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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:
One common correlation function is the radial distribution function which is seen often in statistical mechanics and fluid mechanics. The correlation function can be calculated in exactly solvable models (one-dimensional Bose gas, spin chains, Hubbard model) by means of Quantum inverse scattering method and Bethe ansatz. In an isotropic XY ...
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 .
A different approach to solving this problem was developed by Alessio Zaccone and coworkers and consists in placing the boundary condition right at the boundary layer distance, upon assuming (in a first-order approximation) a constant value of the pair distribution function in the outer layer due to convection being dominant there.
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 most notable problem is that often more than one qualitatively different model will give similar agreement with experimental data. For example, in the case of amorphous silicon, the integral of the first peak in the pair distribution function may imply an average atomic coordination number of 4. This might reflect the fact that all atoms ...