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Advances in chemical physics - Prigogine I.

Prigogine I., Rice S.A. Advances in chemical physics - Advision, 1970. - 167 p.
Download (direct link): advecioninphysical1970.djvu
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The same arguments can be developed for a system which is in a uniform external magnetic field, B. The eigenvectors cannot be made real in this case and the wave functions have the property i|/*(l?) = v|/(—B). It then follows that
Xab(к, ю, В) = Ав%вл(к, о,-В) (41)
ON THE CALCULATION OF TIME CORRELATION FUNCTIONS
83
These symmetry relations are called the Onsager reciprocal relations. Their meaning is best illustrated by reference to the following problem. Suppose the interaction Hamiltonian is
H'(0= -£JM<i№(r,0
i
It then follows from completely equivalent arguments that <B,M> = E Г Л'Ф;,(к, I - ОДкО')
I " — 00
where Фд(к, t) is the after-effect function that relates the jth response at time t to the /th field at all previous times. The reciprocal relations then give
Фл(к, t) = XjXfiuCk, t)
or
Xjii% со) = АДд,/к, со)
According to these relations the response <2?,) produced by a unit pulse of Ft is identical except for sign to the response <2?f> produced by a unit pulse of Fj.
D. The Fluctuation-Dissipation Theorem
The lineshape of the power dissipation function, Q{к, со), is determined by the imaginary part of the susceptibility, %" BB(K a>) as was shown in the previous section.
6(k, со) = V | %"BB(k, со) |FJ2
Since many important dynamical properties of a many-body system are explored through precise lineshape measurements, it is worth studying some of the properties of %"вв(к, со).
According to Eq. (14)
Х*вв(к, со) = хвв(-к, -со) = I dtе~шФвв(—к, t)
Jo
Transforming from tto—t and substitution of Eq. (14) leads to
5C*BB(k, со) = - f ° dt е‘“'Фвв(к, 0 (42)
J - m
84
В. J. BERNE AND G. D. HARP
The imaginary part of the susceptibility can now be found from the difference %BB-%*BB,
Х’в^к, = e'"<IA(<). B.J_> (43)
The quantity <[2?k(7); B_ J_) can be expanded in terms of the complete set of energy eigenstates \n),
<[#k(0, #-k]-> = Z (Pn ~ Pm)(n \Bk\ m)(m \B_k\ n) exp (i(onm t)
nm
where p„ is the Boltzmann factor Q_1 exp (—(3J5r„), co„m = (En — Em)/h, and E„ is the «th energy eigenvalue. The right-hand side of this equation can be simplified by expressing the difference between the Boltzmann factors as
P„ - P„ = [1 - e-s*“™]p„
Substitution of the resulting equation into %"вв(к, ю) and subsequent use of the definition of the delta function yields
X"BB(к, ю) = \ £ [1 - <Грй“'""]р„(и \Вк\ m)(m \В_к\ n) 5(co + co„m)
И nm
Thus from the properties of the delta function it is seen that
Х"м(к, со) = j (1 - e-l«”) f ” dt e'“<Bk(0B-»(0)> (44)
ft — 00
The quantity <2?k(*)2L k(0)> appearing in the integral is a “one-sided ” quantum-mechanical time-correlation function. The quantum-mechanical cross-correlation function CAB(k, t) of the dynamical variables Ak, Bk is defined as
CAB(Kt) = Q[Ak(t),B_k(0))+> (45)
where [ ] + denotes the anticommutator, or symmetrized product,
[a, P]+ = ap + (3a From the preceding section it is easily shown that
(i) CAB(k,t) = CBA(-k,-t)
(ii) CAB(Kt) = yAyBCAB(+k,-t)
(iii) C*AB(Kt) = CAB(-k,t) = yAyBCBA(k,t)
(iv) CAB(k,t) = £a£bCab(~k> 0
ON THE CALCULATION OF TIME CORRELATION FUNCTIONS
85
Combining the results leads to the relations
Слв(к» 0 = CbaQl, t) I
The autocorrelation function satisfies the following relations
^вв(^) 0 = Свв(~k, 0 ~ Свв(к, o = CBB( k,1)
(46b)
and
(46c)
С* вв( k>0 — Свв(к»0
The autocorrelation function is a real even function of к and t.
The Fourier transform, GAB(к, со) of the time-correlation function
plays a very important role in linear response theory. Expansion of GBB(к, ю) in the energy representation and repetition of the same steps that resulted in Eq. (44) leads to the result
This equation is now used to eliminate the integral on the right-hand side of Eq. (44) so that
Since %вв(к, ю) = %вв(-к, со), it follows that CBB(k,t) = CBB(-k,t). Substitution of Eq. (50) into Q(k,a>) yields
The power dissipation is linearly related to GBB{к, со) which is called, for obvious reasons, the power spectrum of the random process Bk. It should be noted that the energy dissipated by a system when it is exposed to an external field is related to a time-correlation function CBB(k, t) which describes the detailed way in which spontaneous fluctuations regress in the equilibrium state. This result, embodied in Eq. (51), is called the fluctuation
Cab(k) O’
— 00
(47)
GM(k,a.) = i[l+ e-l«”] Г *e'“<B,(0B.t(0)> (48)
J — Ю
— 00
(49)
or
Х"вв(к, со) = ^ tanh J
— 00
,00
dte^KBk(t),B_k( o)]+> (50)
(51)
86
В. J. BERNE AND G. D. HARP
dissipation theorem.38 It is a direct consequence of this theorem that weak force fields can be used to probe the dynamics of molecular motion in physical systems. A list of experiments together with the time-correlation functions that they probe is presented in Table I. An experiment which
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