Revista Brasileira de Física, Vol. 17, n? 4, 1987
On Nonequilibriurn Many-Body Systems IV: Response Function Theory
R. LUZZI, A.R. VASCONCELLOS, A.C. ALGARTE
Instituto de Fkica Gleb Wataghin , Universidade Estadual de Campina, Caixa Postal 6165,
Campinas, 13081, SP, Brasil
and
A.J. SAMPAIO
Departamento,&
Fkica, Univenidde Federal do Ceará, 60000, Fortaleza, CE, Brasil
Recebido em '10 de abril de 1987
Abstract
A response f u n c t i o n t h e o r y f o r many-body systems a r b i t r a r i l y
away from e q u i l i b r i u m i s p r e s e n t e d . I t i s based on t h e n o n e q u i l i b r i u m
s t a t i s t i c a l o p e r a t o r method f u l l y d e s c r i b e d i n a p r e v i o u s a r t i c l e [~ev.
B r a s i l . F i s . 1 5 , 106 ( 1 9 8 5 ) l . We p r e s e n t a formal t h e o r y f o r e v a l u a t i o n
o f t r a n s i t i o n p r o b a b i l t i e s and t h e average v a l u e s o f dynamical quant i t i e s i n f a r - f r o m - e q u i l i b r i u m many-body systems under t h e a c t i o n o f
e x t e r n a 1 p e r t u r b a t i o n s . We a l s o d e r i v e a n o n e q u i l i b r i u m thermodynamic
Green's f u n c t i o n s a l g o r i t h m a p p r o p r i a t e f o r t h e c a l c u l a t i o n o f response
f u n c t i o n s and s c a t t e r i n g c r o s s s e c t i o n s i n terms o f a g e n e r a l i z e d f l u c t u a t i o n - d i s s i p a t i o n theorem f o r f a r - f r o m - e q u i l i b r i u m systems.
1. INTRODUCTION
The t h e o r e t i c a l approach t o t h e dynamic o f r e l a x a t i o n phenomena and d e t e r m i n a t i o n o f t r a n s p o r t c o e f f i c i e n t s has been
based f o r a
l o n g t i m e on t h e use o f Boltzmann e q u a t i o n s . Nevertheless,difficuIties
a r i s e when s c a t t e r i n g i s n o t r a r e , and when t h e e x p e r i m e n t a l c o n d i t i o n s
l e a d t h e system t o f a r - f r o m - e q u i 1i b r i u m condi t i o n s . The t h e usual t r a n s p o r t t h e o r y approach, based on p e r t u r b a t i o n a l methods s t a r t i n g f r o m an
e q u i l i b r i u m i n i t i a l c o n d i t i o n , becomes inadequate. These k i n d s o f s i t u a t i o n s i n v o l v e s t r o n g i r r e v e r s i b l e processes, b e g i n f r o m an a r b i t r a r y
n o n e q u i l i b r i u m s t a t e , a r e n o t always d e s c r i b a b l e i n terms o f
well
f i n e d energy i n t e r a c t i o n s , and a r e o f t e n n o n l i n e a r , n o n l o c a l
and non-
-markoffian.
de-
A v i a b l e t h e o r y should i n c o r p o r a t e t h e s e f e a t u r e s f r o m t h e
o u t s e t and nowadays t h e r e e x i s t severa1 approaches
which have been c l a s s i f i e d by Zwanzig.
( ~ f l .I
) .
to
Among
t h i s question,
them t h e
non-
e q u i 1i b r i u m s t a t i s t i c a l o p e r a t o r method (NSOM) d e s c r i bed i n I , deserves
p a r t i c u l a r i n t e r e s t , which a p p l i e s t o hamiltonian
systems
with
very
Revista Brasileira de Fisica, Vol. 17, n9 4, 1987
many degrees o f freedom.
