HAL Id: jpa-00224246
https://hal.archives-ouvertes.fr/jpa-00224246
Submitted on 1 Jan 1984
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of
sci-entific research documents, whether they are
pub-lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
THE APPROACH OF CHAOS IN DEFORMED
NUCLEI
J. Carbonell, F. Brut, R. Arvieu, J. Touchard
To cite this version:
JOURNAL
DE
PHYSIQUEColloque C6, supplCment au n06, Tome 45, juin 1984 page C6-371
THE APPROACH O F CHAOS IN D E F O R M E D NUCLEI
J . C a r b o n e l l , F. B r u t , R. Arvieu and J . ~ o u c h a r d *
I n s t i t u t des Sciences NucZe'aires,
5 3 ,Avenue des Martyrs,
38026GrenobZe Cedex, France
* I n s t i t u t de Physique NucZe'aire, Division de Physique The'orique,
B.P.n o
1, 91406Orsay, France
RQsume - On d e c r i t l ' o r g a n i s a t i o n d e l ' e s p a c e d e p h a s e pour un n e u t r o n dans un p o t e n t i e l d6form4 (A=16). La r b g i o n oh l ' o n t r o u v e l e s t r a j e c - t o i r e s c h a o t i q u e s e s t p r 6 c i s Q e . On c o n s t r u i t numdriquement l a s u r f a c e e n e r g i e - a c t i o n pour l a t o p o l o g i e l a p l u s s i m p l e . On c a l c u l e e n f i n des d n e r g i e s s e m i - c l a s s i q u e s .
A b s t r a c t - The p h a s e s p a c e of t h e t r a j e c t o r i e s of one n e u t r o n i n a de- formed s i n g l e p a r t i c l e p o t e n t i a l i s s k e t c h e d f o r A = 16. The r e g i o n of o c c u r r e n c e of t h e c h a o t i c t r a j e c t o r i e s i s found. The e n e r g y a c t i o n - s u r f a c e i s p l o t t e d f o r t h e s i m p l e s t t o p o l o g y . S e m i - c l a s s i c a l e n e r g i e s a r e d e r i v e d .
I - INTRODUCTION
The s e m i c l a s s i c a l method of q u a n t i z a t i o n W.K.B. o r E.B.K. i s w e l l a d a p t e d t o t h e s y s t e m s which a r e i n t e g r a b l e i n c l a s s i c a l m e c h a n i c s , i . e . which p o s s e s s a s many i n - t e g r a l s of m o t i o n i n i n v o l u t i o n a s i n d e p e n d e n t d e g r e e s of freedom. For t h o s e s y s - tems t h e t r a j e c t o r i e s a r e e s s e n t i a l l y d i v i d e d i n t o two c l a s s e s : t h e p e r i o d i c o r b i t s and t h e q u a s i - p e r i o d i c o n e s . It i s w e l l known t h a t i f t h e m a n i f o l d d e f i n e d by t h e i n t e r s e c t i o n of t h e c o n s t a n t s of motion i s compact and connex t h a t t h e t r a j e c t o r i e s l i e on t o r i / I / i n phase s p a c e . The c l a s s i c a l a c t i o n - a n g l e v a r i a b l e s a r e a d a p t e d t o t h i s s i t u a t i o n : t h e a c t i o n s c a n t h e n b e i n t e r p r e t e d a s t h e r a d i i of t h e t o r i . I n a n a n o t h e r c o m u n i c a t i o n t o t h i s workshop / 2 / we have shown t h a t t h e quantum mechani- c a l s p e c t r u m of t h e s i n g l e p a r t i c l e e n e r g i e s of one n e u t r o n i n a s p h e r i c a l p o t e n t i a l c a n b e u n d e r s t o o d by s t u d y i n g t h e g e o m e t r i c a l p r o p e r t i e s of t h e e n e r g y - - a c t i o n sur-- f a c e .
