CLASSICAL ANALOGUES OF THE RPA OPERATORS AND THE VLASOV EQUATION

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CLASSICAL ANALOGUES OF THE RPA OPERATORS AND THE VLASOV EQUATION

J. da Providência

To cite this version:

J. da Providência. CLASSICAL ANALOGUES OF THE RPA OPERATORS AND THE VLASOV EQUATION. Journal de Physique Colloques, 1987, 48 (C2), pp.C2-45-C2-49.

�10.1051/jphyscol:1987206�. �jpa-00226471�

JOURNAL DE PHYSIQUE

C o l l o q u e C 2 , s u p p l b m e n t au n o 6, T o m e 48, juin 1987

CLASSICAL ANALOGUES OF THE RPA OPERATORS AND THE VLASOV EQUATION

Departamento de ~ f s i c a , Universidade de Coimbra, P-3000 Coimbra, Portugal

Abstract : The linearized Vlasov equation for finite nuclei is discussed.

The nuclear Vlasov equation describes the time evolution OF the classical distributio~~ function f(x,p,t) for a system of fermions interacting through short range forces. The generator h(x,p,t) of the time displacement is itself a functional of the distribution function. Assuming two-body and three body-forces, we have

a t f + axf.

a

- a

P P

where

h(x.p,t)

-

-

The energy of the system is

~ [ f l =

1 &

' i

We observe that the saturation properties of nuclear matter require that v(x,x') is actrative and w(x,xl,x") is repulsive. Canonical transformations play an important role in Vlasov dynamics. The evolution in time is given by a canonical transformation. A state of equilibrium f (x,p) possesses an important property. The inequality

holds for all distribution functions f(x.p) which are related to f (x,p) by a cano- nical transformation. It follows that

where

ho(x.p) = h[fol

By A , B we denote the Poisson bracket of A and B.

The time evolution of systems close to equilibrium is described by the line_

arized Vlasov equation, which reads

a

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987206

C2-46 J O U R N A L DE PHYSIQUE

H e r e , 6 f a n 6 h a r e , r e s p e c t i v e l y , t h e f l u c t u a t i n g p a r t s o f f and h. I t i s c l e a r t h a t

6 h ( x , p , t ) =

I

r

B r i n k e t a 1 / 1 / h a v e s t u d i e d eq. ( 7 ) by a n e l e g a n t F o u r i e r t r a n s f o r m t e c h @ que s i m i l a r t o t h e L a p l a c e t r a n s f o r m method c o n s i d e r e d by Landau 1 2 1 f o r p l a s m a o s c i l a t i o n s . The s o l u t i o n o b t a i n e d by t h e s e a u t h o r s i n v o l v e s a t y p i c a l i n t e g r a t i o n o v e r a s i n g u l a r i t y of t h e form u s u a l l y a s s o c i a t e d w i t h d i s s i p a t i v e t i m e - d e p e n d e n t p r o c e s - s e s . T h i s i s t h e s o u r c e o f t h e a t e n u a t i o n o f s m a l l d i s t u r b a n c e s known as Landau dam- p i n g .

An a l t e r n a t i v e s o l u t i o n o f t h e p l a s m a o s c i l a t i o n s p r o b l e m a s a n o r m a l mode e x p a n s i o n h a s b e e n o b t a i n e d by Van Kampan 131. A n a l o g o u s l y , a c o m p l e t e s e t o f n o r m a l mode g e n e r a t o r s ( c l a s s i c a l i m a g e s o f RPA o p e r a t o r s ) , s u i t a b l e t o e x p r e s s t h e g e n e r a l s o l u t i o n o f t h e n u c l e a r V l a s o v e q u a t i o t i f o r i n f i n i t e s y s t e m s h a s r e c e n t l y b e e n o b t a g ned by P i z a e t a l . 141.

The e q u i v a l e n c e o f t h e Landau and Van Kampen t r e a t m e n t s of t h e i n i t i a l v a l u e p r o b l e m f o r p l a s m a o s c i l a t i o n s h a s b e e n d e m o n s t r a t e d by K . M. Case 151. S i n c e t h e e q u i v a l e n c e o f b o t h t r e a t m e n t s r e m a i n s v a l i d i n t h e n u c l e a r c a s e (D. M . B r i c k and J . d a P r o v i d e n c i a , u n p u b l i s h e d ) , i t f o l l o w s t h a t Landau damping i s j u s t a n i n t e r f e r e n c e e f f e c t b e t w e e n n o r m a l mode a m p l i t u d e s and i s , t h e r e f o r e , s t r o n g l y d e p e n d e n t on tlle i n i t i a l c o n d i t i o n s .

