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NON-MARKOVIAN DIFFUSION OF A QUANTUM PARTICLE IN A FLUCTUATING MEDIUM
P. Reineker, K. Kassner
To cite this version:
P. Reineker, K. Kassner. NON-MARKOVIAN DIFFUSION OF A QUANTUM PARTICLE IN A FLUCTUATING MEDIUM. Journal de Physique Colloques, 1985, 46 (C7), pp.C7-35-C7-39.
�10.1051/jphyscol:1985707�. �jpa-00224955�
JOURNAL DE P H Y S I Q U E
Colloque C7, suppl6ment a u nolO, T o m e 46, o c t o b r e 1985 p a g e C7-35
NON-MARKOVIAN DIFFUSION OF A QUANTUM P A R T I C L E IN A FLUCTUATING MEDIUM
P . R e i n e k e r a n d K. K a s s n e r
Abtei lung Theoretische Physik, Universittit illm, 0-7900 Ulm, F. R. G.
A b s t r a c t - A model f o r d i f f u s i o n o f a quantum p a r t i c l e on a f l u c t u a t i n g l a t t i c e i s c o n s i d e r e d . The H a m i l t o n i a n c o n t a i n s a c o h e r e n t t r a n s f e r m a t r i x e l - ement between n e a r e s t n e i g h b o r s and l o c a l e n e r g y f l u c t u a t i o n s d e s c r i b e d by a d i c h o t o m i c Markov p r o c e s s w i t h c o l o u r e d n o i s e . The d i f f u s i o n c o n s t a n t i s c a l - c u l a t e d up t o t h e f o u r t h o r d e r i n J . The r e s u l t i s compared t o r e c e n t f i n d i n g s o f K i t a h a r a - H a u s and I n a b a . Anderson l o c a l i z a t i o n e m e r g e s i n t h e l i m i t o f s t a t i c f l u c t u a t i o n s .
The dynamics o f e l e c t r o n i c e x c i t a t i o n s i n t e r a c t i n g w i t h v i b r a t i o n s i s o f i m p o r t a n c e i n v a r i o u s f i e l d s o f condensed m a t t e r p h y s i c s 11-31. Examples a r e t h e i n v e s t i g a t i o n o f o p t i c a l and s p i n r e s o n a n c e l i n e s h a p e s , o f r e l a x a t i o n phenomena o r o f c h a r g e and e n e r g y t r a n s p o r t phenomena. Because t h e f u l l quantum m e c h a n i c a l problem is d i f f i c u l t t o t r e a t , i t s H a m i l t o n i a n i s o f t e n r e p l a c e d by a s t o c h a s t i c p r o c e s s / 4 , 5 / modeling t h e v i b r a t i o n s . I n t h e Haken-Strobl model f o r t h e c o u p l e d c o h e r e n t and i n c o h e r e n t ex- c i t o n m o t i o n 16-81 l o c a l and n o n - l o c a l f l u c t u a t i o n s a r e a l l o w e d f o r i n t h e s t o c h a s t i c p a r t and d e s c r i b e d by a G a u s s i a n 6 c o r r e l a t e d ( w h i t e n o i s e ) Markov p r o c e s s . With t h i s H a m i l t o n i a n t h e d i f f u s i o n c o n s t a n t / 7 , 9 , 1 0 / and t h e t i m e dependence o f t h e mean s q u a r e ' d i s p l a c e m e n t o f t h e p a r t i c l e h a v e been d e r i v e d 111-131. The same r e s u l t s h a v e been o b t a i n e d s u b s e q u e n t l y i n 114-161. The i n f l u e n c e o f e x p o n e n t i a l l y d e c a y i n g c o r r e - l a t i o n f u n c t i o n s ( c o l o u r e d n o i s e ) h a s been i n v e s t i g a t e d i n 1 1 7 1 by e x p a n d i n g t h e d i f - f u s i o n c o n s t a n t i n powers o f t h e c o r r e l a t i o n time. I n a r e c e n t a p p r o a c h t h e s t o c h a - s t i c p a r t was r e p r e s e n t e d by a d i c h o t o m i c Markov p r o c e s s 1181 w i t h e x p o n e n t i a l l y de- c a y i n g c o r r e l a t i o n f u n c t i o n s . W i t h i n t h i s model an a p p r o x i m a t e a n a l y t i c a l e x p r e s s i o n f o r t h e d i f f u s i o n t e n s o r was o b t a i n e d by c o n s i d e r i n g o n l y d i a g o n a l e l e m e n t s and t h e i r n e a r e s t n e i g h b o r s i n t h e d e n s i t y m a t r i x 1 1 9 1 , which may b e j u s t i f i e d i n t h e c a s e o f s t r o n g f l u c t u a t i o n s w i t h r a p i d l y d e c a y i n g c o r r e l a t i o n f u n c t i o n s . I n a f o l l o w i n g p a p e r 1 2 0 1 t h e d i f f u s i o n c o n s t a n t was c a l c u l a t e d n u m e r i c a l l y u s i n g t h e d y n a m i c a l c o h e r e n t p o t e n t i a l method. I n t h i s l e t t e r we u s e t h e same model, b u t t h e method o f s o l u t i o n i s q u i t e d i f f e r e n t from t h o s e used i n / 1 9 , 2 0 / b e c a u s e we u s e a n e x p a n s i o n i n powers o f t h e c o h e r e n t p a r t o f t h e H a m i l t o n i a n . The mean s q u a r e d i s p l a c e m e n t i s f i n a l l y d e s c r i b e d by a c o n t i n u e d f r a c t i o n t h e c o n v e r g e n c e b e h a v i o u r o f which i s a n a l y z e d w i t h r e s p e c t t o t h e a m p l i t u d e o f t h e f l u c t u a t i o n s . Anderson l o c a l i z a t i o n i s o b t a i n e d i n t h e l i m i t o f s t a t i c f l u c t u a t i o n s .
