In statistical mechanics, the radial distribution function, (or pair correlation function) g ( r ) {\displaystyle g(r)} in a system of particles (atoms, molecules, colloids, etc.), describes how density varies as a function of distance from a reference particle. If a given particle is taken to be at the origin O, and if ρ = N / V {\displaystyle \rho =N/V} is the average number density of particles, then the local time-averaged density at a distance r {\displaystyle r} from O is ρ g ( r ) {\displaystyle \rho g(r)} . This simplified definition holds for a homogeneous and isotropic system. A more general case will be considered below. In simplest terms it is a measure of the probability of finding one particle at a distance of r {\displaystyle r} away from a given reference particle, relative to that for an ideal gas. The general algorithm involves determining how many particles are within a distance of r {\displaystyle r} and r + d r {\displaystyle r+dr} away from a particle. This general theme is depicted to the right, where the red particle is our reference particle, and the blue particles are those whose centers are within the circular shell, dotted in orange. The radial distribution function is usually determined by calculating the distance between all particle pairs and binning them into a histogram. The histogram is then normalized with respect to an ideal gas, where particle histograms are completely uncorrelated. For three dimensions, this normalization is the number density of the system ( ρ ) {\displaystyle (\rho )} multiplied by the volume of the spherical shell, which symbolically can be expressed as ρ 4 π r 2 d r {\displaystyle \rho \,4\pi r^{2}dr} . Given a potential energy function, the radial distribution function can be computed either via computer simulation methods like the Monte Carlo method, or via the Ornstein–Zernike equation, using approximative closure relations like the Percus–Yevick approximation or the hypernetted-chain theory. It can also be determined experimentally, by radiation scattering techniques or by direct visualization for large enough (micrometer-sized) particles via traditional or confocal microscopy. The radial distribution function is of fundamental importance since it can be used, using the Kirkwood–Buff solution theory, to link the microscopic details to macroscopic properties. Moreover, by the reversion of the Kirkwood–Buff theory, it is possible to attain the microscopic details of the radial distribution function from the macroscopic properties. The radial distribution function may also be inverted to predict the potential energy function using the Ornstein–Zernike equation or structure-optimized potential refinement.
Definition Consider a system of N {\displaystyle N} particles in a volume V {\displaystyle V} (for an average number density ρ = N / V {\displaystyle \rho =N/V} ) and at a temperature T {\displaystyle T} (let us also define β = 1 k T {\displaystyle \textstyle \beta ={\frac {1}{kT}}} ; k {\displaystyle k} is the Boltzmann constant). The particle coordinates are r i {\displaystyle \mathbf {r} _{i}} , with i = 1 , … , N {\displaystyle \textstyle i=1,\,\ldots ,\,N} . The potential energy due to the interaction between particles is U N ( r 1 … , r N ) {\displaystyle \textstyle U_{N}(\mathbf {r} _{1}\,\ldots ,\,\mathbf {r} _{N})} and we do not consider the case of an externally applied field. The appropriate averages are taken in the canonical ensemble ( N , V , T ) {\displaystyle (N,V,T)} , with Z N = ∫ ⋯ ∫ e − β U N d r 1 ⋯ d r N {\displaystyle \textstyle Z_{N}=\int \cdots \int \mathrm {e} ^{-\beta U_{N}}\mathrm {d} \mathbf {r} _{1}\cdots \mathrm {d} \mathbf {r} _{N}} the configurational integral, taken over all possible combinations of particle positions. The probability of an elementary configuration, namely finding particle 1 in d r 1 {\displaystyle \textstyle \mathrm {d} \mathbf {r} _{1}} , particle 2 in d r 2 {\displaystyle \textstyle \mathrm {d} \mathbf {r} _{2}} , etc. is given by
