# Ursell function

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In [statistical mechanics](/source/statistical_mechanics), an '''Ursell function''' or '''connected correlation function''', is a [cumulant](/source/cumulant) of a [random variable](/source/random_variable). It can often be obtained by summing over connected [Feynman diagram](/source/Feynman_diagram)s (the sum over all Feynman diagrams gives the [correlation function](/source/correlation_function)s).

The Ursell function was named after [Harold Ursell](/source/Harold_Ursell), who introduced it in 1927.

==Definition==

If ''X'' is a random variable, the [moment](/source/Moment_(mathematics))s ''s''<sub>''n''</sub> and cumulants (same as the Ursell functions) ''u''<sub>''n''</sub> are functions of ''X'' related by the [exponential formula](/source/exponential_formula):

<math display="block">\operatorname{E}(\exp(zX)) = \sum_n s_n \frac{z^n}{n!} = \exp\left(\sum_n u_n \frac{z^n}{n!}\right)</math>

(where <math>\operatorname{E}</math> is the [expectation](/source/expected_value)).

The Ursell functions for multivariate random variables are defined analogously to the above, and in the same way as multivariate cumulants.<ref>{{Cite journal | last1 = Shlosman | first1 = S. B. | title = Signs of the Ising model Ursell functions | doi = 10.1007/BF01221652 | journal = Communications in Mathematical Physics | volume = 102 | issue = 4 | pages = 679–686 | year = 1986 |bibcode = 1985CMaPh.102..679S | s2cid = 122963530 | url = https://projecteuclid.org/euclid.cmp/1104114545 }}</ref>
:<math>u_n\left(X_1, \ldots, X_n\right) = \left.\frac{\partial}{\partial z_1} \cdots \frac{\partial}{\partial z_n}\log \operatorname{E}\left(\exp\sum z_i X_i\right)\right|_{z_i=0}</math>

The Ursell functions of a single random variable ''X'' are obtained from these by setting {{nowrap|1=''X'' = ''X''<sub>1</sub> = ⋯ = ''X''<sub>''n''</sub>}}.

The first few are given by
<math display="block">\begin{align}
                            u_1(X_1) ={} &\operatorname{E}(X_1) \\[1ex]
                       u_2(X_1, X_2) ={} &\operatorname{E}(X_1 X_2) - \operatorname{E}(X_1) \operatorname{E}(X_2) \\[1ex]
                  u_3(X_1, X_2, X_3) ={} &\operatorname{E}(X_1 X_2 X_3) - \operatorname{E}(X_1) \operatorname{E}(X_2 X_3) - \operatorname{E}(X_2) \operatorname{E}(X_3 X_1) - \operatorname{E}(X_3) \operatorname{E}(X_1 X_2) \\& + 2 \operatorname{E}(X_1) \operatorname{E}(X_2) \operatorname{E}(X_3) \\[1ex]
  u_4\left(X_1, X_2, X_3, X_4\right) ={}
    &\operatorname{E}(X_1 X_2 X_3 X_4) \\
    & - \operatorname{E}(X_1) \operatorname{E}(X_2 X_3 X_4) - \operatorname{E}(X_2) \operatorname{E}(X_1 X_3 X_4) \\
    & - \operatorname{E}(X_3) \operatorname{E}(X_1 X_2 X_4) - \operatorname{E}(X_4) \operatorname{E}(X_1 X_2 X_3) \\
    & - \operatorname{E}(X_1 X_2) \operatorname{E}(X_3 X_4) - \operatorname{E}(X_1 X_3) \operatorname{E}(X_2 X_4) - \operatorname{E}(X_1 X_4) \operatorname{E}(X_2 X_3) \\
    & + 2 \operatorname{E}(X_1 X_2) \operatorname{E}(X_3) \operatorname{E}(X_4) + 2 \operatorname{E}(X_1 X_3) \operatorname{E}(X_2) \operatorname{E}(X_4) \\&+ 2 \operatorname{E}(X_1 X_4) \operatorname{E}(X_2) \operatorname{E}(X_3) + 2 \operatorname{E}(X_2 X_3) \operatorname{E}(X_1) \operatorname{E}(X_4) \\
    & + 2 \operatorname{E}(X_2 X_4) \operatorname{E}(X_1) \operatorname{E}(X_3) + 2 \operatorname{E}(X_3 X_4) \operatorname{E}(X_1) \operatorname{E}(X_2) \\
    & - 6 \operatorname{E}(X_1) \operatorname{E}(X_2) \operatorname{E}(X_3) \operatorname{E}(X_4)
\end{align}</math>

==Characterization==

{{harvtxt|Percus|1975}} showed that the Ursell functions, considered as multilinear functions of several random variables, are uniquely determined up to a constant by the fact that they vanish whenever the variables ''X''<sub>''i''</sub> can be divided into two nonempty independent sets.

==See also==

* [Cumulant](/source/Cumulant)

==References==
{{reflist}}
{{refbegin}}
*{{Citation | author1-link=James Glimm | author2-link=Arthur Jaffe | last1=Glimm | first1=James | last2=Jaffe | first2=Arthur | title=Quantum physics | publisher=[Springer-Verlag](/source/Springer-Verlag) | location=Berlin, New York | edition=2nd | isbn=978-0-387-96476-8 |mr=887102 | year=1987}}
*{{citation|mr=0378683  |last=Percus|first= J. K.|title= Correlation inequalities for Ising spin lattices|journal= Comm. Math. Phys.
|volume= 40 |year=1975|issue=3|pages= 283–308|doi=10.1007/bf01610004|bibcode=1975CMaPh..40..283P|s2cid=120940116|url=https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-40/issue-3/Correlation-inequalities-for-Ising-spin-lattices/cmp/1103860532.pdf}} 
*{{citation|first=H. D. |last=Ursell|title=The evaluation of Gibbs phase-integral for imperfect gases|journal=Proc. Cambridge Philos. Soc.|volume=23 |year=1927|issue=6|pages=685–697|doi=10.1017/S0305004100011191|bibcode=1927PCPS...23..685U|s2cid=123023251 }}
{{refend}}

Category:Statistical mechanics
Category:Theory of probability distributions

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Adapted from the Wikipedia article [Ursell function](https://en.wikipedia.org/wiki/Ursell_function) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Ursell_function?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.