Ursell function

Ursell function

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 Ursell functions) un are related by the exponential formula:

$E(\exp(zX)) = \sum_n s_n \frac{z^n}{n!} = \exp\left(\sum_n \frac{u_nz^n}{n!} \right)$

(where E is the expectation).

The function was named after Harold Ursell, who introduced it in 1927.

References

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