THERMODYNAMICALLY CONSISTENT ENERGY TRANSFER

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THERMODYNAMICALLY CONSISTENT ENERGY TRANSFER

D. Brown, K. Lindenberg, B. West

To cite this version:

D. Brown, K. Lindenberg, B. West. THERMODYNAMICALLY CONSISTENT ENERGY TRANS- FER. Journal de Physique Colloques, 1985, 46 (C7), pp.C7-41-C7-44. �10.1051/jphyscol:1985708�.

�jpa-00224956�

THERMODYNAMICALLY CONSISTENT ENERGY TRANSFER

D.W.

Brown, K. Lindenberg and

B.J.

west*

Department o f Chemistry, U n i v e r s i t y o f C a z i f o m i a , San Diego, La JoZ Za, CaZifornia 92093, U . S. A.

A b s t r a c t - We a d d r e s s t h e problem of t h e thermodynamically c o n s i s t e n t d e s c r i p - t i o n of e n e r g y t r a n s f e r i n condensed media and e x h i b i t r e s u l t s f o r a two-level system. We c o n s t r u c t quantum Langevin e q u a t i o n s which a r e i n g e n e r a l non- Markovian, n o n l i n e a r , and c o n t a i n o p e r a t o r ( r a t h e r t h a n c-number) fluctuations.

Thermodynamic c o n s i s t e n c y i s m a n i f e s t by t h e e x i s t e n c e o f r e l a t i o n s between d i s s i p a t i v e t e r m s and c o r r e l a t i o n f u n c t i o n s of t h e f l u c t u a t i o n s . Under ap- p r o p r i a t e c o n d i t i o n s we a r e a b l e t o r e w r i t e o u r e q u a t i o n s i n t h e form of a s t o c h a s t i c L i o u v i l l e e q u a t i o n (SLE), p e r m i t t i n g a comparison between e x i s t i n g t h e o r i e s and o u r new approach. The approach p r o v i d e s t e m p e r a t u r e dependences f o r t h e SLE c o e f f i c i e n t s and p r e d i c t s temperature-dependent s p e c t r a l a n d t r a n s - p o r t p r o p e r t i e s .

Our u n d e r s t a n d i n g of e x c i t o n t r a n s p o r t i n condensed m a t t e r h a s been g r e a t l y f a c i l i t a t e d by t h e u s e of reduced d e n s i t y m a t r i x e q u a t i o n s i n which t h e e f f e c t s of a b a t h a p p e a r through a d d i t i v e t e n s , each comprised of a p r o d u c t ( o r c o n v o l u t i o n ) of a "bath parameter" ( o r k e r n e l ) w i t h a d e n s i t y m a t r i x element. The p r e m i e r example o f such a n e q u a t i o n i s t h e s t o c h a s t i c L i o u v i l l e e q u a t i o n (SLE)/l/

L

The " b a t h p a r a m e t e r s "

F,

y , and e n t e r t h e s t o c h a s t i c t h e o r y a s t h e c o r r e l a - t i o n s t r e n g t h s of t h e phenomenological f l u c t u a t i n g p o t e n t i a l s modeling t h e e f f e c t s o f a b a t h . S t o c h a s t i c L i o u v i l l e e q u a t i o n s and r e l a t e d g e n e r a l i z e d m a s t e r e q u a t i o n s a r e c a p a b l e of e f f e c t i v e l y a d d r e s s i n g a v a r i e t y of phenomena, p a r t i c u l a r l y when t h e s e b e a r on t h e c o h e r e n c e p r o p e r t i e s o f e x c i t o n t r a n s p o r t / l , 2 / . On t h e o t h e r hand, t r a n s p o r t a l s o i n v o l v e s t h e approach t o e q u i l i b r i u m a t a n ambient t e m p e r a t u r e ,

(*) Permanent a d d r e s s : C e n t e r f o r S t u d i e s of N o n l i n e a r Dynamics, La J o l l a I n s t i t u t e , 3252 Holiday C o u r t , S u i t e 208, La J o l l a , CA 92037 U.S.A.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1985708

