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The pair distribution function describes the distribution of distances between pairs of particles contained within a given volume. [1] Mathematically, if a and b are two particles, the pair distribution function of b with respect to a, denoted by () is the probability of finding the particle b at distance from a, with a taken as the origin of coordinates.
The auxiliary function () is known as the cavity distribution function. [5]: Table 4.1 It has been shown that for classical fluids at a fixed density and a fixed positive temperature, the effective pair potential that generates a given g ( r ) {\displaystyle g(r)} under equilibrium is unique up to an additive constant, if it exists.
The radial distribution function (RDF), also termed the pair distribution function or the pair correlation function, is a measure of local structuring in a mixture. The RDF between components and positioned at and , respectively, is defined as:
In probability theory, the joint probability distribution is the probability distribution of all possible pairs of outputs of two random variables that are defined on the same probability space. The joint distribution can just as well be considered for any given number of random variables.
The interacting pair-distribution functions obtained from CHNC have been used to calculate the exchange-correlation energies, Landau parameters of Fermi liquids and other quantities of interest in many-body physics and density functional theory, as well as in the theory of hot plasmas. [4] [5]
Approximate solutions for the pair distribution function in the extensional and compressional sectors of shear flow and hence the angular-averaged radial distribution function can be obtained, as shown in Ref., [6] which are in good parameter-free agreement with numerical data up to packing fractions .
However, some success has been achieved at estimating the contact value of the pair distribution function for Mie fluids (which consists of particles interacting through a generalised Lennard-Jones potential) and using these estimates to predict the transport properties of dense gas mixtures and supercritical fluids. [15]
If () is a general scalar-valued function of a normal vector, its probability density function, cumulative distribution function, and inverse cumulative distribution function can be computed with the numerical method of ray-tracing (Matlab code). [17]