It i s a prescription admitting
in
principle
a11 many-body problems under a l a r g e c l a s s o f i n i t i a l c o n d i t i o n s ,
whose foundations a r e simple enough t o be r e a d i l y tested.
and
I n I we de-
r i v e d a method based on a v a r i a t i o n a l p r i n c i p l e which a l l o w s one t o o b t a i n a f a m i l y o f NSO's, and t o r e t r i e v e as p a r t i c u l a r cases
Green-Mori and Zubarev,
those
of
F u r t h e r , t h e formalism p e r m i t s one t o d e r i v e
n o n l i n e a r g e n e r a l i z e d t r a n s p o r t equations t h a t d e s c r i b e t h e dynamicsof
r e l a x a t i o n processes t a k i n g p l a c e i n t h e system; t h i s t o p i c
con-
was
sidered i n 3.2
N o t i n g t h a t t h e u l t i m a t e goal o f t h e t h e o r y i s t o
provide
a
comprehension o f t h e u n d e r l y i n g physics r e l a t e d t o t h e r e l a x a t i o n phenomena t h a t can be evidenced i n experiments,
i t needs
w i t h a response f u n c t i o n theory; t h i s i s the s u b j e c t
to
of
be
coupled
the
present
article.
responses
The usual theory t o c a l c u l a t e 1 i near and nonl inear
t o mechanical p e r t u r b a t i o n s i s based on expansions i n terms
of
equi-
l i b r i u m c o r r e l a t i o n functions (of increasing ~ r d e r ) ~
As. i n i t i a l
con-
d i t i o n one takes t h a t o f e q u i l i b r i u m w i t h a thermal r e s e r v o i r , a n d n e x t
t h e e v o l u t i o n o f the system i s s t u d i e d as i f i t were i s o l a t e d from a11
e x t e r n a l i n f l u e n c e s a p a r t from t h e source f i e l d . But i n any experiment
the system r e c e i v i n g energy from the e x t e r n a l sources releases
it
i t s surroundings a t a c e r t a i n pace: a l t h o u g h a t t h e i n i t i a l mmen:
system was i n e q u i l i b r i u m w i t h a thermostat,
to
the
i s dis-
t h i s equilibrium
turbed by the mechanical p e r t u r b a t i o n s . Therefoke, such response funct i o n t h e o r y has i t s own r e g i o n o f a p p l i c a b i l i t y : i t
i s ' when
therrnal
action o f
p e r t u r b a t i o n s which a r i s e s i n the system as a r e s u l t o f the
t h e e x t e r n a l s t i m u l i can be neglected, and a l s o when the're a r e n o feedback mechanisms. Beyond t h e domain o f v a l i d i t y o f such t r e a t m e n t o f t h e
response f u n c t i o n two t y p i c a l s i t u a t i o n s can be d i s t i n g u i s h e d .
them i s when a mechanical p e r t u s b a t i o n i s superimposed on
an
One o f
already
f a r - f r o m - e q u i l i b r i u m system. The second one i s when a strongmechanical
p e r t u r b a t i o n a c t s on a system i n i t i a l l y i n e q u i l i b r i u m .
I n t h i s case a
s t r o n g departure from e q u i l i b r i u m f o l l o w s , and t h e r e a r e l a r g e
inter-
ference e f f e c t s between the mechanical and t h e accompanying thermal perturbations.
Revista Brasileira de Flsica, Vol. 17, no 4, 1987
I n the f i r s t case,
i n the absence o f t h e externa1 p e r t u r b a t i o n
i r r e v e r s i b l e processes develop i n t h e system which a r e d e s c r i b a b l e
in
terms o f e v o l u t i o n equations f o r a b a s i c s e t o f n o n e q u i l i b r i u m thermodynamic v a r i a b l e s , o r macrovariables f o r s h o r t ( c f . 1 and 3 ) . Since the
NSO formalism p r o v i d e s a seemingly powerful method t o o b t a i n a d e s c r i p t i o n o f the macroscopic s t a t e o f such systems,
i t i s reasonable t o p r o -
ceed, w i t h i n i t s framework, t o d e r i v e a response f u n c t i o n t h e o r y based
on c o r r e l a t i o n f u n c t i o n s i n t h e unpertu'rbed
nonequilibriwn s t a t e o f
t h e system. We have proposed an scheme o f t h i s type,4
which
we
de-
s c r i b e i n d e t a i l i n t h i s a r t i c l e i n subsection 2a and i n t h e f i r s t p a r t
o f s e c t i o n 3, adding new r e s u l t s and extensions.
