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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 Zwanzig technique75 proceeds as follows: the variables т and x are so defined that
S = k2x
1 (381)
, = ¥T
Introducing the reduced variables x and x into Eq. (380) leads to
F^=Udxe”x+hlk(k>x) (382)
The long-time behavior of Fs(k, t) is simply
= lim idxeXT A-fti(383)
2m k2^o J x + $Ак(к*х)
X, т const
where the limit is taken such that к -> 0 whereas t -> oo and S -> 0 in such a way that x and т remain constant, k2 acts like a parameter of smallness. It follows that the long time behavior of ^(k, t) is
Fs{k, t) = cj> dx ex~
2ni J x + iAo(0)
Fs(k, t) = е_1/ЗЛо(0)т = e—i/3Ao(o)k2t
Ao(0) = lim lim Г dt e~s\\ eik ’ *| ei(1 ~Рг)и |V eik ‘ *>
S->0 k2^о •'o
From the definition of the projection operator it is easy to evaluate the к2 -> 0 limit. Then
Л0(0) = f dt <V(0) • V(/)> (384)
D-Jf Л(У(»)-У(<)> (385)
J Jo
Fj(k, () = e-*1£* (386)
Thus from the memory function equation we have succeeded in showing that Fs(k, t) satisfies the diffusion equation
-Fs(k,t)= -k2DFs(k,t) (387)
at long times with a diffusion coefficient D. Moreover, we have succeeded in showing that the diffusion coefficient is determined by the velocity autocorrelation function according to Eq. (385). This is a simple example of a Kubo relation.11
In order to apply the memory function formalism to the collective coordinates of Eq. (367), it is necessary to define the dimensionless normalized collective coordinates,
l^ifc) = (PfclPfc) 1/21 РкУ
|C/«> = |e,k-'>
where the classical scalar product is intended. The structure factor S(k) is defined as .F(k, 0), or
S(k) = <p»|p»> = i/ £ e* !"-'-1) (389)
iV \i,m= 1 /
From the preceding formulas we see that the structure factor is related to the pair correlation function
4izn л00
S(k) = 1 + —— Г dr g(2\r)r sin kr (390)
к •'о
The intermediate scattering functions can be expressed in terms of the normalized properties \ Ulk) and | C/2fc>
Жк,/) = ад<Р,*|еш|!7и> (391)
Fs(k,0 = <^2i|e'"|C/n> (392)
F(k, t) and Fs(k, t) consequently satisfy memory function equations with corresponding memories
= (393) where \Ulk} = iL\Uiky, \U2k) — *£|^2fc>> and the projection operators are Plk = and P2k = l^2kX^2fcl>
Consider first the memory function equation for ^(k, t). From Eqs. (167) and (168) it is seen that the short-time behavior of the memory function Ф2*(0 is
®2t(0 = \ *2<У> - j Цо2)2*4 + 5<a2>*2
To second order in the momentum transfer it can be shown that
Ф2*« = 5<1>2>*2Ф(0 + 0(**) (395)
where v[/(0 is the normalized velocity autocorrelation function. Thus for sufficiently small values of k,
Ф2*(0Ц<»2>*2Ф(0 (396)
To get some idea of the values of к for which this approximation may be valid, let us look at the second term in the short-time behavior of Ф2*(0-Note that the term of order A:4 can be neglected if к is such that
,2/24 3 <а*У
or for our conditions к The interesting feature of the approxi-
mate memory function in Eq. (396) is that it will lead to a non-Gaussian Gs(r, t), and may thus provide an approximate method for determining the self Van Hove correlation function, Gs(r, t), for intermediate values of к when it is known that this function deviates from Gaussian behavior. It should be noted that this approximation correctly gives the initial time dependence of <AЯс.м2п(0У only for « = 1, whereas the Gaussian approximation correctly accounts for all of these moments at short times, and the diffusion approximation fails completely at short times.
Fs{k, t) for different values of к is presented in Figure 43; these functions were evaluated using the series expansion for Fs(k, t) discussed in Appendix С and the coefficients a N(t) from the modified Stockmayer simulation. The memory functions for these intermediate scattering functions are presented in Figure 44. Both of these memories were computed using the numerical method outlined in Appendix B. The absolute error in recovering Fs(k, t) from Ф2к(0 was ^0.005 for all times t < 10“12 s. Note that although the two scattering functions are quite different, their normalized memories are very similar. Note further that these normalized memories resemble the velocity autocorrelation function for this simulation (see Figure 10). In addition the approximate memory function, Eq. (396), is used to compute approximate intermediate scattering functions, Fs(k, t). The results of this approximate theory are presented along with the corresponding experimental functions in Figure 47. Note that this approximate theory is better than the Gaussian Gs(r, t) for intermediate values of k.
There is another approach to the problem of determining ^5(к, t). Note that,
Cs(k, 0 = <k • v eik,r |e‘Lt | к • v e(k-r> (397)
Cs(k, 0) = <k • v | к • v> = $k2(v2>
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