I n t h i s communication we want t o d i s c u s s t h e problems which a r i s e when a d e f o r m a t i s n of t h e p o t e n t i a l i s i n t r o d u c e d .We have a l r e a d y b r i e f l y s k e t c h e d / 3 / t h e c l a s s i c a l s i t u a t i o n on which t h e r e h a s a l s o been a t h e s i s by one of u s 141. Our aim w i l l b e f i r s t t o r e v i e w a few r e s u l t s of r e f . 4 , a n d , s e c o n d l y , t o g i v e a n example, t h e s i m p l e s t p o s s i b l e o n e , i n which t h e e n e r g y a c t i o n s u r f a c e h a s b e e n c o n s t r u c t e d and t h e s e m i c l a s s i c a l energsees o b t a i n e d . When t h e Buck-Pilt p o t e n t i a l a l s o u s e d i n r e f . 2 i s g i v e n an e l l i p s o i d a l d e f o r m a t i o n by making t h e t r a n s f o r m a t i o n K, n i 1 ( i n t h e f o l l o w i n g we w i l l u s e JJ = RI/R2; h = 3 0 a t h e t o t a l a n g u l a r mo- mentum i s n o t c o n s e r v e d and t h e r e 1 s no new c o n s t a n t of motion i n p l a c e of L. Howe- v e r L, i s c o n s e r v e d . I f we c o n s i d e r t h e p l a n e t r a j e c t o r i e s ( w i t h L Z = 0 ) of c o n s t a n t e n e r g i e s we need t o e x p l o r e n u m e r i c a l l y t h e p h a s e s p a c e ( 3 d i m e n s i o n a l ) i n o r d e r t o examine t h e dimension of t h e m a n i f o l d on which l i e t h e t r a j e c t o r i e s .
C6-372 J O U R N A L
DE
PHYSIQUEThe numerical and t h e t h e o r e t i c a l works performed d u r i n g t h e l a s t twenty y e a r s on t h e t h e o r y of dynamical systems have l e d t o a q u a l i t a t i v e d e s c r i p t i o n of t h e phase space of q u a s i i n t e g r a b l e systems. The work by Henon and H e i l e s /5/ i s o f t e n quoted a s one of t h e f i r s t numerical experiment on a simple b u t non i n t e g r a b l e system f o r which t h i s d e s c r i p t i o n was d i s c o v e r e d . Let us b r i e f l y mention t h e known r e s u l t s .
Let E be any parameter which c h a r a c t e r i z e s t h e non l i n e a r i t y l i k e t h e e x c i t a t i o n energy, o r t h e deformation o r t h e s i z e of t h e p o t e n t i a l . There e x i s t s a c r i t i c a l v a l u e E which allows t o s e p a r a t e t h e t o p o l o g i c a l flow i n t o two p a r t s
.
a ) f o r E
<
EC t h e system looks mostly l i k e an i n t e g r a b l e one : t h e r e e x i s t s p e r i o d i c o r b i t s and q u a s i - p e r i o d i c o r b i t s . The c h a o t i c o r b i t s e x i s t only a t a microscopic l e v e l . Because of t h e q u a s i - - i n t e g r a b i l i t y and t h e dominance of t h et o r i i n phase space t h e EBK s e m i c l a s s i c a l method of q u a n t i z a t i o n i s p o s s i b l e t h e r e 161. However t h e r e i s a c o n s i d e r a b l e complication of t h e phase space. Indeed every p e r i o d i c o r b i t b i f u r c a t e s and g i v e s r i s e , f o r any i n f i n i t e s i m a l change of E , t o new p e r i o d i c o r b i t s and t o new t o r i . As we w i l l s e e below t h e r e e x i s t s techniques which a l l o w t o l o c a t e t h e v a l u e of E f o r which a given b i f u r c a t i o n o c c u r s . The main problem, s t i l l unsolved t o our knowledge, l i e s i n t h e maximum s i z e of t h e t o r i which surround a given p e r i o d i c o r b i t and of t h e l o c a t i o n of t h e s e p a r a t i x between t h e d i f f e r e n t f a m i l i e s .
