction models). Since that time, Eulerian Vlasov codes are slowly taking their place among the plasma simulation tools To end this introduction we will simply mention two successes o f the Eulerian Vlasov codes the formation o f holes in the Landau problem which, subsequently, drive an oscillation of the bulk o f the plasma and the appearance o f a relativistic beam in the laser plasma interaction Both successes are not suprising since they involve the correct treatment o f particles located in Ihe tail o f the Maxwellian distribution (very badly sampled in the particle codes) Theses two topics will be treated in part 5 The paper is organised as follows:

After this introduction, we presenl in part U the details o f the splitting method and, especially the different possible interpolation schemes (Fourier ,Spline), also the possible boundaries conditions are presented in this paragraph

Part 3 deals with the microstruciure o f the phase space solution and treats in detail the filamentation

problem which was previously mentionned. Two definitions o f entropy as

measure o f the lost o f information are given and the influence of these losts o f informations is carefully studied - wiih the use o f very fine mesh- up to 10' points for 2 D phase space Part 4 deals with the important theorelical and practical problem o f phase space representation Indeed the treatment of the mass o f informations given by the code requires sophisticated graphic treatment

I f the 2D phase space representation is well understood the problem is

widely opened for higher dimensions and we give few hints to beggin a solution In part 5 we present these problems where a very small number o f particles play an important role - the "success" o f the Eulerian Code mentionned just above We present our conclusions and prospectives in part 6

51

2. S P L I T T I N G S C H E M E F O R E U L E R I A N V L A S O V C O D E S

2.1. The s p l i t t i n g scheme:

The eulerian Vlasov code is based on the well-known Splitting scheme introduced by Knorr and Cheng We outline here some physical meaning o f this method in the case o f the study o f an electrostatic plasma. We want to solve the Vlasov equation for the one-dimensional electron plasma distribution function

dt

/(x.v.t):

dx

dv

where the electric field is given self-consistently by the Poisson equation

£-£j)TluW#-l

(2-2)

starting from a given initial conditioner. v,/=0) Note that x, v and / are respectively normalized to the Debye length, the thermal velocity and the inverse plasma frequency
steps), which consists o f treating the convective term and the acceleration term

separately, as previously mentionned The crucial point is the representation o f the electric field Efx.t) by a succession o f Dirac pulses. Therefore, we have to solve Eq.(2.1) in which the electric field Efx,l) has been replaced by E (x.l)

E'(x,IJ

where

given by:

= E(x.OAlJ S{'-l^) n

(2.3)

l

=(n + \)M

Denoting fL, and f * , as the time before and after the Dirac pulse, the integration during the time interval Al from f„ = " A / to t

n M

= ( n + l)At is straighforward and can be divided into

three steps: Step a: For f„ < / < C , , we have the free particle motion solution

f'{x.v)

= r{x-vMj2y)

Step b: Computing the electric field at time equation, we have:

(2.4)

by substituting / " in the Poisson

52 r{x.v)

Step c For l ^ j < l <

= f{x.v-E(x.r^/Al)

(25)

we repeat again step a, to obtain

/ • * * ' ( * . v) = / " ( r - v > A / / 2 . v )

(2.6)

It must be pointed out that substituting successively Eqs. (2 5) and (2 4) into (2 6), we obtain f*'(x,v)

with x-x-v&i/2.

= f"[x-Ai{v-\E(x))Al,v-£(x)til]

(2.7)

Thus we find that Eq. (2 7) is equivalent to the following integrated

equations o f the characteristics o f Eq. (2.1) by comparing the arguments o f f

and the right

side; X, =xfl-*tj=

* „ , - At[v„, - E(x^.,^)Ai

f 2]

g )

The continuum form o f the characteristic equations, is o f course, given by:

r = v

and

v=

E(x(l),i)

(2.9)

Thus, from a numerical point od view, the splitting scheme is equivalent to an integration o f the Vlasov equation along their characteristics and is correct to the second order in A/ From a physical point o f view, the fact that (2.4) and (2 5) give an exact solution o f Eq (2 1) with the replacement o f £ by £ ' provides some interesting information on the size o f A/, as already pointed out in ref [14] For instance, linearizing (2 1) with E equilibrium (i.e. f(x.y,!)-" E (v) 0

a

+ f,(x,y.l)

around an homogeneous

) and performing the usual laplace (I-> s) -

Fourier ( x - * k) transforms yields to:

-tk E/k.si

+ j ^ E

Due to the appearance o f the sum £

'

_

2

^

W

A/ j J— ' ^ ,/u dv s— iki'

s — ikv

(2 10)

on the 1 h.s o f (2 10), the plasma dispersion function

cannot be recovered But, since we know that the frequency spectrum o f the electric field does not exceed a few a> (choosing A/ such that u A / « l p

p

results in a zero contribution of all

terms o f the sum on the I h s o f (2.10), except for n = 0, which recovers the usual Landau dispersion function

53

2.1.Interpolation methods:

N o w , we have to consider the problem o f performing the shifts (2.4) and (2.5) Different

methods can be used. Cubic spline interpolation in the v-space and Fourier

interpolation in x-space (without using Fast Fourier Transform) have been used by Cheng and Knorr [12], which requires, in the x direction, an execution time proportional to N,

(where

is the number o f points in the spatial direction). Because this fact increases the

computational effort when a large number o f points is used, a cubic spline interpolation can be used also in this direction (see Ref [15]) It can be pointed out that during each time step, the Vlasov equation takes on the form of an advective equation (one over x, one over v). On the other hand, the fact that a shift can be expressed in the corresponding Fourier space - by simply changing the phase - suggests the use o f Fast Fourier Transform (FFT).

2.2.1 F o u r i e r i n .1/ Fourier in v interpolation method:

Let us recall briefly the main features o f the Fourier in x/ Fourier in v interpolation method. By performing a Fourier transform in the x space (* being the conjugated position variable). Eq. (2 4) yields to an exact solution in the form;

f { k , v ) = f " ( k , v ) e x p ( - i k v 4 i / 2)

(2.11)

while a Fourier transform in the V space (X being now the conjugate o f the velocity space variable) applied to Eq. (2.5) gives: f-{x.X)^

f{x.X)exp-iXE(x,r )M

(2 12)

tL

Then the successive shifts are achieved in the following way: (a) Free particle motion from /„ -

n

f"{x,v)

(b) Dirac pulse at ' -

>

IIAJ

to

:

1 f"{k.v)—?->exp(-ikv&l/2)

>f'(x,v)

(2.13)

:

f{x.v)^^f{x,X)^ xp<-fk£(x,r e

) & ) ^ ^ r { x . v )

(2.14)

54

Here FT, (respectively FT ) means Fourier transform with respect to i (respectively to v) and V

IFT means inverse Fourier transform. The symbol x means o f course multiplication (i.e. shifting the phase). This last code has been used to study the nonlinear behavior o f B G K waves [16] (the code has been found to be extremely stable over time of order 10 0)~') and this method is !

particularly well adapted to treat the quantum Liouville equation [17] (which gives the time evolution o f the Wigner fmction), to see how the quantum effects change the phase space evolution o f the one-dimensional electrostatic plasma In this case, it must be pointed out that the Fourier transform on the momentum p ( in quantum mechanics we used the momentum variable instead of the velocity ) for a given r (i e applied in the frame o f the splitting scheme) turns the treatment o f the interaction terms into a problem nearly as simple as the classical one

While cubic spline interpolation seems to be well adapted for simulations with periodic and causal boundary conditions, the Fourier interpolation method can lead in the treatment of the velocity space to the occuring of aliasing in the Fourier transform (the Fourier transforms imply that the distribution function is periodic in velocity but, regardless o f its initial gaussian shape, it will be generally evolve to a function that is non periodic Consequently to avoid aliasing a large number o f values o f the distribution function above a given cut-off in velocity Is set equal to zero. But this usual method for controlling this edge noise by filling the grid with zero's near its velocity boundaries, leads to a non-negligible increase o f the memory, together with the computational time)

2.2.2. Flux conservation method.

This method is based on a "fluid" interpretation in which the solution can be written in the form Ax + Bv + C

where / " = f(*,,v,,t,)

(2 15)

denotes the value o f the distribution function at the mesh point ( i j ) .

and A3 and C obey

&«J C=f°-Ax,-

(2 16)

By,

For our purpose It is convenient to calculate rhe total mass M" o f the cell (i,j) at lime t„ t

55

del

dv r

x.v = / " A x A v

(2 17)

By example, in the first step o f the splitting scheme, in which we want to shift the distribution

v A/ function in the x direction o f the quantity Bc ;

positive, the cell ( i j ) has lost the mass SM

U

L

, and assuming that the velocities are

while in the same time it received the quantity

SrV/,_, j from the previous cell (i-1 j ) . We have then

SM

= f

dx\'

dvf"{x,v)

(218)

leading to a value o f the interpolated distribution function o f

/,', = / ; + ^ ( 5 A / , _ „ - 5 M , „ )

(2 19)

Details on this method have been given in Ref [18].

3. M 1 C R O S T R I I C T U R E I N P H A S E SPACE G E N E R A T E D B Y T H E V L A S O V EQUATION:

The use o f noiseless extremely stable eulerian codes on a vector computer allows us to obtain precise information on phase space behaviour o f a one-dimensional electron Vlasov plasma, over a long time, with a large spectrum o f wave numbers. On the other hand, a main property o f solutions o f the Vlasov-Poisson system is that they often contain fomentation leading to the occuring o f a fine microstructure in phase space. The Vlasov equation solution cannot change along a characteristic generated by (2.9), and often the characteristic mix together phase space regions where f exhibits significantly different values and then steep gradients are generated i n / In simulation that are done on a grid in phase space and the grid almost inevitably becomes too coarse as this fine graining develops This fdamentation problem is well known in Vlasov simulation and has been discussed often ,as previously mentionned

Nevertheless, an

1

examination o f the long time behavior (up to ZOOOOJ" in the case o f the study of two-stream instability and B G K waves) in numerical Vlasov simulations have shown the occuring o f a rather smooth structure in the final state in phase space Although these results have been obtained with the Fourier code, they are identical with those obtained with the cubic spline

De-

code. The numerical simulation implies necessarily a discretization o f phase space, which can create a "cleaning" in the temporel evolution o f the tine structure developed by the Vlasov equation. When this fomentation reaches the size o f the elementary cell A i A v , the treatment of the information Is not exactly effected, leading to the phenomenon o f "cleaning" of the distribution function in phase space Numerical simulations have been carried out, in the case o f a large amplitude non linear oscillation in a stable plasma, to investigate the details o f the fomentation mechanism. Furthermore a measure o f the lost o f information is obtained by means o f the entropy o f the system. We choose for the initial condition:

f(x,v,0)=F {v)(] 0

with k\

0

= 0.4

and

+ acoskx) =

I

e

1

(l + a c o j t r )

a = 0.5

To do that, we have used two differents definitions o f the entropy o f the system,

first

Vlasov-Poisson system

The time evolution o f H, (respectively H ) is shown in Fig l a (respectively in Fig.lb) :

using a phase space grid o f N , N = 128 x 128, corresponding to the three differents methods V

o f interpolation (we used the Fourier code, the flux conservation code and the cubic spline code). The curves shows clearly a rapid increase o f the entropy for the three differents interpolation methods

As time increases, the entropy reaches a maximum level at about the

same time that the instability saturates and then shows no tendency for further growth An detailed examination o f the time evolution shows that It takes somewhat longer to obtain the "cleaning" o f the micros!ructure with the spline code and Is related to the treatment o f the small wave lengths, which is numerically quite different in the three codes However, all methods used here exhibits, after the "cleaning", the same state corresponding to values of entropy very close in the asymptotic behavior o f plasma.

The increase o f entropy has been confirmed by different methods o f interpolation and led to the same asymptotic state From a numerical point o f view, a numerical scheme is based on a sampling o f phase space, with a spatial grid spacing Ax and a velocity grid spacing Av. It is clear thai when the micro structure becomes as small as the elementary cell AxAv. the numerical code cannot describe the exact mathematical solution o f the Vlasov equation. However from a physical point o f view, the fine structure generated by the Vlasov equation

57

Fig. 1

Time evolution of the entropy N, (in Figla) and H (in Fig. 1 b) of the system computed using: i

(1) the Fourier code (2) the Flux conservation code (i) the cubic spline code

fi8

has no influence on the "coarse-grained" asymptotic solution In order to check this point, we performed simulations with different phase space sampling, using cubic spline interpolation The first series of experiments (curve denoted 1) was conducted with a phase space grid o f N „ N

V

= 128 x 256 and the numerical results are shown in Fig 2a and Fig.2b, indicating

respectively the time behavior of H, and

As remarked above, the simulation exhibits a

rapid growth o f both forms o f entropy. The help o f a high resolution phase space diagnostics o f a Vlasov code will allow us to give a much more precise explanation. The same simulation is now presented using a phase space sampling o f N , N = 1 2 8 x 4 0 9 6 (curve 2), showing now a V

slow growth o f the entropy in both considered definitions o f entropy The last case (curve 3) corresponds to a higher resolution of N , N = 5 1 2 - 8 1 9 2 , which gives rise, at time tcu - 200, v

to a relative variation o f order 10% in the case ofC(/}example C( /') = flnf.

p

f \ and o f order 20% for the second

(Note that an exact conservation of the entropy o f the system requires

an infinite number of points). Because we can obtain a small relative variation o f entropy, the last experiment is chosen as the "reference solution" and we can admit that the microstructure is well conserved in this case, at least for the evolution lime scales considered here N o w by filtering the distribution function of the "reference solution", an artificial "cleaning" o f the microstructure is introduced inside a given cell A-YAK »

AxAv (thus it is not

necessary to keep all the microstructure during plasma evolution to obtain the asymptotic state) The filtering was done as follows: the distribution function is Fourier transformed both in k and X space, the resulting signal is multiplied by a filter chosen so that one simulates a reduced sampling o f phase space (and consequently an higher elementary cell) and then the resulting function is Fourier transformed back. Thus we start with the "reference solution", which consists in a distribuiion function o f N , N , = S12x 8192, i.e 4,194,304 mesh points in phase space and chose two differents filters in order to obtain an equivalent solution, respectively to a sampling N ^ N . = 128 x 256 in the first case, and N N l

second example

l

= 128 x 4096 in the

Then we calculate the entropy o f system (for both definitions ) using the

reduced distribution functions and make comparisons with the results directly obtained by Vlasov simulations using the same sampling of phase space Although small differences can be observed during the growth o f the entropy, the final state is identical in both cases, indicating thai it is not necessary to keep all the information generated by the filamemation phenomenon to obtain the asymptotic state It must be pointed out that entropy consideration would be a good test to check the validity o f a numerical scheme, and in the considered case here, to check the method o f interpolation used in the splitting scheme Finally, obtaining these insights into the complex behavior of filameniaiion process (and therefore o f the phase space topology) was only possible with an Eulerian Vlasov code, which can provides a great deal o f resolution in phase space

59

0

50

100

0

50

100

Fig,2. Time evolution of the entropy H, fin Fig2a) and H. (in Fig.2b) computed using cubic spline interpolation in the case of a sampling of: (1) N,N = l2Sx256

of the system

v

F

(2) (3)

/V A „ = 128x4096 JVJV„ = 128x8192 i

t

The entropy evolution calculated using a filtered distribution function corresponding to a sampling of N,N = 128 x 256 is noted by + N N = ]2&x 4096 is noted by o V

t

r

60

4 PHASE SPACE R E P R E S E N T A T I O N , M A T H E M A T I C A L T O P O L O G Y AND PHYSICAL RESULTS O FT H E VLASOV EQUATION.

As our first example, we consider the two-stream instability in order to handle the difficult problem o f the phase space representation o f the distribution function

The most

striking advantage of the Vlasov code is the very fine resolution in phase space, capable of describing the vortex fusion process for B G K holes

We begin with a one-dimensional one species (with a fixed homogeneous neutralizing ion background) electron plasma A well-known property o f the plasma is its ability to support BGK equilibrium, consisting o f a stationary spatially periodic structure exhibiting holes (vortices) in phase space Starting with a homogeneous two-stream unstable plasma, we have shown that in a first step, a B G K periodic structure [14] appears with a number o f holes proportional to the length o f the system (the chosen length is twice the marginal length), followed in a second siep. by a coalescence of holes, to end up with a large hole structure.

A first series o f numerical simulation showing the appearence o f a two-hole structure at time tu) = 40 and its subsequent coalescence Is presented in Fig 3 These curves represent, p

at different times, the 3D plot of the distribution function f(x,v) and the corresponding J

-c'

contours. This kind of representation shows clearly the "cleaning" o f the microstructure and the tendency o f vortices to behave as macroparticles before the coalescence (see Ref [16]) However this representation is not adapted when we must descnbe a longer plasma (for example the contour plot give rise to a strong turbulence of contour lines, which is not a physical feature o f plasma)

To check this point, let us consider the phase space representation o f a distribution function f(x,v), solution of the same problem, but for a longer plasma, which exhibits at the beginning o f the evolution (at time tco - 20) about thirty vortices. (In this case the sampling p

o f the distribution function has been chosen to N , N = 2 0 4 8 x 2 5 6 ) . Fig.4 shows the time V

evolution o f this distribution function at different times. Here grey shading has been used to indicate the relatives values o f normalized phase space density between 0 01 and 0 40 (with black for higher values and white for lower values)

The time evolution o f plasma is

characterized by several fusion processes till a three-hole structure is occuring at time \
p

(in fact at time Uu = 40000 the system has evolved towards a two-hole structure). It is the p

inherent phase space representation resolution o f the Vlasov code that allows the presentation o f such detailed structure in phase space, which, in the more common PIC codes, are very coarsely represented

61

PLASMA DISTRIBUTION FUNCTION

Fig. 3. Phase space representation using 3D plot and ihe corresponding f = const contours in the case of the study of the two-s tream instability for a short plasma: we show clearly the appearance of two holes, followed by their coalescence and by the "cleaning" of the microstructure in phase space.

62

T=20

-41

I

1

0

123.1

1

246.2

1

369.3

1 Y

492.5

T=60

Fig. The similar problem for a larger plasma in which thirty holes appear at the beginning of the evolution. In this case the contour plot representation is not adapted Then we have used a grey-shading representation. The time evolution of plasma is characterized by several fusion processes leading to an intermediate state forming a three-vortex structure.

63

Another stricking diagnostics, which allows us to have a more detailed insight into the vortex coalescence process is the behavior o f a population o f several thousand o f test-particles, presented in Fig 5 at the same times The test-particles distribution, initially chosen in a region close to zero (in velocity) - the distribution being uniform along the x direction- have been computed through the data o f the noiseless electric field given by the Vlasov code. The trajectories are then obtained by integrating the equations o f motion(2 9). It is clear that the motion o f trapped electrons in the electric potential is in good agreement with the position o f holes in the grey shading representation o f Fig.4.

Although the grey-shading representation can be improved by the use o f color (several movies concerning the B G K waves, laser-plasma interaction, have been produced in color, given further details o f the mechanism o f the instability), a global view o f the phase space is not possible when the plasma description requires more than two phase space dimension Furthermore we believe that test-particle diagnostics can allows us to achieve a better understanding o f the phase space topology, and that an obvious way, to improve the representation o f the distribution function with more than two phase space variables, is to analyse the plasma by means o f test-particle diagnostics.

To confirm this point, let us give an example o f behavior o f a test-particle distribution in the case o f the study o f the injection o f an electron beam into a two-dimensional and magnetized plasma, in a guiding center approximation (see [19]) In this geometry the electron (and also ion) distribution function fj,x.y,v,?tf)

obeys the usual drift kinetic Vlasov equation

(for more details see Ref [20]) These simulations should be a good starting point for understanding the diffusion processes and therefore the plasma heating mechanism induced by the beam injection

It has been shown that the injection o f high energy electrons into an

inhomogeneous plasma can lead to the appearance o f two types o f instabilities: KelvinHelmholtz induced by the E*.B interaction

drift and the beam plasma instability, and their mutual

Most importantly, the test-particle simulations have indicated the existence o f

electron diffusion mechanism in a given plane of phase space (Without looking at the noiseless behavior o f a test-particle distribution, this feature o f the diffusion process could not have been detected directly in Vlasov simulation since there would have been no way to separate the behavior o f beam electrons with the bulk electron distribution) This feature seems to be due to the existence o f an invariant o f the system v „ - j t D „ / land -const,

where is the electron lt

cyclotron frequency and 6 is the angle between the magnetic field B-(B ,0,B.I s

direction

and the r

64

Fig.5. Behavior of a population of several thousand of test-particles, initialy chosen around v=0 . The trajectories of particles have been integrated using the electric field computed by the Vlasov code. It is clear that the motion of trapped particles is in good agreement with the positions of holes in the grey-shading representation

Fig. 6. Behavior of a population of several thousand of test particles in a 3D representation for different values of the angle 6 (angle between the magnetic field B = (B ,O.B ) and the x direction), in the case of the study of a beam injection into a plasma in a gyrokinetic approximation. The test particle behavior confirms the fact that the anomalous diffusion is characterized by an expansion of particles in a same plane. I

1

For 6=50° the Kelvin-Helmholtz is the dominant instability, For 8=45° For fl=/0°

a strong competition process between both instabilities is observed, The beam plasma instability is dominant.

66

Now we would like 10 point out some o f the salient features related to this type o f representation presented in Fig 6 The test-particle behavior given here confirms the fact that a physical property o f anomalous diffusion is characterized by an expansion o f particles in a plane tilted by angle 8" = - ^ - 8

with respect to the x-y plane. Consider for instance the

distribution o f testparticles plottd in Fig 6a, showing the initial location o f particles (chosen among the electron beam) in a 3D representation o f the phase space For the three different values o f angle 6 , say for 6 = 80° (corresponding to a dominant K H instability), for 6 = 45° (where the plasma heating seems to be higher) and finally for 8 = 10° (the case o f a strong beam plasma instability), particle behavior reveals different mathematical topologies in the diffusion process Furthermore, simulation results indicated a very weak diffusion in a direction perpendicular to 6" Because o f its accuracy and stability we are convinced that a Vlasov gyrokinetic code together with test-particle simulations is indeed a viable approach and it will eventually help us in understanding anomalous transport processes in magnetically confined plasmas

5. R O L E O F M I N O R I T Y P O P U L A T I O N S .

S.I. The non linear Landau damping and phase space holes.