2. NONEQUILIBRIUM CORRELATION FUNCTIONS
l n paper 1 we discussed a formal ism t o d e s c r i b e
the
macro-
scop i c s t a t e o f a n o n e q u i l i b r i u m system: a method t o generate
a
non-
equi 1 i b r i u m s t a t i s t i c a l o p e r a t o r p w ( t ) t h a t should correspond t o given
spec i f i c a t i o n s o f i t s macrostate was proposed, and
th-e
difficulties
i n h e r e n t t o the method, namely, e x i s t e n c e o f a c o n t r a c t e d d e s c r i p t i o n ,
i r r e v e r s i b i l i t y , d e f i n i t i o n o f i n i t i a l c o n d i t i o n s , and c o n s t r u c t i o n o f
a n o n l i n e a r t r a n s p o r t theory, were discussed. The connectionswith proj e c t i o n o p e r a t o r techniques were p o i n t e d o u t , and i n a r t i c l e
III
we
devised a n o n l i n e a r t r a n o p o r t t h e o r y w i t h i n t h e framework o f t h e NSOM.
F o l l o w i n g the method l e t us assume t h a t a s u i t a b l e macroscopic descript i o n o f a system a r b i t r a r i l y away from e q u i l i b r i u m ,
t o be submitted t o
a mechanical p e r t u r b a t i o n , i s provided by a b a s i s s e t o f m a c r o v a r i a b l e s
... ,Qn(t),
Qi ( t ) ,
which a r e t h e averages Q . ( t ) = T r { ~.p ( t )
o f a set o f
a
a w
P1,
Pn The q u e s t i o n o f completeness o f
the
b a s i s s e t o f v a r i a b l e s i s discussed i n a r t i c l e s I and I I I , a n d p a r t i c u dynamical q u a n t i t i e s
..., .
l a r appl i c a t i o n s aregiven in r e f s . 5 and 6. The NSO p (t) i s a
W
funct ional
o f t h e s e t o f dynamical q u a n t i t i e s P. and t h e thermodynamically conju3
gated s e t o f i n t e n s i v e v a r i a b l e s ~ . ( t ) ,j = 1,2,
J
...,n ; i t
describes the
e v o l u t i o n o f t h e system from a g i v e n i n i t i a l macroscopic s t a t e d e f i n e d
by a coarse- grained d i s t r i b u t i o n p
,t
p w ( t ) = exp
{jt0d t '
cg
(to)and i s g i v e n by (Cf. I )
w(t,tl;t,)iog
p
cg
(t+t';tl)l
Revista Brasileira de Física, Vol. 17, no 4, 1987
where
n
We recall that to is an initial time taken in such a way as to
ensure that the system has lost the rnemry o f the details of the micro-
scopic.motion in the first stages of evolution, t o > T after the sysP'
tem has been driven away from equilibrium (Cf.1). Then for t > T it is
v
assumed that a randomization of the microscopic state of the system has
occurred and the set of n variables Q. allows for a descrtption of the
3
macroscopic state of the system. The dimension of the contracted description depends on the scale of time + and therefore it is possible
~ i '
to define different stages of evolution of the system with decreasing
values of n. This has been discussed in articles I and I I I , and can be
seen at work in papers I 1 and V. The coarse-grained statistical operator P
to,^) defines an initial condition for the system after
the
cg
randomization process has taken place, from which the system evolves
under the action of its Hamiltonian and the restrictions imposed by the
function W. This function fixes the initial condition,produces irreversible evolution of the macroscopic state of the system, and makes the
method fully compatible with generalized irreversible thermodynamics,
once the nonequilibrium average value of the negative of the logari thm
of the coarse-grained statistical operator is identified.with the entropy function of general ized i rreversi ble thermodynamics (Cf i ) .