b) f o r E
>
E~ t h e volume of t h e t o r i d e c r e a s e s r a p i d l y w i t h E t o t h e b e n e f i t of t h e c h a o t i c o r b i t s which tend t o occupy t h e whole of phase space. I n t h i s r e g i o n of parameters t h e phase space i s even more complex t h a n i n t h e preceding one. So complex t h a t i t becomes h a r d l y p o s s i b l e t o f i n d t o r i which f u l z i l l t h e s e m i c l a s s i c a l c o n d i t i o n s .The c r i t i c a l v a l u e E, i s o f t e n quoted a s t h e t h r e s h o l d of macroscopic s t o c h a s t i c i - t y . For E
<
€ c i t i s r a t h e r d i f f i c u l t t o f i n d o u t c h a o t i c o r b i t s n u m e r i c a l l y .-
The e v o l u t i o n j u s t d e s c r i b e d has a so c a l l e d "generic" c h a r a c t e r , i . e . it i s q u a l i t a t i v e l y c o r r e c t f o r most of t h e q u a s i . i n t e g r a b l e systems whatever i s t h e p o t e n t i a l o r t h e parameter E .
I n t h e f o l l o w i n g we want t o s k e t c h f i r s t how t h i s behaviour i s seen f o r t h e phase space of L, = 0 t r a j e c t o r i e s i n A = 16, i . e . A = 4.898 and 1
<
JJ<
2 . Fort h e sake of space l i m i t a t i o n we w i l l not d i s c u s s o t h e r v a l u e s of A nor t h e o t h e r v a l u e s of L (which would correspond t o non p l a n a r t r a j e c t o r i e s ) s e e r e f . 4.
I1
-
THE PHASE SPACE OF DEFORMED j60a ) The f i r s t q u e s t i o n i s whether t h e deformed p o t e n t i a l i s i n t e g r a b l e o r n o t . I f t h e non i n t e g r a b i l i t y p r e v a i l s it i s n e c e s s a r y t o l o c a t e t h e t h r e s h o l d i n terms of t h e e x c i t a t i o n energy
n=
1-
o r of t h e deformation JJ.vo
Fig. 1 p r e s e n t s an example of a c h a o t i c t r a j e c t o r y . I n t h e bottom 01 t h e f i g u r e t h e t r a j e c t o r y i s p l o t t e d i n t h e c o n f i g u r a t i o n space (p means x o r y , . The upper p a r t r e p r e s e n t s t h e Poincare s e c t i o n of t h e same t r a j e c t o r y , i . e . t h e s e t of t h e succes- s i v e v a l u e s of i t s i n t e r a c t i o n s ( p) and p r o j e c t i o n s of momentum (pp j on t h e y a x i s (with t h e r e s t r i c t i o n t h a t p,
>
0;. The s e t of p o i n t s so o b t a i n e d c l e a r l y do not form an i n v a r i a n t curve, t h e t r a j e c t o r y belongs t o a manifold of higher dimension The e x i s t e n c e of c h a o t i c t r a j e c t o r i e s i s u s u a l l y taken a s a numerical i n d i c a t i o n t h a t t h e p o t e n t i a l i s not i n t e g r a b l e . The l a r g e v a l u e s of t h e parameters : I.I = 1.5 andrj
= 0.99 ( i . e . n e a r t h e d i s s o c i a t i o n energy) i n d i c a t e t h a t i t has been n e c e s s a r y t o look f o r r a t h e r extreme c o n d i t i o n s i n o r d e r t o observe a chaocic t r a j e c t o r y . In o t h e r words t h e v a l u e s j~ andn
a r e r a t h e r l a r g e . I n t h e c a s e ofappear i n t h e domain