Landau damping is the oldest -and most famous- kinetic effect exhibited by the Vlasov equation

It may be worthwhile to mention, for the youngest readers, that Vlasov in the

linearized treatment of his equation missed an aspect connected to analytical continuation in the complex plane Landau, in 1946. gave a correct -but obscure- treatment o f the problem Consequently a long discussion took place until 1962. some authors claiming that it was a mathematical artefact with no physical aspect until Dawson [5] and independtly Eldridge and Feix showed its existence through computer experiments It is worth to point out that they exhibited Landau damping not on the evolution o f a macroscopic perturbation but in the microscopic fluctuations properties of a plasma at thermal equilibrium ( i e

through an

extension of the Landau theory at a microscopic level) A more direct check was finally given by the Eulerian Codes of Knorr (Fourier, Fourier) and the Hermite Fourier algorithms o f Feix, Grant, Sadowsky and Armstrong These authors notice also that an increase o f the nonlineanty (i.e. the size o f the perturbation) brings a stopping of the damping after some oscillations with possible modulation of the amplitude o f the oscillation and, finally, the establishment o f a steady undamped oscillation A clear physical picture o f this phenomena has waited the appearance of graphical logiciel allowing a representation o f the 2D phase space f(x,v) and in 1988. Bertrand and al [26] showed phase space pictures which, obviously, could not have been obtained by particles codes

67

Fig. 7. Time evolution of the electrostatic energy for the study of the non linear Landau damping. We see clearly that the amplitude of the oscillation is modulated at the beginning of the evolution III! time la> ~ 500 .where we reach a steady amplitude for the oscillation Here we have k\ - 0.3 and a = 0.2. f

0

68

The problem is the following: the plasma is un idi mens ion al. electrostatic with only electrons moving (the ions form the usual homogeneous, motionless, neutralizing background!. The initial condition is

1

/(x.v,0)

= F {v)(\ + acoskx)^-^exp-(\/2)v {\ 0

+ acoskx)

(5 1)

In (5.1) the units have been chosen such that V — 1 (thermal velocity) and u> = ^ . = 1 M

p

D

Two parameters characterize the oscillation: the wave number k o f the initial perturbation indicate the level o f kinetic character o f the problem and a the degree o f nonlinearity -although -ry is certainly a better measure o f the importance of the nonlinear effects We present first the itresults for k = 0.3 and a - 0.2. Fig.7 gives the variation with time o f the electrostatic energy The fundamental oscillation is at frequency close to 2m (since we plot the energy i.e. the f

square o f E). We see clearly that the amplitude of this oscillation is modulated at the beginning until time tco = 500, where we reach a steady amplitude for the oscillation The full p

explanation is provided by the examination o f Fig.8. Only the v>0 half phase space is shown The value o f f is given by grey tints but, in order to see what happens for theses particles in the tail, all parts o f the phase space where / > 0 . 0 2 are in black (notice that at the maximum (V= Oand F (v)=A0) o

and consequently the full spectrum o f grey tints is used to describe an

"hilly area" the highest point o f which are at an altitude equal to 5% o f the height o f the central (v=0) "chain o f mountains" We see clearly on the three first pictures o f Fig.8 the overtaking and the subsequent breaking o f a "phase space wave" followed by the formation o f an hole with a filamentary structure winding around this hole (this structure is quite apprent on the two pictures at time t t i ) = 4 0 and ro) = 100). After this last time the numerical smoothing described in part 2 p

p

takes over, filling the gaps between the filaments, destroying consequently the mathematical topology properties o f the Vlasov equation (the vacuum region o f the hole must

be

continiously connected to the external i> > 8 vacuum) but, in a certain sense, restoring the physical properties of the plasma, since we must remember that the Vlasov model is an asymptotic one where diffusion process in velocity space, have been neglected We have seen in part 3 o f this paper that this numerical diffusion process does not affect the behaviour of the plasma-provided the Av is small enough (here the mesh size is 64 (in jc) x 256 (in v). After to> = 400 a large hole is apparent in phase space which does not evolve any p

more (another is present in the w<0 part o f the phase space not shown in the pictures) Now these holes drive oscillation for the bulk o f the plasma at a frequence a) - kV holes velocity is just F

a

lk

But since the

the phase velocity o f the initial plasma oscillation, we see that the

plasma goes on oscillating at a frequency very close to the initial oscillation The formation o f the holes corresponds to the presence of a small population o f trapped particles, but which play

69

co .t=20.0

w,.t=10.0

&j .t=0.0 p

p

W/

0

5

10

15

20

0

6j..t=i00.0

0

5

10

15

20

0

oj„.t=300.0

5

10

15

5

10

15

20

u..t=200.0

20

0

w„.t=400.0

5

10

15

20

oj .t=500.0 n

e 5

at-.

MM 0

5

10

15

20

0

5

10

15

20

0

5

10

15

20

Fig.8. Phase space representation of the distribution function at different times during evolution for the study of the non linear Landau damping with kX - 0.3 and a -0.2. We see clearly on the three first pictures the overtaking and the subsequent breaking of a "phase space wave ", followed by the formation of an hole. D

70 u .t=0.0

cj .t=10.0

p

0

2 4

6

B 10 12

0 2 4

u .t=30.0

2

4

6

8 1012

0

6

8 10 12

it/Xj

2 4 6

8 10 12

w .t=300.0

p

2 4

0 2

p

0

2 4 6

x/\.

8 10 12

4

6

8 10 12

w .t=100.0

p

w .t=200.0

0

6 8 10 12

(

cj .t=45.0

p

0

tj .t=20.0

p

p

0

2

4

6

8 10 12

£j .t=500.0 p

0 2

4

6

8 10 12

\/\,

Fig.9. For larger k\ the kinetics effects are more important and the wave experiences a more important damping before the appearance of smaller holes. The phase space pictures correspond to kX = 0.3 and a = 0.2. D

D

71

the fundamental role in the stopping o f the Landau damping Notice that the presence o f the holes N„N

was v

obtained

with

a

modest

numerical

effort

(the

treatment

of

= 6 4 x 2 5 6 = 16.384 mesh points) With this number o f particles the particle codes

hardly suggest the holes

To end this paragraph we show that for larger k\

the kinetics effects are more

D

important and the wave experiences a more important damping before the appearance o f smaller holes

Fig.9 gives the phase space pictures for kX -0.5

and a = 0.4 (The same

D

parameters have been used previously in Part 3 to study the problem o f filamentation) We see up to time tti) = 30 the appearance o f the filamentation characteristic o f the free particles p

motions Then, around this time, and correctly centered at the phase velocity, (now equal to 2.6) a hole appears pushing up and down the neighboring filaments and reaching a sort o f asymptotic regime at tu) = 200 (up to the end o f the calculation tto = 500). p

p

Many more interesting results have been obtained with the Vlasov codes, especially concerning the fusion of holes This fusion leads to a wave number divided by two, since we have usually an acceleration o f the new hole, the frequency o f the driven oscillation k V is not t

quite divided by 2 but jumps from (U = 1 15 tool = 0 82 (i e. well below the plasma frequency a characteristic o f the ballistic mode) See Ref [18] and [25] for further informations on the holes fusion

5.2 Periodic simulations o f Stimulated Raman Scattering (SRS), wave action

and

reliitivistic beam generation.

The acceleration o f charged particles by fast (OJ/AC = 1) large amplitude longitudinal electron plasma wave trains has received considerable attention in recent years One o f the potentially most effective ways to produce such a wave uses the forward Stimulated Raman Scattering (SRS) process, long o f interest for laser fusion and current drive As Is well known, this is a stimulated decay o f incident light into a forward-scattered light wave and a forwardgoing plasma wave (necessarily at plasma densitiezs below the pump frequency quarter-critical value). For these

problems, an eulerien relativistic Vlasov code renders possible a detailed

examination o f the low density regions o f phase space For theses problems particle simulations codes suffer from poor statistics This because the PIC codes lack enough simulation particles to display the detailed phase space structure o f the distribution function which is often obtained in those regions o f phase space where panicle and phase velocities are comparable and where the trapping occurs

I f more than one space dimension is involved, only the PIC code is

practicable so far. However, when one space dimension will do, the eulerian Vlasov codes

72 have been found to be a powerful tool for studying in detail the particle acceleration dynamics (relativistic in this case) both in a conveniently short but idealized periodic plasma, and in the more realistic long, open system [21-22) with external sources. Theses codes allow precise measures of action transfer between photons, plasmons and electrons in Raman Scattering processes. Here we present the VJasov simulation results on action evolution for a periodic plasma and comparisons with the canonical three-oscillator model. In order to handle the plasma wave velocities relevant to future particle accelerators, the model equation must at least be relativistic in the acceleration direction (x) . For the linear (y-direction) laser polarization intensities we consider here, the transverse dynamics can be economically included in the nonrelativistic transverse cold-fluid approximation. The VJasov equation in the acceleration (x) direction is thus'

(5 .2)

where y = JI +

p;

is the Lorentz factor. In the y-direction, for reasons of computer economy,

the cold nonrelativistic approximation is used :

dlly

-=-E

dl

(5 .3)

Y

Both equations, together with the usual one-dimensional Maxwell (x) equations, make up the system of equations used in this paragraph. The details of the way in which the equations are advanced and the boundary conditions are imposed have been extensively described elsewhere (see Ref. [21-23)).In our numerical experiments, we use normalized quantities: I,X,Uy, Px are normalized respectively to w~J ,e/w p ,e and me.

In the three-wave parametric decay from the pump "0" to the scattering wave "s" and the electron plasma wave "e", we have, as usual :

(5.4) we assume perfect k matching (since we have a periodic simulation and match mode numbers exactly), while t1w contains the mismatch (if any) . Defining the complex action amplitudes aiD.,.,) (such that the action density is given by

S = aa·), the classic oscillator model is then gi ven by:

73

(5.5)

Here, we have defined C -

—

;

—

To assist in the development and testing o f such basic concepts, the much more limited and less demanding spatially periodic simulations o f evolution from an initial state can be very useful, since now one needs consider the behavior in time only o f the discrete spatial Fourier modes (In the one-dimensional spatially periodic simulations, panicles and waves exciting either spatial boundary enter the other). In the three-oscillator model (5.5 ), this allows one to drop the group-velocity-times-spatial-gradient terms, convert to ordinary coupled differentia] equations and analyze action density conservation Eqs (5.5) yield the Manley-Rowe partition:

CM

=W

+

H

!

= C,(/ = 0) = const

(5.6)

which means photon conservation, and

c

c.M=kf+kf = . ( ' = o ) - 2 r p / K f I f losses are taken equal to zero, Eq disappears, one plasmon is created

(s.7)

(5 7) means that when one pump photon

Indeed the action sum C, generally decreases since It

involves a plasma wave which can accelerate and trap particles and thus loses energy and action Thus the utility o f a periodic simulation is clearly demonstrated since it allows one to use action conservation to separate the effects o f three-wave interaction from (hose due to nonlinear-particle interaction

The convenient wave vector choice is mode 2 for the plasma wave (i.e. A, = 2(2n / L) where L is the length o f the computation box) and mode 4 for the pump with the forward scattered wavenumber k, equal to mode 2)

The plasma was chosen with two electron

temperature components, the majority (95%) component with a 15 keV temperature (high enough for electron Landau damping to subdue the usually growing but here unwanted backward SRS) and a minority (5%) component at lOOkeV (to enhance wave-particle interaction) Numerical parameters are presented in detail in Ref [22)

71

40

SO

120

.016

0

160

200

240

T ••• r • i ' i • l '

40

SO

120

160

200

L

280

• i ' i

240

280

Fig.lO Time evolution of wave actions: (a) pump action S -a a , scattered wave action S —a a' and action sum C . (b) pump S . plasma wave S = a a and action sum C ; obtained using a relativistic Vlasov code for the study of the Stimulated Raman Scattering. The action is transfered back and forth between the pump and daughter waves in a classic fashion with accumulating loss due to one lossy oscillator (the plasma wave). 0

s

s

s

s

t

a

t

a

r

r

t

75

The time behavior o f the action densities from Vlasov simulation ate shown in Fig 10a.b together with the relevant action sum The action is transferred back and forth between the pump and daughter waves in a classic fashion with accumulating loss due to one lossy oscillator (the plasma wave)

As expected by Eq. (5.6) the action sum C, for

electromagnetic wave pair is well conserved (to within 1.5%), while pump-plus-plasma sum C, decreases. In a lossless Vlasov plasma, with a single plasma wave being responsible for the energy transfer to fast electrons, it can be hypothesized that the kinetic energy o f all the electrons above the lower momentum boundary o f the wave separatrix would account for the missing energy, and on division by the electron plasma frequency, for the action C given to t

the electrons.

In Fig. 11 we present the separatrices calculated from the simulation electric potential superimposed on the very accurate phase space representation afforded by the Vlasov code Here grey shading has been used to indicate the relatives values o f normalized phase space density between 5.10

-i

and 1 0

-1

A striking agreement can be seen between the separatrices

and the phase space trapping limits for growing wave: at the frame for Ktt. = 1 5 0 the spiral structure inside the trapping limits reflects the monotonic increase in wave energy After the plasma wave maximum is reached (for toj ~ 180, as seen in Fig. 10b), the marginally trapped p

particles now begin to detrap and are now above the separatrix boundary The fold then rides up over the next separatrix hump as shown in frame t(o = 225 After Kfl = 270 the second p

p

decay o f the pump begins The detrapped particles are now beginning to be retrapped

More

details on this particle acceleration mechanism are available in [22-23] But for our purpose it is convenient to do a careful accounting o f the action C, transfered to electrons

Let us

compute the kinetic energy density o f the particles with a momentum greater than the lower separatrix p ^ =

, divided by the electron plasma wave frequency 0 ) :

lo

e

(i) *

L

In Fig 12, where C, is shown together with the components o f Fig I O b. one can see clearly that adding C, to the C o f Fig. 10b indeed gives a total C , which is well conserved r

r

It must be pointed out that we were able to account very well for the plasma wave action loss because we used the very detailed phase space diagnostics afforded by the eulerian Vlasov code The behavior o f the system can be modelled in an average way by the standard coupled-mode equations with an appropriate chosen ad hoc damping term As discussed in [22], the absorption rate cannot simply be modelled by a linear term T in Eq.(5.5), and it is necessary to take recourse to the noiseless eulerian Vlasov code to compute the action transfer rate, which is a very important parameter for current drive application

76

0

5 xtty/e

10

0

5

10

xa>Jc

Fig.ll We present here the separatrices calculated from the simulation electric potential superimposed on the very accurate phase space representation afforded by the Vlasov code. Here grey shading bas been used lo indicate the relatives values or normalized phase space density betmeen 5 10" and I f f * . A sinking agreement can be seen between the separatrices and the phase space trapping limits for growing wave. !

Fig.12.

Time evolution of wave action: as in FiglOb,

together with C, and C , . where C, r

accounts for particles accelerated above the lower separatrix (see Fig.12) and C, is the sum r

c,+c . t

78

6. C O N C L U S I O N

We

have presented several problems, mostly one dimensional (2D phase spacel s

both in electrostatic and electromagnetic regimes where with typically 10 points an excellent sampling o f the phase space and evolution over hundreds o f plasma unit o f time w"' has been obtained With a particle code rough and noisy but basically correct results can be obtained with a number o f particles as low as 1000 while the Eulerian code will certainly blow up with a 1

mesh o f 10 points These codes imply a minimum size o f the computation It is interesting to notice than for I D plasma the workstations provide today the needed memory size and processors velocity to handle these I D plasma Eulerian Vlasov code J

1

Of course 2D problems imply an effort 10 to 10 times bigger and we reach the level o f tomorrow supercomputer with a few gigawords o f memory and the subsequent velocity Moreover interesting problems requiring around 10' words can be solved implying 3D phase space. Such a problem can be obtained in the gyrokinetic model, a quite important model in the Tokamak's theory In such model the plasma is embedded in a magnetic field

The

motion along the magnetic field is described by a regular 2D phase space The magnetic field being supposed very strong the perpendicular motion is described by the drift velocity (guiding center approximation) and for some problems only one additionnal dimension will be needed 1

and with 100 sampling points for that dimension we arrive at the 10 points indicated above, an easy task for the actual supercomputers

An advantage o f the Eulerian Vlasov codes is that they incorporate easily ideas issued from the analytical results o f the Vlasov equation For example i f we know that we deal with a slight nonlineanty and an inilial condition with one wave length equal to the dimension o f the box we know that the first, second harmonies will be second, third , etc, order effects in the small nonlinear parameter

We will sample consequently with less points the configuration

space What are the problems we must now concentrate on'' We see problems in basically ihree directions The acceleration o f calculations with the use of massively parallel computers To realize the gains in computational power made possible by massively parallel computers generally requires thai algorithms used in traditional serial and vector computers be adapted or redesigned to optimally use a particular parallel architecture

The Eulerian Vlasov code is

probably much easier to realize in such computers, because one does not have to cope with the allocation of the particles between processors, since the Euler elements stay in place Furthermore the efficiency lost o f the Vlasov code for higher dimensions

might be

compensated by the use of massively parallel computers. Both approaches (PIC code and Vlasov code) require a more detailed comparison on the interplay between physics aspects of

79 the model on the one hand, and performance and optimization issues on the other An effort to accomplish this is well in hand and the results w i l l be reported in due course.

The exploitation o f the mass o f results produced in these computations and especially the representation o f phase space for dimensions equal to 3 and higher Why not leave the plane sheet o f paper for a real 3D picture optically produced and animated -our eyes are quite good to apprehend such images- but o f course the higher dimensions are still a problem We would like to end on the problem o f lost o f information, studied in partlll with the introduction o f the concept o f entropy. Can we avoid this lost and conserve strictly entropy and the true nature o f the Vlasov equation, which is to push the value o f f along a trajectory o f the phase space? For that we need to quantify Ax.Av and At such that we do not need to interpolate. I f the positions o f the mesh points are x = j A x and those o f the velocities are c - kAv (with (

t

j . k e Z ) we must have A r - AvAl to have, in step one o f the splitting scheme, a translation bringing each point o f the mesh to another point without any need o f interpolation We consider now the second step (change o f velocity without change o f the position) I f AE is the field quantum, we have Av = A£Ar and finally A r = AE(At)'

This last formula is anoying it

tells us that AE and A r cannot be sampled on a equal footing (i.e as in the above codes where at each point we consider an associated electric field)

To see better the meaning o f this

formula we take a plasma with a field of the order o f the value found in the Debye screen, i e eE- / D = mV,l /D

giving an acceleration of [e / m)E = Vl /D.

We decide to take as AE

10% o f this value (taking into account negative and positive values most o f our fields will be concentrated on 20 quantified value, which is very little N o w taking A / = 0.1 to' we get 1

Av-(v,i

/D)(Q.\{a~p )(\/\Q)

= V, /TOO: we have to consider a few hundred points in the k

velocity space and using now A r = Av(to~' / lo) = V m'J 7 1 0 0 0 = D /1000 and consequently a n

few thousand points in x. It is this unbalance between the r space (oversampled) the velocity space (correctly sampled according our previous criteria) and the electric field (undersampled) which prevents an use o f this "pushing bit" method It is too bad since, otherwise, networks o f cellular automat could very likely give an extremely fast massively parallel treatment May be the method can be checked and improvement will be found

Acknowledgments:

The authors are indebted to the IDRIS center (Institut du Developpement et des Ressources en Informatique Scientifique -Paris -France) for computer time allocation on the Cray-C98 computer

80

REFERENCE

[I]

O Buneman Journal o f Nuclear Energy Part 2, 119 1961

[2]

C.K. Birdsall, W . B . Bridges Journal o f Applied Physics 32,2611 1961

[3]

P.L.Auer, H . Hurwitz. P W. Kilb Physics ofFluidsS,298 1962

[4]

A. Lenard Journal o f Mathematical Physics 2, 682 1961

[5]

J. Dawson Physics o f Fluid 5,445, 1965

[6]

0 C Eldridge, M R Feix Physics o f Fluids 5,1076, 1962

[7]

J L . Rouet, M R Feix Physics o f Fluids 3, 1830 1991

[8]

G. Knorr Naturiforsch 18 A, 1204 1963

[9]

V J. Possov Physics o f Fluids 8, 358, 1965

[10]

F C Grant, M R Feix Physics o f Fluids 10, 696, 1967

[II]

T.P. Armstrong and al Methods in Computationnal Physics Vol 9.29. 1970

[12]

W L Sadowski N A S A SP 153 Symposium on computer simulation o f plasma and many body problem p 133 1967 Ed House for Federal Scientific information Springfielal V A

81

[13]

Cheng, G. Knorr Journal o f Comput. Physics 22, pp 330-351 (1976)

[14]

A. Ghizzo, B . Izrar, P Bertrand, E Fijalkow, M R. Feix, M . Shoucri Physics o f Fluids 31.72 (1988)

[15]

M Shoucri, R. Gagne Journal o f Comput Physics 27, pp 315-322 (1978)

[16]

A Ghizzo, B Izrar, P Bertrand, E. Fijalkow. M R Feix. M Shoucri Physics Letter A 120, 191 (1987)

[17]

N . D . Suh, M R. Feix, P. Bertrand Journal o f Comput Physics V o l 94, N ° 2 , June 1991

[18]

[19]

P. Mineau

PhD Thesis o f University o f Orleans, Orleans France

A Ghizzo, P Bertrand, M . Shoucri, E Fijalkow, M.R. Feix, Journal o f Comput. Physics 108, 105-121 (1993)

[20]

A . Ghizzo, P. Bertrand, M Shoucri, E. Fijalkow, M R. Feix, To be published in Physics o f Fluids B

[21]

A. Ghizzo, P Bertrand, M . Shoucri, T . W Johnston.E. Fijalkow, M R. Feix Journal o f Comput. Physics, V o l 90,N°2, Oct 1990

[22]

T . W . Johnston,, P. Bertrand, A Ghizzo, M Shoucri, ,E Fijalkow, M R. Feix Physics o f F l u i d s B . 4 (8) August 1992

[23]

P. Bertrand, A . Ghizzo, T.W. Johnston, M . ShoucriE. Fijalkow, M R. Feix Physics o f FluidsB 2 (5), May 1990

[24]

P. Bertrand, A. Ghizzo, SJ Karttunen, TJH Pattikangas, RRE Salomaa, M Shoucri Physics o f Fluids B 4 (11), November 1992

[25]

P. Bertrand, A Ghizzo, M.R. Feix, E Fijalkow, P Mineau, N . D . Suh, M . Shoucri Nonlinear phenomena in Vlasov Plasma Ed F Do veil Editions de Physique d'Orsay p i 09 1989

82 MATHEMATICAL MODELS OF ION EXTRACTION AND PLASMA SHEATHS

S. M A S - G A L L I C ( ' ) , P.A. R A V I A R T ( ' ) ( ' ) ( ' ] Centre de Mathemaliques Appliquees, LIRA CNRS 756, Ecole Polytechnique, F-91128 Palaiseau Cedex f ) Centre d'Etudes de Limeil-Valenton, Commissariat a 1'Energie A t o m i q u e . B P 27, F-94195 ViUeneuve Saint Georges Cedex 1

In this paper, we consider several plasma physics problems characterized by the existence of boundary layers or sheaths. These problems can all be viewed as singular perturbation problems for the Vlasov-Poisson system. We review some recent results concerning the asymptotic analysis of these problems and introduce new reduced models which lead to nonstandard free boundary problems.

1.

Introduction

A classical m a t h e m a t i c a l model in plasma physics is provided by the stationary Vlasov-Poisson equations or related equations supplemented w i t h appropriate boundary conditions. T h e mathematical analysis of such boundary value problems is now fairly well understood (cf. [9], [12] for instance]. However the solutions of physical problems are often characterized by the existence of boundary layers which separate zones where the plasma is quasineutral from nonneutral plasma zones or boundaries. This is due to the existence of one or several hidden parameters which make these problems difficult or even impossible to solve numerically. A n alternative consists in deriving reduced or l i m i t problems where the small parameters have been eliminated. On one hand, l i m i t models are often non standard and lead to interesting mathematical problems. On the other hand, the derivation of these models and the analysis of the boundary layers of the original problems are obtained by means or an asymptotic analysis.

83

In t h i s review paper, we w i l l i l l u s t r a t e the above general remarks i n s t u d y i n g a number of physical situations where plasma quasi n e u t r a l i t y breaks down t h r o u g h b o u n d a r y layers called plasma sheaths which arc either free or attached to an electrode. F i r s t , we consider the p r o b l e m of ion e x t r a c t i o n from a neutral plasma we introduce the model i n Section 2 and by scaling we show that i t involves two small parameters. I n Section 3, we give a complete asymptotic analysis of the one dimensional p r o b l e m . the l i m i t p r o b l e m is a free boundary p r o b l e m . We derive in Section 4 a m u l t i d i m e n s i o n a l analogue of this free boundary model which is indeed used i n realistic situations. I n Section 5, we give a numerical a l g o r i t h m of solution of this m u l t i d i m e n s i o n a l p r o b l e m . Next, we consider two classical one dimensional m o d e l used for s t u d y i n g the flow of a plasma towards o. floating fully absorbing wall. I n the first model, the plasma is injected t h r o u g h an emissive electrode : an asymptotic analysis of this problem in given i n Section 6. We study in Section 7 the more difficult case where the plasma is generated by means of a ionization process. We present the corresponding m a t h e m a t i c a l mode! and we describe some open questions concerning the existence and the a s y m p t o t i c behaviour of the solution of the model.

2.

T h e P r o b l e m of I o n E x t r a c t i o n

Given an electrically neutral plasma obtained from a gas by means of an electric discharge, we want to e x t r a c t an ion beam. For e x t r a c t i n g such a beam, the neutral plasma is placed behind an anode held to a zero electric potential and the ions are attracted by a facing highly negative cathode. T h e anode has an aperture such t h a t the plasma is free to enter the interelectrode space. On one hand, the electrons are a t t r a c t e d by the ions w h i c h move towards the cathode b u t , on the other hand, they are repulsed by this cathode. As a consequence, one can encounter the following s i t u a t i o n . electrons a n d ions issued from the plasma form a quasi-neutral zone from the b o u n d a r y of w h i c h the ion beam is extracted. T h e purpose of the first p a r t of t h i s paper is to derive a m a t h e m a t i c a l modeling of this s i t u a t i o n .

We begin by describing the general s t a t i o n a r y p r o b l e m . We assume t h a t the electrons are i n t h e r m o d y n a m i c a l e q u i l i b r i u m and can be considered as an isothermal fluid w i t h a vanishing d r i f t velocity. Hence, denoting by n — n , ( x ) the electron number density, by T the constant temperature of the electron fluid and by <j> - ip[x) the electric p o t e n t i a l , t h e equation of conservation of the m o m e n t u m gives . c

r

Vp

e

= en V<£, ;

P !

