Ursell Function

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:

: $\operatorname(\exp(zX)) = \sum_n s_n \frac = \exp\left(\sum_n u_n \frac\right)$

(where $\operatorname$ is the expectation).

The Ursell functions for multivariate random variables are defined analogously to the above, and in the same way as multivariate cumulants.
:$u_n\left(X_1, \ldots, X_n\right) = \left.\frac \cdots \frac\log \operatorname\left(\exp\sum z_i X_i\right)\_$

The Ursell functions of a single random variable ''X'' are obtained from these by setting .

The first few are given by
:$\begin u_1(X_1) = &\operatorname(X_1) \\ u_2(X_1, X_2) = &\operatorname(X_1 X_2) - \operatorname(X_1) \operatorname(X_2) \\ u_3(X_1, X_2, X_3) = &\operatorname(X_1 X_2 X_3) - \operatorname(X_1) \operatorname(X_2 X_3) - \operatorname(X_2) \operatorname(X_3 X_1) - \operatorname(X_3) \operatorname(X_1 X_2) + 2 \operatorname(X_1) \operatorname(X_2) \operatorname(X_3) \\ u_4\left(X_1, X_2, X_3, X_4\right) = &\operatorname(X_1 X_2 X_3 X_4) - \operatorname(X_1) \operatorname(X_2 X_3 X_4) - \operatorname(X_2) \operatorname(X_1 X_3 X_4) - \operatorname(X_3) \operatorname(X_1 X_2 X_4) - \operatorname(X_4) \operatorname(X_1 X_2 X_3) \\ & - \operatorname(X_1 X_2) \operatorname(X_3 X_4) - \operatorname(X_1 X_3) \operatorname(X_2 X_4) - \operatorname(X_1 X_4) \operatorname(X_2 X_3) \\ & + 2 \operatorname(X_1 X_2) \operatorname(X_3) \operatorname(X_4) + 2 \operatorname(X_1 X_3) \operatorname(X_2) \operatorname(X_4) + 2 \operatorname(X_1 X_4) \operatorname(X_2) \operatorname(X_3) + 2 \operatorname(X_2 X_3) \operatorname(X_1) \operatorname(X_4) \\ & + 2 \operatorname(X_2 X_4) \operatorname(X_1) \operatorname(X_3) + 2 \operatorname(X_3 X_4) \operatorname(X_1) \operatorname(X_2) - 6 \operatorname(X_1) \operatorname(X_2) \operatorname(X_3) \operatorname(X_4) \end$

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

References

*
*
*{{citation, first=H. D. , last=Ursell, title=The evaluation of Gibbs phase-integral for imperfect gases, journal=Proc. Cambridge Philos. Soc., volume=23 , year=1927, issue=6, pages=685–697, doi=10.1017/S0305004100011191, bibcode=1927PCPS...23..685U, s2cid=123023251

Statistical mechanics
Theory of probability distributions