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w h i l e t h e s t a t i o n a r y s o l u t i o n of t h i s and o t h e r e q u a t i o n s based on s t o c h a s t i c poten- t i a l s c o r r e s p o n d t o a n i n f i n i t e t e m p e r a t u r e . T h i s s h o r t c o m i n g of t h e SLE i s w e l l known, and may b e u n d e r s t o o d a s r e s u l t i n g from t h e u n l i m i t e d " h e a t i n g " o f t h e s y s t e m by t h e b a t h . I n a s t o c h a s t i c H a m i l t o n i a n t h e b a t h i s n o t a f f e c t e d by t h e p r e s e n c e of t h e s y s t e m and s o t h e b a l a n c e of h e a t f l u x e s i n t o and o u t of t h e b a t h n e c e s s a r y f o r t h e e q u i l i b r a t i o n o f t h e s y s t e m t o t h e f i n i t e b a t h t e m p e r a t u r e d o e s n o t o c c u r . I n o t h e r words, t h e f l u c t u a t i n g p o t e n t i a l c o n t r i b u t i o n s a r e n o t b a l a n c e d w i t h appro- p r i a t e d i s s i p a t i v e c o n t r i b u t i o n s t o t h e dynamical e v o l u t i o n o f t h e e x c i t o n .

While a f a u l t y e q u i l i b r i u m i n d i c a t e s a f a u l t y f l u c t u a t i o n - d i s s i p a t i o n r e l a t i o n (FDR), t h e a t t a i n m e n t o f p r o p e r e q u i l i b r i u m d o e s n o t imply t h a t f l u c t u a t i o n and d i s - s i p a t i o n a r e p r o p e r l y b a l a n c e d . I n d e e d , a n e q u a t i o n s u c h a s ( 1 ) may b e d r i v e n t o - ward any d e s i r e d e q u i l i b r i u m s o l u t i o n i n a number of ways. To choose t h e thermo- d y n a m i c a l l y c o r r e c t e q u a t i o n i t i s f i r s t n e c e s s a r y t o i d e n t i f y t h e f l u c t u a t i o n s , t h e d i s s i p a t i o n , and t h e p r o p e r FDR t h a t c o n n e c t s them. Although t h i s i d e n t i f i c a t i o n i s q u i t e w e l l u n d e r s t o o d i n a v a r i e t y of c l a s s i c a l $ y s t e m s / 3 , 4 / t h e c o r r e s p o n d i n g quantum problems a r e n o t n e a r l y a s c l e a r / 4 - 6 1 . Thus, i n c o n t r a s t w i t h t h e c l a s s i c a l c a s e , t h e FDR i n quantum systems i s s e n s i t i v e l y d e p e n d e n t on t h e b a t h dependence o f t h e system-bath c o u p l i n g / 4 / and i s w e l l u n d e r s t o o d o n l y when t h i s c o u p l i n g i s l i n e a r i n t h e b a t h v a r i a b l e s 14-81. We r e s t r i c t o u r s e l v e s t o such c o u p l i n g s .

I n t h e f o l l o w i n g we c o n s i d e r t h e H a m i l t o n i a n f o r a dimer i n a phonon b a t h ,

-6

t

w h e r e i n a : ( a i ) c r e a t e s ( a n n i h i l a t e s ) and e x c i t o n on s i t e i ( i = 1 , 2 ) , and bq (bg) c r e a t e s ' ( a n n i h i l a t e s ) a phonon i n mode q. I n s t e a d of f o r m u l a t i n g a t h e o r y I n t e r m s of t h e d e n s i t y m a t r i x , we f o c u s on t h e e v o l u t i o n of t h e H e i s e n b e r g o p e r a t o r s f o r t h e e x c i t o n s and phonons. Such a n approach f o r g o e s some o f t h e

a d v a n t a g e s of a d e n s i t y m a t r i x a p p r o a c h , p r i n c i p a l l y t h e s i m p l i f i c a t i o n of d e a l i n g w i t h e q u a t i o n s of m o t i o n i n a reduced s p a c e ; however, t h e c l a r i t y g a i n e d by d e a l i n g w i t h more e l e m e n t a r y o p e r a t o r s c a n c e l s t h i s d e b t . The e q u a t i o n s o f motion f o r t h e H e i s e n b e r g o p e r a t o r s t a k e t h e form o f quantum Langevin e q u a t i o n s .