= n kT c

c

(2.1)

where p , is the electronic pressure and k is the B o l t z m a n n constant. I t follows from (2..1) t h a t -Vrt, = - ^ W , so t h a t the electron number density is given by the so called B o l t z m a n n equation n

e

= Cexp(-^),

(2.2)

84 On the other hand, the ions are assumed to be cold, monokinetic and noncollisional. Hence their distribution function Ii = li(X, v) is of the form

li(X, v) = ni(x)8(v - Uj(x)) where ni is the number density and Ui the velocity of the ions. Denoting by m, the mass of the ions and supposing for simplicity that they are singly charged, the distribution function Ii satisfies the following Vlasov equation

or equivalently the pair (ni , u;) satisfies the equations of continuity and conservation of momentum 'V.(niU i) = 0 (2 .3) e (2.4) 'V.(niui ® ud + -n;'V¢ = O. mi Finall y the electric potential satisfies Poisson' s equation

- 6.,p

e -(n, - ne). co

=

(2.5)

The mathematical model is defined by t he equat ions (2.1) (or (2.2)) and (2.3) - (2.5) and has to be supplemented with suitable boundary cond itions that we will describe later on. This simple model indeed involves several small parameters and may be viewed as a singular perturbation problem whose direct simulation lea ds to a stiff numerical problem. To make this point clear, we introduce the following scaling of the above equations: gi ven characteristic quantities, length L , density n, velocity u and electric potential ¢, we define the dimensionless variables X'

x

L'

I

nO'

n" n

= --=- ,

=

0:'

e,l,

U,

I

Ui=-:- ,

u

If we denote by

A

D.

= JcokTe

ne 2

the electronic Debye length an d we set

kTe

AD.

<= L '

T/ =

e¢'

2kTe Q= -

--

mi u2

,

the equations (2.1), (2.2) give (d ropping the primes) 1

'V ne = - -ne'V¢ T/

and

¢

ne = C exp (--) T/

(2.6 )

(2.7)

85 w h i l e (2.3) - (2.5) become respectively

V.htnn

V.fnjUi) = 0

(2.8)

® u,-) - ^ A j W = • in

(2.9)

and A<# = J (

n

-

i

(2.10)

I n practical problems, we have £ <

1,

t) -C 1

since on one h a n d the electronic Debye length is usually very small compared w i t h the characteristic dimension of the device and, on the other hand , d> corresponds to a very large electric p o t e n t i a l able to extract a high current ion beam. In a d d i t i o n , we have a =

0(1)

2

since ~rmu , w h i c h can be taken as the kinetic energy of the ions before their ext r a c t i o n f r o m the plasma, is of the same order t h a n the t h e r m a l energy k'f, of the electrons. Moreover, i n many problems, we find 1 Q

<

2 -

Hence, we are faced w i t h a p r o b l e m which involves two small parameters e, IJ which are roughly of the same order. N o t e t h a t the equations (2.8), (2.9) provided w i t h appropriate boundary cond i t i o n s for the pair (rii,U,-) enable us to determine n ; as a function of * .

Using

moreover (2.7), the equation (2.10) becomes a nonlinear elliptic equation involving t w o small parameters.

I n Section 3, we shall give a complete analysis of the one-

dimensional case i n a slab geometry and study the asymptotics as e and 7/ tend to zero.

I n Section 4, we w i l l consider the general m u l t i d i m e n s i o n a l case and intro-

duce a reduced p r o b l e m well suited for numerical s i m u l a t i o n w h i c h appears as a nonstandard free-boundary problem.

3.

A n a l y s i s of the O n e - D i m e n s i o n a l M o d e l

A s s u m i n g now plane symmetry, we consider the one-dimensional case. T h e anode and the cathode are located at x = 0 and x = L respectively. T h e electric p o t e n t i a l is held zero at the anode and at —d> 0 at the cathode. A n e u t r a l plasma reservoir is placed at the anode we denote by n the number density of the electrons and the ions at the anode and by UQ the injection velocity of the ions. Here, B o l t z m a n n ' s equation (2.2) becomes L

0

n, = n

0

expf-^-)

(3.1)

86 while the equations (2.3) - (2.5) give respectively

d dx(n;u;) = 0

d 2 -(n,u;) dx

(3 .2)

e d¢> + -m; n;dx

= 0

(3.3)

and

d2 ¢> e 2 = - (n; - ne). x co They are supplemented with the boundary conditions

(34)

- -d

n;(O) = no,

¢>(O)

= 0,

u;(O) = Uo

(3 .5)

= -¢>L.

(3.6)

¢>(L)

Now, it is a simple matter to obtain n; = n;(¢» from the equations (3.2), (3 .3 ) and (3.5). On one hand, we have

nouo

n j = -- . U;

On the other hand , using (3.2), the equation (3.3) can be written

du; n;(u;dx

e d¢> + --) = m, dx

0

or equivalently

d 1 2 dx ('2 m ;u;

+ e¢»

= O.

This equation simply means that the energy of the ions is conserved during their motion. It yields 1 2 1 2 '2m;u; + e¢> = '2miuO' Assuming the compatibility condition

(3. 7) we obtain

u; = uo(1 -

2e¢>2 )1/2 miuo

since the ions flow from the anode to the cathode and therefore

2e¢> -1 /2 n; = no(1 - - . -2)

(3.8)

m,u o

By replacing He and H; by their expressions (3.1) and (3.8) in Poisson 's equation (3.4), we find that the electric potential is characterized as the solut ion of the nonlinear elliptic boundary value problem

[(1 - dxd2~ = noe co

2e¢>2 )-1/2 - exp (-=t)], m;uo kTe

1 ¢>( O) = 0, ¢>(L) = -¢>L.

x E (0 , L )

(3.9)

87

As i n Section 2, we introduce dimension less variables and the associated small parameters e and tj. We choose here ft = n

, u — uo

0

,i

= 4>L

so t h a t .

UokT,

*D, - \ •

X c

kTc

Dt

= -r~.

"? = —7-1

o -

V not L efa T h e p r o b l e m (3.9) then becomes ( d r o p p i n g again the primes) 1

2kT, — rridug

(3.10) tf(0) = 0,

.#(1) = 1

while the c o m p a t i b i l i t y c o n d i t i o n (3.7) reads * > - ^ .

(3.11)

Let us first consider the existence and uniqueness of a solution of (3.10). We define OJY t o be the positive solution of the equation exp(-) a (a

M

1= 0

(3.12)

Or

~ 0.796). T h e n , we can state

T h e o r e m 1. Assume

0 < cr < 2. T h e n , trie problem (3.10) has a unique nonnega-

tive solution ip. This solution is strictly increasing in [0,1]. Moreover, for 0< o < CIM, this is the unique solution

of (3.10).

T h e proof relies on the existence of a first integral of the second order equat i o n (3.10) obtained by m u l t i p l y i n g i t by j g . For the details, we refer to [1], [2], We have already mentioned t h a t , in practical problems we have a < ^, so t h a t the problem (3.10) has indeed a unique solution which is nonnegativc and s t r i c t l y increasing. Since e and r) are small parameters, we are now looking for the l i m i t of the nonnegative solution 4> — 4>c,n ° f ( 3 1 0 ) as e and n tend to zero. stems from the fact that we have more than one small parameter. that for > 0,

Hence, i t makes sense to consider the three following l i m i t cases (i) i f - 3

* 0 as £,77 —* 0, we o b t a i n (at least formally) & , „ ( i ) - » 4>a{x) = x.

A first difficulty

B u t , we observe

88

A t the l i m i t , we have n , = n , = 0 i n (0,1], This means that the applied electric potential rpr, is not large enough to extract the ions from the neutral plasma n' (ii) I f — j • oo as s, i) —• 0, we find 3 2

- • 0 on [0,a]

for all

a < 1.

Electrons and ions are extracted from the anode and form a quasi-neutral plasma which extends up to a neighborhood of the cathode. T h e function ^ _ , presents i n this neighborhood a boundary layer or a sheath from which the ion beam is extracted. n' ( i i i ) T h e case where —— tends to a finite l i m i t 0 as £ and n tend to zero corree

3 2

sponds to the interesting situation and is to be analyzed i n the sequel. Indeed, one can recover the cases ( i ] and ( i i ] as l i m i t cases of this general s i t u a t i o n . For the sake of simplicity, we will assume from now on that the parameters e and i are related by

£=

fa * 31

(3.13)

so that (3.10) becomes

dV 2

dx

l /?V

/ 5

"'

l + ofr^-expl-J)], "

ze(C,l) (3.14)

# 0 ) = 0, tf(l) = 1. We set

1

; = Observe t h a t

cri u 0

j = 1

(3.15)

0

m, L

2

is the dimensionless ion current density (the scaling factor \f^-fr-

is classical : see

(5) for instance). T h e n , i n t r o d u c i n g the point x defined by s

(3.16) FT

4

and denoting by the nonnegative solution of (3.14), one can prove the following result (see again (l],[2j) T h e o r e m 2, Assume

the hypotheses a < 2 and (3.13). •P„ -

4>o in C"([Q,1]) os t) -

Then, we have 0

(3.17)

89

where

(x) = 0,

0<x< (3.18)

J

I j < X < 1,

2

dx

(1) = 1.

Note t h a t , for j > 5, the p r o b l e m (3.18) can be interpreted as a free b o u n d a r y problem : the pair (xj, nalfo,!])

m

a

y he characterized as the solution of

2

d^

J

<X<1

2

dx

(3.19) & & j J = | P % )

=0,^(13=1.

Hence, we o b t a i n at T - an e x t r a boundary c o n d i t i o n which enables us to determine ;

the free b o u n d a r y i j . Indeed, we can solve e x p l i c i t l y this p r o b l e m (3.19) which gives (3.Ifi) and 1

Mx)

3

= ( T ^ ^ ) ' . z j < x < 1. 1-x/

(3.20)

T h e physical i n t e r p r e t a t i o n of the l i m i t model (3.18) is clear. For j < g, the ion current is not large enough to p u l l the electrons which are thus unable to pass the anode ; i n this case, the ion beam is extracted directly from the original plasma at the anode. O n the other hand, for j > the ion current is sufficiently large t o pull the electrons which enter the interelectrode domain : an electrically neutral zone {0,xj) is created where the electrons are confined and the ion beam is extracted from t h i s n e u t r a l plasma at the point xj. Moreover, the ion emission satisfies CliildLangmuir's /aw, i.e., = 0 (see [5] for a mathematical derivation of this l a w ) . I n order to compare more accurately the original model (3.14) and the l i m i t model (3.19) for j > | , we want t o analyze the behavior of the nonnegative solution fa, of (3.14) for rj small enough.

We begin by i n t r o d u c i n g the point i , of "quasi-

n e u t r a l i t y break d o w n " defined by

- — dx

i„

= max — i). '6(o,il dx'

Hence djfa,

= 0.

3

1

dx

Note t h a t the first equation (3.14) yields

I

d>„ 3

dx T h u s , defining w

a

as the unique positive solution of the equation -

| ( ] + ™

o

r

3

/

I

+

e x p ( - u » „ ) = 0,

(3.21)

90 we obtain t h a t x„ is the unique solution of fc,(i„)

x r,w .

(3.22)

a

T h e n , it is a simple matter to show that x„ —• Xj as 13 —• 0. More precisely, we have x, =

3

X

i

- A»"> + 0( '<),

A =

n

{a + 2)0

[3.23]

Let us now check that the function 4>n presents a free boundary layer of w i d t h 3li

0(ij )

centered at the point

We introduce the change of variable (3.24]

and the function 1/>M) = ^-Mxr,

(3.25)

I

+

0 *€-

Then (3.14) becomes n

= — [ ( 1 + auieipv) ' w,

-

exp (-u> tf>,)] tt

Note t h a t , by (3.22), we have *M

(3.26)

= 1.

Hence, i t appears n a t u r a l to consider the l i m i t problem g

= -L[(l + r W ) -

1

/

J

-

exp

i - u

a

n (3.2T)

lim

0 ( 0 = 0,

0 ( 0 ) = 1.

We can state T h e o r e m 3. The problem (3.27) has a unique positive solution ip € C™(IR).

This

solution is s t r i c t l y increasing and satisfies (3.28) Moreover, there exist three constants B, C, D > 0 such trial (3.29) and

_ m-mpV,)<

^T7i

wfiere f„ denotes the interval [—^r/T, j i j S i ]•

e l t

P(-^i7j)

(3.30)

91

I t follows from (3.25) and (3.29) t h a t the potential <*„ has the following behavior in the quasi-neutral zone ( 0 , * , ) : 4fc)

< w

0

exp [B^-^-Y

0 < x < x„.

(3.31)

3/<

Hence, outside a b o u n d a r y layer of w i d t h 0(n ) located at i , where the p o t e n t i a l d>„ is indeed very small of order 17 exp( — ^ t V ) > 7

=

0(n).

>

N e x t , we note t h a t the dimensionless electron and ion densities are given by n

*

c

= exp ( - ^ ) 5 = (1 + o ^ ) - "

exp (—w >p„) a

J

= (1 +

a w ^ y ^

so t h a t by (3.30)

n (x) r

~

exp

[

X

—W VJ(B

3

1

Thus, n , and n ; present also a b o u n d a r y layer of w i d t h 0 ( r j ' ' ) at x^. I n p a r t i c u l a r , we observe t h a t , except i n the boundary layer, the quasineutrality p r o p e r t y n„ ~ ttj is fairly well satisfied i n (0, x„). On the other hand, we find t h a t , for x > i , , n decreases e x p o n e n t i a l l y to zero i n the boundary layer, so t h a t , except i n this layer, i i , = 0 appears to be a fairly good a p p r o x i m a t i o n of the electron density outside the quasineutral zone. t

As a conclusion, we o b t a i n t h a t the l i m i t m o d e l provides an accurate approxi m a t i o n of the solution of the original model. I n fact, the main p a r t of the error comes out from the position of the free boundary x.i which satisfies only ti-*,-0(n

l f l

)

and is located outside the boundary layer 1

4.

The Multidimensional Case. A Reduced Model

T h e goal of this section is to derive i n the general m u l t i d i m e n s i o n a l case a "reduced m o d e l " which is no longer stiff and can be conveniently used for numerical purposes. As a first step, we go back t o the one-dimensional case and introduce an a p p r o x i m a t e model which is close t o the l i m i t model ( 3 . I S ) and can be easily generalized t o several space dimensions. I n this model, we assume that the inter-elect rode d o m a i n (0, L) consists of two parts :

92

( i ) a subdomain (0, a) filled w i t h a neutral plasma of electrons and ions : the electrons behave as an isothermal fluid w i t h temperature T and a vanishing drift velocity while the ions are cold monokinetic (and therefore noncollisional); r

(ii] a subdomain (a, L) which contains the ion beam : there is no electron and the ions are cold monokinetic and extracted from the neutral plasma at a, i n such a way that the C h i l d - L a n g m u i r ' law holds. In the neutral plasma zone, we thus have rti = n = n t

so that (2.1) gives here dn k

T

' T

di> =

E

n

(

T ,

x

i

A

)

On the other hand, setting u, = u, we obtain from the equations (2.3), (2.4)

J^M-e

(4-2)

-

5

2

M

Observe t h a t we can eliminate the electric potential $ : indeed, using (4.1), the equation (4.3) becomes d , — (nu dx

2

kT dn + —'-— = 0 m , dx c

,

4.4

,

In the beam zone ( o , £ ) , we set n = n , , u = u, so t h a t the equations (4.2) and (4.3) still hold. They have to be supplemented w i t h Poisson's equation

T h e boundary conditions are again n(0) = n , « ( 0 ) = u 0

0

(4.6)

and # 0 ) = 0, 4>(L) - ~4-L

(4.7)

We assume in addition that at the interface a, we have n and u are continuous — (o) - 0

( C h i l d - L a n g m u i r ' s law).

(4.S) (4.9)

Let us check that this model (4.1)-(4.9) is indeed an approximation of the original model. Let us first show n —n , u = u 0

0

in

(0,a).

(4.10)

93 ID fact, we have by (4.2) l

dn dx~

so t h a t (4.4) gives m; Since

m,ul

dx

= — ^ 1, (4.10) follows at once. 2

N e x t , argueing as in Section 3, we find by using (4.2) and (4.3) n = n,(l - - ^ r ) - "

2

in

(0,1).

(4.11)

T h e n , on one h a n d , we o b t a i n f r o m (4.10) and (4.11) 0 = 0

in

(0,a).

(4.12)

On the other hand, using (4.5) and (4.7) - (4.12), we find t h a t d> is solution of the equations _ n o £ _ _ 2 e ^l/a _[ a < x < L dx (4.13) ( 1

1

2

ty.

0(a) = /dx( o ) = O, #(I) = - *

t

In order to prove t h a t the above model is close to the o r i g i n a l model, we pass i n dimensionless variahles. T h e n , under the assumption (3.13), the problem (4.13) becomes d V _ s , a < x < 1 dx =

2

(4.14) * ( o ) = ^ ( « ) = 0,

0 ( 1 ) = 1.

One can prove (cf. [3]) T h e o r e m 4 . /Assume j > | together with the hypotheses o f Theorem enough,

2. For ij small

the problem (4.14) has a unique solution (a","). Moreover,

the function

if we extend

ifP by 0 in ( 0 , a " ) , we have tj,

i>" -"Po

in

C'([0,1])

as

i ) - 0

(4.15)

Hence, for j > 5, the model (4.14), which is i n fact a dimensionless version of the model (4.1) - (4.9), is close to the l i m i t model (3.19) and therefore can be viewed as an a p p r o x i m a t i o n of the original model (3.14). We t u r n now t o the general m u l t i d i m e n s i o n a l case. A l t h o u g h it is an open question to derive the l i m i t model of (2.7) - (2.10) as the parameters s,n tend to zero,

94

i t is, however, a simple matter to extend formally the one-dimensional a p p r o x i m a t e mode] (4.1) - (4.9) to several space dimensions. Denote by S3 an open set of m. which represents the geometrical domain under consideration (the inter-elect rode d o m a i n ) . We assume in our reduced model t h a t d

0 = fl, U E U f l consists of two disjoint open subsets

3

f ! separated by a free interface E ; f ! i is 2

the neutral plasma zone while i f contains the ion beam. 2

In H i , the n e u t r a l i t y assumption yields n = n , — n . Since again the electrons behave as an isothermal fluid w i t h temperature T and zero mean velocity, the density n of the electrons and the ions satisfies the equation e

t

fe^Vn

= enV^ .

(4.16)

On the other hand, the ions being cold and monokinetic, the pair ( n , u = u,) is solution of V.(nu) = 0

(4.17)

V.(nu ® u) + — 7 i V 0 = 0

(4.1S)

By e l i m i n a t i n g the electric potential

e

(4.19)

= 0 (no electron i n f ! ^ ) . T h e n , the pair ( n = n , , u = u , |

satisfies the equations (4.17), (4.18) while 0 is solution of Poisson's equation -Aip

—

—n. So

On the free surface E , we assume the c o n t i n u i t y of the pair (n, u ) . riforeover, since we can suppose t h a t the electric potential vanishes i n Q ] , we have <j> = 0 on E . In a d d i t i o n , we assume t h a t the C h i l d - L a n g m u i r ' s emission law holds for the ions at the boundary E. R e m a r k 1 . Note t h a t , in our model, we assume $ = 0 i n fl\

and therefore o n E

and nevertheless we take into account the V 0 terms in the equations (4.16), (4.17). T h i s is not contradictory since physically ~ 0 i n f ! j b u t the characteristic length \p.

appears t o be very small,

a

To summarise the m u l t i d i m e n s i o n a l reduced model reads as follows : V.(rtu) = 0 k

in

T

V . { r m ® u) + —

Vn = 0

Hi

(4.20)

95

V.(nu) = 0 V.(reu ® u ) +

-Alp

= 0

In

fig

(4.21)

- —71

w i t h the interface conditions ( n , u ) is continuous on E (4.22)

or | * on f*. R e m a r k 2. N u m e r i c a l simulations show t h a t , i n some situations, one can find at a same p o i n t x £ fij ions which have been e m i t t e d at different points of E and have therefore different velocities. Hence, i t is not realistic m a t h e m a t i c a l l y to look for a monokinetic ion d i s t r i b u t i o n of the form / ( x , v ) = n(x)rJ(v - u ( x ) ) ,

x e Oa-

A m a t h e m a t i c a l l y correct setting of the reduced model is obtained by replacing the equations (4.21) by the classical Vlasov-Poisson system i n f l j V . V j c / - — V6. V

v

d

/ = 0

i n Q j x u\

(4.23) — Atj) =

72,

£Q

71 =

/ /dv

J

and r e q u i r i n g t h a t , at the u n k n o w n interface, we have / ( x , v ) = 7i(x)rJ(v - u ( x ) ) ,

x 6 E, v . f < 0

where the pair (re, a) is the trace on E of the solution of the reduced p r o b l e m in fi|. T h i s is indeed the p r o b l e m which is solved numerically (sec Section 5 ) . However, we can s t i l l use the f o r m u l a t i o n (4.21) provided t h a t the functions re a n d u are allowed t o be m u l t i v a l u e d (cf. [6, Section 6J) for a similar s i t u a t i o n . • N o w , i n t h i s m u l t i d i m e n s i o n a l case, i t remains to answer t o the t w o following n o n t r i v i a l m a t h e m a t i c a l questions: (i) T a k i n g i n t o account Remark 2, is the free boundary problem (4.20) - (4.22) well posed ? ( i i ) Does i t a p p r o x i m a t e the original problem (2.1) - (2.5) ?

96

5.

N u m e r i c a l Solution of the R e d u c e d P r o b l e m

We t u r n to the numerical solution of the multidimensional reduced problem (4.20) - (4.22). T h e first step consists i n finding an iterative a l g o r i t h m of solution of this free boundary problem. We begin by observing that the equations (4.20) supplemented w i t h the boundary conditions ( n , u ) given at the inflow boundary

(5.1)

enable us to determine a pair of functions (ft, u ) in the whole c o m p u t a t i o n a l domain f l which coincide w i t h the solution of the reduced problem only i n the quasi-neutral zone til. Hence, solving the reduced problem (4.20) - (4.22) amounts t o solve the free boundary problem (4.21) w i t h the boundary conditions (n,u) = (n,u)

0

= 7J- =

o

(5.2)

n

— o given on 3 f i \ E 5

(for instance)

In practical situations, I ! represents the inflow part of the boundary flfl2 so that the p r o b l e m (4.21), (5.2) makes sense. In particular, we obtain an extra boundary condition for

o{.',o) • x —' 4"o( \ a) the solution of the boundary value p r o b l e m t + I

x

a < x < 1

2

dx

yF+^o'


(5.3)

w i t h the Dirichlet boundary condition at a, we use the Neumann boundary condition as a criterion for characterizing the free boundary a, i.e., we solve the equation

dx

[a;a) = 0

(5.4)

by means of an iterative method. An alternate strategy consists i n defining the function <prj(.;a) : J- - »

3Vw

a < x < 1

2

dx

(5.5) (a; a) = 0,

d> {l;a) = 1 N

97

w i t h the N e u m a n n boundary c o n d i t i o n at a and then in solving instead of (5.4) the equation

(5.6)

N

for d e t e r m i n i n g the free boundary a. N e x t , we observe that the p r o b l e m (5.3) is not always well posed : i t can have zero, one or two solutions. Indeed, as i n [3], one can prove P r o p o s i t i o n 5.

There exist two numbers a,, a

2

w i l h 0 < a, < a

2

< 1 such that

the problem (5.3) has ." (i) no solution for 0 < a < Oj, (ii) a unique solution for a = a ( i i i ) two solutions for aj < a < Q j , 1}

(iv) a unique solution for

Oj < a < 1 .

Moreover, (5.3) has a unique nonnegative solution free boundary given by Theorem 4.

for a > a" where u ' > a

2

is the

R e m a r k 3. I n the case ( i ) , the nonexistence result only means t h a t the VlasovPoisson p r o b l e m w r i t t e n i n physical variables df

e

d6df : r i r = m , dx Ov

ax o?i, dx'

1

0

>

e —n.

=

E o

n =

w i t h the b o u n d a r y conditions

a<x
j

fdv

/(0,11) = n 6(v - u ) , i i > 0, 0

0 ( u ) = O,

0

=

u e fR

f(L, v ) = 0, v < 0

" f e

has no m o n o k i n e t i c solution of the form f(x,v)

= n(x)S(v

- u(i)),

t r ( i ) > 0.