Finally we recall that the evolution of the macroscopic state of the system is described by the set of generalized nonlinear transport equations
described and discussed in article 1 1 1 .
Next, we study two questions: ( a ) a formal theory for evaluatim
of transition probabilities and (b) the calculation of response fun.ctions, both in far-from-equilibrium systems.
.
Revista Brasileira de Física, Vol. 17, n? 4, 1987
2a. Formal Theory o f S c a t t e r i n g f o r N o n e q u i l i b r i u m Systems
Consider t h e c o u p l i n g o f a system w i t h an e x t e r n a l probe.
and H be t h e H a m i l t o n i a n s o f t h e system and o f t h e e x t e r n a l
s
P
r e s p e c t i v e l y , and V t h e i n t e r a c t i o n energy. Then,given t h e t o t a l
H
f u n c t i o n a t time to, the s o l u t i o n o f Schrodinger's equation a t
Let
probe
wave-
t
time
is
I$(t)> = u(t,t,)
I$(to)>
(2)
where U i s t h e e v o l u t i o n o p e r a t o r s a t s f y i n g t h e e q u a t i o n
i r i au/at
and H = H0
+
V(H0 = H
S
+
= HU
HD) i s t h e
U(to,to)
= 1
,
t t a l Hamiltonian.
known p r o r e d u r e s i n s c a t t e r i n g t h e o r y ,
(3
Followingwell
t h e t i m e dependence a s s o c i a t e d
w i t h t h e u n p e r t u r b e d energy o p e r a t o r i s f i r s t removed by i n t r o d u c i n g t h e
o p e r a t o r U t such t h a t
U(t,to)
= Uo(t,to)U'(t,to)
,
(4)
where
and t h e n
The i t e r a t i v e method a l l o w s us t o o b t a i n t h e s o l u t i o n o f eq.(6)
i n the form
L
Considering a s c a t t e r i n g event i n v o l v i n g a t r a n s i t i o n o f
probe between s t a t e s
p ' t o p, w i t h energy t r a n s f e r Ew =
the
h
P
external
- hP ',,the
Revista Brasileira de Física, Vol. 17, n? 4, 1987
t r a n s i t i o n p r o b a b i l i t y a t time t from s t a t e ( + ( t o ) > = ( p l > l ~ ( t o ) > i s
where In> and Ip>, E and Fu, a r e t h e e i g e n f u n c t i o n s and
n
P
o f t h e system and probe Hamiltonians, i.e.
Replacing eq.
with
R
eigenenergies
(8) i n eq. (9) we f i n d
d e f i n e d by t h e i n t e g r a l equation
where we have introduced
eq. (1O) be r e w r i t t e n as
t
~ ( t =)
-L
R
t
j
Itodtl
dttl < $ ( t ~ ) l ~ ' ( t ~ - t ~ )1~$ ( to1 ) > e
t o
F i n a l l y , t a k i n g the average o f eq.
ensemble c h a r a c t e r i z e d by ~ ~ ( wet o) b t a i n
-iw ( & " - t i )
.
(13)
(13) over t h e nonequil i b r i u m
Revista Brasileira de Fisica, Vol. 17,
nP 4, 1987
and the r a t e o f t r a n s i t i o n p r o b a b i l i t y a t time
Q(wlt) =
d
I
t
k(t)lnv
=
A2
becomes
dtl e-iw(t'-t)Tr{~i(tl-t)~(0)
PW(t)
I + C.C.
to
Near e q u i l ibrium, f o r a p e r t u r b a t i o n a d i a b a t i c a l l y
t o =
-
appl i e d a t
and t o lowest o r d e r i n V , eq.(15) becomes
which reproduces well-known r e s u l t s f o r the temperature-dependent
rate
7
o f t r a n s i t i o n probabi l i t y .
