In statistical mechanics, an Ursell function or connected correlation function, is a cumulant of a random variable. It is also called a connected correlation function as it can often be obtained by summing over connected Feynman diagrams (the sum over all Feynman diagrams gives the correlation functions).
If X is a random variable, the moments sn and cumulants (same as the Ursell functions) un are related by the exponential formula:
(where E is the expectation). Formally, the quantities un are functions of the random variable X, in the same way as the expectation:
The Ursell functions for multivariate random variables are defined analogously to the above, and in the same way as multivariate cumulants.
The function was named after Harold Ursell, who introduced it in 1927.
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