From Wikipedia, the free encyclopedia
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.
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 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}}}
Characterization
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
See also
References
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 