I t i s c o n v e n i e n t t o i n t r o d u c e t h e g e n e r a t o r S of t h e d i s t r i b u t i o n f u n c i i o n f l u c t u a t i o n s by

An e q u a t i o n f o r S i s e a s i l y o b t a i n e d . By t h e J a c o b i i d e n t i t y a n d e q . ( 5 ) i t f o l l o w s t h a t { { ~ , 1 f, h 1 = { { S , h 1 , f )

.

f o r e be w r i t t e n ' ^{O 0 0}

T h u s , i n o r d e r t o s o l v e e q . ( 7 ) , i t i s enough t o r e q u i r e

a s

+

t ( 1 2 )

F o l l o w i n g r e f . 1 4 1 we l o o k f o r t h e s t a t i o n a r y s o l u t i o n s o f e q . ( 1 2 ) : i w S + I S , f j - 6 h = 0 .

n n n o ( 1 3 )

H e r e , 6 hn i s o b t a i n e d f r o m 6 h by w r i t i n g 6 f n = S n , f o l f o r 6 f i n e q . ( 8 ) . The e i g e n v a l u e s w i n e q . ( 1 3 ) a r e r e a l . To s e e t h i s , m u l t i p l y by f t h e P o i s s o n b r a c k e t o f e q . ( 1 3 ) a n d S* a n d i n t e g r a t e o v e r t h e whole p h a s e s p a c e .

We f i n d

I t f o l l o w s from ( 4 ) t h a t t h e c h a n g e i n e n e r g y

is positive (more preciseiy, non negative) for any real function S(x,p). Taking, in sequence, S = Sn + ^{:S and S = i(Sn} - ^),s: we prove that the right hand side of eq.

(14) is positive (non negative). Since the normalization integral

is real, it follows that wn cannot be complex. It may be seen that if Sn and wnare some eigensolution and the corresponding eigenvalue, also S* and - w are an eigen- solution and the corresponding eigenvalue.

Finally, it may be seen that "distinct" eigensolutions are orthogonal and may be normalized:

Although it would be more appropriate to label the eigenmodes by continuous parameters, we find it more suggestive to use discrete indices as labels. However, the reader may, if he wishes, understand the Knonecker 6,, as a Dirac &function

6 (n-m) and summations

5

The eigensolutions are complete. Any generator D(x,p)(restricted, at zero temperature, to the Fermi surface ho(x,p) =

5 )

where, if w > 0,

We are now in a position to determine the generator D(x.p,t) which at time t=O sa- tisfies D(x,p,O) = D(x,p), that is, to solve the Vlasov initial value problem. We have

Physically, the coefficients d, in this expansion represent transition amplitudes from the quanta1 ground state 10> to the excited state I n > , induced by the tran- sition operator D,

dn = < n

I D ]

The square of these amplitudes satisfy the energy weighted sum rule

Classicaly, d: d_ measures the energy fraction located at the nth eigenmode

.a L C

The exact solution of the Vlasov equation for finite nuclei appears to be a difficult problem. In principle, the generator of the distribution function fluctua- tions may be expressed as a linear combinationof infinitely many degenerate solutions appropriate to the corresponding (translationaly invariant) extended system. However, the weight function is determined by an awkward boundary condition at the n u c l e a r ~ u ~ ~ face (D. M. Brink, private communication). It seems, therefore, convenient to resort to a suicable approximate scheme. Variational methods appear to be particularly well adapted to this purpose. We will briefly discuss several types of trial generators which have been considered by various authors. It is convenient to split the genera- tor S into the time-even and the time-odd parts,

C2-48

where

JOURNAL DE PHYSIQUE

V a r i a t i o n a l c h o i c e I ( s e e r e f . 1 6 1 ) :

V a r i a t i o n a l c h o i c e I1 ( s e e r e f s . 1 7 . 8 1 ) : Q = @ ( x s t ) + 7 1

&

V a r i a t i o n a l c h o i c e 111 ( s e e r e f . 1 9 1 ) : Q s u c h t h a t

f U ( x , p . t ) = f + 1 Q , f o l + 7 1 {Q, ( Q.£,} } +

. . .

I n t a b l e I we p r e s e n t t h e v a l u e s of t h e e n e r g i e s and t h e p e r c e n t a g e s of the m, sum f o r e l e c t r i c eigenmodes of a n u c l e u s w i t h A=208 a s p r e d i c t e d by v a r i a t i o n a l c a l c u l a t i o n s b a s e d on t h e t r i a l g e n e r a t o r s l i s t e d .

The v a r i a t i o n a l c h o i c e I11 l e a d s t o t h e e l e g a n t f l u i d d y n a m i c a l scheme o f Holzwarth and E c k a r t / l o / . T h i s i s t h e s i m p l e s t c h o i c e which y i e l d s r e a l i s t i c r e s u l t s f o r t h e g i a n t r e s o n a n c e s . The f i e l d s s and u a r e c a n o n i c a l l y c o n j u g a t e p a i r s .

