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.
If X is a random variable, the moments s n u n 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} } 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 ( 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 = X 1 = … = X n
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 ) = E ( X 1 X 2 X 3 ) − E ( X 1 ) E ( X 2 X 3 ) − E ( X 2 ) E ( X 3 X 1 ) − E ( X 3 ) E ( X 1 X 2 ) + 2 E ( X 1 ) E ( X 2 ) E ( X 3 ) u 4 ( X 1 , X 2 , X 3 , X 4 ) = E ( X 1 X 2 X 3 X 4 ) − E ( X 1 ) E ( X 2 X 3 X 4 ) − E ( X 2 ) E ( X 1 X 3 X 4 ) − E ( X 3 ) E ( X 1 X 2 X 4 ) − E ( X 4 ) E ( X 1 X 2 X 3 ) − E ( X 1 X 2 ) E ( X 3 X 4 ) − E ( X 1 X 3 ) E ( X 2 X 4 ) − E ( X 1 X 4 ) E ( X 2 X 3 ) + 2 E ( X 1 X 2 ) E ( X 3 ) E ( X 4 ) + 2 E ( X 1 X 3 ) E ( X 2 ) E ( X 4 ) + 2 E ( X 1 X 4 ) E ( X 2 ) E ( X 3 ) + 2 E ( X 2 X 3 ) E ( X 1 ) E ( X 4 ) + 2 E ( X 2 X 4 ) E ( X 1 ) E ( X 3 ) + 2 E ( X 3 X 4 ) E ( X 1 ) E ( X 2 ) − 6 E ( X 1 ) E ( X 2 ) E ( X 3 ) E ( X 4 ) {\displaystyle {\begin{aligned}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{aligned}}} Percus (1975) 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
Glimm, James ; Jaffe, Arthur (1987), Quantum physics (2nd ed.), Berlin, New York: Springer-Verlag , ISBN 978-0-387-96476-8 MR 0887102 Percus, J. K. (1975), "Correlation inequalities for Ising spin lattices" (PDF) , Comm. Math. Phys. , 40 (3): 283–308, Bibcode :1975CMaPh..40..283P , doi :10.1007/bf01610004 , MR 0378683 , S2CID 120940116 Ursell, H. D. (1927), "The evaluation of Gibbs phase-integral for imperfect gases", Proc. Cambridge Philos. Soc. , 23 (6): 685–697, Bibcode :1927PCPS...23..685U , doi :10.1017/S0305004100011191 , S2CID 123023251 
