# Advances in chemical physics - Prigogine I.

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•'o

f A*j(t) dt ~0.70 x 10-13 s while •'o

r00

\[/(0 dt

Jo

ro°

J A^ J0

0.96 x 10

-13

dt -0.57 x 10-13 s

О 5 10

-13

Time t in 10 s

Fig. 20. Memory functions for ф(0 from the Stockmayer simulation of CO. The approximate memories are based on <a2>/<v2> and Jo ф(0 dt.

0 5 10

-13

t in units of 10 s

Fig. 21. Velocity autocorrelation functions from the Stockmayer simulation of CO.

the exponential memory, and the Gaussian memory I.

183

184

В. J. BERNE AND G. D. HARP

In each case the integral of the approximate correlation function is larger than the integral of the experimental function. Also the difference between the integral of an approximate and the integral of an experimental function is proportional to the magnitude of the long-time behavior of the corresponding experimental memory. In these three examples the neglect of the tail in the experimental memory functions leads to a maximum error of ~23% in the integral of the resulting, approximate autocorrelation function.

Fig. 22. Memory functions for ф(0 from the Stockmayer simulation of CO. The approximate memory is based on <a2>/<w2> and /a2>/<w2>.

Finally, consider the power spectra of the experimental approximate correlation functions which are displayed in Figures 24, 29, and 34. Note that each of these spectra has been normalized to unity at со = 0. Note also that the experimental spectrum from the angular momentum correlation function is much broader than the experimental velocity autocorrelation power spectra. The power spectra of the Gaussian II autocorrelation functions are in much better agreement with the experimental spectra at all frequencies than the power spectra of the other approximate autocorrelation functions.

ON THE CALCULATION OF TIME CORRELATION FUNCTIONS

185

We conclude the following from the above discussion:

(1) The experimental memories for our velocity and angular momentum autocorrelation functions decay initially to approximately zero in a Gaussian fashion.

(2) This initial decay can be adequately approximated by knowing the 2nd and 4th moments of the corresponding autocorrelation function.

Fig. 23. Velocity autocorrelation functions from the Stockmayer simulation of CO, the Gaussian memory based on <a2>/<w2> and <a2}/<v2}, and the short time expansion

of «ко.

(3) The correlation function generated from this approximate memory gives a good approximation to the exact correlation function at least through this latter function’s first minimum.

(4) The power spectrum of this approximate correlation function is in fair to excellent agreement with the experimental spectrum at high frequencies (со $> 1013/s).

186

В. J. BERNE AND G. D. HARP

Fig. 24. Normalized power spectra of ф(0 from the Stockmayer simulation of CO, the exponential memory, and Gaussian memories I and II.

F. Elementary Excitations in Liquids

Many important properties of liquids, solids, and gases can be probed by scattering neutrons off the system in question. The differential scattering crossection in monatomic systems is related to the time Fourier transforms of the intermediate scattering functions3-5,8

F(k, r) = -J- ( £ e " *'r,(0) £ e,k ‘ '•»<'A (365)

N \ i m /

and

F(k, t) and Fs(k, t), it should be noted, are one-sided quantum-mechanical time-correlation functions. We shall be interested in the classical behavior

Time t in Ю ,3s

Fig. 25. Memory functions for ф(0 from the modified Stockmayer simulation of CO. The approximate memories based on <a2>/<w2> and J<? Ф(0-

t in units of 10 ,3s

Fig. 26. Velocity autocorrelation functions from the modified Stockmayer simulation

of CO, the exponential memory and the Gaussian memory I.

187

Fig. 27. Memory functions for ф(/) from the modified Stockmayer simulation of CO. The approximate memory function is based on <o2')/<v2} and <a2>/<i>2>.

TIME t in IO_l3s

Fig. 28. Velocity autocorrelation functions from the modified Stockmayer simulation

of CO, the Gaussian memory based on <a2>/<v2> and <a2>/<w2>, and the short time

expansion of ф(0-

188

ON THE CALCULATION OF TIME CORRELATION FUNCTIONS

189

13

w in 10 /s

Fig. 29. Normalized power spectra of ф(0 from the modified Stockmayer simulation of CO, the exponential memory, and the Gaussian memories I and II, and the continued fraction approximation of Eq. (356)

of these functions. The differential scattering cross section for neutrons turns out to be a linear combination of the two spectral density functions

£(k, со) = —— (+°° dtei(atF(k,t)

27Г * — oo

Ss(k,o>) = ^-J+”<fte!"fs(M)

(366)

The first function, £(к, со) contributes to the coherent scattering, and the second function, £5(к, со), contributes to the incoherent scattering of the neutrons. Neutrons consequently probe the spontaneous fluctuations of the property

Pk(0 =

1

m

1/2

Ее*-

ко

i=i

(367)

190 В. J. BERNE AND G. D. HARP

T- . • ,«-13

Time t in 10 s

Fig. 30. Memory functions for Aj(t) from the modified Stockmayer simulation of CO. The approximate memories are based on <jV2>/<J2> and Jo Aj(t) dt.

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