F u r t h e r , assuming a d i a b a t i c appl i c a t i o n
beginning a t t o =
-
performed eq.(t5)
becomes
h*
t) = 1
with s
+
of
the
perturbation
and a f t e r the v a r i a b l e t r a n s f o r m a t i o n .r = t u - ti s
1'
dr e -i(utis)r
T~IR
i(.r)R(0)pw(t))
'
+
C.C.
, (16)
,a,
+O.
I n s p e c t i o n o f eq. (16) show us t h a t t h e r a t e o f t r a n s i t i o n prob a b i l i t y i n v o l v e s t h e c a l c u l a t i o n o f a time-dependen n o n e q u i l i b r i u m c o r r e l a t i o n o f o p e r a t o r s . Recall i n g t h a t [eq.
w i t h D d e f i n e d by eq.(26d)
(25) f f . i n
i n 1 , i t follows that
fi(t)
i s composedoftwo
p a r t s : one i n v o l v i n g o n l y t h e average over t h e c o a r s e - g r a bned ( r e l a x a t i o n f r e e ) ensemble, p l u s terms t h a t couple t h e mechanical e f f e c t s wi t h
t h e thermal e f f e c t s t h a t develop i n the n o n e q u i l i b r i u m system.
Hence,
u s i n g the r e s u l t s o f a r t i c l e I, we can w r i t e
n(wlt) = f i ( ~ l t )+ nl(wlt)
,
(17)
Revista Brasileira de Física, Vol. 17, no 4, 1987
where
t
=
1
-
h2
dí e - ( w i s ) r < ~ t ( ~ ) ~ ( o )
(t)
I
t >cg + C . C .
,
(1 ~ b )
-03
and eq. ( l 8 b ) a d m i t s a s e r i e s expansion o f i n c r e a s i n g o r d e r i n t h e s t a t -
is t i c a l -entropy production operator o f which
t ) i s a f u n c t i o n a l (Cf
Dw(
I and I I I and see eq. (44) i n s e c t i o n 3 below). F u r t h e r , i t rnust be k e p
i n mind t h a t t h e c a l c u l a t i o n o f eq.(16) depends on t h e s e t o f v a r i a b l e
F.(t)
J
which a r e obtained by s o l v i n g the generalized n o n l i n e a r transpoi
e q u a t i o n s w h i c h g o v e r n t h e i r e v o l u t i o n leqs. (9) and (10) i n I].
2b. Response F u n c t i o n Theory
L e t A be an o p e r g t o r a s s o c i a t e d w i t h a c e r t a i n dynamical quant i t y o f a system c o u p l e d t o an e x t e r n a 1 p e r t u r b a t i o n t h r o u g h an
energy
i n t e r a c t i o n V ( t ) . The average v a l u e o f t h i s q u a n t i t y A a t t i m e t i s
and u s i n g t h e
a ( t ) = <$(to)
T a k i ng t h e average v a l u e o f eq, ( 2 0 ) o v e r t h e nonequi
semble we !Ft
Revista Brasileira de Física, Vol. 17, n? 4, 1987
But because o f eq.
strength,
(8), i n
the l inear approxiniation i n the
interacti-on
i.e.
L
it follows that
where
Using t h e p r o p e r t i e s o f o p e r a t o r Uo,eq. (23) can be r e w r i t t e n as
Assuming a d i a b a t i c a p p l i c a t i o n o f t h e p e r t u r b a t i o n i n t h e r e mote p a s t ( s
-t
+O) we f i n d t h a t
T a k i n g i n t o account t h e s e p a r a t i o n o f p i n t h e f o r m p +pl,eq.
c9
(25) can be r e w r i t t e n i n t h e f o r m
where
Revista Brasileira de Fisica, Vol. 17, no 4, 1987
and, s i m i l a r l y t o t h e case o f e q . ( l 8 b )
[Alt]'
a d m i t s a s e r i e s expansion
i n terms o f i n c r e a s i n g o r d e r i n t h e s t a t i s t i c a l - e n t r o p y p r o d u c t i o n
r a t o r on w h i c h D ( t ) depends (see eqs.
ope-
(26d) and (26e) i n I and eq. (44)
W
i n the next section).
