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 s_n and cumulants (same as Ursell functions) u_n 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

Glimm, James; Jaffe, Arthur (1987), Quantum physics (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-96476-8, MR 887102
Ursell, H. D. (1927), "The evaluation of Gibbs phase-integral for imperfect gases", Proc. Cambridge Philos. Soc 23: 685–697, doi:10.1017/S0305004100011191