The o p e r a t o r s a and b e v o l v e a c c o r d i n g t o t h e coupled s e t of H e i s e n b e r g e q u a t i o n s o f motion

( f o r a i n t e r c h a n g e 1 and 2 ) . The phonon e q u a t i o n b e i n g l i n e a r i n t h e b ' s c a n b e f o r m a l l y i n t e g r a t e d and u s e d t o c l o s e t h e s y s t e m o f e q u a t i o n s i n t h e e x c i t o n oper- a t o r s o n l y . The r e s u l t i n g e q u a t i o n i s non-local i n t i m e and n o n - l i n e a r i n t h e ex- c i t o n o p e r a t o r s , and may b e w r i t t e n

( t ) = - a t

-

i

1

["ljkR(o)

% t

( t ) a , ( t )

- 4

( 0 ) a , ( o ) l a j ( t )

1 2

j ,k, R

w i t h t h e d e f i n i t i o n s

The d e t a i l e d i n t e r p r e t a t i o n of each t e r m i n (5) i s g i v e n e l s e w h e r e / 6 / . Here we wish

I f t h e b a t h i s i n i t i a l l y i n e q u i l i b r i u m a t temperature T t h e n t h e f l u c t u a t i o n s a r e Gaussian,zero-centered, and have t h e c o r r e l a t i o n f u n c t i o n s

The i n t e g r a t e d term i n Eq. ( 5 ) (when s W e t r i z e d / 6 / ) i s t h e n o n l i n e a r d i s s i p a t i v e con- t r i b u t i o n t h a t b a l a n c e s t h e m u l t i p l i c a t i v e f l u c t u a t i o n s F i j ( t ) a j ( t ) (when a l s o sym- m e t r i z e d / 6 / )

-

I n f a c t , Eqs. ( 7 ) and (8) t o g e t h e r c o n s t i t u t e a quantum f l u c t u a t i o n - d i s s i p a t i o n r e l a t i o n 14-61. I n t h e h i g h temperature l i m i t one r e c o v e r s t h e c l a s s i - c a l FDR/4/

r e g a r d l e s s of t h e time s c a l e over which KijkL(t) decays t o zero. A t f i n i t e temper- a t u r e s , t h e f a c t o r coth

( %

R6u ) causes t h e f l u c t u a t i n g o p e r a t o r s Fij ( t ) t o p o s s e s s a f i n i t e c o r r e l a t i o n time which may be s u b s t a n t i a l l y g r e a t e r t h a n t h e time 9

s c a l e over which t h e KijkQ(t) decay. Indeed, i n t h e l i m i t t h a t d i s s i p a t i o n o c c u r s i n s t a n t a n e o u s l y , t h e c o r r e l a t i o n time of t h e f l u c t u a t i o n s i s f i n i t e and i n c r e a s e s w i t h d e c r e a s i n g temperature.

Assuming t h e d i s s i p a t i o n t o be i n s t a n t a n e o u s such t h a t KijkL(t) = KijkR6(t) ( w h i c h i s t h e Markov approximation i n t h e t h e o r y ) , t h e e q u a t i o n s become l i n e a r with t h e a d d i t i o n a l assumption t h a t t h e system i s occupied by a s i n g l e e x c i t o n a t a l l times. The l i n e a r i z e d e q u a t i o n t a k e s t h e form

( t ) = - i J a 2 ( t ) - i

1

^F . ( t ) a . ( t )

1 j 13 3 (10)

which resembles t h e r e s u l t o b t a i n e d from a Haken-Strobl-Reineker s t o c h a s t i c Hamil- t o n i a n / l / . However, t h i s resemblance i s s u p e r f i c i a l : t h e f l u c t u a t i o n h e r e i s an o p e r a t o r w i t h n o n - t r i v i a l commutation p r o p e r t i e s , s o t h a t even t h i s l i n e a r equation provides a g e n e r a l i z a t i o n of previous s t o c h a s t i c t r e a t m e n t s of t h e problem.

