This article deals with the radial distribution function in statistical physics. For the radial probability density function in quantum mechanics, see there.
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 .
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)
Radial distribution function of a Lennard-Jones fluid . The radial distribution function practically takes on the value 0, since the particles interact with a Lennard-Jones potential and thus practically cannot overlap.
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
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