In fact, the p o t e n t i a l <j> associated w i t h any solution of this problem presents a barrier which is large enough for preventing the ions e m i t t e d at a to reach the anode. I n this case, we have t o look for solutions of the form J{x,v)

= n {x)6(v

- u+(x)) + n_(x)6{v

+

- u-(x)),

u

+

> 0, u _ < 0

O n the other hand, for u i < a < a", the electric potential still presents a barrier which is now too small for preventing the ions to reach the anode.

•

2

Since a ' — Q| = 0 ( i j ' ' ) , i t follows from Proposition 5 t h a t the function a —1 0 (n;a) D

1

is n o t everywhere defined in suitable neighborhoods of a" .

Hence, the

p r o b l e m (5.4) appears rather t r i c k y to solve by means of an iterative m e t h o d .

98 Conversely, we have T h e o r e m 6. The problem (5.5) has a unique solution ipfj[.;u) lor ail 0 < a < 1. Moreover, for j > ^ and n small enough, the (unction a —> di^(a; a) is strictly concave and strictly increasing. Therefore, the function a —» i>tt(a; a] is defined for all 0 < a < 1 and the problem (5.6) can be solved by Newton's iterative method : for instance, s t a r t i n g from a — 0, Newton's method converges monotojiica//y to a". For detailed proofs, we refer t o [3j. Let us t u r n again to the multidimensional problem (4.21), (5.2). The above analysis suggests to use the Dirichlet condition 0 = 0 on E as a criterion for finding the free boundary S. More precisely, given S , we define ( n * , u. 1 "the s o l u t i o n ' of the equations (4.21) i n fl£ supplemented w i t h the boundary conditions 3 5

(n*,u*) = (fi.u)

^ dv

= 0

0 = T h e n , a new iterate E *

+ 1

on E*

(5.7)

onE'

a

on a f l J \ E

4

of the free boundary E is derived in such a way t h a t +l

W* ls'+i|!
(S*(«) = 0 or more generally as some "mean" of this surface and S* One can also t h i n k of a Newton's type a l g o r i t h m for solving the reduced model . research efforts i n this direction are i n progress. I t remains to introduce a numerical discretization of the equations (4.20) - (4.22) in order to obtain an implementable method. Let us only sketch this point. The system (4.20), which is identical to the isothermal gas dynamics equations, is solved by a classical second order finite volume m e t h o d . On the other hand, we solve the Vlasov-Poisson equations (4.23) (see Remark 2) by means of a coupled particle-finite element method. In particular, the ion d i s t r i b u t i o n function / is approximated by Ax,v) = X> (x-x,)®tT(v-u,) 0

11

where a, > 0 , x , £ f l , u , £ IR denote respectively the weight, the position and the velocity of the particle i (cf. [10]) for instance for a mathematical discussion of the particle a p p r o x i m a t i o n ) . The hyperbolic system (4.20) and Vlasov's equation (4.23) 2

99

are b o t h solved by means of an unstatioriary m e t h o d which consists in solving the time-dependent

equations

d kT — (nu) + V.fnu ® u) + — V n = 0 1

at

mi

and | ( Ot and l e t t i n g ( t e n d t o + 0 0 .

+

v

.V ,/--^V0.V / = u mi :

v

For the details, we refer to a f o r t h c o m i n g paper [4]

devoted t o the numerical solution of the m u l t i d i m e n s i o n a l reduced model.

6.

A Plasma-Sheath Model

We end this review paper by discussing briefly t w o one-dimensional problems which are characterized by the existence of boundary layers which connect a quasin e u t r a l plasma zone w i t h either a fixed boundary or a nonneutral zone. We consider in this section a plasma-sheath model (cf. [11]) which can be studied w i t h the same a s y m p t o t i c techniques as in Section 3 : an emissive w a l l (cathode) at J = 0 injects a neutral plasma towards a fully absorbing floating wall (anode) at x = L . The d i s t r i b u t i o n functions / „ and / , of the electrons and ions respectively and the electric potential 6 satisfy the coupled Vlasov-Poisson equations df e dt- Bf ~ra ^~ + — m T ax~ Ov ~ °< c

v

t

x e (0,£),u e R ,

(6.1)

c

df, v-= ax _

t

dA

df,

-T- - 5 -

(6.2)

= 0.

m , dx Ov

= _(«,--„.),

+

/ °°

n.

f dv, B

a m e.i.

(6.3)

A t the cathode, we specify the d i s t r i b u t i o n functions of the injected particles while there is no particle injected at the anode. / . ( O . u ) - ,("),

v > 0,

},{L, v) = 0, v < 0

(6.4)

= 9<{v),v > 0 ,

/ . ( £ , " ) = 0, v < 0

(6.5)

9

M0,v)

T h e functions g and o; satisfy the n e u t r a l i t y constraint e

ftX/

/CO

g dv ~ j e

a

dv.

gi

We specify t h e electric p o t e n t i a l at the cathode 0(0) = 0

(6.6)

(6.7)

100

On the other hand, particle losses t o the absorbing floating anode are balanced by the injection of plasma at the cathode. Hence, we w r i t e that the o u t w a r d current density vanishes at the anode 1°° vf (L,v)dv Jo

=

c

'vML,v)dv.

Jo

16.81

We look for a solution [f„fi,4>) of the problem (6,1) (6.8) w i t h a strictly decreasing and therefore negative electric potential. Given such a potential <j>, the equations (6.1), (6.2) w i t h the boundary conditions (6.4), (6.5) can be solved by the method of characteristics to give 2e

v>

<j>{x) ,

v

2e —{(L))

> -,

(6.9) 0, and 2e f,(x,v)

2e

=

(6.10) 0,

T h e n , it follows from the above formulas (6.9) and (6.10) that

/

vf,(L,v)dv=

j j —

Jo

vg (v)dv, e

JJ-ffciw

/ Jo

— I Jo

vf,(L,v)dv

vg,{v)dv.

Thus, assuming that g is a s t r i c t l y decreasing function w i t h c

J.

= j Jo

vg dv > J. = / Ja c

VQcdv,

(6-11)

we define tpi, > 0 to be the unique solution of a. = f fT,— "9,dv.

(6.12)

J

so that the equation (6.8) becomes =

(6.13)

-PL-

Next, setting _

I™

vg (v)dv r

IVsj*<-

vg,{u)du vg {v)dt t

(6.14)

101

f°°

vgi(v)dv (6.15)

and F(0) = - ( , ( 0 ) e n

n c

(0)),

(6.16)

0

we o b t a i n from the equations (6.3), (6.9), (6.10) t h a t 0 is solution of the nonlinear boundary-value problem -

^

= F(0)

in

(0,1), (6.17)

0 ( 0 ) = 0,

0(L) = - 0 . t

A g a i n this p r o b l e m involves several small parameters. Let us introduce a scaling of the above equations. We set : 1/J

]?v* dv 9t

(6.18)

2e

Jo

Observe t h a t u is the thermal velocity of the injected electrons. T h e n , we define the dimensionless variables x = y,

v = -,

= —, o

n

a

= e,i,

(6.19)

and the small parameters £ = - v .

(t =

—

(6.20)

TO,

where 1

eonifii 2fie

2

a — e,i,

is again the electronic Debye length. Assume t h a t g , a

is of the form (6.21)

T h e n , the problem (6.17) becomes ( d r o p p i n g (he primes)

(6.22) 0 ( 0 ) = 0,

0(1) = o

where 0 3

Jo

vgj(v)dv

J

V " + i)0 2

\ ^

/oo

2

vA - 0

/oo ug du = y c

/fS° ^

e

and a = —0 ' 0 / , is the unique solution of y

/ o

«g (u)du

vg,dv.

+

vg vg(v)dv (v)dv\

• ' v ^ Vv

t

1

e

~

(6.23)

102

One can prove (see [2]] T h e o r e m 7. There exists a number /in > 0 such that, for a l l 0 < u < pa a n d a l l e > 0, (he problem (6.22) has a unique nonnegafive solution increasing.

0

which is

I i U

strictly

Moreover, for u < /la, the equation PM

= 0

(6-24)

has a unique solution $oj, and we have lim0

S l t

, = 0o,

(6.25]

u

in L ' f O , 1), 1 < p < oo, a n d i n C ° ( [ a , l - a]) for a i l o > 0. Hence the solution 0

t J I

presents two boundary layers at the cathode x = 0 and

the anode x = 1, Outside the boundary layers, the plasma is quasineutral and the electric p o t e n t i a l 0 „ is a p p r o x i m a t i v e ^ constant and equal to a V . ;

u

Here again, one can investigate the structure o f these b o u n d a r y layers. Indeed, there exists a constant B > 0 such t h a t 1

- 0o.J < o m a x ) e x p ( - B | ) , exp[-B -^)} T h e boundary layers have therefore w i d t h s o f order 0(e)

(6.26)

or o f order 0 ( A D J when

going back to the physical variables.

7.

A P l a s m a - S h e a t h Model with Ionization Let us finally consider a more sophisticated model plane model introduced by

Tonks and L a n g m u i r [13] (see also [8, Chapter 3]).

In this model, a plasma is

generated between two parallel plane electrodes at the same potential a n d a distance 2a from each other.

We assume s y m m e t r y of the properties of the plasma w i t h

respect to the median plane located at i = 0 and we set 0 ( 0 ) = 0. T h e electrons of the plasma are assumed to behave as an isothermal fluid w i t h temperature T, and zero mean velocity. Hence, TI, is again given by Bolzmann's equation (3.1) where no denotes the electron number density at the median plane. On the other hand, the ions are generated by means o f an ionization process : classically the ion generation rate is o f the form g = 9(0) = firjexp(7 — )

(7.1)

where v is n positive constant and 7 characterizes the ionization process. Suppose t h a t the ions are formed w i t h zero velocity and move to the electrodes w i t h no collisions.

Suppose i n a d d i t i o n t h a t the electric potential 0 decreases from

the

median plane to the wall at x — a. T h e n an ion formed at £ > 0 reaches the point x > £ w i t h a velocity v such t h a t

- m , V + e0( ) = 0(O. I

e

103

As a consequence, we o b t a i n t h a t the ion density n , is given by (7.2)

and the p o t e n t i a l 0 is solution of the initial-value problem -n exp( 0

— ) } , (7.3)

#G) = p(6)=Q. ax

As above, we scale the problem : we choose a characteristic

L = and a characteristic

'

2

k

T

length

'

potential e

and we set •x = Lx\

0 — —00',

g — i"iag'

I f we introduce again the electronic Debye length s hT a

c

riot

1

and the parameter

the equations (7.3) become ( d r o p p i n g the primes)

dx'

-i:

g{
-exp(-0)

yl{x)-4>(y)

(7.4)

0(0) = ^ ( 0 ) = 0 where 9(0) = e x p ( - 7 0 ) .

(7.5)

In practice, £ is a small parameter and i t makes sense to consider the so-called plasma approximation

which consists in neglecting the t e r m

or equivalently i n assuming electric neutrality n

c

g(4>{y))dy ^[x)-^y)

i n the first equation (7.4)

= n,-, i.e.,

= exp(-0).

(7.6)

104

Concerning the plasma equation (7.6), one can prove (cf. [7]) L e m m a 8. There exists a number i

0

> 0 with the following

(i) (fie equation (7.6) has a unique nonnegative that 0(0) — 0. Moreover, we have

properties:

solution A defined on (O.^oj such

(ii) this solution cannot be extended beyond x • ( i i i ) the limit value 0 = 4>(x ) is independent of the ion generation rate y. P r o o f . Let us only sketch the proof. Instead of looking for a function 0 : x —> (x) solution of (7.6), we look for the inverse function x : 0 —* l ( 0 ) . T h e n E q . (7.6) becomes 0

O

0

-dip — exp( — ( Next, we observe that the transform S defined for 0 > 0 by

s u m

ft ftth) = / M h ^10(0-0)

is invertible and its inverse transform T = 5 " ' is given for 0 > 0 by

rW(0)4wo)

+

v

^/;j^^.

Using this property w i t h

f(4) = ifyffl$'m<

M0) = exp(-0),

we obtain

Together w i t h the initial condition x(0) = 0, this enables us to determine the function x(6) for 0 > 0. I n a d d i t i o n , we note that x'(d>) vanishes at the point 0o unique solution of 1 \f$a

f*° exp(-i/>) Jo

\/0o


— 4>

so that the function T ( 0 ) is s t r i c t l y increasing in [O,0o] and s t r i c t l y decreasing in ( 0 , + c o ) . Setting x o

0

= i ( 0 ) , we obtain that the inverse function 0 ( a ) is indeed D

defined and s t r i c t l y increasing i n the interval (0, z ] and cannot be extended beyond 0

Xo-

•

105

After having characterized the solution 6 of the plasma a p p r o x i m a t i o n l i m i t model (7.6), i t remains to study the i n i t i a l value problem (7.4) for small but non zero values of the parameter e. I n this d i r e c t i o n , we conjecture t h a t the problem (7.4) has a unique non t r i v i a l solution 6 defined for all x > 0 and e

I

ip(x)

uniformly in [ 0 , x ] ,

[

+oo

for x > xo-

0

A l t h o u g h the asymptotic behaviour of the function 4 as t tends to zero has been analyzed heuristically by the physicists, a precise m a t h e m a t i c a l study of the corresponding asymptotics needs t o be developed. C

Acknowledgments T h e authors would like t o t h a n k N . Ben A b d a l l a h , B . B o d i n and J. Segre for fruitful discussions.

References [1] N . Ben A b d a l l a h , S. Mas-Gallic and P.-A. R a v i a r t , Analyse modele d ' e x t r a c t i o n d'ions, C.R. Acad. Sci. Paris

asymptotique

d'un

3 1 6 , Serie I (1993), 779-7S3.

[2] N . Ben A b d a l l a h , S. Mas-Gallic and P.-A. R a v i a r t (in p r e p a r a t i o n ) . [3] B . B o d i n and P.A. Raviart (in p r e p a r a t i o n ) . [4) B . B o d i n , E . Puertolas, P.-A, R a v i a r t and J. Segre (in preparation). [5] P. Degond and P.-A. R a v i a r t , An asymptotic Vlasov-Poisson

analysis of the

one-dimensional

system : the C h i l d - L a n g m u i r law, A s y m p t o t i c Anal.

4 (1991),

187-214. [6] P. Degond and P.-A. R a v i a r t , On the paraxial approximation Vlasov-Maxwell

system, Math. Models Meth. in Appl.

of the

stationary

Sci 3 (1993), 513-562.

[7] G . A . E m m e r t , R . M . W i e l a n d , A . T . Mense and J . N . Davidson, Electric and presheath in a coliisionless,

sheath

Unite ion temperature plasma, Phys. Fluids 23

(1980), 803-812. |8] A . T . Forrester, L a r g e I o n B e a m s (John W i l e y , 1988). [9] C. Greengard and P.-A. R a v i a r t , A Boundary Value Problem for the Stationary Vlasov-Poisson Equations : the Plane Diode, Comm. Pure Appl. Maths. X L H I (1990), 473-507. [10] S. Mas-Gallic and P.-A. R a v i a r t , A Particle Method Systems, Numer. Math. 5 1 (1987), 323-352.

for First-order

Symmetric

106

[11] S.E. Parker, R.J. Procassini and C . K . Birdsall, A suitable boundary {or bounded

plasmas simulation

without sheath resolution,

J.

condition

Comput.Phys.

104 (1993), 41-49. [12] F . Poupaud, B o u n d a r y Value Problem for the Stationary tem, Forum Math. 4 (1992), 499-527.

Vlasov-Maxwell

Sys-

[13] L . Tonks and 1. Langmuir, Genera/ Iheory of the plasma of an arc, Phys.

Rev.

3 4 (1929), 876-922.

109

TRANSPORT EQUATIONS F O R QUANTUM PLASMAS

GIOVANNI MANFREDI and MARC R. FEIX Centre National de la Recherche Scieniifique CNRS/PMMS, 3A Avenue de la Recherche Scienafique, 45071 Orleans cedex \ FRANCE

We present the Schrodinger and Wigner-Poisson systems as tools for [he modelling of quantum plasmas. For each of them, a numerical code, based on a splitting scheme, is proposed. The codes are used to simulate two typical situations : the quantum Brillouin flow (electron gas confined by a uniform magnetic field), and the one-dimensional an harmonic oscilator (with special concern for the choice of [he initial condition and the nature of the classical and quantum phase space).

1. I n t r o d u c t i o n - m o d e l l i n g q u a n t u m p l a s m a s 1

T h e study o f q u a n t u m plasmas is a relatively y o u n g subject. I t is however b e c o m i n g of w i d e r a n d w i d e r i m p o r t a n c e , due l o the o n g o i n g m i n i a t u r i z a t i o n o f electronic devices. I t w i l l p r o b a b l y b e c o m e a key concept i n (he design of future components. W h a t d o w e exactly m e a n by a "quantum plasma" ? Classically, a many-body system is r e f e r r e d t o as a plasma w h e n long-range, electromagnetic interactions are d o m i n a n t . T h i s occurs w h e n i o n i z e d particles can subsist and their density is sufficiently high. H o w e v e r , w h e n the density is still higher, the D e Broglie wavelength o f the charge carriers can b e c o m e o f the same o r d e r as the interparticle distance

i n this case

q u a n t u m effects must be taken i n t o account. A s s o o n as w e a t t e m p t to m o d e l a q u a n t u m plasma, a certain n u m b e r o f assumptions are necessary t o o b t a i n a tractable system o f equations. I n the following, we shall make s o m e r a t h e r drastic s i m p l i f i c a t i o n s , and only take i n t o consideration the very basic p h e n o m e n a . T h e resulting m o d e l , however, is not trivial, and throws some light on the p e c u l i a r i t y o f q u a n t u m effects on the dynamics of a plasma. T h e first m o d e l that w e describe is the so-called Schrodinger-Poisson system, w h i c h reads as :

no

at

2m

.

.

where ^ is ihe reduced Planck constant, e and m are respectively the electron charge and mass, E ^ and

0

is the vacuum dielectric constant, N is the t o t a l n u m b e r of electrons, and

V arc the wavefunction and the electrostatic p o t e n t i a l . T h e second t e r m i n the

r.h.s. o f the Poisson equation represents a motionless ionic background, w h i c h confines the otherwise free electrons : i n the f o l l o w i n g it w i l l be often o m i t t e d , and the confinement o b t a i n e d by some other t e r m (e.g. a magnetic

field).

I n some sense, the m o d e l (1.1) is the quantum-mechanical analogue o f the classical Vlasov-Poisson m o d e l . M o s t o f the assumptions of the two models are the same : collisions are neglected, only electrostatic interactions are taken i n t o account, a oneparticle wavefunction is used. Two

m o r e assumptions are specifically quantum-mechanical. First, we neglect all

effects due t o q u a n t u m statistics ; the electrons are considered as spinless particles. This means that for extremely high densities or l o w temperatures o u r m o d e l is of course b o u n d t o fail. Secondly, the E q s . ( l . l ) only allow us to deal w i t h pure states : in o t h e r words, all the electons are supposed t o be i n the well-defined q u a n t u m state i. I n order t o w o r k w i t h mixed states, one should use the density m a t r i x formalism, w h i c h f u r t h e r complicates the m o d e l . I n fact, it w i l l become

apparent

from

the

f o l l o w i n g discussion that m i x e d states can be conveniently treated t h r o u g h the W i g n e r formalism. A l t h o u g h simple, the Schrodinger-Poisson m o d e l contains the two m a i n ingredients for a q u a n t u m plasma, namely, long-range, self-con sis tent interactions and a quantummechanical law o f m o t i o n . M o r e o v e r , it is particularly s t r a i g h t f o r w a r d to numerically, as w e shall see i n the next section. W e stress, en passant, Schrodinger equation requires

treat

that the

a lower numerical cost than the classical Vlasov

equation. T h i s is due, o f course, to the fact that the wavefunction only depends o n half of the phase space variables, whereas classically the w h o l e phase space must be discretized. I n a sense, therefore, an outcome of the uncertainty p r i n c i p l e is t o simplify the n u m e r i c a l treatment of q u a n t u m problems. H o w e v e r , this advantage is lost w h e n dealing w i t h the semiclassical l i m i t o f E q s . ( l . l ) . I n this case, a semiclassical initial c o n d i t i o n (i.e. an i n i t i a l c o n d i t i o n that corresponds

to an a r b i t r a r y phase space

d i s t r i b u t i o n ) must be represented by a stroDgly oscillating wavefunction, thus r e q u i r i n g a huge number of mesh p o i n t s .

1 3

T h i s corresponds to the w e l l k n o w n fact that the

classical l i m i t implies large q u a n t u m numbers. T h e second representation

mathematical m o d e l that we consider of quantum

mechanics.

Such

makes use of the W i g n e r

a representation

plugs a

quantum

mechanical p r o b l e m i n ihe familiar phase space environment. T h e r e are, of course, some fundamental differences, but, what is m o r e i m p o r t a n t for o u r purposes, the n u m e r i c a l t r e a t m e n t of the q u a n t u m phase space w i l l t u r n out t o be no m o r e difficult than that o f the corresponding classical one. M a n y good review articles on the W i g n e r 5

formalism already exist i n the l i t e r a t u r e , ' '

4 , 7

t o w h i c h the interested reader may refer

111

for further details. Here, we shall just recall the essential concepts. The Wigner function is defmed as follows:

f(x,p) : _1_Jp(X->./2,x+>./2)exp( -ip ,\/},)d>. 211 },

(1.2)

where p(x,y) is the density matrix. We recall that, for a pure quantum state, the density matrix can be written p(x,y) = 1/>'(x) ..p(y) ; for a mixed state it is given by a sum of terms like the previous one. The function f(x,p) possesses nearly all the properties of a phase space density - it is real, normalized to unity, and, when integrated over x or p gives the correct marginal distributions, e.g. :

Jf(x,p)dp

: p (x,x)

spatial density

(1.3)

However, the Wigner function fails to be an actual probability distribution, because it usually takes on negative values. The Wigner function obeys the following evolution equation (we only deal with one-dimensional systems) :

af +.!!.... af

at

m

ax

- 2!e},2JJ

[v(x-~ ,t)-v(x+~,t)]exp(i(p-p ')>'/}')x

(1.4)

xf(x,p ', t)dp 'd>. The Wigner function can be used, just as in classical statistical mechanics, to calculate the mean values of any dynamical variable A (x,p,t) :

:

JJf(x,p,t)A(x,p,t)dxdp

Note however that, since some terms in A may not commute, it is necessary to establish a well-defined correspondence rule between classical variables and quantum operators (Weyl's rule).s If the potential V(x,t) is obtained from the Poisson equation: 1I

V : -NeJ - f(x,p,t)dp £0

+ -Ne

(1.5)

£0

then the Eqs.(1.4)-(1.5) constitute the Wigner-Poisson model. The same assumptions and limitations mentioned for the Schrodinger-Poisson model are also valid for the Wigner-Poisson one. However, the latter allows us to work both with pure states and mixtures, as it is apparent from Eq.(1.2) . In fact, the Wigner function contains the same information as the density matrix. Therefore, the initial condition can be chosen in a wider class than in the Schrodinger representation, and the choice can rely more easily on intuition, thanks to the familiar phase space formalism. However, it must be stressed that not every function f(x,p) can be associated, through the defmition (1.2), with a density matrix p(x,y) which is positive

112

definite - consequently, not every function f(x,p) is an admissible i n i t i a l c o n d i t i o n for the W i g n e r equation. W e shall give some m o r e indications about this delicate question in Section 3.2. Finally, we point out that the W i g n e r equation has the same c o m p u t a t i o n a l complexity as the Vlasov one - this is the price t o pay i n o r d e r to deal w i t h m o r e general quantum states. For

the W i g n e r - P o i s s o n m o d e l some analytical results can be obtained by the

linearization

procedure

c u r r e n t l y used

for the

classical

plasma.

A l l the

linear

phenomena are i n fact described by the q u a n t u m dispersion relation, w h i c h we n o w calculate, f o l l o w i n g the m e t h o d given i n Ref. 8. W e linearise (1.4) by p u t t i n g /

=

F(p)

+f,(x,p,l) , w h e r e / , is considered as a small p e r t u r b a t i o n around a homogeneous

F(p).

