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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>
In recent years, some attention has been given to develop
==Experimental==
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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==
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* Widom, B. (2002). Statistical Mechanics: A Concise Introduction for Chemists. Cambridge University Press.
* McQuarrie, D. A. (1976). Statistical Mechanics. Harper Collins Publishers.
▲==See also==
▲* [[Ornstein–Zernike equation]]
▲* [[Structure factor|Structure Factor]]
[[Category:Statistical mechanics]]
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