Equations of e v o l u t i o n may b e s i m i l a r l y developed f o r t h e q u a d r a t i c o p e r a t o r s

aij

= a 1 a j and s o l u t i o n s obtained v i a time-ordered cumulant expansions. T r u n c a t i o n a t t h e second cumulant y i e l d s t h e weak-coupling s o l u t i o n s of t h e l i n e a r i z e d e q u a t i o n s f o r

a

i n t h e d e l o c a l i z e d b a s i s

l+>

= ( I I > f 1 2 > ) / f i :

< a + + ( t ) > : exp(-Qot) --

++

[l-exp(-Qot)] [ l + tanh (6J)I [<o,+(o)'+<a--(0)>] ('la)

from which i t i s c l e a r t h a t t h e expected number of e x c i t a t i o n s i n each s t a t e ap- proaches t h e proper thermal e q u i l i b r i u m v a l u e . (The c o e f f i c i e n t s Q,, A , and

r

^{a r e}

temperature dependent and g i v e n e x p l i c i t l y i n terms of t h e parameters appearing i n t h e Hamiltonian; c f . r e f . 6) The d e n s i t y m a t r i x e q u a t i o n s corresponding t o e q u a t i o n s (11) i n t h i s weak coupling l i m i t a r e o b t a i n e d through t h e correspondence pmn(t) = Tr[W(O)a,(t)] wherein W(0) i s t h e i n i t i a l d e n s i t y m a t r i x f o r t h e composite system comprised of t h e dimer and bath, and Tr i n d i c a t e s t h e complete t r a c e over t h e dimer B bath space. The r e s u l t i n g e q u a t i o n s are171

with p-- = -p,+, p+ = p+-* and c o e f f i c i e n t s given by

y = 2g1212J[coth(BJ)

+

^{l ] , y ' =} Z K ~ ~ ~ ~ . J [ C O ~ ~ ( R J ) - l ]

C7-44 JOURNAL

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PHYSIQUE

E q u a t i o n s of t h e form (12) a r e a common r e s u l t of t h e o r i e s based o n reduced d e n s i t y m a t r i c e s l l , 7-111. These t h e o r i e s d i f f e r a t t h i s l e v e l of approximation o n l y i n t h e v a l u e s t a k e n by t h e c o e f f i c i e n t s and i n t h e p h y s i c a l i n t e r p r e t a t i o n s g i v e n t h e v a r i o u s t e r m s . The r e s u l t

n ' ⁼

0 a p p e a r s i n t h e weak-coupling l i m i t s i n c e

n l / n

i s o f t h e o r d e r o f t h e c o u p l i n g of t h e e x c i t o n t o t h e h e a t b a t h . T h i s d i f f e r e n c e i n o r d e r of

n

and

n '

c a n be shown t o f o l l o w a s w e l l i n t h e weak c o u p l i n g l i m i t s of o t h e r t h e o r i e s based on a dynamical d e s c r i p t i o n o f t h e b a t h / 8 , 1 0 , 1 1 / . Moreover a d i r e c t c a l c u l a t i o n o f t h e o p t i c a l spectrum from e q u a t i o n s (12) shows t h a t

n ' / n

must v a n i s h a s T - t O i n o r d e r t h a t t h e spectrum remain p o s i t i v e d e f i n i t e , a s r e q u i r e d on p h y s i c a l g r o u n d s / 7 / .

The o p t i c a l spectrum which f o l l o w s from t h e above weak-coupling r e s u l t s con- s i s t s of two L o r e n t z i a n s c e n t e r e d a t t h e s h i f t e d p o s i t i o n s w = f 2 J weighted w i t h Bose f a c t o r s a t t h e ambient t e m p e r a t u r e :

w h e r e i n Z i s t h e p a r t i t i o n f u n c t i o n w i t h r e s p e c t t o J . The w i d t h s o f t h e s e p a r a t e l i n e s a r e g i v e n by t h e decay r a t e

I'

o f t h e o f f - d i a g o n a l d e n s i t y m a t r i x e l e m e n t s and a r e i n d e p e n d e n t o f t h e r a t e s of e n e r g e t i c e q u i l i b r a t i o n y , y ' .

