# Radial distribution function

The **radial distribution function** (abbreviation **rdf** ) with the symbol between two **types of **particles A and B describes the frequency with which one finds a particle of type B at a distance from a particle of type A, based on the frequency that two particles of an ideal gas are found present at this distance. The radial distribution function is therefore dimensionless .

## determination

To determine the radial distribution function, as in Figure 1, count the number of particles of type B (blue) in the spherical shell with radius and thickness around a particle of type A (dark red). This gives a histogram . If this histogram is normalized accordingly, the radial distribution function is obtained. In molecular dynamics or Metropolis importance sampling following formula: . The histogram entry, which is assigned to the distance , is divided by the bin volume and the number of samples ( ), resulting in a mean density in the bin. This mean density is then compared with the density of an ideal gas .

## definition

In the NVT ensemble , the radial distribution function can also be derived from the 2N point probability density ( locations and speeds)

can be obtained for a Hamilton function.

By integrating locations and all speeds from the 2N-point probability density, the 2-point probability density is obtained

This is normalized with , where the mean particle number density is:

In the Thermodynamic Limes :

- .

In a homogeneous system is

## Pair distribution function

The **pair distribution function** (also **pair correlation function** ) does not only depend on the distance , but also on the angles and because of ( spherical coordinates ) . The (static) pair correlation function is given by:

This result is obtained from the calculation of the (collective) Van Hove correlation function by inserting the definition of the density , integrating it with and then evaluating it with. It should be noted that

## Applications

With the help of the radial distribution function, the structure factor can be determined by Fourier transformation .

The radial distribution function plays an important role in the Kirkwood-Buff theory .

In a homogeneous system, the pair correlation function indicates the “ potential of mean force ” , which is defined by the assignment (with the Boltzmann constant ).

## References and comments

- ↑ Molecular Modeling: Principles and Applications, Pearson Education, 2001, ISBN 0582382106 , page 310 ff, Google Books
- ↑ with ,
- ↑ In homogeneous systems applies: . If you choose , you get