# Advances in chemical physics - Prigogine I.

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The total directional-correlation function is determined by averaging the above formula over all possible diffusional paths between collisions and all sets of deflection angles. The specific F*s and x’s may be quite different for each path. According to the model that we are studying, successive binary collisions are uncorrelated. Each diffusional step starts out with knowledge of the preceding diffusional steps, and each deflection angle is uncorrelated with successive or preceding deflection angles x^-i or lj+i- This assumed lack of correlation leads to the statistical independence of successive paths and deflection angles, and the resulting directional correlation are

0(0 = e-»°'t [<»o<cos XT>Y f dt, FT(t -(„)••• f dt, FT(t2 -

n=0 J0 J0

A,«= f"dt^h-tJF'itJ

n=0 J0 J0

166

В. J. BERNE AND G. D. HARP

where FT(t) and FR(t) represent the path averaged translational and rotational diffusional motions, and <cos % r> and <cos %R> are the averages of cos %T and cos x* over all collisions. The above formulas for D(t) and Dj(t) can be reduced to the integral equation.

D(t) = e"“ofFr(0 + co0<cos xr> f dtt e-“o(f-,l)Fr(f —

•'o

Dj(t) satisfies a completely analogous equation. This equation can be solved in terms of Laplace transforms,

B(s) = FT(s + (o0) + g>0 <cos tT)FT{s + сo0)D(s)

or

Likewise

ад. FT(.s+*o)

В A*) =

1 - co0<cos %TyFT(s + co0) FR(s + co0)

®0<cos %R)FR(s + co0)

Consistent with our model is the assumption that FT(t) and FR(t) are the ordinary weakly coupled or Brownian motion exponentials.

Fr(0 = e-P*lfl FR(t) = е-Ря*|г|

where P5 and ря5 are the translational and rotational friction coefficients due to the soft forces, Then FT(s) = (s + P)-1, FR(s) = (s + рл)-1. It follows directly that

D{t) = exp (-{P + co0[l - <cos %r>]}k|)

Dj(t) = exp (- {P7 + co0[l - <cos Xя)]}kl)

The Rice-Allnat model predicts an exponential directional correlation functions with time constants which are additive in the weak soft force and the hard force. When no soft forces are present D(t) reduces to

D(t) = exp (-coot1 ~ <cos ХГ>]0

which is precisely the form of the velocity correlation function discussed by Longuett-Higgins and Pople. These authors evaluated <cos xr> from the Boltzmann equation. Similarly Ds(t) turns out to be in the absence of soft forces,

Dj{t) = exp (-coot1 ~ <cos X*>]0

ON THE CALCULATION OF TIME CORRELATION FUNCTIONS

167

It is a trivial matter to evaluate <cos from the Boltzmann equation. This gives the rotational diffusion coefficient in gases as a function of pressure and temperature.

Thus our stochastic model predicts a monotonic decay of D(t) and Dj(t). This may be valid for gases, but it is incorrect for liquids. From our computer experiments we saw that there are negative regions in both D(t) and Dj(t) in liquids. Thus we must search for better models of the liquid state.

B. Memory Function Theory of Linear and Angular Momentum Correlations

The first attempt to account for the structure of the empirically determined velocity autocorrelation function using the memory function

was based on the simple ansatz that the memory function depends on a single relaxation time34; that is

where a is the reciprocal of the relaxation time, <й2> is the mean square acceleration, and <y2> is the mean square velocity of a labeled molecule. In this discussion computer-generated values of <я2> are used. Alternatively it is quite possible to determine <й2> over a narrow range of temperatures and densities using isotope separation data.

The single relaxation time approximation corresponds to a stochastic model in which the fluctuating force on a molecule has a Lorentzian spectrum. Thus if the fluctuating force is a Gaussian-Markov process, it follows that the memory function must have this simple form.64 Of course it would be naive to assume that this exponential memory will accurately account for the dynamical behavior on liquids. It should be regarded as a simple model which has certain qualitative features that we expect real memory functions to have. It decays to zero and, moreover, is of a sufficiently simple mathematical form that the velocity autocorrelation function,

*♦(0.

<a| ei(1-p)Lt|a> <^>

(287)

(288)

(289)

168

В. J. BERNE AND G. D. HARP

can be determined analytically from the memory function equation. That the exponential form of the memory function can never be the exact memory function follows from the fact that it has odd derivatives at the initial instant and, furthermore, it has moments, \i2n, which do not exist for n ^ 1. The corresponding power spectrum of the velocity will be non-Lorentzian with finite moments, v2n, for n < 1, and infinite moments for n > 1. It should be noted that this non-Lorentzian power spectrum is a considerable improvement over more traditional theories according to which the power spectrum of the velocity is Lorentzian (vida Brownian motion). A Lorentzian power spectrum has finite moments only for n = 0 whereas the exponential memory function leads to a velocity power spectrum which has finite moments for n< 1. It is therefore quite profitable to study the properties of the exponential memory.

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