T h e r e f o r e , a c c o r d i n g t o s e c t i o n s 2a, b t h e c a l c u l a t i o n o f t r a n s i t i o n p r o b a b i l i t i e s and o f response f u n c t i o n s i n v o l v e
calculations of
nonequilibrium c o r r e l a t i o n functions. This i s i n general
a
di ff icult
mathematical t a s k , w h i c h can be more e a s i l y performed u s i n g t h e f o r m a l ism o f n o n e q u i l i b r i u m thermodynamic Green f u n c t i o n s d e s c r i b e d i n t h e n e x t
section.
3. NONEQUILIBRIUM THERMODYNAMIC GREEN'S FUNCTIONS
We d e s c r i b e i n t h i s s e c t i o n a Green's f u n c t i o n
formalism
for
n o n e q u i l i b r i u m s t a t i s t i c a l systems, which i s a n a t u r a l g e n e r a l i z a t i o n o f
t h e e q u i li b r i u m thermodynamic Green's f u n c t i o n f o r m a l ism.'
be two o p e r a t o r s i n t h e Heisenberg r e p r e s e n t a t i o n . We
L e t A and B
define
the
re-
t a r d e d Green's f u n c t i o n
+
where rl =
and
isfy
or
-
s t a n d s f o r a n t i c o m m u t a t o r o r commutator o f o p e r a t o r s A
B , and 0 i s H e a v i s i d e ' s s t e p f u n c t i o n . T h i s Green's f u n c t i o n s s a t the
equation o f motion
I n eq.(29)
and i n what f o l l o w s ,
@,a
w i t h o u t s u b s c r i p t i s t h e commutator
o f q u a n t i t i e s A and B.
I n t r o d u c i n g a F o u r i e r t r a n s f o r m o f t h e Green's f u n c t i o n (28) i n
Revista Brasileira de Física, Vol. 17, no 4, 1987
We also define the nonequilibrium correlation functions
,
(32a)
FBA(r;t) = <B A(T) It> = T~-{BA ( T ) P ~ ( ~1 ) ,
(32b)
F ~ ~ ( T ;=~ <)A ( T ) B J ~ >= T~-{A(T)B pw(t) 1
and using a complete set of eigenfunct ions of H , In> , wi th eigenenergies
E , we find
n
Introducing the nonequilibrium spectral density functions
and
we obtain that
and
Revista Brasileira de Flsica, Vol. 17, n? 4, 1987
with s
-t
+O.
Eqs. (36) a r e a g e n e r a l i z a t i o n o f t h e f l u c t u a t i o n - d i s s i p a t i o n
theorem t o systems a r b i t r a r i l y away f r o m e q u i l i b r i u m .
c o n d i t i o n s , where
Near e q u i l i b r l u m
p w ( t ) i s r e p l a c e d b y , t h e c a n o n i c a l d i s t r i b u t i o n , one
r e c o v e r s t h e w e l l known r e s u l t e
where @ = l / k T , and i t s h o u l d be n o t e d t h a t o u r $:(a)
JBA(*).
is
Zubarev's
8
We a r e now i n c o n d i t i o n s t o r e w r i t e t h e e x p r e s s i o n s
for
r a t e o f t r a n s i t i o n p r o b a b i l i t y and t h e response f u n c t i o n s o f s e c t i o n
i n terms o f n o n e q u i l i b r i u m thermodynamics Green's F u n c t i o n s .
eqs.
