Ursell function explained

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 s_n and cumulants (same as the Ursell functions) u_n are functions of X related by the exponential formula:

\operatorname{E}(\exp(zX)) = \sum _{n} s _{n} (z ^{n})/(n!) = \exp\left(\sum _{n} u _{n} (z ^{n})/(n!) \right)

(where

\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.^[1]

u _{n\left(X} _{1,} \ldots, X _{n\right)} = \left. (\partial)/(\partial z _{1}) … (\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 .

The first few are given by

\begin{align} u _{1(X} _{1)} = {} &\operatorname{E}(X _{1)} \\ u _{2(X} _{1,} X _{2)} = {} &\operatorname{E}(X _{1} X _{2)} - \operatorname{E}(X _{1)} \operatorname{E}(X _{2)} \\ 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)} \\ 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}}

Characterization

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_i can be divided into two nonempty independent sets.

See also

• Cumulant

Notes and References

1. Shlosman . S. B. . Signs of the Ising model Ursell functions . 10.1007/BF01221652 . Communications in Mathematical Physics . 102 . 4 . 679–686 . 1986 . 1985CMaPh.102..679S . 122963530 .