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1985707
C7-36 JOURNAL DE PHYSIQUE
The H a m i l t o n i a n o f t h e model H = H. + H l ( t ) i s g i v e n by
In> d e s c r i b e s a s t a t e l o c a l i z e d a t s i t e n. I n t h e c o h e r e n t p a r t H. o f t h e Hamilto- n r a n J d e s c r i b e s t h e p a r t i c l e t r a n s f e r bEtween n e i g h b o r i n g s i t e s ;
n
r u n s o v e r a l l s i t e s ,2
o v e r n e a r e s t n e i g h b o r s o n l y . h n ( t ) i n t h e s t o c h a s t i c p a r t Hl(t) i s a c l a s - s i c a l f l u c t u a t i n g q u a n t i t y w i t h < h ( t ) > Z 0 ( a n g u l a r b r a c k e t s d e n o t e s t o c h a s t i c a v e r - a g i n g ) and c o r r e l a t i o n f u n c t i o n s"
w i t h tl 2 t2 2 t 3 , t 4 ,
...,
t is c h a r a c t e r i s t i c 1181 f o r a d i c h o t o m i c Markov p r o c e s s . PA d e s c r i b e s t h e s t r e n g t h o f t h e f l u c t u a t i o n s and y t h e d e c a y r a t e .
Using t h e d i s e n t a n g l e m e n t t h e o r e m f r o m t h e e q u a t i o n o f motion f o r t h e d e n s i t y o p e r a - t o r o f t h e p a r t i c l e
6
= - i [ ~ , p ] e q u a t i o n s d e t e r m i n i n g t h e L a p l a c e t r a n s f o r m o f t h e mean s q u a r e d i s p l a c e m e n t <R -2 ( S ) > a r e d e r i v e d ( f o r d e t a i l s see 1 2 1 , 2 2 1 ) .E x p l i c i t l y we o b t a i n
^2 2 2
< R ( s ) > = 2 J / s C
a $ ' %
(S) a a 'a f a r --
A
where t h e m a t r i x K (S) i s d e t e r m i n e d f r o m
^ ^
{S
+
;(S)} w ( s ) = 1.
(5)0
T h i s e q u a t i o n h a s t h e s t r u c t u r e o f e q u a t i 2 n s d e t e r m i n i n g G r e e n ' s f u n c t i o n s w i t h p l a y i n g t h e r o l e o f a s e l f - e n e r g y . After
z
h a s been o b t a i n e d up t o s e c o n d o r d e r i n J , ( 5 ) h a s been s o l v e d a n a l y t i c a l l y 1 2 2 1 f o r a l i n e a r c h a i n . The d i f f u s i o n c o n s t a n t , c o r r e c t t o f o u r t h o r d e r i n J , i s c a l c u l a t e d from1 2 -2
D = l i m s <R ( S ) > (6)
s-to
and r e p r e s e n t e d i n F i g . 2 .
F o r more c o m p l i c a t e d l a t t i c e s , however, it is more c o n v e n i e n t t o d e r i v e r e c u r r e n c e r e l a t i o n s f o r t h e F e a n s q u a r e d i s p l a c e m e n t start!ng from ( 5 ) ( w i t h t h e s e c o n d - o r d e r a p p r o x i m a t i o n f o r :(S) ) and t h e d e f i n i t i o n o f <R'(s)>. To t h a t end it i s u s e f u l t o d e f i n e ( a l l t h e sums r u n o v e r n e a r e s t n e i g h b o r s )
It i s o b v i o u s t h a t <R -2 ( S ) > = R o ( s ) . We f u r t h e r m o r e d e f i n e l a t t i c e sums
5, =
al,a2,
C . . . , a 2 n 6 ( 0 , a +a -1 -2 +...
