# Advances in chemical physics - Prigogine I.

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Until now there have been no simulations done on liquids whose constituents possess internal degrees of freedom. We have therefore undertaken a series of computer studies of the simplest liquids of this type: liquids made up of the diatomic molecules carbon monoxide and nitrogen. There were a number of compelling reasons for making these studies:

(1) To obtain a realistic and detailed picture of how individual molecules rotate and translate in these classical fluids.

ON THE CALCULATION OF TIME CORRELATION FUNCTIONS

69

(2) To examine in detail some of the time-correlation functions that enter into the theories of transport, light absorption, and light scattering and neutron scattering.

(3) To see how well simulations based on various proposed potentials reflect physical reality.

(4) To test various stochastic assumptions for molecular motion that would simplify the N-body problem if they were valid. Molecular dynamics is far superior to experiment for this purpose since it provides much more detailed information on molecular motion than is provided by any experiment or group of experiments.

These studies are to be regarded as “experiments” which probe time-correlation functions. They provide the raw data against which various dynamical theories of the liquid state can be checked. These studies provide insight into the microscopic dynamical behavior of real diatomic liquids for both the experimentalist and theoretician alike.

There have been a number of attempts to calculate time-correlation functions on the basis of simple models. Notable among these is the non-Markovian kinetic equation, the memory function equation for time-correlation functions first derived by Zwanzig33 and studied in great detail by Berne et al.34 This approach is reviewed in this article. Its relation to other methods is pointed out and its applicability is extended to other areas. The results of this theory are compared with the results of molecular dynamics.

Linear response theory is reviewed in Section II in order to establish contact between experiment and time-correlation functions. In Section III the memory function equation is derived and applied in Section IV to the calculation of time-correlation functions. Section V shows how time-correlation functions can be used to guess time-dependent distribution functions and similar methods are then applied in Section VI to the determination of time-correlation functions. In Section VII a succinct review is given of other exact and experimental calculations of time-correlation functions.

П. LINEAR RESPONSE THEORY

A. Linear Systems

When a system of molecules interacts with a weak radiation field the i nteraction Hamiltonian in the dipole approximation is

H' = — Г d3rM(r) • E(r, 0

V

(1)

70

В. J. BERNE AND G. D. HARP

where E(r, t) is the classical electric field at the space-time point (r, t) and M(r) is the electric polarization at the point r.

М(г) = £ цт 5(r - rm)

m

Here Ц m is the electric dipole operator and г m the center of mass position of molecule m. The Hamiltonian can also be written as

H' = - £ • E(rm, 0

m

There is a completely analogous development for a system of nuclear spins interacting with a time-dependent magnetic field polarized along the x axis.

It is a fact that when a system interacts with a weak probe the interaction Hamiltonian can often be written as

H' = - j d3rfi(r)F(r, t) (2)

Here

= (3) with Bm a molecular, property and rm the position of particle m. F{r, t) is a field which acts on the property Ё(г) at the space-time point (r, t), much as the electric field at the space-time point (r, t) acts on the dipole moments in the neighborhood of the point r. F(r, t) depends only on the properties of the probe. [ ]+ denotes the anticommutator.

More generally there may be a set of different forces ^((r, t) acting on the molecular system so that

H' = -JJdVi,(r)F,(r,0

i

This form of the interaction potential between a system and a probe is quite ubiquitous. We shall therefore restrict our attention to the study of how a system responds to the adiabatic turning on of a Hamiltonian of the form given by Eq. (2).

It is convenient to assume from the outset that in the absence of the probing field Fthe expectation value of the observable Ё is zero. In the presence of the probe F, <5> is in general not zero, because the system is “ driven ” by the force F. This also applied to other properties of the system which in the absence of the probe are expected to be zero. The perturbation thus “induces” certain properties of the system to take on nonzero expectation values. If the perturbation is sufficiently weak it produces a

ON THE CALCULATION OF TIME CORRELATION FUNCTIONS

71

linear response in the system. In the linear regime, doubling the magnitude of F simply doubles the magnitude of the induced responses. A simple example of linear response is Ohm’s law.

J = <7 E

according to which the current induced in a medium is linear in the electric field E (although not necessarily in the same direction as E because of possible anisotropies in the conductivity tensor a).

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