I t must be p o i n t e d out that, because o f the nonlinear relation between * and f,

it is impossible t o consider the strictly equivalent linear p r o b l e m i n the Schrodinger f o r m a l i s m . A u n i f o r m ionic density is present i n the Poisson equation , and the b o u n d a r y conditions are p e r i o d i c i n t h e * d i r e c t i o n . A f t e r linearizing (1.4), we take the F o u r i e r t r a n s f o r m over x ; subsequent integration over J and p' is trivial, and yields

(1.6)

W e then take the F o u r i e r transform of (1.6) over ( (u is the conjugate variable) and, by use o f the Poisson equation, obtain the quantum dispersion relation :

i• " y(f^/2)-fQ.-u/2)^ w

f

k

where u

1

= (jVe^/mtJ

f

J

U(mu

=

0

m

-kp)

is the plasma frequency.

Now, w e give the explicit f o r m o f the dispersion relation in the t w o l i m i t cases o f a hot and cold plasma. For the hot plasma, let us take for F(p) a maxwellian w i t h t h e r m a l velocity v . N o w supposing that Wt/rnv, rn

b

is small (a hot plasma is semiclassical), we

can develop F i n T a y l o r serie. Finally, w e arrived at the desired relation:

. ' . . • ^ J * * * * ^ tp

T h e cold plasma case ( v Eq.(1.7) assuming F(p)

a

***

as,

* 0 ) can be handled directly ( w i t h o u t any expansion) i n

= S(p), and yields the exact relation :

u

!

2

< ^ + r, *-*/"*"'

2

(1.9)

W e see consequently that, for a cold plasma, the q u a n t u m c o r r e c t i o n is i n general the d o m i n a n t one.

113 2. N u m e r i c a l Methods T h e above m e n t i o n e d results on the linear theory are virtually all I hat can be o b t a i n e d by purely analytical methods for b o t h models. T o explore the vast r e a l m of n o n l i n e a r p h e n o m e n a , one has t o resort t o n u m e r i c a l computations. A p p l i e d physicists have been interested since l o n g i n the s o l u t i o n o f the stationary Schrodinger-Poisson system, i n o r d e r t o d e t e r m i n e the energy levels i n a gas o f correlated charged particles''

1 0

I t is a n o n - l i n e a r eigenvalue p r o b l e m , w i t h very interesting features, b u t

we w i l l not deal w i t h it here. R a t h e r w e shall concentrate on time-dependent situations, for w h i c h the existing l i t e r a t u r e is m u c h n a r r o w e r . Several numerical methods for the 3,

S c h r o d i n g e r e q u a t i o n are already available " •

a

- most are based either o n spectral

m e t h o d s or on finite differences schemes. I n fact, b o t h o f t h e m can be i m p l e m e n t e d i n the f r a m e o f a s p l i t t i n g scheme, w h i c h w i l l be described i n details later on. Generally speaking,

spectral

methods

are

convenient w h e n dealing w i t h

relatively simple

geometries (e.g. p e r i o d i c b o u n d a r i e s ) , a n d have also the advantage o f b e i n g available a n d o p t i m i z e d i n standard software libraries. F i n i t e difference ( a n d finite elements) methods, o n the o t h e r hand, are m o r e apt t o simulate complicated geometries. Some a l t e r n a t i v e methods, a l t h o u g h not very successful, make use o f a hydrodynamic version o f q u a n t u m mechanics. T h e y w i l l be quickly reviewed at the e n d o f the next section, m o s t l y t o p o i n t out the technical p r o b l e m s they face. The

use

o f the W i g n e r equation

to simulate numerically q u a n t u m

transport

p h e n o m e n a is, o n the contrary, q u i t e recent. T o our knowledge, the literature o n this subject is still very l i m i t e d

6 , 1 3

: w e shall propose here a splitting scheme, w h i c h makes

use o f a Fast F o u r i e r T r a n s f o r m ( F F T ) step. Such a m e t h o d has the remarkable p r o p e r t y o f b e i n g quite s i m i l a r t o the splitting scheme used for the Vlasov equation, so that the n u m e r i c a l t r e a t m e n t o f q u a n t u m plasmas turns out to be of the same order o f difficulty as that o f classical plasmas. I n analogy w i t h p a r t i c l e codes for classical plasmas, a few q u a n t u m particle methods have recently been p r o p o s e d for the W i g n e r e q u a t i o n . " '

15

H o w e v e r , for this k i n d of

codes, the n u m e r i c a l effort d r a m a t i c a l l y increases w h e n we pass f r o m classical to q u a n t u m physics. T h i s is i n fact n o surprise, since the very concept o f deterministic trajectory loses its sense i n q u a n t u m mechanics. W e w i l l not, however, deal w i t h this k i n d o f codes i n the present paper. F i n a l l y , w e have decided t o o m i t any detailed analysis o f the n u m e r i c a l s o l u t i o n o f the Poisson

e q u a t i o n . I t is a vast subject, w h i c h is treated i n details i n many 16,

c o m p r e h e n s i v e w o r k s o n n u m e r i c a l analysis and plasma s i m u l a t i o n s . " Besides, i n the specific situations that w e shall investigate i n a subsequent section, only the oned i m e n s i o n a l f o r m o f the Poisson e q u a t i o n w i l l be used, w h i c h can easily be solved by s t a n d a r d methods. T h e reader interested i n the s o l u t i o n o f the Poisson e q u a t i o n i n m o r e c o m p l e x geometries can refer t o the above m e n t i o n e d w o r k s . 2.1. T h e Schrodinger equation T h e S c h r o d i n g e r e q u a t i o n for a one-particle wave function, can be w r i t t e n as :

114

i^L dt

= - i i i <-V(x,t)i 2

(x,t)

(£+V)0

(2,1)

where K and V are respectively the kinetic and potential energy operators, and we have taken for simplicity all n u m e r i c a l constants equal to one. T h e potential V(x,l) obeys the Poisson equation (1.1). T h e splitting m e t h o d consists i n separating the potential and kinetic terms i n Eq.(2.1) : 1 8 , !

(2.2) f

dt and then solving i n sequence each of the above equations for a time-step Ar. Formally, the result is :
(() (2.3)

4{t*hi)

= expl-iJ"* VdrW ,((*A 0 i

w h e r e ^ , indicates the intermediate result, obtained after application of the kinetic operator. N o t e that, i n view o f an application t o the coupled Schrodinger-Poisson system, we have taken into account a possible lime-dependence of the potential. A s a first r e m a r k , we note that a scheme based on Eqs. (2.3) is not symmetric, since the kinetic step always precedes the potential one. I t can be shown that such an asymmetric scheme is only correct to first order i n ar. W e therefore split the energy operator i n three, rather than t w o , terms, as follows :

i S , ( i + A ( / 2 ) - exp(-i'/?A t/2)
2

1

ii (r+A 0 = exp(-iRr>

t/2)i

;

t/2)

(2.4)

{r+A t/2)

where ^ , and $ , are intermediate results. I n other words, we first apply the kinetic operator for half a time-step, then the potential operator for one time-step, finally again the kinetic operator for half a t i m e step. I t can be shown that this trick provides a scheme that is exact t o second o r d e r i n AT. T h e block of operations expressed by Eqs. (2.4) is then repeated a large n u m b e r of times to give the evolution o f the wavefunction. In order t o solve the second step i n Eqs. (2.4), one needs the knowledge of the potential, and thus has t o solve the Poisson equation just before. N o t i c e however that

115

the potential operator only changes the phase of tP, whereas its modulus is left unchanged. Since the Poisson equation depends only on the square modulus of the wavefunction, the potential is not affected by a phase change in tP, so that, in the second of Eqs. (2.4), V can actually be taken out of the integral, to yield tP2 = exp(iVM)tP l in perfect analogy with the kinetic terms. A typical code would thus be represented by the following flow-chart : Initial condition

Kinetic term : M /2

Poisson solver

Potential term : M

Kinetic term : M/2 As a matter of fact, the two kinetic steps are strictly identical and can be performed together as a single step of duration M. The attentive reader would object that, by doing this, we eventually come back to the original, asymmetric scheme (2.3), which we said to be exact only to fIrst order in M. Actually, over two loops as given in the above flow-chart, the two schemes are identical and both second order, so that our discussion might seem superfluous. However, if other terms are included in Eq.(2.1), one must be aware that only symmetric schemes are second order accurate. For example, if we express the kinetic operator in cylindrical coordinates, we have

K

~ -~2 [~r ~ar [r~) ~r2 ~l ar aip +

2

'"

K +K r

..

and a possible splitting scheme is the following

which is symmetric. Again we can perform the two kinetic steps in the r direction together, thus reducing the number of steps per loop from fIve to four. However, the

116

K steps cannot be compacted i n one single step, i f we want the scheme to be second o r d e r accurate. T h i s is o f course due t o the fact that non-cartesian coordinates are not independent. A s w e shall see i n Sect. 3 . 1 , the use of the F F T for the radial variable w i l l provide a cheaper scheme. t

T h e most popular way t o solve explicitly the Eqs.(2.2) is to w o r k i n F o u r i e r space for the kinetic t e r m , and i n real space for the potential one. T h e n , the t w o steps are reduced t o m u l t i p l y i n g the wavefunction (or its F o u r i e r t r a n s f o r m ) by a phase factor. A l t e r n a t i v e l y , the kinetic t e r m can be discretized by a finite differences scheme. I n one spatial d i m e n s i o n , w e can use the C r a n k - N i c o l s o n scheme," w h i c h reads as

.90

_ . 0"*'

_

-i'

af (2.5) ^ 1tfj-i~

2

1 d*

IIP

2

2

A *

m

j

T7

2

2 where 4 "

2

2

i ( f * x , HA f) . Such a scheme is second order accurate b o t h i n 01 and 2

in Ar, and conserves exactly the L n o r m o f 0. 22. T h e W i g n e r e q u a t i o n T h e splitting scheme can be usefully applied also to the n u m e r i c a l s o l u t i o n of the W i g n e r equation (1.4).

8, 1 8 , 2 0 , 2 1

W e again split the equation i n t o three terms, a n d first

solve the kinetic part :

ar

(2-6)

m dx

over half a time-step, then the potential part

at

2* y,

II

x-~A-v

exp(i(£>-»')*/*)"

2

*f(x,p',t)dp'd\

(2,7) = 0

over At, and finally again the kinetic part over Af/2. T h e Poisson equation is solved just before E q . ( 2 . 7 ) , so that the flow-chart is identical to the one we gave for the Schrodinger-Poisson system. T h e kinetic step (2.6) can be solved i n a n u m b e r o f ways. I n fact, it possesses the f o l l o w i n g explicit s o l u t i o n

f(x,p,t+t>

t) = / U - p A

t/m,p,t)

(2,8)

117

which represents a translation of example, by interpolating / with a equation. Another method would in the x variable, which yields the f(k,p,t+L

(p/m)at in the x direction. This can be done, for cubic spline, a method currently used for the Vlasov consist in taking the Fourier transform of Eq.(2.6) following explicit solution :

t) = / ( * j v ) e x p ( - i A p A t/m)

(2.9)

k being the Fourier conjugate of*. O n e then comes back to the ordinary space by the inverse Fourier transform. A s to the potential step (2.7), we notice that the integral has the form of a convolution product. Therefore, we perform the Fourier transform in p space ( i is the conjugate variable) to obtain the solution :

/ ( x , A , r + A r)

f(x,x,t)exp

Af

(2.10)

Notice, at this point, that the Fourier transform on p turns the treatment of the potential term into a problem nearly as simple as the classical one ; this is really the method to treat the Wigner equation. T h e nonlocal character of the quantum interaction appears in a very clear way. Although we enforce the concept of an entity present at point (x,p) of the phase space, the distribution function at a given x interacts with the entire field. Finally, we point out that the Schrodinger equation is easier to solve numerically than the Wigner one, since the former depends only on the configuration variables, while the latter is expressed in the phase space. However, in the Schrodinger formalism one can only deal with pure quantum states, but not with mixtures, which require the use of the Wigner picture. T h i s advantage is lost when dealing with quasi-classical states, since their wavefunction is violently oscillating and a very fine mesh is required for a correct discretization. 23. A hydrodynamic model It has been known for a long time that the Schrodinger equation can be written in a form that reminds one of the equations of hydrodynamics." This is done by separating the amplitude A(x,t)

and the phase S(x,c) :

=A{x,l)exp[iS(x,t)\

(2.11)

A and S being real functions. W e then introduce the density n(x,t) and the average velocity u(x,t), with :

"(x,t)

- A

1

,

u(x,t)

= A i £ m dx

(2.12)

With this definitions, the Schrodinger equation is equivalent to the following system :

118

— +— (nu) at ax

0

au

l

(2.13) au

at *U- ax

aV

dp

m ax

dx

w i t h an exotic 'pressure", which is a purely q u a n t u m effect :

l aV" P

- - l r 7 -

n»*

dx

(2.14)

2

O n e can easily identify the first o f Eqs.(2.13) w i t h the c o n t i n u i t y equation, and the second w i t h the E u l e r equation for a compressible ideal fluid. Therefore, it has been proposed t o solve the system (2.13) by m a k i n g use o f standard hydrodynamic codes, 2

2J

i n c l u d i n g particle codes, *' '

25

However, the technical difficulty lies i n the treatment of

the pressure t e r m , w h i c h contains the t h i r d derivative o f the density. T h i s k i n d of codes is o f very l i m i t e d use nowadays, but we felt it interesting to m e n t i o n it, since it provides an alternative representation o f q u a n t u m

mechanics.

3, A p p l i c a t i o n s 3.1. The B r i l l o u i n flow T h e theory of nonneutral plasmas confined by magnetic fields is well k n o w n i n the frame of classical physics.

26

T h e simplest situation that one can imagine is given by an

electron gas which is infiniteiy long in t h e z direction, i m m e r s e d i n a u n i f o r m magnetic field parallel to the z axis. N o ions are present i n the system. I t is assumed that, apart f r o m the external magnetic field, only electrostatic self-cons is tent interactions

are

relevant. I t is easy to prove that there exists a family o f e q u i l i b r i u m configurations for w h i c h the electron gas possesses azymuthal symmetry, and rotates around the z axis w i t h constant angular velocity (i.e. as a solid b o d y ) . T h e B r i l l o u i n flow corresponds to a special case o f such e q u i l i b r i u m configurations. Since we shall be interested i n the q u a n t u m mechanical version of the B r i l l o u i n flow, we begin by recovering the m a i n classical results by means of a H a r a i l t o n i a n theory, w h i c h can be directly translated in the quantum f r a m e . subjected to a magnetic field B = rolA

20

T h e H a m i l t o n i a n of a particle

and an electrostatic field V is w r i t i e n as :

H = ? -

e

A

}

* y e

(3.1)

2m where e and m

are respectively the electron charge and mass, A

is the vector

potential, and P the canonical m o m e n t u m , connected to the usual kinetic m o m e n t u m p

(mass times velocity) by the relation :

119

P =

p+eA

(3.2)

I t is convenient to use cylindrical coordinates (r, % z), r b e i n g the radius, 9 the a z y m u t h a l angle, a n d z the d i r e c t i o n parallel t o the magnetic field. W h e n the magnetic field

is u n i f o r m , the vector p o t e n t i a l can be chosen as follows : A,

= A

r

= 0

;

A

•

t

Br/2

(3.3) 2 7

T h e n , the H a m i l t o n i a n (3.1) can be w r i t t e n i n the f o l l o w i n g w a y :

Pj

2

Pi

ma. ,

» .P. +eV{r, ) v

2m

2

2m!

(3.4)

8

w h e r e w e have i n t r o d u c e d the cyclotron frequency u

f

= eB/m.

T h e equations of

m o t i o n read as :

r 9

PJm • PJmr*-u

J2 (3.5)

P, P

2

Pf/mr'-mu

r/4-edV/dr

-eav/3v

A dot stands for d e r i v a t i o n w i t h respect t o I . I f the

p o t e n t i a l V does n o t depend

momentum

on the

angle, P .

the canonical

angular

is a constant o f the m o t i o n . T h e r a d i a l dependence o f V is given by the

Poisson e q u a t i o n , i n w h i c h w e assume the electron density t o be radially u n i f o r m and equal to n

0

( a n d , o f course, not depending o n the angle) :

Er Sr

e„

* ar

(3.6)

N o t e that, i n t w o dimensions, the "electrons" w e refer to are i n fact infinitely long charged w i r e s , parallel t o the z axis. T h e parameter N then represents the n u m b e r o f real electrons per meter. I n t e g r a t i o n o f Eq.(3.6) yields :

(3.7) 2e

where u

p

= (we'rij/mtj '

is the plasma frequency. U s i n g (3.7), a little algebra on

120

the Eqs,(3.5) leads to the f o l l o w i n g equation for the radius :

2

r = r[a + u ,a

w h e r e It

i

+Wp/2)

is the angular velocity. E q u i l i b r i u m

(3-8)

configurations can be found by

setting equal to zero the t e r m i n parenthesis i n Eq.(3.8). I t is a second degree algebraic equation i n n, w i t h the solutions :

1

(3,9)

T h e B r i l l o u i n flow corresponds t o Ihe case i n which the t w o solutions coincide, i.e. when

-

c

(3-10)

2",

and therefore -u /2

(3.11)

t

F r o m (3.11) and the second o f Eqs,(3.5), one i m m e d i a t e l y realizes that the B r i l l o u i n flow is characterized by the relation P

r

= 0. Strictly speaking, the B r i l l o u i n flow is not

an attractor. I n fact, by plugging Eq.(3.11) i n Eq.(3.8), we obtain r = 0 . I n order to assure the approach to the radial e q u i l i b r i u m , one

must

require

that the radial

velocity be zero at a certain time. I f we slightly p e r t u r b this c o n f i g u r a t i o n by i n t r o d u c i n g a small radial velocity, the system will oscillate around the e q u i l i b r i u m configuration, but never reach it again. H o w e v e r , we shall prove, i n the following, that in a weaker sense the B r i l l o u i n flow does behave as an attractor, w h e n some conditions are verified. W e n o w t u r n to the q u a n t u m mechanical case. B y expressing the Laplacian operator in cylindrical coordinates, the Schrodinger-Poisson system (1.1) becomes

1 9 ' 90 St

2m~ r 3r ma

1 9

W_ \

r

,

J

2

2

r ij

:

l a tf

ar (

22

a

w

*eVi

(3.12)

2

1 BV

r dr A further assumption consists i n considering a system w i t h a z i m u t h a l symmetry, which is obtained by p u t t i n g

3/d
does not

121

depend

on


a system w i t h average P , equal t o zero : i n o t h e r words,

the q u a n t u m B r i l l o u i n flow must be given by a wavefunction w i t h azymuthal symmetry. T h i s s i m p l e fact has a considerable n u m e r i c a l consequence : we just need t o w o r k i n one spatial d i m e n s i o n . I n a d d i t i o n , a little dimensional analysis on the system (3.12) shows the existence of the dimensionless p a r a m e t e r r :

r =

(3.13) 2

eN

w h i l e the characteristic length and t i m e are given by Ihe f o l l o w i n g relations :

L

2

c

, 1

, 2

= y,^ m- ' e^N- l

•

T

=

e

fet^-W'

(3.14)

W e shall therefore solve n u m e r i c a l l y the f o l l o w i n g system, expressed i n dimensionless variables :

1

. 90 St

3 ( dif

2r dr

2

+_r 0

+K0

Sr (3.15)

rj~r\

17

A s we p o i n t e d out before, the B r i l l o u i n flow i m p l i e s that $ depends only o n r. M o r e o v e r , the spatial density must be u n i f o r m over a b o u n d e d d o m a i n

n(r)

= !c (r)f

n{r)

- 0

= n,

0

S

r < R

e

(3.16)

r > R„

H o w e v e r , it is not difficult t o convince oneself that, i n q u a n t u m mechanics, such a configuration

is not

an e q u i l i b r i u m . I n fact, due

wavefunction the support o f w h i c h is 0 < r < R

B

t-/R . B

to the uncertainty p r i n c i p l e , a

possesses a m o m e n t u m o f the o r d e r

T h i s radial m o m e n t u m acts as a p e r t u r b a t i o n , and w i l l start radial oscillations

that never d a m p away. T h i s p e r t u r b a t i o n disappears w h e n R

B

goes to infinity ( c o l d

plasma), w h i c h requires the magnetic field to be extremely weak, i.e. u —t 0. I n this c

case, we recover the classical behavior, relation

as

one

could

have

expected

from

the

(3.13), since the l i m i t u - . 0 corresponds t o ti - . 0. t

B e f o r e t u r n i n g t o n u m e r i c a l simulations, let us calculate the B r i l l o u i n radius R , B

defined i n E q . ( 3 . 1 6 ) . F r o m the n o r m a l i z a t i o n c o n d i t i o n l\l> \ rdr

n,Rt

- 2

w h i l e f r o m the fundamental r e l a t i o n (3.10) one has :

= 1 we get :

(3.17)

122

Ne n. 2

(3.18)

Eqs.(3.17) and (3.18) yield ihe result :

!

e /V

R.

*0 R _£

(3.19)

-

2

r

L.

T h e last equality, obtained t h r o u g h Eq.(3.14). expresses *! in a dimensionless f o r m . fl

T h e B r i l l o u i n mean radius is therefore : "a A

B

-

j r\* frdr

=

(3.20)

±R

B

I n the first simulation, we follow the t i m e evolution of the mean radius .

The

initial c o n d i t i o n is a gaussian wavefunction :

(r,0) = - e x p

r

(3.21)

T h e simulations confirm that, i n the complete self-consistent

plus magnetic

field

p r o b l e m , there exists t w o fundamental lengths, namely, the B r i l l o u i n mean radius R

g

see Eq.(3.20) - and the characteristic length L „ given i n Eq.(3.14), w h i c h is taken equal to one i n our unities. N o t e that L

c

contains h, and therefore is somewhat a

measure o f the importance o f q u a n t u m effects. T h e different regimes of such a system w i l l depend on the r a t i o o f the above typical lengths. I n the simplest case, w h e n the self-cons is tent interaction is absent, one can prove that the Ehrenfest t h e o r e m is exactly verified : thus the mean radius follows the classical trajectory, oscillating at the cyclotron frequency. W h e n w e t u r n on the self-consistent interaction, the dimensionless parameter r appears, as w e l l as (he B r i l l o u i n mean radius

R. g

L e t us start w i t h the semi-classical case, for which the magnetic field is weak, and therefore :

r
(3,22)

The_simulations show that the mean radius < r > is attracted" towards a value close to R , B

and then oscillates around this value. I f the initial c o n d i t i o n is very close to the

B r i l l o u i n flow, the a m p l i t u d e of these oscillations is very small, and the plasma is

123

almost at e q u i l i b r i u m . I n this sense, the classical B r i l l o u i n f l o w can be considered an a t t r a c t o r , even t h o u g h , however, the oscillations never d a m p away since the system is conservative. Several simulations i n this r e g i m e are shown i n F i g . 1. N o w , i f we increase the value o f r ( s t r o n g m a g n e t i c fields), w e shall have :

R

3

< L

= 1

(3.23)

m e a n i n g that q u a n t u m effects have become i m p o r t a n t . I n this case, the simulations show that the B r i l l o u i n f l o w n o m o r e acts as an attractor. Indeed, this is not surprising, since w e are t r y i n g t o confine the plasma on a length smaller than the typical length o f its g r o u n d state. F i g . 2 shows the e v o l u t i o n o f the mean radius for different initial conditions, i n the case o f a large value o f r . T h e B r i l l o u i n m e a n radius R

s

is not

approached, not even i n average, and the p l a m a oscillates at a frequency that is close t o T (since i n this case u = R ,

=

» u ) . I f w e start f r o m an i n i t i a l c o n d i t i o n for w h i c h p

the plasma leaves this configuration, and < r ( f ) > oscillates around a

B

value larger than

R.

In

have s h o w n

summary,

we

g

that,

i n the

semiclassical

regime, a

nonneutral,

a z y m u t h a l l y s y m m e t r i c plasma approaches, at least i n the average, the c o n f i g u r a t i o n given by the B r i l l o u i n flow. H o w e v e r , the fluctuations persist for an indefinitely l o n g t i m e . I n the q u a n t u m r e g i m e , q u a n t u m effects prevent the plasma to be confined inside the B r i l l o u i n radius, so that the B r i l l o u i n flow is n o m o r e approached. T h e physical s i t u a t i o n treated so far is actually a very particular one, for w h i c h the wavefunction is independent o f the angle : as we have seen, this case corresponds t o a plasma

r o t a t i n g w i t h angular

velocity u / 2 . T h e r e f o r e , our initial c o n d i t i o n is c

necessarily "close" to the classical B r i l l o u i n flow, and it is not so surprising that, at least i n the semiclassical l i m i t , this c o n f i g u r a t i o n acts as an attractor. I n order to avoid the above l i m i t a t i o n , one has t o solve the complete, bidimensionat Schrodinger-Poisson system (3.12). Before d o i n g that, we consider the ease w h e r e the wavefunction can be w r i t t e n as follows :

* [ f , « , | ) -
(3.24)

w h e r e s is an integer. T h e wavefunction (3.24) represents a state w i t h well-defined canonical angular m o m e n t u m , equal to P . = to. B y injecting the ansatz (3.24) i n t o the Eqs.(3.12), one obtains a system identical t o (3.15), w i t h , i n the 2

e q u a t i o n , one m o r e t e r m o f the f o r m (s /2r)^

Schrodinger

, acting as a repulsive p o t e n t i a l .