+a -2n ) which d e s c r i b e t h e number o f ways o f r e t u r - n i n g t o t h e o r i g i n a f t e r 2n n e a r e s t n e i g h b o r s t e p s and o b t a i n ( d e t a i l s o f t h e d e r i - v a t i o n w i l l b e p u b l i s h e d i n /22/ )S u b t r a c t i n g two consecutive equations we have unxRn
-
(x-on4J)- 4'
= 0w i t h an = (2n+l) ~ , + ~ / ( 2 n + 3 ) The f u n c t i o n s rp(J,s), x ( J , s ) and $ ( J , s ) have been c a l c u l a t e d i n 1221. For t h e f o l l o w i n g c a l c u l a t i o n o f t h e d i f f u s i o n constant, however, we need and g i v e them below f o r S = 0 only. A t t h e moment we have
t g
know t h a t x(J,s) contains terms independent o f J whereas $ ( J , s ) i s p r o p o r t i o n a l t o J and t h u s a s m a l l q u a n t i t y f o r s m a l l J . The system o f equations (9) i s solved by a continued f r a c t i o n 1231. Using (8) f o r n=O, t h e f a c t t h a t gl=z (number o f nearest neighbors), t h e con- n e c t i o n between Ro(s) and t h e mean square displacement, and (6) t h e d i f f u s i o n con- s t a n t i s given byw i t h
2 2 y-3 qO = l i m q ( J , s ) = J 'l
s+o
1 2
xo
= l i m x(J,s) = 2l' + zqC -
2 ($2 27 Y }s+O 0 2 A
+73y2+4a2
where
r-'
= (y/A2+y-l). To evaluate (10) i n N-th order approximation we have t o know ol...aN f o r t h e l a t t i c e under c o n s i d e r a t i o n which a r e determined by t h ec ~ . . . < ~ + ~ .
I t i s easy t o g i v e c o m b i n a t o r i a l formulae f o r these l a t t i c e sums. For a l i n e a r chain we have ( = ( Zn ) and f o r a square l a t t i c e C = (2:)2. I n t h e case o f a simple cubic o r a bodyncentePed c u b i c l a t t i c e t h e an may Be c a l c u l a t e d up t o N = 30 w i t h i n some seconds o f CPU t i m e on a minicomputer. For a f a c e centered cubic l a t t i c e , however, one has t o evaluate 8 nested sums which makes t h e c a l c u l a t i o n r a t h e r t i m e consuming.
Truncating t h e continued f r a c t i o n s by n e g l e c t i n g t h e 4J0 term i n t h e second denomi- n a t o r o f (10) g i v e s
T h i s r e s u l t may be compared w i t h expressions (12,13) by K i t a h a r a and Haus 1171 and by Inaba 1191, r e s p e c t i v e l y ,
a f t e r expanding a l l r e s I t s up t o f o u r t h order i n J. For a l i n e a r chain (z=2) t h e
8 .
terms p r o p o r t i o n a l t o J I n (11,13) a r e i d e n t i c a l , whereas i n (12) a term
-
y-l i smissing. I n t h e expression p r o p o r t i o n a l t o J~ b o t h i n (12) and i n (13) terms are missing.
For z
>
2 o n l y (11) and (13) can be compared. The expressions do n o t agree because I n a b a ' s r e s u l t i s p r o p o r t i o n a l t o z whereas (9) contains a c o n t r i b u t i o n p r o p o r t i o n a l t o z2. The reason f o r t h i s d i f f e r e n c e i s t h a t i n I n a b a ' s treatment a l l t h e memory of jumps o f t h e d i f f u s i n g p a r t i c l e beyond nearest neighbors i s t l o s t , whereas our r e s u l t c o n t a i n s memory e f f e c t s o f closed paths c o n s i s t i n g o f f o u r jumps ( t o a near- e s t neighbor each).JOURNAL DE PHYSIQUE
F i g . 1 - D / ( J a 2 ) a s a f u n c t i o n o f y / J f o r s e v e r a l v a l u e s o f A/J f o r a l i n e a r c h a i n ( z = 2 ) .
.
K i t a h a r a - Haus / 1 5 / ; ---: I n a b a / 1 7 / ; -:e q . ( 9 ) o f t h i s p a p e r . S i n g l e p o i n t s : s o l u t i o n o f t h e s t o c h a s t i c S c h r o d i n - g e r e q u a t i o n / 1 7 / .
