In statistical mechanics, an Ursell function or connected correlation function, is a cumulant of a random variable. It can often be obtained by summing over connected Feynman diagrams (the sum over all Feynman diagrams gives the correlation functions). The Ursell function was named after Harold Ursell, who introduced it in 1927.
Definition If X is a random variable, the moments sn and cumulants (same as the Ursell functions) un are functions of X related by the exponential formula:
E ( exp ( z X ) ) = ∑ n s n z n n ! = exp ( ∑ n u n z n n ! ) {\displaystyle \operatorname {E} (\exp(zX))=\sum _{n}s_{n}{\frac {z^{n}}{n!}}=\exp \left(\sum _{n}u_{n}{\frac {z^{n}}{n!}}\right)}
(where E {\displaystyle \operatorname {E} } is the expectation). The Ursell functions for multivariate random variables are defined analogously to the above, and in the same way as multivariate cumulants.
u n ( X 1 , … , X n ) = ∂ ∂ z 1 ⋯ ∂ ∂ z n log E ( exp ∑ z i X i ) | z i = 0 {\displaystyle 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}}
The Ursell functions of a single random variable X are obtained from these by setting X = X1 = ⋯ = Xn. The first few are given by
u 1 ( X 1 ) =
E ( X 1 ) u 2 ( X 1 , X 2 ) =
E ( X 1 X 2 ) − E ( X 1 ) E ( X 2 ) u 3 ( X 1 , X 2 , X 3 ) =
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