The p r e c i s e l o c a t i o n of t h e t h r e s h o l d n e c e s s i t a t e s an extremely d e t a i l e d study of t h e Poincare p l a n e such a s F i g . 4 of Ref. 3 which shows t h a t q~ N 0.95 f o r j.~ = 1 . 5
w h i l e F i g . 4 of t h e p r e s e n t paper shows t h a t qr:
-
>
0.95 f o r p = 1.4. I n o t h e r words t h e c h a o t i c r e g i o n corresponding t o t h e Lz = 0 t r a j e c t o r i e s of A = 16 i s l i m i - - t e d t o a r a t h e r t h i n s k i n of v a l u e s of rl of width Aq ~ 0 . 0 5 f o r p>
1.3. F i g . 1- A c h a o t i c
t r a j e c t o r y of t h e defor- med w e l l f o r A = 16. The upper. p a r t i s t h e Poin- c a r 4 s e c t i o n of t h e t r a - j e c t o r y r e p r e s e n t e d i n t h e lower p a r t i n c o n f i - g u r a t i o n space.JOURNAL
DE
PHYSIQUE I F i g . 2-
The upper p a r t i s t h e PoincarG s e c t i o n { p ,pp} of t h e deformed w e l l f o r p = 1 . 2 ; i4
TI
= 0.99. The f o u r lower f i g u --
I r e s r e p r e s e n t L i s s a j o u s l i k eI
t r a j e c t o r i e s found a t t h e p l a c ei
i n d i c a t e d above.A numerical procedure has been developed i n Ref. 4 which a l l o w s t o study t h e proper- t i e s of t h e mapping i n t h e v i c i n i t y of t h e l i n e a r p e r i o d i c t r a j e c t o r i e s . The t r a - j e c t o r i e s which a r i s e through t h e b i f u r c a t i o n s of t h e l i n e a r t r a j e c t o r i e s a r e L i s - s a j o u s - l i k e f i g u r e s . Some a r e r e p r e s e n t e d on F i g . 2. These curves a r e surrounded by t o r i which form i n t h e PoincarG plane i s l a n d s of dif'erent s i z e s . Some of t h e
i s l a n d s ( t h e m a j o r i t y !) a r e s o t i n y t h a t t h e y cannot be d e t e c t e d numerically, how- ever a few i s l a n d s c a n always be d e t e c t e d l i k e i n F i g . 2 . The c e n t e r of t h e i s l a n d s i s t h e p l a c e of t h e p e r i o d i c t r a j e c t o r y . The v a l u e of t h e parameter where t h e L i s - s a j o u ~ f i g u r e i s degenerate with t h e l i n e a r t r a j e c t o r y marks t h e b i r t h of t h i s f i - gure. It i s fouad by t h e f o l l o w i n g method :
Let M p . be t h e non l i n e a r mapping i n t h e { p , p p } PoincarG plane which makes t h e
iteration between t h e kth and t h e ( k + l ) t h I n t e r s e c t i o n s : M (0 )k= ( g p I k + , . PP
Let us c o n s i d e r t h i s mapping near t h e o r i g i n which i s a f i x e d p o i n t of t h e mapping. If p and pp a r e very small M c o i n c i d e s w i t h i t s l i n e a r p a r t . It i s known t h a t
P.
i T / ~ r M p l
<
2 t h e f i x e d p o ~ n t i s e l l i p t i c a l while i f ( ~ r M ~ />
2 t h e f i x e d p o i n t i s h y p e r b o l i c . A b i f u r c a t i o n o c c u r s wheneverIT^
M ~ (,
o r any of i t s power/
Tr Mp n \ , equal 2 . ( I t can b e proved/
4 / t h a t i n our p o t e n t i a l T r M Z -2, the- r e f o r e t h e s i n g u l a r v a l u e i s + 2 ) . I f Tr M i s w r i t t e n a s 2 c o sa
.
f o r y ~ r M p l<
2,P
t h e v a l u e of
a
i s a continuous f u n c t i o n of t h e parameters p and f).