F i g . 2 - S e v e r a l a p p r o x i m a n t s o f t h e c o n t i n u e d f r a c t i o n (8) f o r t h e d i f f u s i o n c o n s t a n t a s com- p a r e d t o t h e s o l u t i o n / 2 0 / o f ( 2 , 4 ) f o r a l i n e a r c h a i n . For A / J = 1 t h i s s o l u t i o n d o e s n o t e x i s t o v e r t h e whole r a n g e o f y / ~ .
... .
a p p r o x i m a t i o n ( 9 ) . Only t h e t h i r d and f i f t h a p p r o x i - m a n t s a r e p l o t t e d f o r A / >>
1.F i g . 1 i l l u s t r a t e s t h e d i f f e r e n c e between t h e t h r e e r e s u l t s i n t h e c a s e o f a l i n e a r c h a i n . It shows t h e r e d u c e d d i f f u s i o n c o n s t a n t D / ( a J ) c o r r e s p o n d i n g t o (11-13) a s a 2 f u n c t i o n of t h e r e d u c e d s w i t c h i n g r a t e y / J f o r s e v e r a l v a l u e s o f A / J . The f i g u r e shows t h a t a p p r o x i m a t i o n (12) becomes r a t h e r p o o r f o r s m a l l v a l u e s o f y / J and t h a t t h e o t h e r a p p r o x i m a t i o n s a g r e e p r e t t y w e l l f o r l a r g e v a l u e s o f A / J . The s i n g l e p o i n t s t o g e t h e r w i t h c h a r a c t e r i s t i c e r r o r b a r s r e p r e s e n t s o l u t i o n s o f t h e s t o c h a s - t i c S c h r o d i n g e r e q u a t i o n o b t a i n e d by I n a b a 1 2 0 1 .
F i g . 2 shows (11) and a d d i t i o n a l a p p r o x i m a n t s o f t h e c o n t i n u e d f r a c t i o n (10) t o g e t h e r w i t h t h e s o l u t i o n o b t a i n e d from ( 5 ) . The f i g u r e shows t h a t f o r A/J = 1 t h e l a t t e r s o l u t i o n d o e s n o t e x i s t o v e r t h e whole r a n g e o f y / J . I n t h e same i n t e r v a l t h e con- t i n u e d f r a c t i o n d i v e r g e s a s i s s e e n f r o m t h e b e h a v i o u r o f t h e p l o t t e d a p p r o x i m a n t s . The f i g u r e shows a l s o t h a t f o r A / J 2 1 . 5 t h e r e i s c o n v e r g e n c e o v e r t h e w h o l e r a n g e o f y / J and t h a t a l r e a d y t h e f i f t h a p p r o x i m a n t g i v e s a r a t h e r good d e s c r i p t i o n . A r a t h e r c r u d e e s t i m a t e 1 2 2 1 i n o r d e r t o i n v e s t i g a t e t h e c o n v e r g e n c e b e h a v i o u r of t h e c o n t i n u e d f r a c t i o n (10) shows t h a t it c o n v e r g e s f o r A ~ / J ~ > z ( z + l ) i n d e p e n d e n t o f t h e s w i t c h i n g r a t e y / J . For a l i n e a r c h a i n ( z = 2 ) we e x p e c t c o n v e r g e n c e f o r
A/J
>
V@ 2 . 4 . From F i g . 2 we s e e t h a t t h e r e i s p e r f e c t c o n v e r g e n c e e v e n f o r A/J 2 1 . 5 .l I
I n t h e c a s e o f c o n v e r g e n c e we c a n c o n s i d e r t h e l i m i t y/J+O, i . e . we a p p r o a c h t h e c a s e of a l a t t i c e w i t h s t a t i c random p o t e n t i a l f l u c t u a t i o n s (Anderson p r o b l e m / 2 4 / ) . The f i g u r e s show t h a t t h e d i f f u s i o n c o n s t a n t v a n i s h e s f o r y / J -t 0 . T h i s means t h a t i n t h e c a s e o f s t a t i c f l u c t u a t i o n s t h e p a r t i c l e i s immobile, i . e . we o b t a i n Anderson l o c a l i z a t i o n / 2 4 / . To t h e knowledge o f t h e a u t h o r s t h i s i s t h e f i r s t p r o o f o f Ander- s o n l o c a l i z a t i o n s t a r t i n g f r o m a time d e p e n d e n t s t o c h a s t i c model w i t h c o l o u r e d n o i s e . F o r l a r g e enough v a l u e s o f A / J t h e c o n t i n u e d f r a c t i o n r e s u l t ( 1 0 ) f o r t h e d i f f u s i o n c o n s t a n t c o n v e r g e s a l s o f o r h i g h e r d i m e n s i o n a l l a t t i c e s and shows Anderson l o c a l i z a - t i o n f o r y / J + 0 . T h e s e i n v e s t i g a t i o n s a r e r e p r e s e n t e d i n d e t a i l i n / 2 2 / .
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