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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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To proceed it is necessary to evaluate the single relaxation time, a-1, which appears in Eq. (288). For this purpose it is important to note the relationship between the normalized velocity autocorrelation function and the self-diffusion coefficient, D,
D = K»2> Г dt ф(Г) = Ke2>fe (290)
J0
where \J/(0) is the Laplace transform, \J/(S), of \|/(f) evaluated at 5 = 0. Eq. (290) is the Kubo relation for the diffusion coefficient. It can be derived from the Einstein relation,
D = lim (291)
t-*00 61
in which AR(f) is the displacement of the tagged particle during the time t,
AR(0 = f dtx V(^)
•'o
Substitution of this into the Einstein relation yields
D = \ lim - f dt2 f dty <v(^) • v(f2)>
J t-*oo t •'o "O
The velocity is a stationary stochastic process so that
<v(?i) * v(f2)> = <v(0) • \(t2 - tj)} = <у2Ж'2 ~t\)
Substitution of this into Eq. (291), followed by an integration by parts, yields
Z) = <^lim f' <*,(1—^W,) (292)
3 t-*ao 0 \ 4
ON THE CALCULATION OF TIME CORRELATION FUNCTIONS
169
Л 00
If the integral I dtt ^хИЛ) exists, the limit can be taken so that
1 KT
0 = -02П(0) = — ^(0) (293)
3 M
The friction coefficient, y, is so defined that
KT My
KT
D = — (294)
from which it follows that,
Y-1 = W>) (295)
The Laplace transform of the memory function equation d\|/
~~dt=\0 ЛА*(ТЖГ_Т)
subject to the conditions \|/(0) = 1, \J/(0) = 0 leads to
>KS) = [S + ^(5)]-1 (296)
where is the Laplace transform of the memory function. It follows
directly that ([/(0) = [^(0)]-1, and consequently from Eq. (296) that
У = **(0) (297)
The exponential memory function has the property that
= (298)
from which it follows that
Y \<у2>/
(299)
Consequently if <az) and у are known, the single relaxation time, a 1, can be determined. In terms of the velocity autocorrelation function, a is
/;<**« ото)
Thus the single relaxation time approximation to the memory function is
{(a2s> г00 П
^ “Р - ^ fo ' (301)
170
В. J. BERNE AND G. D. HARP
To find the velocity-correlation function corresponding to this memory substitute Eq. (298) into Eq. (296) and then find the inverse Laplace transform
(S + a)
v ' (302)
*(£) =
Laplace inversion then yields
S(S + а) +<л2>/<у2>
i|/(0 =
1
[5+es"f-^_es+f]
S+ - S_
where S± are the roots of the equation [S2 + aS + <a2)/<y>2 = 0],
1/2
(303)
(304)
Depending on the value of <л2>, <y2>, and y, these roots can be real or complex. Explicitly, if
Z><2 0^[<a2>/<y>]1/2
the roots are complex and i|/(f) will oscillate. In this case
S± = — — [1 + * A,]
(305)
(306)
where X = [ — 1 + 4«a2)/<y2))a *]1/2. Then
\|/(0 = exp~“/2f |cos (y ^ ^ sin f j
The power spectrum of the velocity-correlation function is consequently
4 S+S_(S+ + SL)
G(co) =
(307)
(S+2 + (o2)(S- 2 + со2)
and goes asymptotically as 1/co4. This is why v2n does not exist for n ^ 2. The exponential approximation will be discussed later in this section.
This initial attempt to compute the time-correlation function was followed by a study of the Gaussian memory function with no significantly new results.67 The Gaussian memory, adjusted to give the correct diffusion coefficient, is found in exactly the same way as the exponential memory. It turns out to be
.<vz} К
dt' i|/(0
(308)
ON THE CALCULATION OF TIME CORRELATION FUNCTIONS
171
The major advantage of this memory function is that all of its moments are finite. The corresponding velocity correlation function cannot be determined analytically, but must be studied numerically. More will be said about this approximation later.
Prior to our computer experiments little, if indeed anything, had been reported about the full-time evolution of the angular momentum autocorrelation function of diatomic molecules in gases and liquids. The relaxation of nuclear spins is determined by the coupling of the spins to the rotational and translational motions of the molecules in the system. For nuclei with spin 1 /2, the spin-rotation interaction of a linear molecule leads to an interaction Hamiltonian of the form (— cl • J) where I is the spin angular momentum of the nucleus, J is the angular momentum of the molecule, and с is the spin rotation coupling constant. When this is the only part of the Hamiltonian leading to nuclear spin relaxation, the spin relaxation time, Tj, is
where co0 is the Larmour precession frequency.8 In liquids the angular momentum autocorrelation function decays on a time scale.of the order of 10"12 which is many orders of magnitude shorter than typical precessional periods (l/oo0 ~ 10”6 s). Thus the integral above is to an excellent approximation j*^ dt <J(0) • J(0>.
As we saw in the previous sections, the normalized angular momentum autocorrelation function, Aj(t),
satisfies the memory function equation with memory
where N is the torque acting on the molecule.
Consider the unit vector u(t) pointing in the direction of the molecular axis of a diatomic molecule at time t. The angle that this vector makes with u(0) is denoted by 0(0- According to Debye63 the rotational diffusion coefficient, DR, is
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