Radial distribution function: Difference between revisions

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

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

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

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

[[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>.]]

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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.

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>.

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]].

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>

== Definition ==

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.

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

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=== Energy equation ===

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}}}}

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== Higher-order correlation functions ==

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>

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{{reflist}}

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

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

[[Category:Statistical mechanics]]