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)}$ ${\ displaystyle k _ {\ mathrm {B}}}$

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}})}$