Although t h e v a r i a t i o n a l c h o i c e I p r o v i d e s a p p a r e n t l y a r i c h e r p a r a m e t r i z g t i o n of t h e d i s t r i b u t i o n f u n c t i o n , i t d o e s n o t seem t o c o n s t i t u t e a n improvement over c h o i c e 111. The r e a s o n f o r t h i s may be r e l a t e d t o t h e f a c t t h a t t h e f i e l d s @ , b a n d sk c a n n o t be grouped i n t o c a n o n i c a l l y c o n j u g a t e p a i r s . T h i s i m p l i e s t h a t t h e i n f o r - m a t i o n c o n t a i n e d i n $k(l i s t o some e x t e n t r e d u d a n t , a s f a r a s i t d o e s n o t a f f e c t r m c h t h e dynamics.

The v a r i a t i o n a l c h o i c e I1 g i v e s a good d e s c r i p t i o n n o t o n l y of t h e g i a n t r e s o n a n c e s , b u t a l s o of t h e low l y i n g modes. T h i s f e a t u r e i s c o n n e c t e d w i t h a p r o p e r t r e a t m e n t o f t h e s u r f a c e dynamics. C l a s s i c a l l y , c o n t a c t f o r c e s a r e u n a b l e t o produce s u r f a c e t e n s i o n , which may t h e r e f o r e be r e g a r d e d a q u a n t a 1 e f f e c t . A c c o r d i n g l y , s u r f a c e modes come o u t a t z e r o e n e r g y . I f a r e a s o n a b l e v a l u e of t h e s u r f a c e t e n s i o n i s added by hand t o t h e model 1111, s a t i s f y i n g v a l u e s f o r t h e e n e r g i e s of t h e s u r f a c e modes a r e o b t a i n e d ( s e e l a s t column of t a b l e I ) .

The f i e l d s Q and a r e c a n o n i c a l l y c o n j u g a t e , r e s p e c t i v e l y , K O t h e f i e l - d s W and X k p .

..-

V a r i a t i o n a l s o l u t i o n s of t h e l i n e a r i z e d Vlasov e q u a t i o n f o r h o t n u c l e i have a l s o been o b t a i n e d by J. P. d a P r o v i d h n c i a / 1 2 / .

T a b l e I -

-- - - -- - -

C h o i c e I1 + surf2

C h o i c e I C h o i c e I1 C h o i c e 111 ce tension u = 1.0 17

MeV fm-2

k k n m l e d g e m e n t s : The a u t h o r i s g r a t e f u l t o C. F i o l h a i s f o r b r i n g i n g t o h i s a t t e n t i o n t h e p a p e r o f K . M. C a s e . He a l s o a c k n o w l e d g e s f i n a n t i a l s u p p o r t f r o m I n s - t i t u t o N a c i o n a l d e I n v e s t i g a ~ a o C i e n t i f i c a and ~ u n d a ~ g o C a l o u s t e G u l b e n k i a n .

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/ 2 / L . D. Landau, S o v i e t P h y s i c s JETP 2 (1957) 920; 5 (1957) 101.

/ 3 / N . G . Van Kampen, P h y s i c a (1955) 149.

141 M. C. Nemes, A.F.R.T. P i z a and J . d a P r o v i d e n c i a , Van Kampen Waves i n F e r m i S y s tems a n d t h e Random P h a s e A p p r o x i m a t i o n , U n i v e r s i d a d e d e ^{S ~ O} P a u l o , p r e p r i n t ( 1 9 8 7 ) .

/ 5 / R. M. C a s e , Ann. P h u s . (N.Y.) 7 (1959) 349.

161 J. P. d a P r o v i d e n c i a and G. ~ o i z w a r t h , Nucl. P h y s . (1983) 59.

/ 7 / J . d a P r o v i d e n c i a , L. B r i t o a n d C. d a P r o v i d e n c i a , I1 Nuovo C i m e n t o , 87A (1985) 248.

1 8 1 L. P. B r i t o a n d C. d a P r o v i d e n c i a , Phys. Rev. C32 (1985) 32.

/ 9 / J . P . d a P r o v i d e n c i a , V a r i a t i o n a l D e r i v a t i o n o f N u c l e a r F l u i d Dynamics, J. Phys.

G . , t o b e p u b l i s h e d .

1101 G. H o l z w a r t h and G. E c k a r t . N u c l . P h y s . A325 (1979) 1.

/ I l l L. P. B r i t o , Phys. L e t t . B

177

1'121 J. P. d a P r o v i d e n c i a , N u c l e a r F l u i d Dynamics o f Hot N u c l e i , J . Phys. G . , t o be p u b l i s h e d . The S c a l i n g Approach f o r F i n i t e T e m p e r a t u r e s , J . P h y s . G . , t o b e p g b l i s h e d .