It i s then e a s y t o f i n d t h e p o i n t s wherea
= 2 ~ r !! f o r which T r M; = 2. At each of t h e s eF i g . 3 - E a c h c u r v e r e p r e s e n t s t h e va- l u e s of q and JJ where e q u a t i o n (2) i s s a t i s f i e d f o r v a l u e s of "indicated a t n t h e b o t t o m . M i s t h e mapping i n t h e v i c i n i t y of t g e l i n e a r t r a j e c t o r y a l o n g t h e l o n g ( z ) a x i s . Or, F i g . 3 t h e p o i n t s where T r M ( p , q ) = 2 c o s 271 P (2) 1 1
a r e p l o t t e d w i t h JJ and q a s c o o r d i n a t e s a x i s f o r =
5,
+,
-i; and-
a s w e l l a s t h e5
c u r v e s w i t h m = n . The l a t t e r c u r v e s d e f i n e r e g i o n s i n which T r
>
2 where no b i f u r c a t i o n s a r e p o s s i b l e . The P o i n c a r e s e c t i o n shown i n Big. 231
r
s r a e s5
5
this
s i t u a t i o n . P e r i o d i c t r a j e c t o r i e s c o r r e s p o n d i n g r e s p e c t i v e l y t o2
=-,-,
n 2 3
T 9 T a r e
met when g o i n g away from t h e c e n t e r of t h e mapping f o r JJ = 1.2. The l o c a t i o n of t h e c o r r e s p o n d i n g b i f u r c a t i o n s a r e made by o r d e r of d e c r e a s i n g e n e r g y on F i g . 3 . The c h a o t i c r e g i o n s a r e n o t r e p r e s e n t e d o n F i g . 3 which h a s been o b t a i n e d o n l y by a l o c a l s t u d y of t h e mapping n e a r p = 0.I n t h e r e g i o n where t h e c e n t e r of t h e mapping b i f u r c a t e s t h e c e n t e r i s a n e l l i p t i c p o i n t l i k e f o r t h e harmonic o s c i l l a t o r . It t u r n s o u t t h a t i n t h i s r e g i o n t h e r e i s a v e r y b r o a d r a n g e of v a l u e s of p a r a m e t e r s f o r which t h e t o r i a r o u n d t h e L i s s a j o u s c u r v e s a r e v e r y s m a l l . It i s p o s s i b l e t o s a y t h a t i n t h i s r e g i o n t h e t o p o l o g y i s t h a t of t h e harmonic o s c i l l a t o r . An example i s g i v e n w i t h F i g . 4 where no o t h e r t o r u s i s s e e n f o r q
<
0.85 b u t t h o s e d e f i n e d by t h e i n v a r i a n t c u r v e s which look l i k e o v a l s s u r r o u n d i n g t h e c e n t e r .I n F i g . 4 t h e t o r i i s s u e d from b i f u r c a t i o n s of f i r s t g e n e r a t i o n a r e s e e n o n l y f o r
q = 0.90. I f q
<
0.85 t h e p i c t u r e of t h e P o i n c a r e map r e s e m b l e s much t h e l e f t u p p e r s i d e of F i g . 4 .F i g . 3 shows t h a t t h i s p i c t u r e i s v a l i d l o c a l l y , i n d e e d two p a r t s of phase s p a c e w i t h a c o m p l e t e l y d i f f e r e n t t o p o l o g y a r e shown a s t h e dashed a r e a i s s u e d from
JJ = 2 and t h e o t h e r one around JJ = 1.
A p r e l i m i n a r y c o n c l u s i o n i s t h a t f o r 1 . 3
<
JJ<
2 t h e r e i s a r e g i o n where t h e topo- l o g y of t h e harmonic o s c i l l a t o r i s dominant and t h e r e f o r e t h e motion c a n b e quan t i z e d w i t h t h e r u l e s used f o r t h e harmonic o s c i l l a t o r .T,.