W e n o w t u r n to the fully b i d i m e n s i o n a l case. T h e previous considerations suggest that the B r i l l o u i n flow is not necessarily approached w h e n the initial c o n d i t i o n has a n o n z e r o canonical m o m e n t u m . I n fact, i n o r d e r t o prepare an i n i t i a l c o n d i t i o n that is far f r o m the B r i l l o u i n flow, one must i n t r o d u c e a large canonical m o m e n t u m , w h i c h means a wavefunction violently oscillating i n * : as stated above, the presence o f high a z y m u t h a l modes s h o u l d act as a repulsive t e r m . T h e b i d i m e n s i o n a l results are obtained by solving the system (3.12), by means o f the s p l i t t i n g m e t h o d described i n the previous section. I n this case, the m e t h o d combines

124

b o t h a finite difference

and a F F T scheme. T h e Schrodinger e q u a t i o n i n (3.12) is split

as follows :

. 30

1 3 r

ill .

1

!

30

*l

— + —

2 »
dr

dr

2

r i

8

(3.25)

Vi

at T h e second o f Eqs.(3.25) has an exact solution. T h e first of Eqs.(3.25) can be F o u r i e r transformed i n the angle variable (A is the conjugate o f
H

(r X,t) t

_ 1

at

1

3

3

p

0 1

-_A0 2

2 7 3r

r'

(3.26)

2

+

r 0 8

T h e n equation (3.26) is solved for each A using a finite difference, C r a n k - N i c o l s o n scheme i n the r a d i a l variable. F i n a l l y one comes back t o the real space by p e r f o r m i n g the inverse F o u r i e r transform. T h i s k i n d of scheme allows us t o split the original equation i n just t w o steps, as it was done i n the one-dimensional case. T h e

Poisson

equation i n (3.12) is solved by a similar method, taking the F o u r i e r t r a n s f o r m i n % and using a finite difference scheme i n r. W e now present the results for the bidimensional simulations. T h e i n i t i a l c o n d i t i o n is the f o l l o w i n g : 0 (r,#> , 0 )

2

2

const.rexy(-r j4a )

exp[i t r c o s f j * ) ]

(3.27)

A s expected, i n the semiclassical regime ( f < 1), the m e a n radius oscillates around a value larger than R

B

( F i g . 3 ) . I n order t o study the approach towards the B r i l l o u i n

configuration, we plot, i n F i g , 4, the evolution of Ihe f o l l o w i n g mean quantity :

-rdrdt

(3.28)

a it w h i c h represents the angular

kinetic energy, and is zero for the B r i l l o u i n

(azymuthal s y m m e t r y ) . F r o m F i g . 4, we see that its value decreases, and

flow finally

oscillates a r o u n d a nonzero value. I n other words, Ihe plasma "makes some effort"' to approach the B r i l l o u i n configuration, by g e t t i n g r i d o f some o f the angular dependence o f the wavefunction, although this is not completely achieved. T h i s is apparent also f r o m Fig. 5, w h i c h gives a contour plot for the density | * (r,g> )f at different times. T h e angular dependence o f the density never damps away.

0 40 time

0

80

Fig. 1 . Evolution of for the azymuthally symmetric plasma, with T • 0.3. The initial condition is given by Eq.(3.21) with : (a) a = 8 ; (b) a 2 ; (c) a = 3.55. The broken line indicates R indicates L - 1

B

c

, the dotted line

126

Ol

.

0

J

60 time

120

Fig. 3 ; Evolution of for the bidimensional plasma, with T 0.3. The initial condition is given by Eq.(3.27) with : a = 4 ; £ 0.2 ; (a) s = 8 ; (b) 5 = 6 : (c) s = 0 (azymuthally symmetric case). The broken line indicates R

a

, the dotted line indicates L = 1. c

0.8

0.0 I 0

. 60 time

120

Fig. A : Evolution of the angular kinetic energy (3.28) in the same cases as in Fig. 3 (a) and (b).

128

3.2. T h e a n h a r m o n i c o s c i l l a t o r i n the W i g n e r f o r m a l i s m I n this section, we are g o i n g t o deal w i t h an example o f numerical s o l u t i o n o f the W i g n e r equation. T h o u g h , the self-cons is tent t e r m w i l l be neglected ; we shall simply investigate the e v o l u t i o n of an

initial wavepacket

i n a given external p o t e n t i a l .

Simulations of the complete Wigner-Poisson system can be found elsewhere.

8

It is a trivial exercise t o show that for a quadratic potential, the W i g n e r e q u a t i o n is strictly identical t o the Vlasov ( L i o u v i l l e ) o n e .

28

For a m o r e complicated potential, the

first q u a n t u m c o r r e c t i o n is of second o r d e r i n K H o w e v e r , w h e n the potential has the form o f a q u a r l i c p o l y n o m i a l :

V(x)

l

I m u x*+hmtx*

(3.29)

only the first c o r r e c t i o n t e r m survives, while all the others are strictly equal to zero.

6

T h e r e f o r e , the W i g n e r equation takes the following form :

3

tf+l.£-{ ix+mtx )M. ma

dt

A

m

3X

dp

= - t m t x M 4 dp

(3.30) 1

little d i m e n s i o n a l analysis on the previous equation shows the existence o f a

dimensionless

parameter

H

= .

1

m u

(3.31) i

W e shall always lake m = a = e = 1 , and let H vary, as a measure o f the i m p o r t a n c e of quantum effects. I n the following, we w i l l be concerned w i t h s i m u l a t i n g the e q u a t i o n (3.30) by means of the n u m e r i c a l code previously described. T h e reasons for considering this simplified m o d e l are many : firstly, the Eq.(3.30) is considerably simpler than the complete W i g n e r equation ; secondly, the q u a r l i c potential (3.29) is the first c o r r e c t i o n to h a r m o n i c m o t i o n , and occurs whenever we want t o take i n t o account nonlinear effects a r o u n d an e q u i l i b r i u m ; finally, the numerical solution can be m o r e accurate since the Eq,{3.30) is local i n x, and we need not interpolate Ihe potential. First of all, we investigate numerically some elementary properties o f the W i g n e r equation. O n e of the m a i n puzzles, w h e n posing a quantum-mechanical p r o b l e m i n phase space, is the choice o f the initial c o n d i t i o n . O f course, not every function of two variables corresponds to a possible q u a n t u m state, represented by a positive-definite density matrix. A n operational c r i t e r i o n to lest whether an arbitrary d i s t r i b u t i o n 5

function represents a pure state exists, but unfortunately it cannot be generalized to the case o f a m i x t u r e . I n t u i t i o n suggests that a "good" d i s t r i b u t i o n function should not be too peaked, nor vary too rapidly over a region o f area approximately equal t o K 5

M a t h e m a t i c a l l y , this is expressed by the f o l l o w i n g necessary c o n d i t i o n :

129

1

jjf

(x,p)dxdp

<-

(3.32)

t,)

w h e r e w e have assumed lhat / is n o r m a l i z e d t o unity. T h e equality sign i n (3.32) corresponds t o p u r e states. U n l u c k i l y , the c o n d i t i o n (3.32) is not sufficient f o r / t o be an admissible q u a n t u m state. O n the o t h e r hand, the W i g n e r equation i n itself does not prescribe any p a r t i c u l a r choice o f the i n i t i a l c o n d i t i o n , w h i c h may w e l l be a "forbidden" one, not representing any q u a n t u m state. W h a t happens w h e n the i n i t i a l c o n d i t i o n does not respect the necessary c o n d i t i o n (3.32) ? I n o r d e r t o answer this question, w e study the e v o l u t i o n o f a gaussian

exp

x

_

p'

(3.33)

~2~?~2~o~\

i n a q u a r t i c p o t e n t i a l . T h i s i n i t i a l state is an admissible one as l o n g as o o i

p

i

H/2,

Figs. 6, 7 a n d 8 show the e v o l u t i o n o f the kinetic energy for different values o f the q u a n t u m p a r a m e t e r H. N o t e that for H = 0 (classical case) the system approaches a stationary s o l u t i o n . A s w e shall see later o n , this is due to the filamentation

finer

and

finer

in the phase space, w h i c h u l t i m a t e l y reaches the level o f the mesh size.

H o w e v e r , the r e l a x a t i o n towards a stationary state is "physical" a n d does not depend on the step of the g r i d . A s H increases, the kinetic energy starts oscillating w i t h o u t d a m p i n g . W h e n o J>, * H/2.

t

the oscillations are so v i o l e n t that the kinetic energy

becomes either negative or greater t h a n the t o t a l energy ( w h i c h indicates a negative p o t e n t i a l energy). N o t e however that the t o t a l energy is still constant, since this is a p r o p e r t y o f the W i g n e r e q u a t i o n , irrespective o f the i n i t i a l c o n d i t i o n . T h u s , w h e n we t r y to violate the H e i s e n b e r g p r i n c i p l e by concentrating the d i s t r i b u t i o n function over a r e g i o n smaller than \

w e arrive at the absurd situation i n w h i c h the m e a n value o f

a positive q u a n t i t y becomes negative. A second p o i n t concerns the reversibility o f the W i g n e r equation. O f course, f r o m a strictly m a t h e m a t i c a l point o f view, the W i g n e r equation, just as the L i o u v i l l e e q u a t i o n for classical systems, is c o m p l e t e l y reversible. H o w e v e r , as w e p o i n t e d out before, the classical phase space tends t o become m o r e and m o r e i n t r i c a t e d , so that a small e r r o r or p e r t u r b a t i o n prevents the possibility o f inversing the dynamics and thus r e c o v e r i n g the i n i t i a l c o n d i t i o n . I t was noticed by M . V . B e r r y

2 5 ,x , l i

the case for the

effects

filamentation

quantum

phase space : i n fact, q u a n t u m

that this is not prevent

the

to reach the m i n i m u m unit cell o f area o. B e r r y was m a i n l y interested

i n the behavior o f chaotic dynamical systems, b u t we can easily prove that this is t r u e also for ours, w h i c h is o f course c o m p l e t e l y integrable. Firstly, w e c o n f i r m Berry's results by a t i m e - r e v e r s a l c o m p u t e r experiment. T h e initial c o n d i t i o n is the f o l l o w i n g

(*+*o)' f)

•

4n a a

exp -XL 2a]

•-exp

exp 2o \

2o\

(3.34)

130 with.t

0

= 1.8 ,a

x

= .5 , o

p

= 1.4. T h i s i n i t i a l state evolves i n the quartic potential up

to r = 15, then the velocities are reversed and the system goes back t o its initial state. N o w , due to the discretization, some i n f o r m a t i o n gets lost d u r i n g the e v o l u t i o n , and therefore

the system w i l l not r e t u r n exactly t o its initial c o n d i t i o n . H o w e v e r , the

situation is very different for the classical and quantum eases. T h e classical evolution (H = 0 ) is shown i n F i g . 9. A t t i m e t = 15, the filamentation has already reached the mesh size, and the time-reversal is n o m o r e able t o b r i n g back the system to the i n i t i a l c o n d i t i o n . For the q u a n t u m evolution, we choose a pure state, w i t h H = 1.4 ( F i g . 10). T h e f i l a m e n t a t i o n is m u c h less i m p o r t a n t , and the initial c o n d i t i o n is recovered quite precisely. N o t e that, o f course, we used the same mesh i n b o t h cases. N o w that we have established that q u a n t u m dynamics is, i n some sense, "more reversible" than classical dynamics, it w o u l d be useful to define a m a t h e m a t i c a l object which somewhat measures these different degrees o f reversibility. I n o l h e r words, we need a definition of entropy c o m m o n to both the classical and q u a n t u m phase space. T h e standard definition o f entropy as the mean value of - l o g / cannot be used, since / c a n take negative values. The entropy should be defined as a functional of the f o r m

S\f\

=\\G(f)dxdp

(3.35)

F u r t h e r m o r e , we ask G to be a concave function o f / , and, of course, we require 5 to be constant i n t i m e for a closed H a m i l t o n i a n system.

31

Given the above constraints, it

is n o l difficult to show that the most natural definition o f S is the following:

S = 1 - 2 * -hjjf

(x,p)dxdp

(3.36)

I n fact, the integral i n the Eq.(3.36) is a constant o f the m o t i o n for the W i g n e r equation. T h i s is easily shown by m u l t i p l y i n g Eq.(1.4) by f/2, 2

the phase space. The l.h.s. gives simply (d/dt)llf dxdp,

X ' V x - — 2 ;

-

-441 exp I 4

iP'P'x h

' \dxfdx V\x-t 4* V J J

and integrating over all

while the r.h.s, yields :

f(x,p',t)f{x,p,t)dp'dpdxdx

j dp'exp

Jrfpexpllfil/^O

=

(3.37)

=0

The i n t e g r a n d i n Eq.(3.37) is an o d d function i n X, so that i n t e g r a t i o n from -» t o + « yields zero, thus dS/dt

= 0. I n fact, the integration in x was not necessary to e l i m i n a t e

the r.h.s. o f the W i g n e r equation. I f we only integrate i n p, we get the continuity e q u a t i o n for the local entropy s(x,i) :

following

131

i£+i£ =0 at

(3.38)

Bx

w h e r e the local e n t r o p y s and the e n t r o p y flux / have been defined as follows

s(x,t)

"J/4. -2^|/Vp

T h e e n t r o p y (3.36) is a measure o f the purity of a q u a n t u m state. I n fact, 0 < S < 1, and S = 0 i f / represents a pure state. W e n o w m a k e use o f o u r e n t r o p y as a measure o f the loss o f i n f o r m a t i o n i n the n u m e r i c a l s o l u t i o n o f the W i g n e r equation. A s usual, the initial c o n d i t i o n is a gaussian, a n d the p o t e n t i a l a q u a r t i c p o l y n o m i a l . F i g . l l shows the e v o l u t i o n o f the entropy, for various values o f the parameter H. I t is apparent that the i n f o r m a t i o n is lost m o r e and m o r e slowly as H increases. M o r e o v e r , S increases m o n o t o n i c a l l y w i t h the t i m e for every value o f H. L e t us n o w come back t o the previous time-reversal experiments. I n Fig. 12,

the

e n t r o p y is p l o t t e d against t i m e ; at ( = 15 the velocities are reversed, and the system evolves backwards i n t i m e . H o w e v e r , there is no discontinuity i n the e v o l u t i o n of S at the t i m e / = 15 : the e n t r o p y goes on increasing b o t h i n the q u a n t u m and i n the classical cases. T h i s indicates that o u r e n t r o p y t r u l y measures the loss of i n f o r m a t i o n due to the n u m e r i c a l discretization.

Fig. 6 : Evolution of Ihe kinetic energy of a wavepacket in a quartic potential (3.29). The initial condition is given by Eq.(3.33) with o 1 I Op = \A ;H - 0 (full line) ; H - 1 (doited line). t

4.0

0

15 time

30

Fig. S : Same as Fig. 6 with H • 5 The kinetic energy now takes on negative values

Fig. 9 : Evolution of a wavepacket (3.34) in a quarlic potential; r„ = 1.8 ; 0, - 0.5 ; i7p 1,4 ; classical case (H = 0). The dynamics is reversed at I = 15.

Fig. 9 : conLinued

135

Fig. 10 : Same as Fig. 9 focH = 1.4 (quantum pure state). We show ihe posirive pari of Ihe Wigner function.

Fig. 10 : continued.

0.24

0.16L 0

, 12 time

-

J 24

Fig. 11 : Evolution of 1 - It J ff'dxdp tor: (a) H = 0 ; (b) H = 0.5 ; (c)H - 1 ; (d) H - 2.4 (pure stale).

138

4. C o n c l u s i o n T h e firs! purpose of Ihe present paper was to investigate the properties of a gas of electrically interacting particles where q u a n t u m effects cannot be neglected. A certain number

of

rather

drastic

assumptions

were necessary i n order

to o b t a i n a

m a t h e m a t i c a l m o d e l suitable for analytical and numerical investigations. W e described t w o o f such possible models, the Schrodinger and Wigner-Poisson systems, and, for each of t h e m , presented an efficient numerical a l g o r i t h m based on a splitting scheme. B o t h are colli si onless, one-particle models, and neglect the second i m p o r t a n t q u a n t u m effect, the spin. H o w e v e r , they contain the two m a i n ingredients for a quantum plasma, i.e. long-range interactions and a quantum equation o f m o t i o n . N u m e r i c a l l y , ihe Schrodinger representation is cheaper, but the W i g n e r one permits to treat a wider class o f q u a n t u m states, in the familiar phase space formalism, and is of considerable help to understand h o w the classical l i m i t is recovered. Certainly, the study o f realistic problems, connected to the design o f m i c r o e l e c i r o n i c devices, w o u l d require m o r e sophisticated models. I n particular, one w o u l d want to investigate three-dimensional geometries, endowed w i t h m o r e complicated boundary conditions. I n that case, the W i g n e r representation must be r u l e d out, since it w o u l d imply the discretization of a six-dimensional phase space. A n o t h e r , m o r e delicate point, is the m o d e l i n g of dissipative effects. I n solids, the electrons are confined by the presence o f an underlying ionic, motionless lattice : collisions between

the electrons

and the lattice are

nonelastic, and represent

a

dissipative p h e n o m e n o n , w h i c h i n general cannot be neglected. H o w e v e r , there is no universal consensus on h o w to deal w i t h dissipation i n quantum mechanics, A deeper understanding o f these issues w i l l probably constitute a significative advance i n the technology o f future semiconductor devices. A second purpose o f this paper was to analyse the role o f initial conditions i n the W i g n e r equation. T a k i n g an i n i t i a l state that corresponds to a given ii (i.e., a pure state) does not add anything new, and i t is simpler i n that case to solve Schrodinger equation. M o r e interesting are the initial conditions connected

the to a

m i x t u r e , or even m o r e general ones. A n interesting result was the i n t r o d u c t i o n o f an entropy w h i c h measures the loss o f i n f o r m a t i o n due to the numerical scheme. T h i s entropy gives a precise meaning to the observation, made by Berry, that classical systems develop an intricated phase space structure, a n d thus loose small scale i n f o r m a t i o n m o r e quickly than the q u a n t u m ones. A last remark concerns the good properties of the E u l e r i a n codes (i.e. phase space codes) i n theoretical problems where simple, l o w dimensional geometries can be used. U n f o r t u n a t l y , particle codes (which, i n the classical case, allow a r o u g h treatment o f such complex geometries) are, for q u a n t u m problems, as costly as the E u l e r i a n codes, and presumably m o r e noisy. T h e creation o f a good W i g n e r code for t w o and three dimensions is, to o u r knowledge, still an open and i m p o r t a n t question.

139

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13. N . C . K l u k s d a h l , A . M . K r i m a n , D . K . F e r r y and C. Ringhofer, Self-consistent of the resonant-tunneling

wave

Phys. 96 (1992) 2077-2084. study

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14. A . A r n o l d a n d P A . M a r k o w i c h , Quantum

transport models for semiconductors,

in

Applied and Industrial Mathematics, ed. R. Spigler ( K l u w c r A c a d e m i c Publishers, 1991), p p . 301-307. 15. A . A r n o l d a n d F. N i e r , Numerical

analysis of the deterministic panicle

method

applied to the Wigner equation, Math, of Comp. 58 (1992) 645. 16. R . W . H o c k n e y a n d J . W . E a s t w o o d , Computer simulations using particles ( A d a m H i l g e r , 1988). 17. W . F , A m e s , Numerical methods f o r p a r t i a l diiTerential equations ( A c a d e m i c Press, 1977). 18. N . N . Y a n e n k o , The method of fractional steps ( S p r i n g e r - V e r l a g , 1971). 19. W . H . Press, B P. Flannery, S.A. T e u k o l s k y and W . T . V e t t e r l i n g , Numerical recipes ( C a m b r i d g e U n i v e r s i t y Press, 1986). 20. C . Z . C h e n and G , K n o r r , Tlte integration of Ihe Vlasov equation in space, J. Comput.

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Phys. 24 (1976) 330-351. Vlasov equation, J. Comput.

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Phys. 24 (1977) 445-449.

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of the time-

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141

Mathematical Theory of Kinetic Equations for Transport Modelling in Semiconductors F . Poupaud Laboratoire J.A. Dieudonne, U.R.A. 168 du C.N.R.S., Universite de Nice, Pare Valroae, F - 06108 Nice Cede* 2, France

Abstract We present a review of mathematical results available for kinetic models for semiconductors. These models i r e mainly based on Vlasov-PoissonBoltzmann system of equations. However the transport of particles (electrons and holes) is not purely classical but takes into account quantum effects due to the periodicity of the semiconductor lattice. Although the mathematical analysis is almost complete from the point of view of existence and uniqueness of solutions, i t remains a lot of open problems concerning the connection between quantum, kinetic and fluid levels of modelling.

