Distribution and Correlation Functions

In Section 7.1 we have defined a stochastic process as a time series, z(Z), of random variables. If observations are made at discrete times 0 Zi Z2. t, then the sequence z(Z/) is a discrete sample of the continuous function z(Z). In examples discussed in Sections 7.1 and 7.3 z(Z) was respectively the number of cars at time Z on a given stretch of highway and the position at time Z of a particle executing a one-dimensional random walk. 

We can measure and discuss z(Z) directly, keeping in mind that we will obtain different realizations (stochastic trajectories) of this function from different experiments performed imder identical conditions. Alternatively, we can characterize the process using the probability distributions associated with it. P(z, Z) /z is the probability that the realization of the random variable z at time Z is in the interval between z and z +- dz. P2(z2t2 zi fi )dzidz2 is the probability that z will have a value between zi and zi + dz at Zi and between Z2 and Z2 -F t/z2 at t, etc. The time evolution of the process, if recorded in times Zo, Zi, Z2, - - , Zn is most generally represented by the joint probability distribution Piz t , , z iUp. Note that any such joint distribution function can be expressed as a reduced higher-order function, for example. 

As discussed in Section 1.5.2, it is useful to introduce the corresponding conditional probabilities. For example. 

In practice, numerical values of time correlations functions are obtained by averaging over an ensemble of realizations. Let z (Z) be the th realization of the random function z(Z). Such realizations are obtained by observing z as a function of time in many experiments done under identical conditions. The correlation function C(Z2, Zi) is then given by 

If the stochastic process is stationary, the time origin is of no importance. In this case Pi(zi,Zi) = Pi(zi) does not depend on time, while 7 2(z2Z2 ziZi) = P2(z2, Z2 — Zi zi, 0) depends only on the time difference AZ21 = Z2 — Zi. In this case the coiTelation function C(Z2, Zi) = C(AZ2i) can be obtained 

In the absence of time correlations, the values taken by z t) at different times are independent. InthiscaseP(z t z it i . .. zoto) = 

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The experimental and theoretical descriptions of liquid structure are most conveniently achieved in terms of distribution functions. This is because there is short-range structure in the liquid, but at large distances from the point of reference, the distribution of molecules is random. In this section, the fundamental aspects of distribution and correlation functions, especially, the pair correlation function, are outlined. 

On the other hand since the dynamics may be simulated at any chosen temperature or range of temperatures (rather than at 0 K, as is formally the case in static simulations), MD results may be processed to provide many experimentally interesting quantities—diffusion parameters, thermodynamic functions, radial distribution, and correlation functions—the last of which may be related to measured relaxation data. As a result the two techniques should be regarded as complementary. With the current rapid expansion in computing power, it is likely that at least some of the limiting caveats described will soon be eliminated. 

Distribution and Correlation Functions.—We consider a single spherical particle with position r and velocity v at time t in a concentrated dispersion of mean number density p. The distribution function measures the probability of finding a particle (the same or another particle) with position r" and velocity v" at time t". The osmotic equation of state is related to a time-averaged distribution function that depends on r alone, whereas the dynamic behaviour depends on time-dependent functions. A basic premise of statistical mechanics is that a time-average is equivalent to an ensemble average at fixed time the ensemble average is denoted by angular brackets (...). 

In this section we define the molecular distribution and correlation functions we need for die discussion of the e-liquid surface, and set out the more important equations that relate these functions to eadi other and to die thermodynamic properties. We restrict ourselves to states of equilibrium but not necessarily to homogeneous ones we therefore introduce at once an arbitrary extmnal potential Y which acts independendy on each molecule and whidi is a function of position x within a vessel of fitmd shiqie and volume V. The molecules are assumed to be without internal structure, of spherical symmetry, and all of one species. None of diese restricdons is important at this stage, but without than die notation becomes intolerably dumsy eadi can usually be lifted when die occasion arises. 

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