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DAMPING OF COLLECTIVE MOTION, MEMORY EFFECTS AND ENTROPY
K. Gütter
To cite this version:
K. Gütter. DAMPING OF COLLECTIVE MOTION, MEMORY EFFECTS AND ENTROPY. Jour-
nal de Physique Colloques, 1987, 48 (C2), pp.C2-55-C2-57. �10.1051/jphyscol:1987208�. �jpa-00226473�
JOURNAL DE PHYSIQUE
Colloque C2, supplbment au n o 6, Tome 48, juin 1987
DAMPING OF COLLECTIVE MOTION, MEMORY EFFECTS AND ENTROPY
Institut fiir Theoretische Physik 11, Gliickstrasse 6 , 0-8520 Erlangen, F. R. G.
ABSTRACT
Une L t i 6 o r i e de r k p o n s e l i n & : ~ i r e d i s s i p a t i v e e s t presentee q u i c o n t i e n t des e f f e t s de mtmolre p a r on terme de c o l l l s l o r ~ dkpendant de l ' b n e r y l e . Le nomhre de p a r t i - c u l e s e t 1 ' k r i e r q i e s o n t c o n s e r v h s , uric en t r o p i e d & f i r ~ i e propremen t augmen t e v e r s s a v a l e u r d ' e q u i l i b r e . ILes e q u a t i o n s s o r ~ t rk s o l u e s p o u r uri modele s i m p l e $ deux n i v e a u x .
A d i s s i p a t i v e l i n e a r response t h e o r y i s p r e s e n t e d i n c o r p o r a t i n q memory e f f e c t s by a n energy dependent collision term. P a r t ~ c l e number and c n e r q y a r e conserved, a s u i t a b l y d e f l n c d e n t r o p y i n c r e a s e s t o i l s e c l ~ l i l i b r ~ u m v a l u e . The e q u a t i o n s a r e s o l v e d i n a s i m p l e t w o - l e v e l model.
DISSIPATIVE LINEAR RESPONSE TH€OR_Y_
We want t o d e r i v e an e q u a t l o n d c s c r l b ~ n q damped c o l l e c t i v e rnotion o f a f e r m l o n many-body system near t h e r m a l e q u l l i b r l ~ i m . To t h ~ s end we s t a r t from t h e e q u a t l o n o f motron (LoM: f o r t h e o n e - p n r t l c l e c j e n s ~ t y o p e r a t o r f=<a+a> ( t h e H a m ~ l t o n i a n i s assumed t o be H
=
a + l a + -la+a+vaa):4
d 1
1-f
=
[ h ( f ) , f ]+ ?
t r 2 [ v , g ]d t ( 1 )
where h ( f ) = t
+
t r 2 { v l m f } i s t h e s e l f - c o n s l s t e r i l . m e a n - f i e l d H s m i l t o n i a n and g=
<a+a+aa> - d { f m f } j s t h e t w o - p a r t i c l e correlation.f - o r g we use t h e a p p r o x i m a l j o n
where
?:=
I-f andU(
t , s ) : = T e x p {-i y d t h ( F ) } (mean-f i e l d t l m e e v o l u t i o no p e r a t o r ) . 5
We a r e I n t e r e s t e d i n d e v i a t i o n s f r o m e q u l l l h r i u m , s o we s u b l r a c t t h e s t a t i o n a r y e q u a t l o n from F.q. (11, a p p r o x i m a t i n g f o
=[
l+exp ~ ( h ~ - ( ~ ) ] - l . I n o r d e r t o e v a l u a t e t.he i n t e g r a l o v e r t h e s y s t e m ' s h i s t o r y w c assume(')The c o l l e c t l v e f r e q u e n c y (I: may he complex, t h e i m a g i n a r y p a r t d e s c r ~ b l r l g t h e damping o f Ltle collective m o t i o n . O c c a s i o n a l l y we distinguish on- and o f f - d l a g o n a l e l e m e n t s o f f ( + ) ( w i t h r e s p e c t t o t h e 1 1 0 e i g e n b a s ~ s ) :
f ( + )
-. -.
L' r + , f o ] - +
0./ \
( l ) The upper i n d e x \ + ' d e n o t e s h e r e and i n t h e following t h e component a e x p ( - i w t )
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987208
C2-56
J O U R N A L D E PHYSIQUE
The r e s u l t i n g €OM may b e w r i t t e n as
w
f ( + )=
~ f ( + )+
L ( w ) f(+I
where ~ f ( + ) : = [ h ( + ) , f o ]
+
[ h o , f ( + ) ] i s t h e "RPA part." o f t h e EoM, and~ ( w ) f ( + ) : =
7
1 t r 2 [ v , g ( + ' ] 1 s t h e " c ~ ~ l i s i o n p a r t " .The e x p l ~ c ~ t form o f L ( w ) is g i v e n e.g. ~ n Ref. 1.
Memory e f f e c t s r e s u l t i n a s t r o n g w-dependence o f t h e c o l l i s i o n term. The mairi i n - g r e d i e n t i s t h e i n t e g r a l
where t h e E ' S a r e e i g e n v a l u e s o f h,. The i n t e g r a l h a s t o be p e r f o r m e d a f t e r t h e summation o v e r t h e i n t . e r m e d i a t e s t a t e s I n o r d e r t o converge a t t-*
--.