(32a),
(35a) and eq. (16) becomes
F i na 1 1y , because o f eq. (36a) we f i nd
the
2
First,using
Revistá arasileira de Flsica, Vol. 17, n? 4, 1987
the
interac-
t i o n s t r e n g t h ãnd any o r d e r i n t h e r a t e o f s t a t i s t i c a l - e n t r o p y
i.e.
t h e r a t e o f t r a n s i t i o n p r o b a b i l i t y , i n any o r d e r
produc-
tion,
i s g i v e n by t h e i m a g i n a r y p a r t o f t h e sum o f
in
t h e - r e t a r d e d non-
e q u i l i b r i u m thermodynamic Green f u n c t i o n s o f t h e s c a t t e r i ng o p e r a t o r
taken
w i t h c o m n t u t a t o r and a n t i c o m n u t a t o r . We r e c a l l t h a t .the Greeds
f u n c t i o n s o f eq.
(39)
depend on t h e macroscopic s t a t e
of
the
system
,
t h r o u g h v a r i a b l e s F .(t) and t h e r e f o r e t h e e q u a t i o n f o r t h e Green's f u n c 3
t i o n s must be s o l v e d i n c o n j u n c t i o n w i t h t h e g e n e r a l i z e d m n l i n e a r t r a n s p o r t e q u a t i o n s f o r t h e s e t f . ~ ~ ( t(Cf.3).
))
Next, g o i n g o v e r eq. (25), assuming an i n t e r a c t i o n
the form V(t) =
-Wt
-2 e
1
h B ( O ) + C.C.,
energy
where 1 i s a c o u p l i n g
of
constant
and B an h e r m i t i a n o p e r a t o r , we f i n d t h a t
=
h < < A ; ~ l w t i s , t > > - + C.C.
2iR
where we used t h e d e f i n i t i o n o f t h e r e t a r d e d Green's f u n c t i o n , eq. ( 2 8 ) .
Hence,
t h e l i n e a r response f u n c t i o n t o a harmonic p e r t u r b a t i o n
i s g i v e n by a r e t a r d e d n o n e q u i l i b r i u m thermodynamic Green's
function
dependent on t h e macroscopic s t a t e o f t h e system c h a r a c t e r i z e d bymacrovariables
~ . ( t( o)r & . ( t ) ) .
3
3
C l o s i n g t h i s s e c t i o n , we n o t e t h a t s i n c e t h e n o n e q u i 1 i b r i u m
Green's f u n c t i o n s o f eq, (28) a r e d e f i n e d as nonequi 1 i b r i u m averages
of
+ p ' , and t h a t p ' a d m i t s
Pcg
a s e r i e s expansion i n terms o f e v e r i n c r e a s i n g o r d e r i n t h e s t a t i s t i c a l
dynamical q u a n t i t i e s , r e c a l l i n g t h a t p
w
=
e n t r o p y - p r o d u c t i o n o p e r a t o r , we can wr i t e
whe r e
Revista Brasileira de Física, Vol. 17, n9 4, 1987
Writing
(44)
) :D
where
( t ) i s d e f i n e d i n 3, we o b t a
for
the
Green's f u n c t i o n s i n t h e form
w i t h t h e terms o f t h e series o n t h e r i g h t hand s i d e s a t i s f y i n g t h e equat ions
Whenever any t r u n c a t i o n p r o c e d u r e process
is
i'ntroduced
a p p r o x i m a t e l y s o l v e these e q u a t i o n s , c a r e should be taken
if
terms o f t h e same o r d e r i n t h e i n t e r a c t i o n s t r e n g t h s . For example,
we s t a r t w i t h eq.(46) where t h e t e r m i n v o l v i n g
to
t o maintain
@,H] leads, a s
it
is
w e l l known, t o an i n f i n i t e s e t o f c o u p l e d e q u a t i o n s o f e v e r i n c r e a s i n g
o r d e r i n t h e i n t e r a c t i o n s t r e n g t h s , a t r u n c a t i o n procedure t h a t c l o s e s
t h e system i n second o r d e r i n t h e i n t s r a c t i o n s t r e n g t h s
i n c l u s i o n o f t h e Green's f u n c t i o n s o f eqs.
requires
the
n
= 2,
(47) w i t h n = 1 and
b u t n e g l e c t i n g t h e i n t e r a c t i o n s i n t h e H a m i l t o n i a n i n eq.