1. I n t r o d u c t i o n S e m i c o n d u c t o r device m o d e l l i n g requires the knowledge of the d y n a m i c of charged particles i n semiconductor lattices. T h e free particles are of t w o types. F i r s t there are electrons w h i c h belong t o the c o n d u c t i o n band a n d second electrons w h i c h bel o n g t o the upper valence b a n d . Since the valence band is almost f u l l , i t is more convenient t o consider a lack o f electrons and the d y n a m i c o f these vacancies. T h i s leads t o the concept o f holes w h i c h can be considered as quasi-particles w i t h posi t i v e charges. T h e n the key p o i n t w h i c h allows t o construct devices w i t h m a n y different physical properties is t h a t the i n t r i n s i c c o n c e n t r a t i o n level o f electrons and holes is controled by the level o f c o n c e n t r a t i o n o f i m p u r i t i e s w h i c h are added t o the s e m i c o n d u c t o r l a t t i c e . For instance i f y o u a d d a t o m s o f donors i n the c r y s t a l , these a t o m s w i l l give electrons t o the c o n d u c t i o n b a n d . Conversely i f you a d d acceptors, the c o n c e n t r a t i o n o f holes w i l l increase. T h e concentration level o f i m p u r i t i e s w h i c h becomes charged by g i v i n g or t a k i n g an electron is usualy named the d o p i n g profile. T h e n the t r a n s p o r t processes for electrons and holes are governed by t w o phen o m e n a : free flights a n d collisions. D u r i n g the free flights, electrons (resp. holes) m o v e w i t h a velocity v (k) (resp. v (k)) which depends on t h e i r wave-vector it. T h i s wave-vector is accelerated by an electric field. Hence the kinetic of particles depends s t r o n g l y on the f o r m o f the f u n c t i o n k —* t i „ ( i ) . A t this p o i n t , q u a n t u m effects are taken i n t o account because the functions v„ (k) depends on the struct u r e o f the bands. T h i s s t r u c t u r e is d e t e r m i n e d by s o l v i n g an eigenvalue p r o b l e m for Schrodinger equations. These eigenvalues E (k) are the energies of electrons n

p

i P

iP

n

p

142

and holes. T h e y are related t o velocities t h r o u g h the relations

v , w = l^kE . {ky n P

n p

(i.i)

T h i s connection between q u a n t u m physics and kinetic models w i l l be clarified i n Section 2. Let us m e n t i o n t h a t the electric field depends on the applied bias and on the d o p i n g profile ( w h i c h are a - p r i o n k n o w n ) b u t also on the concentration o f electrons and holes t h r o u g h Poisson's equation. T h u s nonlinear phenomena occur. T h e collisions involve several mechanisms. F i r s t there are interactions between particles and the device which play the role of a t h e r m a l b a t h . These interactions are collisions w i t h i m p u r i t i e s and phonons. Phonons are n o t h i n g b u t representations as quasi-particles o f v i b r a t i o n s of the crystal. These collisions are p r e d o m i n a n t on the other mechanisms. They tend to thermalize the particles at the fixed t e m p e r a t u r e o f the device w i t h a vanishing velocity mean-value. T h e second i n i m p o r t a n c e phenomena is the generation-recombination process. I t occurs i f an electron goes from the valence band t o the c o n d u c t i o n band (generation of a pair electron-hole) or conversely i f i t goes from the conduction band t o the valence band ( r e c o m b i n a t i o n o f a pair electron-hole). T h e energy necessary t o this t r a n s i t i o n is given or taken by a phonon via a collision. F i n a l l y there are b i n a r y colisions which are modelled exactly as i n gas dynamics. T h i s last mechanism is often neglected i n semiconductors theory compared t o the previous ones. I t becomes significant o n l y for very high concentration o f particles. Moreover the m a t h e m a t i c a l analysis o f these collisions processes lead t o the same k i n d o f technics and difficulties as i n gas dynamics. Therefore i n the remainder o f this paper we w i l l not deal w i t h b i n a r y collisions. We only refer t o [7, 14, 37], There are three levels of m o d e l l i n g which take i n t o account all these physical phenomena. T h e most accurate is the q u a n t u m one where the particles are represented by wave functions s o l v i n g Schrodinger equations. T h e H a m i l t o n i a n incorporates potentials due t o the semiconductor l a t t i c e , C o u l o m b interactions, the applied bias, particles-phonons interactions, and so on ... A l t h o u g h such models begin to be used for analysing w i t h numerical s i m u l a t i o n s small p a r t of devices where q u a n t u m effects are i m p o r t a n t [9, 38, 55, 66], up to now i t is impossible t o m o d e l a complete device at t h i s level. T h e coursest models are based on d r i f t diffusion equations which are equations of parabolic t y p e solved by the concentration of particles. I t is c e r t a i n l y the most used ( a n d usefull) equations to s i m u l a t e real devices. We refer for the physical background of these models t o [57, 6 1 , 62], I n t h i s field, the m a t h e m a t i c a l and numerical analysis is almost complete, see [23, 28, 34, 32, 58, 59]. B u t even i f the drift-diffusion equations give very good results for components whose t y p i c a l length scale is o f order o f a few micrometers, these models are no more v a l i d for s u b m i c r o n devices, [56]. Therefore more sophisticated models are studied. T h e y are based on e l e c t r o - h y d r o d y n a m i c equations [or energy balanced equations) which are o b t a i n e d f r o m drift-diffusion equations by a d d i n g balanced equations for the flux a n d the energy o f particles. I n the last few years, numerical investigations o f these models have increased very m u c h , see [60] for instance. B u t only few m a t h e m a t i c a l results are available due to the h y p e r b o l i c i t y o f these nonlinear system of equations. L e t us only m e n t i o n [12, 18, 50, 69]. B u t one of the m a i n p r o b l e m concerning electro-

143

h y d r o d y n a m i c m o d e l s is t h e i r j u s t i f i c a t i o n . Indeed non physical effects occur when s o l v i n g these equations as the cooling o f particles for instance. Moreover these equations need fitted coefficients w h i c h depend s t r o n g l y o f the p a r t i c u l a r device which you simulate. Between the t w o previous levels o f m o d e l l i n g , there is the k i n e t i c one. T h e d y n a m i c of particles is described by the e v o l u t i o n of d i s t r i b u t i o n functions, f (t,x,k) for electrons, f (t,x,k) for holes, ( „ for negative charges, for positive charges), w h i c h depend on the t i m e , ( > 0, on the p o s i t i o n , x £ SI, Q being an open set o f R representing the device geometry and on the wave vector k. T h i s wave vector belongs t o a torus B o f R w h i c h is defined as follows. L e t a y ) , o r j j , o ^ r ) be a basis of R T h e n the c r y s t a l l a t t i c e is defined by n

p

p

3

3

3

£ -

+n

)j

( 2

2

3

T h e d u a l basis vectors « W , a( >, a' ' a and the d u a l l a t t i c e L

+ a, j \

2

2

3

£ 2Z}

(1.2)

are d e t e r m i n e d by the e q u a t i o n

m

( 0

j,, j ,j

3) 3

a< >-2*t)( ,

1,2,3

£,m=

m

(1.3)

( ' r e c i p r o c a l l a t t i c e ' ) reads

T h e basic p e r i o d cell o f the l a t t i c e L is denoted by 3 C .:-{^( <-j, |0
i

l

1

!

(1.5)

3

I n physics b o o k s ( [ 5 , 53, 5 4 ] ) , the B r i l l o u i n zone B is usally the W i g n e r - S e i t z cell of the d u a l l a t t i c e . B u t a l l the physical q u a n t i t i e s have the p e r i o d i c i t y o f L

w.r.t.

it. T h e n because o f m a t h e m a t i c a l convenience we prefer t o define B as ; 3

B := K / £ *

(1.6)

T h e n the semiclassical B o l t z m a n n e q u a t i o n o f semiconductors reads !/„((,!,*)

+ v {k)

•V,/„(i,x,ifc) + | V J S M

n

Qn(M(t,z,k)-r

o

n„(f ,f )(t,x,k), n

P

• V f (t,x,k) k

=

n

o, i e n , * e B ,

(1.7)

for electrons and s i m i l a r l y ^f (t,x,k) P

+ v (k).V f (i,x,k)-^V U(t,x)-V f (t,x,k) fl

I

p

c

t

Qp(/ )(i.i,'t)-l-/i ,(/ ,,/ )(i,i,fc), P

J

f

n

=

p

O O , x£Sl,

keB,

(1.8)

for holes. I n the above equations q is the elementary charge, A is the reduced Planck constant. T h e p o t e n t i a l U solves Poisson's e q u a t i o n - A U(t, t

x) =

- n(t, x) + pit,

x))

(1-9)

144

where f r is the p e r m i t i v i t y of the semiconductor, C is the ( k n o w n ) d o p p i n g profile. T h e concentrations o f electrons, n , and holes, p, are given by n(t,x) = ~ (

/„(«,«,*)«,

^*.*> = i

/

&&*t*)

rffc

(1-10)

F i n a l l y the collisions operators are integral operators w . r . t k w h i c h model collisions w i t h phonons and i m p u r i t i e s and generation r e c o m b i n a t i o n p r o cesses respectively. T h e y are given by

QnAa)

=

I * » , ( * . *0

- 9)M .

- *(i-

n r

fiK,)

<"*'.

( U l )

JB

Rn[g,h)=

f yfk,k')((l-g){l-h')M M -gh')dk\ n

(1.12)

p

JB

f y,k',k)((l-g')(l-h)M^M -g'h)dk'.

Rp(h,g)=

(1.13)

p

JB

I n the previous expressions we have used the convention t h a t the functions w i t h a ' are taken at k' and w i t h o u t a ' are taken at k. T h e m a x w e l l i a n functions M are given by n

M„,(*) - K

BJ)

**p f - ^ r )

p

(1-14)

where Jf ,|> are constants o f n o r m a l i z a t i o n and is the B o l t z m a n n constant, 6 the (constant) t e m p e r a t u r e of the device. T h e kernels rr„ , 7 satisfy n

p

0„,„

7 EC"(flx5)-,
r r , , 7 > 0, n

p

= a- , {k\k). n p

(1.15) (1.16)

They are determined by q u a n t u m scattering theory. I t is the purpose o f t h i s paper t o give a survey of m a t h e m a t i c a l results for the system o f equations (1.7), (1.8), (1.9). I n Section 2, we focus on the j u s t i f i c a t i o n of B o l t z m a n n equations from q u a n t u m physics. I n this field there are more open problems t h a n m a t h e m a t i c a l theorems. We give few results and a ( n o n exhaustive) list o f unsolved problems. I n Section 3, the classical question of existence and uniqueness of solutions is studied. We also give fundamental properties of collisions operators. T h e n some of the m a i n a p p r o x i m a t i o n s used by ingeniors and physicists are discussed. F i n a l l y Section 4 is devoted to the fluid a p p r o x i m a t i o n of kinetic models. Some rigorous derivations of drift-diffusion equations are given. Some attemps to o b t a i n electro-hydrodynamic models are presented.

2. F r o m q u a n t u m t o k i n e t i c m o d e l s W h e n conduction electrons move i n a s o l i d , then q u a n t u m effects o f the ions located at the crystal l a t t i c e points have t o be taken i n t o account i n the equations of m o t i o n . O n a fully q u a n t u m mechanical level o f description t h i s is done by

145

i n c o r p o r a t i n g a l a t t i c e - p e r i o d i c p o t e n t i a l V (generated by the l a t t i c e ions) i n the (effective o n e - p a r t i c l e ) H a m i l t o n - o p e r a t o r for a c o n d u c t i o n electron

// = - — X

+ f,

(2.1)

Here m denotes the mass o f the electron and ft the Planck constant. T h e electron wave f u n c t i o n # then satisfies the I V P for the Schrodinger e q u a t i o n 3

ihjp, = Bj>,

x e IR ,t

> 0

(2.2)

w i t h the q u a n t u m p o s i t i o n density

N o t e t h a t i n t h i s m o d e l no exterior ( n o n - p e r i o d i c ) field influences the m o t i o n of the considered p a r t i c l e a n d we also do n o t take i n t o account the C o u l o m b i n t e r a c t i o n a m o n g electrons neither interactions w i t h phonons a n d / o r i m p u r i t i e s of the l a t t i c e . Usually, the i n i t i a l state 0 / is n o t k n o w n exactly a n d , thus, corresponding s t a t i s t i c a l m i x t u r e s (so called m i x e d states) w i t h given o c c u p a t i o n p r o b a b i l i t i e s for the set o f possible states have t o be considered. I t is well k n o w n t h a t the H a m i l t o n operator (2.1) w i t h the l a t t i c e periodic p o t e n t i a l V induces a direct s u m d e c o m p o s i t i o n o f the space o f states L (St ) corresponding t o the eigenspaces o f the F l o q u e t - eigenvalues o f H (see e.g. [53]). These eigenvalues {Em}me j v , called energy b a n d s ' i n solid state physics, are periodic functions o f the wave-vector it defined i n the B r i l l o u i n zone B o f the crystal l a t t i c e . O n a semi-classical level o f d e s c r i p t i o n , the Schrodinger e q u a t i o n is usually replaced by the semiclassical L i o u v i i l e e q u a t i o n for the phase space ( i.e. position-wave vector space) density / = f(x,k,t). W h e n no exterior field and no collisions are present and w h e n the electron is k n o w n t o move i n the n - t h b a n d (i.e. the states of the s t a t i s t i c a l m i x t u r e belong t o the n - t h F l o q u e t eigenspace) then the semiclassical L i o u v i l l e e q u a t i o n reads 2

3

1

with v (fc) = jV E (k) n

t

(2.5)

n

subject t o an i n i t i a l c o n d i t i o n

/(< = 0) = //, zeiR , 3

fceS.

(2.6)

In the physical l i t e r a t u r e the equations o f m o t i o n x = v (k), n

hk

=0,

(27)

(2.8)

146

which correspond t o the semiclassical L i o u v i l l e equation (2.4) are usually derived by t r a c k i n g the m o t i o n of wavepackets of the Schrodinger e q u a t i o n (2.2). T h i s s t a t i o n a r y phase m e t h o d does, however, not lead to a rigorous j u s t i f i c a t i o n of the semiclassical L i o u v i l l e equation and o f the semiclassical m o m e n t s . A rigorous d e r i v a t i o n of the semiclassical t r a n s p o r t equation (2.4), (2.6) was given by P. G e r a r d i n [19]. His very elegant approach is based on m i c r o l o c a l analysis of pseudodifferential operators. For a b o u n d e d sequence i n I? he constructs " t h e semiclassical measure" which describes the oscillations of the sequence. He proves t h a t the measure associated t o the sequence of solutions o f (2.2) satisfies equations s i m i l a r t o (2.4), (2.6). I n fact, i t can be shown (cf. [24]) t h a t the semiclassical measure is n o t h i n g b u t the l i m i t o f the W i g n e r t r a n s f o r m o f the density m a t r i x corresponding to (2.2). T h i s approach does, however, not p r o v i d e exactly the equations (2.4), (2.6) and need asumptions o n the i n i t i a l wave-functions which are not very natural. I n [30] a different approach based e n t i r e l y o n kinetic equations (cp. [24] and [29] for the v a c u u m case) is proposed. Using the above mentioned eigenspace decompos i t i o n o f the space of states, the I V P for the Schrodinger equation is r e f o r m u l a t e d as a denumerable system o f W i g n e r - t y p e equations. T h e basis for t h i s r e f o r m u l a t i o n lies i n the a p p r o p r i a t e definition o f a W i g n e r - t y p e function for each band. These Wignerseries are defined i n analogy (i.e. using Fourierseries instead o f Fouriertransf o r m ) t o the non-crystal 'whole space case' (which is presented e.g. i n [63], [64] ) and have s i m i l a r properties as the semiclassical d i s t r i b u t i o n functions (solutions of the semiclassical L i o u v i l l e equations ) . I n p a r t i c u l a r they allow the calculation o f the macroscopic densities i n analogy t o the semiclassical k- space m o m e n t s (1.10). T h e semiclassical L i o u v i l l e equation (2.4) (2.6) is then obtained by i n t r o d u c i n g an a p p r o p r i a t e scaling (analogously t o the one used i n [19]), which assumes t h a t the period of the p o t e n t i a l V is ' o f order h' (after a p p r o p r i a t e scaling) and by carrying o u t the l i m i t h —> 0. T h e convergence o f the q u a n t u m p o s i t i o n - , current- and momentum-densities can then be concluded from the convergence properties of the Wignerseries. We r e m a r k t h a t t h i s approach (just as the one o f P. G e r a r d ) is not l i m i t e d t o a one-band a p p r o x i m a t i o n . I n i t i a l wave-functions which belong t o the direct s u m of a r b i t a r y many eigenspaces are admissible. However, for the sake of s i m p l i c i t y , we present here the result only for one band. Let us now go deeper i n details T h e crystal lattice L , its basic period cell C-i. the reciprocal l a t t i c e L and the torus B are s t i l l defined by (1-2), ( 1 4 ) , (1.5), (1-6). For the f o l l o w i n g let V — V(x) be a real-valued p o t e n t i a l o n R w i t h the properties 3

3

(Al)

V£L™(R ),

V is L-

For a S (0, o ] , o 0

0

periodic, i.e. V(x + u) = V(x)

2

which we w i l l regard b o t h on L (CL) 1

3

R

Vu £ L .

> 0 fixed, we define the H a m i l t o n i a n

»*;=

from N

a.e.

—2"A

i

t

+t(J) 2

3

and o n L ( i R )

(2.9) O b v i o u s l y , H° is o b t a i n e d

by the rescaling o f the position variable x —t J , where a is the scaled

147

P l a n c k constant. For k € B we define t h e operator H'(k) as H' w i t h t h e p e r i o d i c i t y conditions : *(* + ft) = e'>

,

3

i €

fff ,

(2.10)

i

^ ( * + M ) = «''"' |^(a ), * ei^,/*,€£,<=l,2,8 (2.11) OZ( OX l I t is well k n o w n [53] t h a t each H'(k) has a complete set o f eigenfunctions ip — V (x,k),m e W , ( o r t h o n o r m e d i n L (C }) w i t h eigenvalues Ei(fc) < E (k) < E (k) < -•• < E _ i { i ) < E (k) < ••• (counted w i t h m u l t i p l i c i t i e s ) . A s a consequence o f t h e m i n - m a x f o r m u l a , for every fixed m g N the f u n c t i o n E — E (k) is lipschitz o n B, see [19]. T h e set {E (k)\k^B}CR (2.12) !

2

m

L

3

m

2

m

m

m

m

l

1

is called t h e m - t h (energy) band o f f / and the eigenfunction * is called t h e m — th B l o c h - f u n c t i o n . O b v i o u s l y , the set { * ( . , k) } is o r t h o n o r m e d and t o t a l in L\C ). m

m

L

T h e n t h e c r u c i a l p o i n t is t h a t we can use the previous spectral decomposition o f the o p e r a t o r s H {k) t o o b t a i n a B l o c h decomposition o f the operator H, see [53], T h e o r e m X I I I . 9 8 , a n d consequently o f the operators H" by rescaling i —> ¥ . T h i s is a consequence o f t h e fact t h a t any f u n c t i o n defined o n R can be decomposed as k-quasiperiodic functions. Indeed let us defined t h e following spaces l

3

L \ = {u€Ll,(M

3

x S ) Hi

(

i i ( i

T

^ ^ e S (

I

= {u e Ll.s.t.Vue

, i )

. t . / o r ^ i ) ,

a

(2.13)

L\)

(2.14)

provided w i t h the norms 2

N I ° . * = (i5T /

1

WUM dxdkfl\

H u t l L ^ d h l l ^ + IIVull^) '

2

(2.15)

P r o p o s i t i o n 2 . 1 The following map $ = iP(x)~v.

is an isomorphism

2

3

from L (R )

v

= {x k)^iP v

l

= u(x,k)=

^ ^ ( i + ^ e - ^ "

3

onto L \ and from H'(R ) 1 = ii>{x) = ~

f /

\B\

JB

onto Hi wkose

u(x,k)dk

(2.16)

inverse

(2.17)

T h i s p r o p o s i t i o n is j u s t a consequence o f Parseval's f o r m u l a . W e now set :=c<-H (^,jt) m

3

, i € R ,k€B.

(2.18)

148

T h e scaling argument mentioned above a n d Proposition 2.1 i m p l i e s t h a t t h e subspace SS

2

:= | y « r ( f c ) * S ( i , / t } < i t

3

treL (fl)|

|

(2.19)

3

of L (R ) is i n v a r i a n t under the action o f H" Indeed i t is isomorph t o the space V£ {o~(k)^l%(i,k)dk | rr £ L ( f l ) } whose functions are eigenvectors o f the operators H (k). is called t h e n - t h band space. O b v i o u s l y 5 ° a n d S%, are o r t h o g o n a l for m , ^ m since the spaces V£ a n d V£ are o r t h o g o n a l . T h e action o f the H a m i l t o n i a n i n the subspaces is described i n : 2

a

l

2

t

a

L e m m a 2.1 Lei ip £ S ° . Then H ip =

3

t

£ S° and £4,0,^(1,+ ^ )

ujftere f n ( / j ) are the Founercoefficients

(2.20)

of E (k) n

We now consider t h e Schrodinger equations

ftljjrtf

-

— W t + V t y t .

< / £ ( * , ( = 0) = uf(t),

x € & , t > Q z e R

3

(2.22) (2.23)

for i 6 I V , where we prepare the i n i t i a l d a t a w° as follows : (A2)

/ uffiPitf*

=

Viiyii

e W,

wf£

& 2

3

T h e sequence w f o f i n i t i a l d a t a is o r t h o n o r m e d i n L (R ) a n d therefore the solutions ip"(.,i) r e m a i n o r t h o n o r m e d for a l l times / > 0. Moreover ip£(.,i) £ S° for all / > 0 follows f r o m L e m m a 2 . 1 . W e also conclude t h a t the 1VP for ip" can be w r i t t e n as 3

= £ft,(u)0?(* + afi,t),

I e JR ,(>0

a

= 0) = ^ , ( i ) , r e JR

3

(2.24)

(2.25)

For the definition o f t h e m i x e d state densities we prescribe a sequence o f o dependent o c c u p a t i o n probabilities AJ* o f the sequence o f states wf We shall use t h e f o l l o w i n g assumptions (cp [29]) (A3)

A? > 0

We

W,

.

0

independent o f a.

149

T h e t o t a l charge density is defined b y DO

n

"(z.*):=EA?|tf[>,0r

(2.26)

W e set u p t h e density m a t r i x

z°(r,

*,i) = £

3

V ^ V , <Wf ( s , * ) .

r, s £ H , ( > 0

(2.27)

and define t h e " Wignerserie" w" by t h e Fourier serie : a

w (x,k,t)

a

:= £

z (x +

* e

, A € B.

T h e i n i t i a l W i g n e r f u n c t i o n wf(x, k) is defined by replacing tpf(.,t) wave f u n c t i o n s wf (.) i n (2.27) , i.e. i u f = t u ° ( t - 0 ) .

(2.28)

by the i n i t i a l

I t is t h e it - m o m e n t s w h i c h give Wignerseries a physical m e a n i n g . F r o m the definition (2.28) a n d f r o m t h e Fourier inversion f o r m u l a we o b t a i n the zeroth order k - moment

±j °( w

X t k

,t)dk

= "( ,t). n

(2.29)

x

B S i m i l a r l y t h e flux a n d t h e energy density can be c o m p u t e d f r o m m o m e n t s o f ui, see [30], W e focus n o w o n properties o f Wignerseries. For the f o l l o w i n g analysis we define t h e space o f test-functions o n R% x B :

U<=L ( i n analogy t o t h e analysis o f t h e whole space case i n [24] ) . i t is an easy exercise t o show t h a t B is a separable Banachalgebra. We equip 6 w i t h the n o r m IMI : = l « l £ l M . . / 0 l l l - < « S )

a n d we easily o b t a i n L e m m a 2 . 2 Let ( A l ) - ( A 3 ) hold.

IK(0llo-

<

Then

£ A ?

<

D

Vt>0.

1-1

A s o m e w h a t better b o u n d , i.e. an L

2

-estimate, o n the Wignerseries can be

o b t a i n e d b y also assuming (A4)

X

2

^ H%i( i)

^

D

i

" h e r e D is independent o f a.

T h e n as i n [29] for t h e whole space case we have

150

L e m m a 2.3 Let

( A l ) - ( A 4 ) hold.

Then

IKWIIl^KB)

<

K

(2.30)

I t is well k n o w n t h a t the 'full space' Wigner t r a n s f o r m o f an a r b i t r a r y positive definite density m a t r i x is not non-negative everywhere. However, an a p p r o p r i a t e averaging of the W i g n e r f u n c t i o n over sufficently large phase space regions smoothes out those oscillations which are due t o the uncertainty p r i n c i p l e and gives a nonnegative function (see [64], [63], [24], [29]). We have the series-analogue of this so called H u s i m i - t r a n s f o r m a t i o n .

L e m m a 2.4

Set +

F"(|);=£e^*l** '**,

(2TCT)S

and define the Bvsimi

• function

(2.31)

2a

by Q

w% : = ( u ) * Then l e g = wf](x,k,t}

k€B

is non-negative

t

F")

*, G"

(2.33)

almost everywhere

on R

3

x B x (0,oo).

T h e most i m p o r t a n t consequence of L e m m a 2.4 lies i n the fact t h a t weak l i m i t s of Wignerseries are non-negative : L e m m a 2.5

Let fj,f

be accumulation

points of wj

weak r - weak and, resp., L ° ° ( ( 0 , oo); B')— weak on R? x B,

fi>0 m the sense of

/ > 0

»

and, resp., w" in the B~ — topologies.

3

on R

Then

x B x (O.oo),

(2.34)

measures.

We now derive an evolution equation for w" :

L e m m a 2.6

at

The Wignerserie

w" solves the initial value

rri

problem:

a 3

for x € R , w°{x,k.t

= 0) - « £ ( » , - * ) . ,

3

x € R,

k e B

k e B, t > 0 (2.36)

151

T h e p r o o f is a s i m p l e c o m p u t a t i o n using the definitions ( 2 . 2 7 ) , ( 2 . 2 8 ) , the Schrodinger e q u a t i o n ( 2 . 2 2 ) , (2.23) and the representation ( 2 . 2 0 ) , (2.21) o f the H a m i l t o n operator a c t i n g on SS. N o w let us f o r m a l l y derive the semi-classical l i m i t a —• 0 o f the W i g n e r e q u a t i o n ( 2 . 3 5 ) . We o b t a i n for the l i m i t /

w h i c h gives the free-streaming semiclassical Liouville-equations because the Fourier coefficients o f V £ ( i ) are n

i£ (u)u n

In order t o r i g o u r o u s l y o b t a i n t h i s result, we have to use some r e g u l a r i t y result on the energy b a n d E . I t is a well k n o w n p r o p e r t y o f these energy bands E„ t h a t there exists a closed set F C B o f measure zero, such t h a t E is an a n a l y t i c f u n c t i o n i n B — F. We refer t o [67] for details. Using t h i s and L e m m a 2.2 we prove i n [30] : n

n

T h e o r e m 2 . 1 Let ( A l ) , ( A 2 ) , ( A 3 ) hold and let a € (0, oro] be a sequence with limit zero.