The total number of particles is huge, so that P ( N ) {\displaystyle P^{(N)}} in itself is not very useful. However, one can also obtain the probability of a reduced configuration, where the positions of only n < N {\displaystyle n<N} particles are fixed, in r 1 … , r n {\displaystyle \textstyle \mathbf {r} _{1}\,\ldots ,\,\mathbf {r} _{n}} , with no constraints on the remaining N − n {\displaystyle N-n} particles. To this end, one has to integrate (1) over the remaining coordinates r n + 1 … , r N {\displaystyle \mathbf {r} _{n+1}\,\ldots ,\,\mathbf {r} _{N}} :
P ( n ) ( r 1 , … , r n ) = 1 Z N ∫ ⋯ ∫ e − β U N d 3 r n + 1 ⋯ d 3 r N {\displaystyle P^{(n)}(\mathbf {r} _{1},\ldots ,\mathbf {r} _{n})={\frac {1}{Z_{N}}}\int \cdots \int \mathrm {e} ^{-\beta U_{N}}\,\mathrm {d} ^{3}\mathbf {r} _{n+1}\cdots \mathrm {d} ^{3}\mathbf {r} _{N}\,} . If the particles are non-interacting, in the sense that the potential energy of each particle does not depend on any of the other particles, U N ( r 1 , … , r N ) = ∑ i = 1 N U 1 ( r i ) {\textstyle U_{N}(\mathbf {r} _{1},\dots ,\mathbf {r} _{N})=\sum _{i=1}^{N}U_{1}(\mathbf {r} _{i})} , then the partition function factorizes, and the probability of an elementary configuration decomposes with independent arguments to a product of single particle probabilities,
Z N = ∏ i = 1 N ∫ d 3 r i e − β U 1 = Z 1 N P ( n ) ( r 1 , … , r n ) = P ( 1 ) ( r 1 ) ⋯ P ( 1 ) ( r n ) {\displaystyle {\begin{aligned}Z_{N}&=\prod _{i=1}^{N}\int \mathrm {d} ^{3}\mathbf {r} _{i}e^{-\beta U_{1}}=Z_{1}^{N}\\P^{(n)}(\mathbf {r} _{1},\dots ,\mathbf {r} _{n})&=P^{(1)}(\mathbf {r} _{1})\cdots P^{(1)}(\mathbf {r} _{n})\end{aligned}}}
Note how for non-interacting particles the probability is symmetric in its arguments. This is not true in general, and the order in which the positions occupy the argument slots of P ( n ) {\displaystyle P^{(n)}} matters. Given a set of positions, the way that the N {\displaystyle N} particles can occupy those positions is N ! {\displaystyle N!} The probability that those positions ARE occupied is found by summing over all configurations in which a particle is at each of those locations. This can be done by taking every permutation, π {\displaystyle \pi } , in the symmetric group on N {\displaystyle N} objects, S N {\displaystyle S_{N}} , to write ∑ π ∈ S N P ( N ) ( r π ( 1 ) , … , r π ( N ) ) {\textstyle \sum _{\pi \in S_{N}}P^{(N)}(\mathbf {r} _{\pi (1)},\ldots ,\mathbf {r} _{\pi (N)})} . For fewer positions, we integrate over extraneous arguments, and include a correction factor to prevent overcounting, ρ ( n ) ( r 1 , … , r n ) = 1 ( N − n ) ! ( ∏ i = n + 1 N ∫ d 3 r i ) ∑ π ∈ S N P ( N ) ( r π ( 1 ) , … , r π ( N ) ) {\displaystyle {\begin{aligned}\rho ^{(n)}(\mathbf {r} _{1},\ldots ,\mathbf {r} _{n})&={\frac {1}{(N-n)!}}\left(\prod _{i=n+1}^{N}\int \mathrm {d} ^{3}\mathbf {r} _{i}\right)\sum _{\pi \in S_{N}}P^{(N)}(\mathbf {r} _{\pi (1)},\ldots ,\mathbf {r} _{\pi (N)})\\\end{aligned}}} This quantity is called the n-particle density function. For indistinguishable particles, one could permute all the particle positions, ∀ i , r i → r π ( i ) {\displaystyle \forall i,\mathbf {r} _{i}\rightarrow \mathbf {r} _{\pi (i)}} , without changing the probability of an elementary configuration, P ( r π ( 1 ) , … , r π ( N ) ) = P ( r 1 , … , r N ) {\displaystyle P(\mathbf {r} _{\pi (1)},\dots ,\mathbf {r} _{\pi (N)})=P(\mathbf {r} _{1},\dots ,\mathbf {r} _{N})} , so that the n-particle density function reduces to ρ ( n ) ( r 1 , … , r n ) = N ! ( N − n ) ! P ( n ) ( r 1 , … , r n ) {\displaystyle {\begin{aligned}\rho ^{(n)}(\mathbf {r} _{1},\ldots ,\mathbf {r} _{n})&={\frac {N!}{(N-n)!}}P^{(n)}(\mathbf {r} _{1},\ldots ,\mathbf {r} _{n})\end{aligned}}} Integrating the n-particle density gives the permutation factor N P n {\displaystyle _{N}P_{n}} , counting the number of ways one can sequentially pick particles to place at the n {\displaystyle n} positions out of the total N {\displaystyle N} particles. Now let's turn to how we interpret this functions for different values of n {\displaystyle n} . For n = 1 {\displaystyle n=1} , we have the one-particle density. For a crystal it is a periodic function with sharp maxima at the lattice sites. For a non-interacting gas, it is independent of the position r 1 {\displaystyle \textstyle \mathbf {r} _{1}} and equal to the overall number density, ρ {\displaystyle \rho } , of the system. To see this first note that U N = 0 {\displaystyle U_{N}=0} in the volume occupied by the gas, and ∞ {\displaystyle \infty } everywhere else. The partition function in this case is
Z N = ∏ i = 1 N ∫ d 3 r i 1 = V N {\displaystyle Z_{N}=\prod _{i=1}^{N}\int \mathrm {d} ^{3}\mathbf {r} _{i}\ 1=V^{N}}
from which the definition gives the desired result
ρ ( 1 ) ( r ) = N ! ( N − 1 ) ! 1 V N ∏ i = 2 N ∫ d 3 r i 1 = N V = ρ . {\displaystyle \rho ^{(1)}(\mathbf {r} )={\frac {N!}{(N-1)!}}{\frac {1}{V^{N}}}\prod _{i=2}^{N}\int \mathrm {d} ^{3}\mathbf {r} _{i}1={\frac {N}{V}}=\rho .}
In fact, for this special case every n-particle density is independent of coordinates, and can be computed explicitly ρ ( n ) ( r 1 , … , r n ) = N ! ( N − n ) ! 1 V N ∏ i = n + 1 N ∫ d 3 r i 1 = N ! ( N − n ) ! 1 V n {\displaystyle {\begin{aligned}\rho ^{(n)}(\mathbf {r} _{1},\dots ,\mathbf {r} _{n})&={\frac {N!}{(N-n)!}}{\frac {1}{V^{N}}}\prod _{i=n+1}^{N}\int \mathrm {d} ^{3}\mathbf {r} _{i}1\\&={\frac {N!}{(N-n)!}}{\frac {1}{V^{n}}}\end{aligned}}} For N ≫ n {\displaystyle N\gg n} , the non-interacting n-particle density is approximately ρ non-interacting ( n ) ( r 1 , … , r N ) = ( 1 − n ( n − 1 ) / 2 N + ⋯ ) ρ n ≈ ρ n {\displaystyle \rho _{\text{non-interacting}}^{(n)}(\mathbf {r} _{1},\dots ,\mathbf {r} _{N})=\left(1-n(n-1)/2N+\cdots \right)\rho ^{n}\approx \rho ^{n}} . With this in hand, the n-point correlation function g ( n ) {\displaystyle g^{(n)}} is defined by factoring out the non-interacting contribution, ρ ( n ) ( r 1 , … , r n ) = ρ non-interacting ( n ) g ( n ) ( r 1 … , r n ) {\displaystyle \rho ^{(n)}(\mathbf {r} _{1},\ldots ,\,\mathbf {r} _{n})=\rho _{\text{non-interacting}}^{(n)}g^{(n)}(\mathbf {r} _{1}\,\ldots ,\,\mathbf {r} _{n})} Explicitly, this definition reads g ( n ) ( r 1 , … , r n ) = V N N ! ( ∏ i = n + 1 N 1 V ∫ d 3 r i ) 1 Z N ∑ π ∈ S N e − β U ( r π ( 1 ) , … , r π ( N ) ) {\displaystyle {\begin{aligned}g^{(n)}(\mathbf {r} _{1},\ldots ,\,\mathbf {r} _{n})&={\frac {V^{N}}{N!}}\left(\prod _{i=n+1}^{N}{\frac {1}{V}}\!\!\int \!\!\mathrm {d} ^{3}\mathbf {r} _{i}\right){\frac {1}{Z_{N}}}\sum _{\pi \in S_{N}}e^{-\beta U(\mathbf {r} _{\pi (1)},\ldots ,\,\mathbf {r} _{\pi (N)})}\end{aligned}}} where it is clear that the n-point correlation function is dimensionless.
Relations involving g(r)
Structure factor The second-order correlation function g ( 2 ) ( r 1 , r 2 ) {\displaystyle g^{(2)}(\mathbf {r} _{1},\mathbf {r} _{2})} is of special importance, as it is directly related (via a Fourier transform) to the structure factor of the system and can thus be determined experimentally using X-ray diffraction or neutron diffraction. If the system consists of spherically symmetric particles, g ( 2 ) ( r 1 , r 2 ) {\displaystyle g^{(2)}(\mathbf {r} _{1},\mathbf {r} _{2})} depends only on the relative distance between them, r 12 = r 2 − r 1 {\displaystyle \mathbf {r} _{12}=\mathbf {r} _{2}-\mathbf {r} _{1}} . We will drop the sub- and superscript: g ( r ) ≡ g ( 2 ) ( r 12 ) {\displaystyle \textstyle g(\mathbf {r} )\equiv g^{(2)}(\mathbf {r} _{12})} . Taking particle 0 as fixed at the origin of the coordinates, ρ g ( r ) d 3 r = d n ( r ) {\displaystyle \textstyle \rho g(\mathbf {r} )d^{3}r=\mathrm {d} n(\mathbf {r} )} is the average number of particles (among the remaining N − 1 {\displaystyle N-1} ) to be found in the volume d 3 r {\displaystyle \textstyle d^{3}r} around the position r {\displaystyle \textstyle \mathbf {r} } . We can formally count these particles and take the average via the expression d n ( r ) d 3 r = ⟨ ∑ i ≠ 0 δ ( r − r i ) ⟩ {\displaystyle \textstyle {\frac {\mathrm {d} n(\mathbf {r} )}{d^{3}r}}=\langle \sum _{i\neq 0}\delta (\mathbf {r} -\mathbf {r} _{i})\rangle } , with ⟨ ⋅ ⟩ {\displaystyle \textstyle \langle \cdot \rangle } the ensemble average, yielding:
where the second equality requires the equivalence of particles 1 , … , N − 1 {\displaystyle \textstyle 1,\,\ldots ,\,N-1} . The formula above is useful for relating g ( r ) {\displaystyle g(\mathbf {r} )} to the static structure factor S ( q ) {\displaystyle S(\mathbf {q} )} , defined by S ( q ) = ⟨ ∑ i j e − i q ( r i − r