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===The potential of mean force===
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:
{{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>.
===The energy equation===
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{{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 | pmid = | pmc = |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|url=http://www.sciencedirect.com/science/article/pii/0375960174908470|journal=Physics Letters A|language=en|volume=49|issue=3|pages=197–198|doi=10.1016/0375-9601(74)90847-0|issn=0375-9601}}</ref>
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 |doi=10.1103/PhysRevE.97.062104|arxiv=1804.03452 }}</ref>
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