(47) f o r n=2.
Revista Brasileira de Física, Vol. 17, nP 4, 1987
4. CONCLUDING REMARKS
We have shown t h a t t h e n o n e q u i l i b r i u r n
statistical operator
method described i n paper I p r o v i d e s the framework t o b u i l d a response
f u n c t i o n t h e o r y f o r many-body systems a r b i t r a r i l y awayfrom e q u i l i b r i u m .
The formalism provides responde f u n c t i o n s o r s c a t t e r i n g cross s e c t i o n s
f o r a measurement o p e r a t i o n which i n v o l v e s c h a r a c t e r i s t i c time i n t e r v a l s much longer than t h e r e l a x a t i o n time f o r microprocesses,
T
v'
i .e.
f o r time i n t e r v a l s t h a t a r e c o n s i s t e n t w i t h t h e c o n t r a c t e d d e s c r i p t i o n
o f t h e macroscopic s t a t e o f the system i n terms o f
dynamical q u a n t i t i e s Pj, j = 1,2,
...,n,
and
the
the b a s i c
set
of
n thermodynamically
conjugated v a r i a b l e s F . ( t ) . An accornpanying n o n e q u i 1 i b r i um t h e r m o d
dynamic Green's f u n c t i o n s a l g o r i t h m was described i n s e c t i o n 3 .
The t h e o r y developed i n t h i s paper can be considereda n a t u r a l
e x t e n s i o n of Kubo's formalism, and i t provides t h e l i n e a r an n o n l i n e a r
responses o f many-body systems a r b i t r a r i l y away
terms o f c o r r e l a t i o n f u n c t i o n s c a l c u l a t e d
in
frorn e q u i 1 i b r i u m i n
the
nonequi 1 i b r i u m
ensemble c h a r a c t e r i z e d by t h e MSO p w ( t ) . Thus, i t d e s c r i bes the e f f e c t
o f a mechanical p e r t u r b a t i o n on a system which i s e v o l v i n g i n an i r r e v e r s i b l e way governed by the n o n l i n e a r generalized t r a n s p o r t equations
discussed i n I I I . I n a forthcoming paper we a p p l y the r e s u l t s presented
here t o t h e study o f o p t i c a l responses i n h i g h l y e x c i t e d plasma i n semiconductors?
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-Body ProbZem,
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Revista Brasileira de Física, Vol. 17, n? 4, 1987
5.
A.C.Algarte,
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on
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H.
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A.
t o be p u b l i s h e d , and s h o r t communication
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s i c s of Semiconductors ( S tockho 1m, 1986) , ed i t e d by O . Eiigs t r6m ( ~ o r l d
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9. See a l s o A.J.Sampaio,
t h e s i s ( UNICAMP ,
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(1 983) ; A.R.Vasconcellos
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J.Raman Spectrosc.10,
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B27, 3874 (1 983) ; C.
Machado, M. Sc. t h e s i s (UNICAMP, 1985, unpubl i s h e d )
.
Resumo
E apresentada uma t e o r i a da função r e s p o s t a a p r o p r i a d a p a r a
t r a t a r sistemas de m u i t o s c o r p o s a r b i t r a r i a m e n t e a f a s t a d o s do- e q u i l í b r i o , c o n s t r u i d a usando o método do operador e s t a t í s t i c o de nao-equi1.ib r i o p r e v i a m e n t e d e s c r i t o ( ~ e v . B r a s i 1. F i s . l S , 106 ( 1 9 8 5 ) ) .
Derivamos
um a l g o r i t m o de funções de Green termodinâmicas de n ã o - e q u i l Í b r i o , assim como uma g e n e r a l ização, d e n t r o do arcabouço d e s t a s ú l t i m a s , do t e o rema de f l u t u a ç ã o - d i s s i p a ç ã o p a r a sistemas l o n g e do equ i 1 í b r i o .