The g e n e r a l i z e d m a s t e r e q u a t i o n i m p l i c i t i n Eqs. (12) and (13) may be s t a t e d t

f ( t ) =

-

²

/

d ~ W(t-T)P(T) (15)

0

i n which P = P I , - P2, and t h e t r a n s i t i o n k e r n e l W =

W12

=

WZ1

i s g i v e n by

T h i s memory k e r n e l h a s t h e two-channel form f a m i l i a r from t h e l i t e r a t u r e r e l a t i n g t o t h e SLE 12,121. The appearance o f t h e same parameter

I'

as t h e c o e f f i c i e n t of 6 ( t ) and t h e r a t e c o n s t a n t o f t h e e x p o n e n t i a l component r e c a l l s t h e form of t h e SLE memory due t o n o n - l o c a l f l u c t u a t i o n s o n l y ; however, t h e p a r a m e t e r i n v o l v e s c o n t r i b u t i o n s from b o t h l o c a l and non-local i n t e r a c t i o n s w i t h t h e b a t h . T h i s i s c o n s i s t e n t w i t h dynamical a p p r o a c h e s t o t h e problem based o n e q u a t i o n s f o r d r e s s e d e x c i t a t i o n s / 9 / .

Phase r e l a t i o n s between f l u c t u a t i o n s a t d i f f e r e n t s i t e s have been i g n o r e d i n t h e c a l c u l a t i o n s d e s c r i b e d h e r e i n . Such r e l a t i o n s b r i n g a b o u t memory e f f e c t s due t o t h e p r o p a g a t i o n of b a t h modes a t f i n i t e v e l o c i t i e s . Memory e f f e c t s i n t u r n p r e v e n t t h e e x a c t l i n e a r i z a t i o n o f t h e dynamical e q u a t i o n s . The n a t u r e of t h e r e s u l t a n t non- l i n e a r dynamics w i l l b e e x p l o r e d e l s e w h e r e .

REFERENCES

/ I / R e i n e k e r , P., i n E x c i t o n Dynamics i n M o l e c u l a r C r y s t a l s and A g g r e g a t e s , S p r i n g e r T r a c t s i n Modern P h y s i c s , Vol. 94 ( S p r i n g e r V e r l a g , B e r l i n 1982).

/ 2 / Kenkre, V. M . , i n E x c i t o n Dynamics i n M o l e c u l a r C r y s t a l s and A g g r e g a t e s , S p r i n g e r T r a c t s i n Modern P h y s i c s , Vol. 94 ( S p r i n g e r V e r l a g , B e r l i n 1982).

/ 3 / West, Bruce J. and Lindenberg, K a t j a , i n P r o c e e d i n g s of t h e Workshop on F l u c t u - a t i o n s and S e n s i t i v i t y i n Nonequilibrium Systems, ( S p r i n g e r V e r l a g , B e r l i n 19841, ed. by V. Kondepudi and W. Horsthemke, and r e f e r e n c e s t h e r e i n .

/ 4 / C o r t g s , E., West, B. J . , and L i n d e n b e r g , K., J . Chem. Phys.

82

(1985) 2708.

/ 5 / L i n d e n b e r g , K a t j a and West, Bruce J., Phys. Rev. A s (1984) 568.

/ 6 / West, Bruce J. and Lindenberg, K a t j a , t o be p u b l i s h e d i n J. Chem. Phys.

/ 7 / Brown, David W . , Lindenberg, K a t j a and West, Bruce J . , t o be p u b l ~ s h e d i n J.

Chem. Phys.

/ 8 / Grover, M. and S i l b e y , R., J . Chem. Phys.

2,

(1971) 4843.

/ 9 / Rakovsky, R. and S i l b e y , R., Mol. Phys.

2

(1973) 61.

/ l o / Rahman, T. S., Knox, R. S. and Kenkre, V. M . , Chem; Phys.

2

(1979) 197.

/11/ Munn, R. W. and S i l b e y , R., Mol. C r y s t . L i q . C r y s t .

11

(1980) 131 1121 Kenkre, V. M., Phys. Rev. B

12

(1975) 2150.