C6-376
JOURNAL
DE
PHYSIQUE ,. . ... . .--
..-
-- - - - . ., :. .:-
F+g. 4 - PoincarC sec-q=
0.8s
'7
=
0.30
t l o n d f o r j~ = 1.4 a t / s e v e r a l e n e r g i e s . - - - - F i g . 5-
Two t r a j e c t o r i e s w i t h d i f f e r e n t t o p o l o g i e s i n c o n f i g u r a t i o n space (up- p e r p a r t ) and t h e i r Poinca- r6 s e c t i o n (lower p a r t ) . Here j~ = 1.06 and q = 0.7 f o r both t r a j e c t o r i e s .I11
--
QUANTIZATION I N THE CASE OF THE SIMPLEST TOPOLOGYI f t h e m a j o r i t y of t h e o r b i t s of t h e deformed p o t e n t i a l a r e s i m i l a r t o t h o s e of Fig. 4 f o r q = 0.80, they can b e q u a n t i z e d w i t h t h e same r u l e s a s t h e harmonic o s c i l l a t o r . Let pp and pz be t h e p r o j e c t i o r s o f t h e momentum of t h e p a r t i c l e on t h e
p and z a x i s w h i l e i t i n t e r s e c t s t h e p and z a x i s r e s p e c t i v e l y and l e t Cp and Cz b e t h e s e t of v a l u e s of p and z taken d u r i n g t h e s e i n t e r s e c t i o n s , t h e a c t l o n s a r e d e f i n e d a s
Using K.A.M. theorem, "most o f t h e t r a j e c t o r i e s l i e o n t o r i " , and s i n c e t h e t o p o l o - gy ( a t l e a s t t h e "macroscopic ones" I ) i s t h a t o f t h e harmonic o s c i l l a t o r we a r e a b l e t o d e f i n e a s u r f a c e e n e r g y a c t i o n :
C6-378
JOURNAL
DE
PHYSIQUEF i g . 6 i l l u s t r a t e s i n a v e r y r e m a r k a b l e way t h e power of t h e K.A.M. theorem. It c a n b e compared t o F i g . 7 which r e p r e s e n t s t h e e n e r g y a c t i o n s u r f a c e E(Ir,!L) of 160 f o r p = 1 . B o t h F i g . 6 and 7 s h a r e t h e same p r o p e r t y and giv.e t h e same messa- ge : e x c e p t n e a r r1-0.90 t h e e n e r g y a c t i o n s u r f a c e i s p l a n a r and t h e s p e c t r u m i s harmonic. T h i s p r o p e r t y i s c o n s e r v e d even i n t h e non i n t e g r a b l e s i t u a t i o n !
However n e a r q - 0 . 9 0 t h e s u r f a c e i s non p l a n a r f o r p = 1 t h i s means t h a t t h e r e a r e n o n l i n e a r i t y i n t h e m o t i o n . F o r p = 1 . 4 t h i s n o n l i n e a r i t y p r o d u c e s a d e s t r u c - t i o n o f t h e e n e r g y a c t i o n s u r f a c e and t h e i m p o s s i b i l i t y o f u s i n g p r o p e r l y t h e EBK method.
The comparison of t h e s e m i c l a s s i c a l and quantum e i g e n v a l u e s o f IJ = 1 a n d 1.4 i s made i n T a b l e 1 . T h i s t a b l e shows t h a t t h e r e i s no e s s e n t i a l d i f f e r e n c e between t h e two c a s e s e x c e p t i n t h e c h a o t i c r e g i o n .
I V
-
CONCLUSIONI n c o n c l u s i o n i t seems t h a t b e l o w t h e t h r e s h o l d o f s t o c h a s t i c i t y i t i s p o s s i b l e t o u s e p r o p e r l y t h e s e m i c l a s s i c a l EBK method of q u a n t i z a t i o n and t h a t t h i s method i s a b l e t o e x p l a i n t h e p r o p e r t i e s o f t h e s p e c t r u m w i t h t h e same q u a l i t y a s f o r t h e i n t e g r a b l e c a s e . However i n o r d e r t o r e a l i z e w h o l l y t h i s program we n e e d t o f a c e t h e c h a n g e s of t o p o l o g i e s l i k e t h o s e o f F i g . 5 . ( i n t e g r a b l e )
/
( n o t i n t e g r a b l e )I
-20.71 8 - 0 . 3 6 4 - 8.901 - 0 . 8 3 2 - 1.615 c h a o s T a b l e 1 REFERENCES/ I / ARNOLD V . I . , Mkthodes Mathkmatiques d e l a Mbcanique C l a s s i q u e , E d i t i o n s MIR - Moscou (1967)
/ 2 / CARBONELL J . , BRUT F . , TOUCNARD J. and ARVIEU R . , communication t o t h i s workshop
/ 3 / CARBONELL J . a n d ARVIEU R., N u c l e a r f l u i d dynamics, (1983) T r i e s t e 141 / 4 / CARBONELL J. t h e s e de 36me C y c l e
-
U n i v e r s i t k de G r e n o b l e (1983)ISN 83-07
