Radial distribution function

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 . ${\displaystyle g_{AB}(r)}$ ${\displaystyle r}$

determination

Figure 1: Scheme for determining the rdf

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 . ${\displaystyle r}$ ${\displaystyle dr}$ ${\displaystyle \left(\lim _{dr\to 0}\right)}$ ${\displaystyle {\text{rdf}}(R)=\left({\frac {H(R)}{{\text{num}}\cdot V(R)}}\right)/\rho _{0}}$ ${\displaystyle R}$ ${\displaystyle V(R)}$ ${\displaystyle {\text{num}}}$ ${\displaystyle \rho _{0}=N/V}$

definition

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

${\displaystyle p_{N}({\vec {r}}^{N},{\vec {v}}^{N})={\frac {\exp(-\beta \cdot {\mathcal {H}}({\vec {v}}^{N},{\vec {r}}^{N}))}{Z_{N}(V,T)}}}$

can be obtained for a Hamilton function. ${\displaystyle {\mathcal {H}}}$

By integrating locations and all speeds from the 2N-point probability density, the 2-point probability density is obtained ${\displaystyle N-2}$ ${\displaystyle p_{N}^{(2)}(r_{1},r_{2}).}$

This is normalized with , where the mean particle number density is: ${\displaystyle {\frac {N!}{(N-2)!}}{\frac {1}{\rho ^{2}}}}$ ${\displaystyle \rho =N/V}$

${\displaystyle g_{N}(r_{1},r_{2})={\frac {N!}{(N-2)!}}{\frac {1}{\rho ^{2}}}\cdot p_{N}^{(2)}(r_{1},r_{2})}$

In the Thermodynamic Limes :

${\displaystyle \lim _{N\to \infty ,V\to \infty ,{\frac {N}{V}}={\text{const}}}g_{N}(r_{1},r_{2})=g(r_{1},r_{2})}$.

In a homogeneous system is

${\displaystyle g(r_{1},r_{2})=g(r_{1}-r_{2})=:g(r)}$

Pair distribution function

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. ${\displaystyle r=0}$

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: ${\displaystyle g_{AB}({\vec {r}})}$ ${\displaystyle r}$ ${\displaystyle {\vec {r}}={\vec {r}}(r,\theta ,\phi )}$ ${\displaystyle \theta }$ ${\displaystyle \phi }$

${\displaystyle g({\vec {r}})={\frac {V}{N^{2}}}\left\langle \sum _{i\neq j}\delta ({\vec {r}}-({\vec {R}}_{i}-{\vec {R}}_{j}))\right\rangle .}$

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 ${\displaystyle G({\vec {r}},t):={\frac {V}{N}}\langle \rho ({\vec {\tilde {r}}},{\tilde {t}})\rho ({\vec {\tilde {\tilde {r}}}},{\tilde {\tilde {t}}})\rangle }$ ${\displaystyle \rho ({\vec {\tilde {r}}},{\tilde {t}})=\sum _{i=1}^{N}\delta ({\vec {\tilde {r}}}-{\vec {\tilde {R}}}_{i}(t))}$ ${\displaystyle {\vec {\tilde {r}}}}$ ${\displaystyle t=0}$ ${\displaystyle G({\vec {r}},0):=\delta ({\vec {r}})+{\frac {N}{V}}g({\vec {r}})}$

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 ). ${\displaystyle g({\vec {r}})}$ ${\displaystyle w({\vec {r}})}$ ${\displaystyle g({\vec {r}}){\overset {!}{=}}\exp \left(-{\frac {w({\vec {r}})}{k_{\mathrm {B} }\cdot T}}\right)}$ ${\displaystyle k_{\mathrm {B} }}$

References and comments

1. ↑ Molecular Modeling: Principles and Applications, Pearson Education, 2001, ISBN 0582382106 , page 310 ff, Google Books
2. ↑ with , ${\displaystyle {\vec {r}}={\vec {\tilde {r}}}-{\vec {\tilde {\tilde {r}}}}}$ ${\displaystyle t={\tilde {t}}-{\tilde {\tilde {t}}}}$
3. ↑ In homogeneous systems applies: . If you choose , you get ${\displaystyle g({\vec {r}}_{1},{\vec {r}}_{2})=g({\vec {r}}_{1}+{\vec {h}},{\vec {r}}_{2}+{\vec {h}})}$ ${\displaystyle {\vec {h}}=-{\vec {r}}_{2}}$ ${\displaystyle g({\vec {r}}):=g(\underbrace {{\vec {r}}_{1}-{\vec {r}}_{2}} _{\overset {!}{={\vec {r}}}},{\vec {0}})}$