Tken there ezist subsequences

of { i u ° } , { u ' / } (denoted by tke same

suck that W

J

^ i

u , < * ^ / > 0 The limits f = ffx, k,t)

f, n

>o

B'

weak

*,

(2.37)

L™((0,cc);B*)weafe«,VmeJV

(2.38)

in D ' ( ^ x (B - F) x [ 0 , c o ) ) of ihe

are solutions ^f

in

+ V E(k)V,f

= t),

t

(2.40)

satisfy (for the considered

„"^ ---LJ f(.,k,.)dk n

IVP's (2.39)

/(( = 0 ) = A , Also, the densities

symbol)

subsequence):

c o

0

3

-

mL ((0,co);C (ffe ) )weafc .,

(2.41)

B

N o t e t h a t the s o l u t i o n o f ( 2 . 3 9 ) , (2.40) reads /(*,***)«//(»-V*E(*)i,ft);

* e K

3

,kSB-F,

t> 3

Hence i t is u n i q u e l y d e t e r m i n e d except on the negligeable set St

0. x F.

(2.42) However,

we r e m a r k t h a t t a k i n g o u t the zero- Lebesgue-measure-set F m a y have i m p o r t a n c e since the measures //(as,.) m a y be s u p p o r t e d i n F.

N o assertion is made on the

e v o l u t i o n o f t h i s p a r t of the i n i t i a l measure / / . A stronger result can be proven i f the a s s u m p t i o n ( A 4 ) on the o c c u p a t i o n p r o b a b i l i t i e s is added. T h e n , thanks to L e m m a 2.3, the l i m i t i n g Wigner-measure is an !

i - f u n c t i o n a n d , thus, absolutely continous w i t h respect t o the Lebesgue measure. T a k i n g o u t the set F then is o f no i m p o r t a n c e a n y m o r e . We o b t a i n

152

T h e o r e m 2 . 2 Let the assumptions (Al) (A4) sequence with limit 0. Then there exist subsequences same symbol) such that f, w°

2^! /

in

hold and let a £ (0, do] he a of {w"}, (/) (denoted by the

a

in

Z , ( J ? x B) weakly

m

2

L ((0,ca);L (F?

(2.43)

x B)) weak . 3

(2.44) 3

The limits fi and f are nonnegative a.e on IR . x B and, resp. , R x B x ( 0 , 0 0 ) . The function f is the unique solutions in D'iR . x B x [0, oo)) of the IVP (2.39) ( 2 . 4 0 ) . Also, the concentration satisfies 3

n"

—

n : = - L / f(k)dk

M

!

in L ((0,co);L (fl^))

weak *

(2.45]

B T h e previous analysis shows t h a t there is a b i g difference between the v a c u u m case as i t studied i n [24, 29], and the crystal case. Hence the d e r i v a t i o n o f the L i o u v i l l e equation is s t r a i g h t f o r w a r d for electrons m o v i n g i n v a c u u m and i t is possible t o take i n t o account C o u l o m b interactions. O n the contrary the Vlasov e q u a t i o n w h i c h i n adimensionnal variables reads ^f (t, ,k) n

1

+ u (k)-V:f (t,x,k) n

+ V U(t,x)V f„(t,z,k)

n

I

k

=Q

has no rigorous j u s t i f i c a t i o n for electrons m o v i n g i n a crystal even i f the exterior p o t e n t i a l U is assumed t o be k n o w n and s m o o t h . T h e H a m i l t o n i a n corresponding to this equation is

ff° .=

~ A V(-) + U(z) I a and the m a i n difficulty is t h a t the Floquet-eigenspaces V° are no more i n v a r i a n t . Let us m e n t i o n t h a t a d e r i v a t i o n of the semiclassical Vlasov e q u a t i o n has been o b t a i n e d in [11], b u t the s t a r t i n g p o i n t was a phenome no logical q u a n t u m m o d e l which has s t i l l t o be justified, see [2, 10]. I

+

Other difficulties has t o be overcome t o rigorously derive models w i t h collisions terms, as i n ( 1 . 7 ) . A n a t e m p t i n t h i s d i r e c t i o n is due to F . Nier. I n [39], the t r a n s p o r t o f electrons i n vaccuum is studied i n presence o f a p o t e n t i a l barrier created for instance by an i m p u r i t y . Even i f the i n i t i a l energy of electrons are less t h a n the p o t e n t i a l barrier, some of t h e m are t r a n s m i t t e d due to t u n e l l i n g effects. I n t h i s s i t u a t i o n , i t is interesting to analyze the semiclassical l i m i t . F . Nier proves i n [39] t h a t the d i s t r i b u t i o n o f electrons solves a L i o u v i l l e equation w i t h a collision t e r m localized at the p o t e n t i a l barrier i n the l i m i t fi —• 0. T h i s is the first m a t h e m a t i c a l j u s t i f i c a t i o n of w h a t is named the golden rule i n solid states physics books. However it remains a lot o f difficulties i n order to rigorously j u s t i f y (1-7). T h e first step w o u l d be t o consider a density o f i m p u r i t i e s instead of one i m p u r i t y i n order to delocalized the collision t e r m . T h e second one w o u l d t o be t o transpose t h i s v a c u u m analysis to the crystal case.

3. B a s i c p r o p e r t i e s o f B o l t z m a n n e q u a t i o n s o f

semiconductors

153

For the sake o f s i m p l i c i t y , i n the following we w i l l consider only t r a n s p o r t equations for electrons We p o i n t o u t t h a t adding equations for holes and generationr e c o m b i n a t i o n t e r m s d o n o t provide any s u p p l e m e n t a r y difficulties i n the m a t h e m a t i c a l analysis. A l l the results presented i n t h i s Section are available for the whole system ( 1 . 7 ) , ( 1 . 8 ) , ( 1 . 9 ) ( w i t h slight changes). We refer t o [43, 48] for details. We first focus on properties o f collision operators defined by (1.11). As i n gas d y n a m i c s , c f [7], the d e t e r m i n a t i o n o f e q u i l i b r i u m d i s t r i b u t i o n functions, i .e. o f the nullspace o f the collision operator, is closely related to entropies inequalities. We have P r o p e r t y 3 . 1 [43} For any non decreasing function bution function f s.t.

\ € C ( S t ) and for any

distri-

we get -

f

<3
dk=

JB

{ o(i-

f)(\-f')M M' (h-h') n

(x{h)-x(ti))dk'>b

n

JB

with 1 -

f)M ' n

T h e p r o o f o f t h i s P r o p e r t y is j u s t a c o m p u t a t i o n using the non negativeness and the s y m e t r y of the cross section tr. T h a n k s to this P r o p e r t y we o b t a i n P r o p o s i t i o n 3 . 1 [0]

Assume

that the measure a satisfy <7>0

then the nullspace

of Q N{Q)={0
is determined

(3.1)

feC°(B),

< ? ( / ) = 0}

by

£

^ <

1

+

e x p (

'^)" '-°°'°°''-

T h e d i s t r i b u t i o n functions of the n u l l space of Q are so-called F e r m i - D i r a c d i s t r i b u t i o n s . T h e proof follows from P r o p e r t y 3 . 1 . Indeed i f a > 0 the e n t r o p y i n e q u a l i t y i m p l i e s t h a t for any f u n c t i o n / i n the nullspace the f u n c t i o n h is constant. Therefore / is a F e r m i - D i r a c d i s t r i b u t i o n . T h e above p r o p o s i t i o n leads to several remarks. F i r s t we p o i n t o u t t h a t the r e q u i r e m e n t / £ C°(B) is n o t necessary i f we assume for instance t h a t the t o t a l cross section A is bounded : \fk)

=

f o-(k,k')M (k')dk\ n

A£l°°(fl).

(3.2)

JB l

Indeed i n t h i s case, the collision operator is lipschitz i n L (B) t o the set o f measurable functions w i t h values i n [0, 1].

and can be p r o l o n g e d

154

Second the assumption (3.1) is very s t r o n g . I t w o u l d be enough to assume t h a t any couple (it, j t ' ) can be j o i n e d by a p a t h where
k s.t. a(k, * i ) , ff(ti, k ),o-(k ,

t

p

2

k') > 0. (3.3)

n

T h i r d , the conclusion o f P r o p o s i t i o n 3.1 is sometimes false. For instance the scattering rate cr w h i c h is used for m o d e l l i n g electron-phonon interactions is the following pft

fe


fl*.

~E

n

+

M

™p(f^)

+

6(E'„-E -hu) p(-pi-)), n

(3.4)

eX

where Q is a s m o o t h s y m e t r i c positive function and hw a constant (the energy of phonons). I n this case M a j o r a n a , [26, 27], has proved t h a t other functions belong to the nullspace of Q. P r o p o s i t i o n 3.2 [27] Assume space of Q is determined by N(Q)={

Ihe cross section c is given by (3.4)

! l+p(£ (t))exp

+

. , [ffi)

n

peC(K ;[0,coj,

then the null-

pishw-peuodic}-

T h i s strange phenomena does n o t appear i n physics books and is s t r o n g l y related to the nature o f the interactions. Indeed the collision processes which correspond to (3.4) do n o t m i x e d the energy groups : ( H S ,

E (k) n

= £„ modutohu},

£„e[0,fiw).

(3.5)

T h e physical relevance of the previous result is questionable. Indeed t h i s result is not

stable by p e r t u r b a t i o n s of collision mechanisms.

interactions occur w i t h different energies Kui, n u

2

For instance i f two phonon

corresponding to the collision

operators Q\, Q2, i t is p r e t t y easy using P r o p e r t y 3.1 t o prove t h a t the nullspace oiQ-Qi+Q-2 N{Q)

is

= N(Q )nNlQ ) l

2

= {

— T J T ^ , l+p(£ (il)exp^

+

P € C(ffi ;[0,oo]),

n

p is hull and hiij^-periodic}. Therefore the nullspace is only spanned by F e r m i - D i r a c d i s t r i b u t i o n s i f f w i / t i l j is i r r a t i o n a l ! T h e d e t e r m i n a t i o n of e q u i l i b r i u m d i s t r i b u t i o n functions is also i m p o r t a n t concerning the diffusion a p p r o x i m a t i o n of the B o l t z m a n n equation. T h e results w i l l differ considerably depending on the conclusion of P r o p o s i t i o n 3.1 or of P r o p o s i t i o n 3.2. T h i s w i l l be discussed in the next Section. We focus now on the question of existence o f solutions. T h e first step is to study the Cauchy p r o b l e m on the whole space (51 — K ) . I t reads 3

j f {t,x,k) ( n

+

• V*f*®,

+ PlJ&»,

ffj

•

ft-*)

=

155

Q«(M(t,*,k),

O O , i e f l , keB,

(3.6)

w i t h n ( t , z ) a n d <2„ s t i l l g i v e n by ( 1 . 1 0 ) , ( 1 . 1 1 ) . These equations are completed by the i n i t i a l c o n d i t i o n A(t

= 0,x,fc) = /

o

( M )

(3.7)

We have T h e o r e m 3 . 1 # 5 / -4ssnme lo

On € W °(B), If the initial

distribution

function

l

h € W '°°(K lAen the problem (3.6),

3

(3.7)

( f l x B).

satisfies U

x B ) f l W \R?

xB),

0 < /o < 1 a.e.,

has a unique solution f in the space 1

I™( (0,T) ;W >™(K which

, 0 Q

n, € W

3

M

x 6)rW (.ffc

3

x B) )

sattsfies

0 < / < 1 a.e.,

3

/ e w ' ™n

T ) x 2R x B ) ) , /J e H

/ 2 , o o

((0,r) x

3

R ). 3

T h e p r o o f is based on c o n s t r u c t i n g a c o n t r a c t i v e m a p P i n C ° ( (0, T ) ; L ' ( J i x B ) ) (or e q u i v a l e n t l y considering a Cauchy sequence i n t h i s space). T h i s m a p is defined as follows 9 = P(f).

9 solves

i s + *.(*) - Vrf + f « ( i , *) • v 01

i 9

+ A(/) = /i(/) t> o, x €fi,fee B , ff

fi FF

(( = 0 , M ) - / o ( M )

e € fi, J t € B , ( 3 . 8 )

with

A(/)=

/
+ rM )dk',

n

n

p(f)=

JB

f


n

JB

T h e n i t is easy t o prove t h a t the m a p P let the set {/eI~((0,T)x K

3

x B), 0 < / <

l,||/||L-((o,T);i'(*'xfl))


i n v a r i a n t (for a convenient choice o f the constant K). N e x t d i f f e r e n t i a t i n g (3.8) w . r . t . x and k and using classical l o g a r i t h m estimates on the second derivatives o f U (cf [65]), we prove t h a t a set •(ll/llL-l(0.T-);lV'.'OlVi.~(iR3 S)] < K

K]

156

is also i n v a r i a n t . I t follows f r o m these estimates t h a t the m a p P is locally i n t i m e contractive. L e t us p o i n t o u t t h a t for this p r o b l e m we do not have the classical difficulty of b o u n d i n g m o m e n t s o f / w . r . t . it as i t appears i n the m a t h e m a t i c a l analysis o f Vlasov-Poisson problems. Indeed the fc-space is bounded c o n t r a r y to the classical velocity space. T h i s explains w h y we do not have t o use the t r i c k y technics o f [25, 41]. We now discuss some extensions of t h i s result. T h e first one is to get r i d o f the a s s u m p t i o n o f r e g u l a r i t y o f a„ w h i c h is obviously not satisfied for the electronphonon cross section (3.4). I t seems not to be a great difficulty. Indeed i n [43] the regularity o f c is only needed t o prove properties of the t y p e n

< c ( i + ||v»/iit-). Such an estimate is also true for u = o~ph (3-4), as i t can be verified by tedious computations. T h e second question is w h a t happens when the i n i t i a l d i s t r i b u t i o n f u n c t i o n is only supposed t o satisfy the n a t u r a l bounds n

3

0 < /o < 1,

f €L\R xB). Q

T h e n considering a regularized sequence w h i c h converges to the i n i t i a l d a t a and using results of T h e o r e m 3 . 1 , u n d o u b t e d l y provide weak solutions for the Cauchy p r o b l e m . T h e way t o control nonlinear terms is given by averaging lemmas, [13, 20]. B u t i n t h i s case the question o f uniqueness is an open p r o b l e m (as for the VlasovPoisson system). We p o i n t out t h a t averaging lemmas can be used o n l y i f the cross section r r is s m o o t h (at least belongs t o some If space). B u t for the cross section given by (3.4), the collision operator do n o t any more average w . r . t . energies. I n t h i s case there is no p r o o f of existence o f weak solutions or o f the weak s t a b i l i t y of sequences of classical solutions. n

T h e s i t u a t i o n is the same considering t i m e dependent b o u n d a r y value p r o b l e m First let us r e m a r k t h a t i n realistic devices the b o u n d a r y conditions for the p o t e n t i a l are of m i x e d t y p e : D i r c h l e t conditions on O h m i c contacts, N e u m a n n conditions on i n s u l a t i n g boundaries. I t leads t o a loss o f r e g u l a r i t y for the p o t e n t i a l . T h e n the characteristics of the equation (3.6) m a y n o t to be uniquely defined (even i n the weak sense of [15]). Therefore uniqueness results seem not to be possible t o o b t a i n . O n the c o n t r a r y weak solutions can be constructed [42] as for Vlasov-Poisson o r V l a s o v - M a x w e l l systems [ 1 , 4, 22], I t is also possible t o prove existence o f s t a t i o n a r y weak solutions for b o u n d a r y value problems. T h e uniqueness is doubtefull even for strong solutions and some counterexamples are given i n [47]. Technics to prove existence has first been developped for the V l a s o v - M a x w e l l system, cf [47], I t consist i n using Schauder's fixed p o i n t T h e o r e m for a m a p on electrostatic potentials. T h e p r o b l e m reads " . ( * ) • 7x/» + { V I / { ( , I ) I

- A(/(aO = - ( C ( r ) - »(*)), r e

V*/„ =Q (/„)(z,it), n

x 6 fi.

«(*) = A 4 IT

i 6 S ! , H B ,

/ JB

M- ) x

fc

*•

(3.9)

( ) 310

157

c o m p l e t e d w i t h b o u n d a r y c o n d i t i o n s as for instance /„(*,

*) = {x,fc),x £ dQ,

€ B , s.t. u „ ( * ) • i/(ar) < 0

(3.11)

where v is the o u t w a r d n o r m a l to i i , u(x)

- uo(x),

z e an.

(3.12) 3

T h e n we define a m a p 7" as follows. A p o t e n t i a l U - U(x),U for a fixed a > 0 we first solve the nonlinear e q u a t i o n

£ C ( Q ) being given,

<*/„ + v„(k) .v»/„ + | v f / ( , i ) v / =g„(/ )(*,t),

l e a s e e , (3.13)

r

1

i

n

B

c o m p l e t e d w i t h b o u n d a r y c o n d i t i o n s (3.11). For the p r o b l e m (3.11), (3.13), existence a n d uniqueness o f solutions follow f r o m a m o n o t o n y p r o p e r t y o f collision operators. We define the sign f u n c t i o n sg by = - 1 for ( < 0,

sg(t)

= 1 for ( > 0,

sg(t)

so{0) = 0.

W e have P r o p o s i t i o n 3.3 [46], Let f and g be continuous

f (Q (f) -<3n(fl)) n

JB

function

from B into [0,1],

tken

*9U-9)dk<0.

O f course such a p r o p e r t y can be used for measurable f u n c t i o n using a regularized f u n c t i o n sg. We refer t o [40] for details. T h e n T(U) is defined by solving (3.10), (3.12) and by r e g u l a r i z i n g the result i n order t o o b t a i n a C f u n c t i o n . T h e r e g u l a r i z i n g m a p is p a r a m e t r i z e d by a and tends to i d e n t i t y when a goes t o zero. 2

T h e key e s t i m a t e is o b t a i n e d by considering upper solutions o f the B o l t z m a n n e q u a t i o n (3.13). Indeed we have

for a convenient constant u depending o n l y on the d a t a rf\tio- T h i s estimate s t i l l depend on the u n k n o w n U. B u t i f we define Un as the s o l u t i o n o f (3.10), (3.12) w i t h n = 0, t h e n the m a x i m u m p r i n c i p l e i m p l i e s t h a t T(U) < UQ. Therefore the convex set S - { U € / f ' ( f i ) , U < Co} is i n v a r i a n t for T. B u t i f U lies i n S, we o b t a i n e d a u n i f o r m e s t i m a t e by replacing U w i t h U i n ( 3 . 1 4 ) . I t follows t h a t n is u n i f o r m l y b o u n d e d w h i c h is enough t o o b t a i n e d a compactness p r o p e r t y for T i n the space / f ' ( Q ) . T h e c o n t i n u i t y o f T follows f r o m a weak s t a b i l i t y o f the s o l u t i o n o f ( 3 . 9 ) . T h e c o n t r o l o f the nonlinear terms o f is s t i l l o b t a i n e d by averaging lemmas o f [13]. I t r e m a i n s t o pass t o the l i m i t a —> 0 t o o b t a i n 0

0

2

T h e o r e m 3 . 2 [40} Let a G L (B x B),let $ be bounded by a Fermi-Dirac distribution and let tlo £ H ^ (dCl), then there is at least one weak solution of the problem (3-9), (3.10), (3.11), (3.12). 1

2

158

We discuss now various a p p r o x i m a t i o n s w h i c h are often used at the kinetic level. One o f these is called the non degeneracy assumption. I t consists i n neglecting f compared t o 1 i n the expression of the collision operator. T h u s i t becomes linear and reads L(f)=

( JB

MJW»f - M „ / )

df

(3.15)

For such m o d e l we loose the o p r i o n estimate 0 < / < 1 b u t on the other hand, we do not have any more t o use averaging lemmas t o c o n t r o l nonlinear terms of the type / / ' • Hence, for instance T h e o r e m 3.2 becomes available for non s m o o t h cross section a, cf [49], We have only to assume (3.2) which i n practice is always satisfied. T h i s non degeneracy assumption can be m a t h e m a t i c a l y justified by i n t r o d u c i n g the scaling / = af ,a — 0, see [ I T ] , new

A n other c o m m o n l y used a p p r o x i m a t i o n is the parabolic band a p p r o x i m a t i o n . T h e n B is replaced by R and E„(k), v (k) by \hk\ /2m-„, hk/m^. m is the so called effective mass. Let us remark t h a t we recover the classical relation ship E = m' \v \ /2. T h i s a p p r o x i m a t i o n lead to m a t h e m a t i c a l difficulties, for instance a b o u n d for m o m e n t s o f / w . r . t . k is no more free. I n this context, using weighted Sobolev spaces as i n [65], Mustieles has o b t a i n e d existence and uniqueness of classical solutions for the Cauchy p r o b l e m i n 2 dimensions, [35], and existence o f weak solutions i n dimension 3, [36]. There is no d o u b t t h a t i t w o u l d be possible t o extend the result of the dimension 2 i n dimension 3 using the technics of [25, 41], b u t up t o now i t has n o t been done. For stationary problems, the control on m o m e n t s o f / are given by the upper solutions (3.14) and the result o f T h e o r e m 3.2 is s t i l l v a l i d , see [40, 47, 49], We p o i n t o u t t h a t up t o now the parabolic b a n d a p p r o x i m a t i o n has no m a t h e m a t i c a l j u s t i f i c a t i o n . A hint w o u l d be t o introduce the scaling k = ok ,a —> 0. 3

2

n

n

2

n

n

n

ntm

4. F r o m k i n e t i c t o f l u i d

models

One of the most used models of t r a n s p o r t processes i n semiconductors are based on so-called drift-diffusion equations. These equations govern the e v o l u t i o n o f the concentration o f particles. T h e y are of parabolic t y p e . I n this Section we give rigorous derivations of these models s t a r t i n g f r o m the B o l t z m a n n equation. A p a r t from its interest i n itself, t h i s derivation allow t o precise the d o m a i n of v a l i d i t y o f d r i f t diffusion models, t o give a way t o c o m p u t e fluid coefficients from kinetic datas and t o analyse b o u n d a r y layers. S t a r t i n g from equation (3.6), (3.10) we introduce the f o l l o w i n g scaling

'new

=

,

Xnevj — f~ >

Jo

nevJ

= ——.

A>o

V

«*. - 9t„,;E„^

k

with Vo

no -

E

=

(4.1)

E

V=

M

MO)

=

0

"/to

-O . n

r

(4.3)

159

7 is i n t e r p r e t e d as the r e l a x a t i o n t i m e . I t ' s a measurement of the mean t i m e of free flights of electrons between collisions. Vn is a characteristic t h e r m i c velocity and t h u s f — VrjT is the mean free p a t h . A diffusion a p p r o x i m a t i o n o f (3.6) is o b t a i n e d by assuming the scaled mean free p a t h vanishes or = l/L —• 0. Let us also m e n t i o n t h a t i n ( 4 . 2 ) we have assumed t h a t the p o t e n t i a l energy qU and the mean k i n e t i c energy E„ have the same order o f m a g n i t u d e as the t h e r m i c energy t f l S . T h i s correspond t o an a s s u m p t i o n o f l o w electric field. A t the end o f t h i s Section we w i l l i n t r o d u c e an o t h e r scaling w h i c h tries to take i n t o account h i g h fields effects. Now we choose the t i m e scale s.t. T — a r, a reference c o n c e n t r a t i o n a

2

a

C

and K

B

0

3

= (4TT C )''

2

=

?

2

t o finally o b t a i n

0

c

0

~ } °

+ «< M * ) - V - /

0

+ V (7

Q

r

U" = 'C(x)-n<-[t,x))*~

V

n°= X

\\

t

/ ") = «„(/»)

f JB

f°dk.

(4.4)

(4.5)

T h e p r o b l e m is now t o d e t e r m i n e the l i m i t o f ( / " , U") when a goes t o zero. F o r m a l l y we o b t a i n f* —* / ' , w i t h Q (f°) — 0- Therefore the d e t e r m i n a t i o n o f the nullspace o f Q„ is i m p o r t a n t . As we already m e n t i o n e d i n Section 3 the result differ i f we are i n the s i t u a t i o n o f P r o p o s i t i o n 3.1 or P r o p o s i t i o n 3.2. We first assume the cross section is s m o o t h and p o s i t i v e n

1,eo

rr„ £ W (B

x B),

( I n fact i t w o u l d be enough t o impose c r

r r ( M ' ) > 0.

(4.6)

n

2

m

G L ).

I n t h i s case the nullspace o f Q„ is

spanned by F e r m i - D i r a c d i s t r i b u t i o n s denoted by

T h e n we have T h e o r e m 4 . 1 [21] Let (f, 3.1 with the Cauchy data f(t f

a

U)

be the solution

of ( 4 . 4 ) , (4-5) as given by

= 0) —