Radial distribution function

In statistical mechanics, the radial distribution function, (or pair correlation function) g ( r ) {\displaystyle g(r)} in a system of particles (atoms, molecules, colloids, etc.), describes how density varies as a function of distance from a reference particle. If a given particle is taken to be at the origin O, and if ρ = N / V {\displaystyle \rho =N/V} is the average number density of particles, then the local time-averaged density at a distance r {\displaystyle r} from O is ρ g ( r ) {\displaystyle \rho g(r)} . This simplified definition holds for a homogeneous and isotropic system. A more general case will be considered below. In simplest terms it is a measure of the probability of finding one particle at a distance of r {\displaystyle r} away from a given reference particle, relative to that for an ideal gas. The general algorithm involves determining how many particles are within a distance of r {\displaystyle r} and r + d r {\displaystyle r+dr} away from a particle. This general theme is depicted to the right, where the red particle is our reference particle, and the blue particles are those whose centers are within the circular shell, dotted in orange. The radial distribution function is usually determined by calculating the distance between all particle pairs and binning them into a histogram. The histogram is then normalized with respect to an ideal gas, where particle histograms are completely uncorrelated. For three dimensions, this normalization is the number density of the system ( ρ ) {\displaystyle (\rho )} multiplied by the volume of the spherical shell, which symbolically can be expressed as ρ 4 π r 2 d r {\displaystyle \rho \,4\pi r^{2}dr} . Given a potential energy function, the radial distribution function can be computed either via computer simulation...