So A i s Lo b e c o n s i d e r e d as a d i s t r i b u t i o r ) i n t h i s sense.CONSERVED QUANT I T IES, ENTROPY
P a r t i c l e nt~rnhcr c o n s e r v a t i o n i s p r o v e d i m m e d i a t e l y by t r a c i n g o v e r t h e EoM:
Energy c o n s e r v a t i o n i s a b i t more complicated. I t i s g u a r a n t e e d by t h e chosen a p p r o x i m a l i o r i ( 2 ) :
To f i n d an e x p r e s s i o n f o r t h e e n t r o p y we s t a r t f r o m t h e g e n e r a l von Neumann d e f i n i - t i o n S
=
- ~ r { p l o g p } where p i s t h e f u l l s t a t i s t i c a l o p e r a t o r . F o r p we make an a n s a t z compatible w i t h t h e c o n s e r v a t i o n p r o p e r t i e s ( 5 ) and ( 6 ) :We s i m u l a t e t h e n o n - e q u i l i b r i u m d i + s l r i b u t i o n by I h e e q u i l i b r i u m d i s t r i b u t i o n i n an e x t e r n a l one-body p o t e n t i a l Y
=
a ya. N o t e l h a t p c o n t a i n s t w o - p a r t i c l e c o r r e l a - t i o n s v i a H w h l c h i s n e c e s s a r y t o a c c o u n t f o r e n e r g y c o n s e r v a t i o n . y i s d e l e r m i n e d b y t h e c o n d i t i o r i T r {pa+a}=
f .We want t o c o n s i d e r t h e correlations o n l y i n t h e EoM and i n c o n n e c t i o n wit.h e n e r g y conservation, so we may n e q l e c t ttiem i n c a l c u l a t i n g y. Expanding Lhe above c o n d i t i o n y i e l d s
so we g e t f o r t h e e n t r o p y
and, i n s e r t i n g 18) a n d ( 3 ) :
6s = -0 Re(oj)
~ ~ ( c c , c + ! ~ o \
e 2Irn(w)t The t o t a l l y dissipated h e a t is1 dS
Q
=
- { d t3 =
R e ( o ) t r ( [ C , C + ] f o } w h i c h i s j u s t t h e RPA e n e r g y o f t h e c o l l e c t i v e m o t i o n .A TWO-LEVEL MODEL
( 2 ) I n o r d e r Lo s t u d y t h e p r o p e r t i e s o f t h e EoM ( 4 ) we use a stochastic 2 - l e v e l model
,
a q e n e r a l l z a t l o n o f t h e L l p k i n model (gef.2:. The e i g e n v a l u e s o f ho a r e randomly l o c a t e d around two e n e r g l e s t- I n Gausslan d l s t r l b u t l o n s w i t h e q u a l
2
w i d t h s a and equal numbers o f s l a t e s W > l . The H l l b e r t space i s
]e= c2@~'.
The interaction 1 s assumed t o s e ~ a r a t e : v=
SwM where t h e m a t r l x elements o f M a r e l u s t-
random phases +1. W l t h t h e a n s a t z f ( + )
=
? ( + ) a 1 we average t h e EoM o v e r t h e C -space y i e l d i n ga
where
R ,
C(U) a r e t h e avaraged RPA and collision p a r t s . The d i s t r i b u t i o n A ( z ) now becomes a w e l l - d e f i n e d f u n c t i o nwhere @ i s t h e e r r o r f u n c t i o n .
E q . ( l l ) i s a & - d i m e n s i o n a l q e n e r a l i z e d nonlinear e i g e n v a l u e p r o b l e m w h i c h may be s o l v e d by some i t e r a t i o n method. There a r e two s o l u t i o n s o f L q . ( l l ) w i L h
o
=O, c o r r e s p o n d i n g t o t h e two c o n s e r v e d q u a n t i t i e s E and N. The o t h e r two s o l u t i o n s l i e s y m m e t r i c a l l y w i t h r e s p e c t t o t h e i m a g i n a r y a x i s . S i n c e z ( z ) i s peaked a t zero,t h e c o l l i s i o n t e r m shows resonances a t
w =
e and 2e. F i g . 1 shows a t y p i c a l p l o t o f Im;cb) vs. R e ( o ) o f t h e s o l u t i o n w i t h Re(wj>O, g a i n e d b y v a r y i n g t h e l r ~ t e r a c t i o r l s t r e n q t h . The l p - l h resonance i s s l ~ p p r e s s e d b u t because o f i t s finltr w i d t h g o i t s t a i l i s seen a t w = 1.3e. Near w=
2e t h e damplng o f t h e c o l l e c t i v e m o l i o n i s maximal due t o t h e c o u p l l n q t o 2p-2h e x c i t a t i o n s .1 ) H. R e i n h a r d t , P.-G. R e i n h a r d and K . Goeke, P t > y s . L e t t . 1518 ( 1 9 8 5 ) 177 2 ) H.J. C i p k i n , N. Mechkov and A.J. G l i c k , Nucl.Phys. 62 (1985) 188
0
43
-.
( 2 ) T h i s model has been p r o p o s e d i n an u n p u b l i s h e d work o f P.-G. R e i n h a r d , H.L. Yadav and C . T o e p f f e r , 1985
/-.
3-
E
M
-.01
-.
0 2-
.-
-
.-
--
R e a l and i m a g i n a r y p a r t o f t h e s o l u t i o n w o f E q . ( l l ) . The p a r a m e t e r s a r e :
