Pair distribution function

The pair distribution function describes the distribution of distances between pairs of particles contained within a given volume. Mathematically, if a and b are two particles in a fluid, the pair distribution function of b with respect to a, denoted by g a b ( r → ) {displaystyle g_{ab}({vec {r}})} is the probability of finding the particle b at distance r → {displaystyle {vec {r}}} from a, with a taken as the origin of coordinates. The pair distribution function describes the distribution of distances between pairs of particles contained within a given volume. Mathematically, if a and b are two particles in a fluid, the pair distribution function of b with respect to a, denoted by g a b ( r → ) {displaystyle g_{ab}({vec {r}})} is the probability of finding the particle b at distance r → {displaystyle {vec {r}}} from a, with a taken as the origin of coordinates. The pair distribution function is used to describe the distribution of objects within a medium (for example, oranges in a crate or nitrogen molecules in a gas cylinder). If the medium is homogeneous (i.e. every spatial location has identical properties), then there is an equal probability density for finding an object at any position r → {displaystyle {vec {r}}} : where V {displaystyle V} is the volume of the container. On the other hand, the likelihood of finding pairs of objects at given positions (i.e. the two-body probability density) is not uniform. For example, pairs of hard balls must be separated by at least the diameter of a ball. The pair distribution function g ( r → , r → ′ ) {displaystyle g({vec {r}},{vec {r}}')} is obtained by scaling the two-body probability density function by the total number of objects N {displaystyle N} and the size of the container: