Radial distribution function: Difference between revisions

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{{Short description|Description of particle density in statistical mechanics}}

In [[computational mechanics]] and [[statistical mechanics]], a '''radial distribution function''' (RDF), ''g''(''r''), describes how the density of surrounding matter varies as a function of the distance from a distinguished point.

{{Use American English|date=January 2019}}

{{Use mdy dates|date=January 2019}}

[[File:Rdf schematic.svg|thumb|250px|right|calculation of <math>g(r)</math>]]

[[File:Lennard-Jones Radial Distribution Function.svg|thumb|300px|Radial distribution function for the [[Lennard-Jones potential|Lennard-Jones model fluid]] at <math>\textstyle T^* = 0.71, \; n^* = 0.844</math>.]]

In [[statistical mechanics]], the '''radial distribution function''', (or '''pair correlation function''') <math>g(r)</math> in a system of [[particles]] (atoms, molecules, colloids, etc.), describes how density varies as a function of distance from a reference particle.

Suppose, for example, that we choose a molecule at some point O in the volume. What is then the average density at some point P at a distance r away from O? If <math>\rho=N/V</math> is the average density, then the mean density at P ''given'' that there is a molecule at O would differ from &rho; by some factor g(r). One could say that the radial distribution function takes into account the correlations in the distribution of molecules arising from the forces they exert on each other:

If a given particle is taken to be at the origin ''O'', and if <math>\rho = N/V</math> is the average number density of particles, then the local time-averaged density at a distance <math>r</math> from ''O'' is <math>\rho g(r)</math>. This simplified definition holds for a [[homogeneous]] and [[isotropic]] system. A more general case will be considered below.

<center>

(mean local density at distance r from O) = <math>\rho</math>g(r) (1)

</center>

In simplest terms it is a measure of the probability of finding a particle at a distance of <math>r</math> 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 <math>r</math> and <math>r+dr</math> 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.

As long as the gas is '''dilute''' the correlations in the positions of the molecules that g(r) takes into account are due to the potential <math>\phi</math>(r) that a molecule at P feels owing to the presence of a molecule at O. Using the Boltzmann distribution law:

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 <math>( \rho )</math> multiplied by the volume of the spherical shell, which symbolically can be expressed as <math>\rho \, 4\pi r^2 dr</math>.

<center>

<math>g(r) = e^{-\phi(r)/kT} \,</math> (2)

</center>

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 equation|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]].

If <math>\phi(r)</math> was zero for all r - i.e., if the molecules did not exert any influence on each other g(r) = 1 for all r. Then from (1) the mean local density would be equal to the mean density <math>\rho</math>: the presence of a molecule at O would not influence the presence or absence of any other molecule and the gas would be ideal. As long as there is a <math>\phi(r)</math> the mean local density will always be different from the mean density <math>\rho</math> due to the interactions between molecules.

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.<ref>{{cite journal |last1=Shanks |first1=B. | last2 = Potoff | first2 = J. | last3 = Hoepfner | first3 = M. |title=Transferable Force Fields from Experimental Scattering Data with Machine Learning Assisted Structure Refinement |journal=J. Phys. Chem. Lett. |date=December 5, 2022 |volume=13 |issue=49 |pages=11512–11520 |doi= 10.1021/acs.jpclett.2c03163|pmid=36469859 |s2cid=254274307 }}</ref>

When the density of the gas gets higher the so called low-density limit (2) is not applicable anymore because the molecules attracted to and repelled by the molecule at O also repel and attract each other. The correction terms needed to correctly describe g(r) resembles the [[virial equation]], it is an expansion in the density:

== Definition ==

<center>

Consider a system of <math>N</math> particles in a volume <math>V</math> (for an average [[number density]] <math>\rho =N/V</math>) and at a temperature <math>T</math> (let us also define <math>\textstyle \beta = \frac{1}{kT}</math>; <math>k</math> is the [[Boltzmann constant]]). The particle coordinates are <math>\mathbf{r}_{i}</math>, with <math>\textstyle i = 1, \, \ldots, \, N</math>. The [[potential energy]] due to the interaction between particles is <math>\textstyle U_{N} (\mathbf{r}_{1}\, \ldots, \, \mathbf{r}_{N})</math> and we do not consider the case of an externally applied field.

<math>g(r)=e^{-\phi(r)/kT}+\rho g_{1}(r)+\rho^{2}g_{2}(r)+\ldots</math> (3)

</center>

The appropriate [[Ensemble average|averages]] are taken in the [[canonical ensemble]] <math>(N,V,T)</math>, with <math>\textstyle Z_{N} = \int \cdots \int \mathrm{e}^{-\beta U_{N}} \mathrm{d} \mathbf{r}_1 \cdots \mathrm{d} \mathbf{r}_N</math> the configurational integral, taken over all possible combinations of particle positions. The probability of an elementary configuration, namely finding particle 1 in <math>\textstyle \mathrm{d} \mathbf{r}_1</math>, particle 2 in <math>\textstyle \mathrm{d} \mathbf{r}_2</math>, etc. is given by

in which additional functions <math>g_{1}(r), \, g_{2}(r)</math> appear which may depend on temperature <math>T</math> and distance <math>r</math> but not on density, <math>\rho</math>.

{{NumBlk|:| <math> P^{(N)}(\mathbf{r}_1,\ldots,\mathbf{r}_N ) \, \mathrm{d} \mathbf{r}_1 \cdots \mathrm{d} \mathbf{r}_N = \frac{\mathrm{e}^{-\beta U_{N}}}{Z_N} \, \mathrm{d} \mathbf{r}_1 \cdots \mathrm{d} \mathbf{r}_N\, </math>.|{{EquationRef|1}}}}

Given a [[potential energy]] function, the radial distribution function can be found via computer simulation methods like the [[Monte Carlo method]]. It could also be calculated numerically using rigorous methods obtained from [[statistical mechanics]] like the [[Perckus-Yevick approximation]].

The total number of particles is huge, so that <math> P^{(N)}</math> in itself is not very useful. However, one can also obtain the probability of a reduced configuration, where the positions of only <math>n < N</math> particles are fixed, in <math>\textstyle \mathbf{r}_{1}\, \ldots, \, \mathbf{r}_{n}</math>, with no constraints on the remaining <math>N-n</math> particles. To this end, one has to integrate ({{EquationNote|1}}) over the remaining coordinates <math>\mathbf{r}_{n+1}\, \ldots, \, \mathbf{r}_{N}</math>:

==Importance of g(r)==

: <math> 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 \, </math>.

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, <math display="inline">U_N(\mathbf{r}_1,\dots,\mathbf{r}_N)=\sum_{i=1}^N U_1(\mathbf{r}_i)</math>, then the partition function factorizes, and the probability of an elementary configuration decomposes with independent arguments to a product of single particle probabilities,

g(r) is of fundamental importance in thermodynamics for macroscopic thermodynamic quantities can be calculated using g(r). A few examples:

<math display="block"> \begin{align}

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{align} </math>

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 <math> P^{(n)}</math>matters. Given a set of positions, the way that the <math> N</math> particles can occupy those positions is <math> N!</math> 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]], <math> \pi</math>, in the [[symmetric group]] on <math> N</math> objects, <math> S_N</math>, to write <math display="inline"> \sum_{\pi\in S_N} P^{(N)}(\mathbf{r}_{\pi (1)},\ldots,\mathbf{r}_{\pi (N)}) </math>. For fewer positions, we integrate over extraneous arguments, and include a correction factor to prevent overcounting,<math display="block"> \begin{align}

''The virial equation for the pressure:''

\rho^{(n)}(\mathbf{r}_1,\ldots,\mathbf{r}_n)

:<math>p=\rho kT-\frac{2\pi}{3}\rho^{2}\int d r r^{3} u^{\prime}(r) g(r, \rho, T) </math> (4)

&=\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{align} </math>This quantity is called the ''n-particle density'' function. For [[Indistinguishable particle|indistinguishable]] particles, one could permute all the particle positions, <math> \forall i, \mathbf{r}_i\rightarrow \mathbf{r}_{\pi(i)}</math>, without changing the probability of an elementary configuration, <math> P(\mathbf{r}_{\pi(1)},\dots,\mathbf{r}_{\pi (N)})=P(\mathbf{r}_{1},\dots,\mathbf{r}_{ N})</math>, so that the n-particle density function reduces to <math display="block"> \begin{align}

\rho^{(n)}(\mathbf{r}_1,\ldots,\mathbf{r}_n)

&=\frac{N!}{(N-n)!}P^{(n)}(\mathbf{r}_1,\ldots,\mathbf{r}_n)

\end{align} </math>Integrating the n-particle density gives the [[Permutation|permutation factor]] <math> _NP_n</math>, counting the number of ways one can sequentially pick particles to place at the <math> n</math> positions out of the total <math> N</math> particles. Now let's turn to how we interpret this functions for different values of <math> n</math>.

For <math>n=1</math>, 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 <math>\textstyle \mathbf{r}_1</math> and equal to the overall number density, <math>\rho</math>, of the system. To see this first note that <math>U_N = \infty </math> in the volume occupied by the gas, and 0 everywhere else. The partition function in this case is

''The energy equation:''

: <math> Z_N = \fracprod_{E}{NkT}i=\frac{31}{2}+^N\frac{int\rho}mathrm{2kTd}^3\int d mathbf{r}_i \,4\pi r1=V^{2} u(r)g(r, \rho, T) N</math> (5)

from which the definition gives the desired result

: <math>\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.</math>

In fact, for this special case every n-particle density is independent of coordinates, and can be computed explicitly<math display="block"> \begin{align}

''[[Compressibility equation|The compressibility equation]]:''

\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\\

:<math>kT\left(\frac{\partial \rho}{\partial p}\right)=1+\rho \int d r [g(r)-1] </math> (6)

&=\frac{N!}{(N-n)!}\frac{1}{V^n}

\end{align}</math>For <math>N\gg n</math>, the non-interacting n-particle density is approximately <math>\rho^{(n)}_\text{non-interacting}(\mathbf{r}_1,\dots,\mathbf{r}_N)= \left(1-n(n-1)/2N+\cdots \right)\rho^n\approx \rho^n</math>.<ref>{{cite journal |last1=Tricomi |first1=F. |last2=Erdélyi |first2=A. |title=The asymptotic expansion of a ratio of gamma functions |journal=Pacific Journal of Mathematics |date=1 March 1951 |volume=1 |issue=1 |pages=133–142 |doi=10.2140/pjm.1951.1.133 |doi-access=free }}</ref> With this in hand, the ''n-point correlation'' function <math> g^{(n)}</math> is defined by factoring out the non-interacting contribution{{Citation needed|date=September 2022}}, <math display="block">\rho^{(n)}(\mathbf{r}_{1}, \ldots, \, \mathbf{r}_{n}) = \rho^{(n)}_\text{non-interacting}g^{(n)}(\mathbf{r}_{1}\, \ldots, \, \mathbf{r}_{n}) </math>Explicitly, this definition reads <math display="block">\begin{align}

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{align} </math>where it is clear that the ''n''-point correlation function is dimensionless.

== Relations involving ''g''(''r'') ==

==Experimental==

It is possible to measure g(r) experimentaly with [[neutron scattering]] or [[x-ray scattering]] diffraction data. In such an experiment, a sample is bombarded with neutrons or x-ray which then diffract to all directions. The average molecular density at each distance can be extracted in according to [[Snells law]]: r=wavelength/sin(scattered angle), where r is the distance the neutron traveled during diffraction.<br />

For an example of RDF experiment see [http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JCPSA6000125000001014508000001&idtype=cvips&gifs=yes Eigen vs. Zundel structures in HCl solution, 2006]

=== Structure factor ===

==Formal derivation==

The second-order correlation function <math>g^{(2)}(\mathbf{r}_{1},\mathbf{r}_{2})</math> 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]].<ref>{{cite book | last1 = Dinnebier | first1 = R E | last2 = Billinge | first2 = S J L | title = Powder Diffraction: Theory and Practice | url = https://archive.org/details/powderdiffractio00redi | url-access = limited | publisher = Royal Society of Chemistry | edition = 1st | date = 10 Mar 2008 | pages = [https://archive.org/details/powderdiffractio00redi/page/n492 470]–473 | language = en | doi = 10.1039/9781847558237 | isbn = 978-1-78262-599-5}}</ref>

If the system consists of spherically symmetric particles, <math>g^{(2)}(\mathbf{r}_{1},\mathbf{r}_{2})</math> depends only on the relative distance between them, <math>\mathbf{r}_{12} = \mathbf{r}_{2} - \mathbf{r}_{1} </math>. We will drop the sub- and superscript: <math>\textstyle g(\mathbf{r})\equiv g^{(2)}(\mathbf{r}_{12})</math>. Taking particle 0 as fixed at the origin of the coordinates, <math>\textstyle \rho g(\mathbf{r}) d^3r = \mathrm{d} n (\mathbf{r})</math> is the ''average'' number of particles (among the remaining <math>N-1</math>) to be found in the volume <math>\textstyle d^3r</math> around the position <math>\textstyle \mathbf{r}</math>.

Consider a system of ''N'' particles in a volume ''V'' and at a temperature ''T''. The probability of finding molecule 1 in <math>d \rm{r}_{1}</math>, molecule 2 in <math>d \rm{r}_{2}</math>, etc., is given by

We can formally count these particles and take the average via the expression <math>\textstyle \frac{\mathrm{d} n (\mathbf{r})}{d^3r} = \langle \sum_{i \neq 0} \delta ( \mathbf{r} - \mathbf{r}_i) \rangle</math>, with <math>\textstyle \langle \cdot \rangle</math> the ensemble average, yielding:

<center>

<math> P^{(N)}(\rm{r}_{1},\ldots,\rm{r}_{N}) d \rm{r}_{1},\cdots, d \rm{r}_{N}=\frac{e^{-\beta U_{N}}d\rm{r}_{1} \cdots d \rm{r}_{N}}{Z_{N}} \, </math> (7)

</center>

{{NumBlk|:| <math>g(\mathbf{r}) = \frac{1}{\rho} \langle \sum_{i \neq 0} \delta ( \mathbf{r} - \mathbf{r}_i) \rangle = V \frac{N-1}{N} \left \langle \delta ( \mathbf{r} - \mathbf{r}_1) \right \rangle</math>|{{EquationRef|5}}}}

where <math>Z_{N}</math> is the configurational integral. To obtain the probability of finding molecule 1 in <math>d \rm{r}_{1}</math> and molecule ''n'' in <math>d \rm{r}_{n}</math>, irrespective of the remaining ''N-n'' molecules, one has to integrate (7) over the coordinates of molecule ''n'' + 1 through ''N'':

where the second equality requires the equivalence of particles <math>\textstyle 1, \, \ldots, \, N-1</math>. The formula above is useful for relating <math>g(\mathbf{r})</math> to the static structure factor <math>S(\mathbf{q})</math>, defined by <math>\textstyle S(\mathbf{q}) = \langle \sum_{ij} \mathrm{e}^{-i \mathbf{q} (\mathbf{r}_i - \mathbf{r}_j)} \rangle /N</math>, since we have:

<center>

: <math>

<math> P^{(n)}(\rm{r}_{1},\ldots,\rm{r}_{n}) =\frac{\int \cdots \int e^{-\beta U_{N}}d\rm{r}_{n+1} \cdots d \rm{r}_{N}}{Z_{N}} \, </math> (8)

\begin{align}

</center>

S(\mathbf{q}) &= 1 + \frac{1}{N} \langle \sum_{i \neq j} \mathrm{e}^{-i \mathbf{q} (\mathbf{r}_i - \mathbf{r}_j)} \rangle = 1 + \frac{1}{N} \left \langle \int_V \mathrm{d} \mathbf{r} \, \mathrm{e}^{-i \mathbf{q} \mathbf{r}} \sum_{i \neq j} \delta \left [ \mathbf{r} - (\mathbf{r}_i - \mathbf{r}_j) \right ] \right \rangle \\ &= 1+ \frac{N(N-1)}{N} \int_V \mathrm{d} \mathbf{r}\, \mathrm{e}^{-i \mathbf{q} \mathbf{r}} \left \langle \delta ( \mathbf{r} - \mathbf{r}_1 ) \right \rangle ,

\end{align}

</math>

and thus:

: <math>S(\mathbf{q}) = 1 + \rho \int_V \mathrm{d} \mathbf{r} \, \mathrm{e}^{-i \mathbf{q} \mathbf{r}} g(\mathbf{r})</math>, proving the Fourier relation alluded to above.

This equation is only valid in the sense of [[Distribution (mathematics)|distributions]], since <math>g(\mathbf{r})</math> is not normalized: <math>\textstyle \lim_{r \rightarrow \infty} g(\mathbf{r}) = 1</math>, so that <math>\textstyle \int_V \mathrm{d} \mathbf{r} g(\mathbf{r})</math> diverges as the volume <math>V</math>, leading to a Dirac peak at the origin for the structure factor. Since this contribution is inaccessible experimentally we can subtract it from the equation above and redefine the structure factor as a regular function:

Now the probability that ''any'' molecule is in <math>d \rm{r}_{1}</math> and ''any'' molecule in <math>d \rm{r}_{n}</math>, irrespective of the rest of the molecules, is

: <math>S'(\mathbf{q}) = S(\mathbf{q}) - \rho \delta (\mathbf{q})= 1 + \rho \int_V \mathrm{d} \mathbf{r} \, \mathrm{e}^{-i \mathbf{q}\mathbf{r}} [g(\mathbf{r}) - 1]</math>.

Finally, we rename <math>S(\mathbf{q}) \equiv S'(\mathbf{q})</math> and, if the system is a liquid, we can invoke its isotropy:

<center>

{{NumBlk|:| <math>S(q) = 1 + \rho^ \int_V \mathrm{(n)d}( \rmmathbf{r}_ \, \mathrm{1e},^{-i \ldots,mathbf{q}\rmmathbf{r}_{n} [g(r) - 1] = 1 + 4\pi\rho\frac{N!1}{(N-n)!q} \cdotint P^{(n)}(\rmmathrm{r}_{1d} r \, r\ldots, \rmmathrm{r}_{nsin}(qr) \,[g(r) - 1]</math> (9).|{{EquationRef|6}}}}

</center>

=== Compressibility equation ===

For ''n'' = 1 the one particle distribution function is obtained which, for a crystal, is a periodic function with sharp maxima at the lattice sites. For a (homogeneous) liquid:

Evaluating ({{EquationNote|6}}) in <math>q=0</math> and using the relation between the isothermal [[compressibility]] <math>\textstyle \chi _T</math> and the structure factor at the origin yields the [[compressibility equation]]:

{{NumBlk|:| <math>\rho kT \chi _T= kT\left(\frac{\partial \rho}{\partial p}\right)=1 + \rho \int_V \mathrm{d} \mathbf{r} \, [g(r) - 1] </math>.|{{EquationRef|7}}}}

=== Potential of mean force ===

<center>

It can be shown<ref name="Chandler1987">{{cite book |author=[[David Chandler (chemist)|Chandler, D.]] |year=1987 |title=Introduction to Modern Statistical Mechanics |publisher=Oxford University Press |section=7.3}}</ref> that the radial distribution function is related to the two-particle [[potential of mean force]] <math>w^{(2)}(r)</math> by:

<math> \frac{1}{V} \int \rho^{(1)}(\rm{r}_{1})d \rm{r}_{1}=\rho^{(1)}=\frac{N}{V}=\rho \,</math> (10)

{{NumBlk|:| <math> g(r) = \exp \left [ -\frac{w^{(2)}(r)}{kT} \right ] </math>.|{{EquationRef|8}}}}In the dilute limit, the potential of mean force is the exact pair potential under which the equilibrium point configuration has a given <math>g(r)</math>.

</center>

=== Energy equation ===

It is now time to introduce a correlation function <math> g^{(n)}</math> by

If the particles interact via identical pairwise potentials: <math>\textstyle U_{N} = \sum_{i > j = 1}^N u(\left | \mathbf{r}_i - \mathbf{r}_j \right |)</math>, the average [[internal energy]] per particle is:<ref name="HansenMcDonald2005">{{cite book |author=[[Jean-Pierre Hansen|Hansen, J. P.]] and McDonald, I. R. |year=2005 |title=Theory of Simple Liquids |edition= 3rd |publisher=Academic Press}}</ref>{{rp|Section 2.5}}

{{NumBlk|:| <math>\frac{\left \langle E \right \rangle}{N} = \frac{3}{2} kT + \frac{\left \langle U_{N} \right \rangle}{N} = \frac{3}{2} kT + \frac{\rho}{2}\int_V \mathrm{d} \mathbf{r} \, u(r)g(r, \rho, T) </math>.|{{EquationRef|9}}}}

=== Pressure equation of state ===

<center>

Developing the [[virial equation]] yields the pressure equation of state:

<math>\rho^{(n)}(\rm{r}_{1},\ldots,\rm{r}_{n})=\rho^{n}g^{(n)}(\rm{r}_{1},\ldots,\rm{r}_{n}) \, </math> (11)

{{NumBlk|:| <math>p = \rho kT - \frac{\rho^2}{6} \int_V \mathrm{d} \mathbf{r} \, r g(r, \rho, T) \frac{\mathrm{d} u(r)}{\mathrm{d} r}</math>.|{{EquationRef|10}}}}

</center>

=== Thermodynamic properties in 3D ===

<math>g^{(n)}</math> is called a correlation function since if the molecules are independent from each other <math>\rho^{(n)}</math> would simply equal <math>\rho^{n}</math> and therefore <math>g^{(n)}</math> corrects for the correlation between molecules.

The radial distribution function is an important measure because several key thermodynamic properties, such as potential energy and pressure can be calculated from it.

For a 3-D system where particles interact via pairwise potentials, the potential energy of the system can be calculated as follows:<ref name=softmatter>{{cite book|last=Frenkel|first=Daan; Smit, Berend|title=Understanding molecular simulation from algorithms to applications|year=2002|publisher=Academic Press|___location=San Diego|isbn=978-0122673511|edition= 2nd}}</ref>

From (9) it can be shown that

: <math>PE=\frac{N}{2}4\pi\rho\int^{\infty}_0r^2u(r)g(r)dr ,</math>

where ''N'' is the number of particles in the system, <math> \rho </math> is the number density, <math> u(r)</math> is the [[pair potential]].

The pressure of the system can also be calculated by relating the 2nd [[virial coefficient]] to <math> g(r)</math>. The pressure can be calculated as follows:<ref name=softmatter/>

<center>

: <math>P = \rho kT-\frac{2}{3}\pi\rho^2\int_{0}^{\infty}dr\frac{du(r)}{dr}r^3g(r)</math>.

<math>g^{(n)}(\rm{r}_{1},\ldots,\rm{r}_{n})=\frac{V^{N}N!}{N^{n}(N-n)!}\cdot\frac{\int \cdots \int e^{-\beta U_{N}}d\rm{r}_{n+1},\cdots,\rm{r}_{N}}{Z_{N}} \, </math> (12)

</center>

Note that the results of potential energy and pressure will not be as accurate as directly calculating these properties because of the averaging involved with the calculation of <math>g(r)</math>.

In the theory of liquids <math>g^{(2)}(\rm{r}_{1},\rm{r}_{2})</math> is of special importance for it can be determined experimentally using [[X-ray diffraction]]. If the liquid contains spherically symmetric molecules <math>g^{(2)}(\rm{r}_{1},\rm{r}_{2})</math> depends only on the relative distance between molecules, <math>\rm{r}_{12}</math>. People usually drop the subscripts: <math> g(r)=g^{(2)}(r_{12})</math>. Now <math> \rho g(r) d\rm{r}</math> is the probability of finding a molecule at '''r''' given that there is a molecule at the origin of '''r'''. Note that this probability is not normalized:

== Approximations ==

<center>

For dilute systems (e.g. gases), the correlations in the positions of the particles that <math>g(r)</math> accounts for are only due to the potential <math>u(r)</math> engendered by the reference particle, neglecting indirect effects. In the first approximation, it is thus simply given by the Boltzmann distribution law:

<math>\int_{0}^{\infty}\rho g(r) 4\pi r^{2} dr = N-1 \approx N </math> (13)

{{NumBlk|:| <math>g(r) = \exp \left [ -\frac{u(r)}{kT} \right ] </math>.|{{EquationRef|11}}}}

</center>

If <math>u(r)</math> were zero for all <math>r</math> &ndash; i.e., if the particles did not exert any influence on each other, then <math>g(r) = 1 </math> for all <math>\mathbf{r}</math> and the mean local density would be equal to the mean density <math>\rho</math>: the presence of a particle at ''O'' would not influence the particle distribution around it and the gas would be ideal. For distances <math>r</math> such that <math>u(r)</math> is significant, the mean local density will differ from the mean density <math>\rho</math>, depending on the sign of <math>u(r)</math> (higher for negative interaction energy and lower for positive <math>u(r)</math>).

In fact, equation 13 gives us the number of molecules between r and r + d r about a central molecule.

As the density of the gas increases, the low-density limit becomes less and less accurate since a particle situated in <math>\mathbf{r}</math> experiences not only the interaction with the particle at ''O'' but also with the other neighbours, themselves influenced by the reference particle. This mediated interaction increases with the density, since there are more neighbours to interact with: it makes physical sense to write a density expansion of <math>g(r)</math>, which resembles the [[virial equation]]:

As of current, information on how to obtain the higher order distribution functions (<math>g^{(3)}(\rm{r}_{1},\rm{r}_{2},\rm{r}_{3})</math>, etc.) is not known and scientists rely on approximations based upon [[statistical mechanics]].

{{NumBlk|:| <math>g(r) = \exp \left [ -\frac{u(r)}{kT} \right ] y(r) \quad \mathrm{with} \quad y(r) = 1 + \sum_{n=1}^{\infty} \rho ^n y_n (r)</math>.|{{EquationRef|12}}}}

This similarity is not accidental; indeed, substituting ({{EquationNote|12}}) in the relations above for the thermodynamic parameters (Equations {{EquationNote|7}}, {{EquationNote|9}} and {{EquationNote|10}}) yields the corresponding virial expansions.<ref name="Barker:1976">{{Cite journal | last1 = Barker | first1 = J. | last2 = Henderson | first2 = D. | doi = 10.1103/RevModPhys.48.587 | title = What is "liquid"? Understanding the states of matter | journal = Reviews of Modern Physics | volume = 48 | issue = 4 | pages = 587 | year = 1976 |bibcode = 1976RvMP...48..587B }}</ref> The auxiliary function <math>y(r)</math> is known as the ''cavity distribution function''.<ref name="HansenMcDonald2005" />{{rp|Table 4.1}} It has been shown that for classical fluids at a fixed density and a fixed positive temperature, the effective pair potential that generates a given <math>g(r)</math> under equilibrium is unique up to an additive constant, if it exists.<ref>{{Cite journal|last=Henderson|first=R. L.|date=1974-09-09|title=A uniqueness theorem for fluid pair correlation functions|journal=Physics Letters A|language=en|volume=49|issue=3|pages=197–198|doi=10.1016/0375-9601(74)90847-0|bibcode=1974PhLA...49..197H|issn=0375-9601}}</ref>

==References==

In recent years, some attention has been given to develop pair correlation functions for spatially-discrete data such as lattices or networks.<ref>{{cite journal |last1=Gavagnin |first1=Enrico |title=Pair correlation functions for identifying spatial correlation in discrete domains |journal=Physical Review E |date=4 June 2018 |volume=97 |issue=1 |page=062104 |doi=10.1103/PhysRevE.97.062104|pmid=30011502 |arxiv=1804.03452 |bibcode=2018PhRvE..97f2104G |s2cid=50780864 }}</ref>

#D.A. McQuarrie, Statistical Mechanics (Harper Collins Publishers) 1976

== Experimental ==

==External links==

One can determine <math>g(r)</math> indirectly (via its relation with the structure factor <math>S(q)</math>) using [[neutron scattering]] or [[x-ray scattering]] data. The technique can be used at very short length scales (down to the atomic level<ref>{{Cite journal | last1 = Yarnell | first1 = J. | last2 = Katz | first2 = M. | last3 = Wenzel | first3 = R. | last4 = Koenig | first4 = S. | title = Structure Factor and Radial Distribution Function for Liquid Argon at 85 K | doi = 10.1103/PhysRevA.7.2130 | journal = Physical Review A | volume = 7 | issue = 6 | pages = 2130 | year = 1973 |bibcode = 1973PhRvA...7.2130Y }}</ref>) but involves significant space and time averaging (over the sample size and the acquisition time, respectively). In this way, the radial distribution function has been determined for a wide variety of systems, ranging from liquid metals<ref>{{Cite journal | last1 = Gingrich | first1 = N. S. | last2 = Heaton | first2 = L. | doi = 10.1063/1.1731688 | title = Structure of Alkali Metals in the Liquid State | journal = The Journal of Chemical Physics | volume = 34 | issue = 3 | pages = 873 | year = 1961 |bibcode = 1961JChPh..34..873G }}</ref> to charged colloids.<ref>{{Cite journal | last1 = Sirota | first1 = E. | last2 = Ou-Yang | first2 = H. | last3 = Sinha | first3 = S. | last4 = Chaikin | first4 = P. | last5 = Axe | first5 = J. | last6 = Fujii | first6 = Y. | doi = 10.1103/PhysRevLett.62.1524 | title = Complete phase diagram of a charged colloidal system: A synchro- tron x-ray scattering study | journal = Physical Review Letters | volume = 62 | issue = 13 | pages = 1524–1527 | year = 1989 | pmid = 10039696|bibcode = 1989PhRvL..62.1524S }}</ref> Going from the experimental <math>S(q)</math> to <math>g(r)</math> is not straightforward and the analysis can be quite involved.<ref>{{Cite journal | last1 = Pedersen | first1 = J. S. | doi = 10.1016/S0001-8686(97)00312-6 | title = Analysis of small-angle scattering data from colloids and polymer solutions: Modeling and least-squares fitting | journal = Advances in Colloid and Interface Science | volume = 70 | pages = 171–201 | year = 1997 }}</ref>

* http://www.ccr.buffalo.edu/etomica/app/modules/sites/Ljmd/Background1.html

It is also possible to calculate <math>g(r)</math> directly by extracting particle positions from traditional or confocal microscopy.<ref>{{Cite journal | last1 = Crocker | first1 = J. C.| last2 = Grier | first2 = D. G.| doi = 10.1006/jcis.1996.0217 | title = Methods of Digital Video Microscopy for Colloidal Studies | journal = Journal of Colloid and Interface Science | volume = 179 | issue = 1| pages = 298–310| year = 1996 | bibcode = 1996JCIS..179..298C}}</ref> This technique is limited to particles large enough for optical detection (in the micrometer range), but it has the advantage of being time-resolved so that, aside from the statical information, it also gives access to dynamical parameters (e.g. [[diffusion constant]]s<ref>{{Cite journal | last1 = Nakroshis | first1 = P. | last2 = Amoroso | first2 = M. | last3 = Legere | first3 = J. | last4 = Smith | first4 = C. | title = Measuring Boltzmann's constant using video microscopy of Brownian motion | doi = 10.1119/1.1542619 | journal = American Journal of Physics | volume = 71 | issue = 6 | pages = 568 | year = 2003 |bibcode = 2003AmJPh..71..568N }}</ref>) and also space-resolved (to the level of the individual particle), allowing it to reveal the morphology and dynamics of local structures in colloidal crystals,<ref>{{Cite journal | last1 = Gasser | first1 = U. | last2 = Weeks | first2 = E. R. | last3 = Schofield | first3 = A. | last4 = Pusey | first4 = P. N. | last5 = Weitz | first5 = D. A. | title = Real-Space Imaging of Nucleation and Growth in Colloidal Crystallization | doi = 10.1126/science.1058457 | journal = Science | volume = 292 | issue = 5515 | pages = 258–262 | year = 2001 | pmid = 11303095|bibcode = 2001Sci...292..258G | s2cid = 6590089 }}</ref> glasses,<ref>M.I. Ojovan, D.V. Louzguine-Luzgin. Revealing Structural Changes at Glass Transition via Radial Distribution Functions. J. Phys. Chem. B, 124 (15), 3186-3194 (2020) https://doi.org/10.1021/acs.jpcb.0c00214</ref><ref>{{Cite journal | last1 = Weeks | first1 = E. R. | last2 = Crocker | first2 = J. C. | last3 = Levitt | first3 = A. C. | last4 = Schofield | first4 = A. | last5 = Weitz | first5 = D. A. | title = Three-Dimensional Direct Imaging of Structural Relaxation Near the Colloidal Glass Transition | doi = 10.1126/science.287.5453.627 | journal = Science | volume = 287 | issue = 5453 | pages = 627–631 | year = 2000 | pmid = 10649991|bibcode = 2000Sci...287..627W }}</ref> gels,<ref>{{Cite journal | last1 = Cipelletti | first1 = L. | last2 = Manley | first2 = S. | last3 = Ball | first3 = R. C. | last4 = Weitz | first4 = D. A. | title = Universal Aging Features in the Restructuring of Fractal Colloidal Gels | doi = 10.1103/PhysRevLett.84.2275 | journal = Physical Review Letters | volume = 84 | issue = 10 | pages = 2275–2278 | year = 2000 | pmid = 11017262|bibcode = 2000PhRvL..84.2275C }}</ref><ref>{{Cite journal | last1 = Varadan | first1 = P. | last2 = Solomon | first2 = M. J. | doi = 10.1021/la026303j | title = Direct Visualization of Long-Range Heterogeneous Structure in Dense Colloidal Gels | journal = Langmuir | volume = 19 | issue = 3 | pages = 509 | year = 2003 }}</ref> and hydrodynamic interactions.<ref>{{Cite journal | last1 = Gao | first1 = C. | last2 = Kulkarni | first2 = S. D. | last3 = Morris | first3 = J. F. | last4 = Gilchrist | first4 = J. F. | title = Direct investigation of anisotropic suspension structure in pressure-driven flow | doi = 10.1103/PhysRevE.81.041403 | journal = Physical Review E | volume = 81 | issue = 4 | pages = 041403 | year = 2010 | pmid = 20481723|bibcode = 2010PhRvE..81d1403G }}</ref>

Direct visualization of a full (distance-dependent and angle-dependent) pair correlation function was achieved by a [[scanning tunneling microscope|scanning tunneling microscopy]] in the case of 2D molecular gases.<ref>{{Cite journal|last1=Matvija|first1=Peter|last2=Rozbořil|first2=Filip|last3=Sobotík|first3=Pavel|last4=Ošťádal|first4=Ivan|last5=Kocán|first5=Pavel|title=Pair correlation function of a 2D molecular gas directly visualized by scanning tunneling microscopy|journal=The Journal of Physical Chemistry Letters|volume=8|issue=17|pages=4268–4272|doi=10.1021/acs.jpclett.7b01965|pmid=28830146|year=2017}}</ref>

[[Category: Statistical mechanics]]

[[Category: Mechanics]]

[[Category: Physical chemistry]]

== Higher-order correlation functions ==

[[it:funzione di distribuzione radiale]]

It has been noted that radial distribution functions alone are insufficient to characterize structural information. Distinct point processes may possess identical or practically indistinguishable radial distribution functions, known as the degeneracy problem.<ref>{{Cite journal|last1=Stillinger|first1=Frank H.|last2=Torquato|first2=Salvatore|date=2019-05-28|title=Structural degeneracy in pair distance distributions|url=https://aip.scitation.org/doi/10.1063/1.5096894|journal=The Journal of Chemical Physics|volume=150|issue=20|pages=204125|doi=10.1063/1.5096894|pmid=31153177|bibcode=2019JChPh.150t4125S|s2cid=173995240|issn=0021-9606|doi-access=free|url-access=subscription}}</ref><ref>{{Cite journal|last1=Wang|first1=Haina|last2=Stillinger|first2=Frank H.|last3=Torquato|first3=Salvatore|date=2020-09-23|title=Sensitivity of pair statistics on pair potentials in many-body systems|journal=The Journal of Chemical Physics|volume=153|issue=12|pages=124106|doi=10.1063/5.0021475|pmid=33003740|bibcode=2020JChPh.153l4106W|s2cid=222169131|issn=0021-9606|doi-access=free}}</ref> In such cases, higher order correlation functions are needed to further describe the structure.

Higher-order distribution functions <math>\textstyle g^{(k)}</math> with <math>\textstyle k > 2</math> were less studied, since they are generally less important for the thermodynamics of the system; at the same time, they are not accessible by conventional scattering techniques. They can however be measured by [[coherent scattering|coherent X-ray scattering]] and are interesting insofar as they can reveal local symmetries in disordered systems.<ref>{{Cite journal | last1 = Wochner | first1 = P. | last2 = Gutt | first2 = C. | last3 = Autenrieth | first3 = T. | last4 = Demmer | first4 = T. | last5 = Bugaev | first5 = V. | last6 = Ortiz | first6 = A. D. | last7 = Duri | first7 = A. | last8 = Zontone | first8 = F. | last9 = Grubel | first9 = G. | doi = 10.1073/pnas.0905337106 | last10 = Dosch | first10 = H. | title = X-ray cross correlation analysis uncovers hidden local symmetries in disordered matter | journal = Proceedings of the National Academy of Sciences | volume = 106 | issue = 28 | pages = 11511–4 | year = 2009 | pmid = 20716512| pmc = 2703671|bibcode = 2009PNAS..10611511W | doi-access = free }}</ref>

== See also ==

* [[Ornstein–Zernike equation]]

* [[Structure factor|Structure Factor]]

== References ==

{{reflist}}

* Widom, B. (2002). Statistical Mechanics: A Concise Introduction for Chemists. Cambridge University Press.

* McQuarrie, D. A. (1976). Statistical Mechanics. HarperCollins Publishers.

[[Category:Statistical mechanics]]

[[Category:Mechanics]]

[[Category:Physical chemistry]]