Ursell function: Difference between revisions
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minor expansion with ref |
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connected [[Feynman diagram]]s (the sum over all Feynman diagrams gives the [[correlation function]]s). |
connected [[Feynman diagram]]s (the sum over all Feynman diagrams gives the [[correlation function]]s). |
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If ''X'' is a random variable, the [[Moment (mathematics)|moment]]s ''s''<sub>''n''</sub> and cumulants (same as Ursell functions) ''u''<sub>''n''</sub> are related by the [[exponential formula]]: |
If ''X'' is a random variable, the [[Moment (mathematics)|moment]]s ''s''<sub>''n''</sub> and cumulants (same as the Ursell functions) ''u''<sub>''n''</sub> are related by the [[exponential formula]]: |
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: <math>E(\exp(zX)) = \sum_n s_n \frac{z^n}{n!} = \exp\left(\sum_n u_n \frac{z^n}{n!} \right)</math> |
: <math>\operatorname E(\exp(zX)) = \sum_n s_n \frac{z^n}{n!} = \exp\left(\sum_n u_n \frac{z^n}{n!} \right)</math> |
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(where |
(where E is the [[Expected value|expectation]]). Formally, the quantities ''u''<sub>''n''</sub> are functions of the random variable ''X'', in the same way as the expectation: |
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:<math>u_n= u_n(X). </math> |
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The Ursell functions for multivariate random variables are defined analogously to the above, and in the same way as multivariate cumulants.<ref>{{cite doi|10.1007/BF01221652}}</ref> |
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The function was named after [[Harold Ursell]], who introduced it in 1927. |
The function was named after [[Harold Ursell]], who introduced it in 1927. |
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==References== |
==References== |
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{{reflist}} |
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*{{Citation | author1-link=James Glimm | author2-link=Arthur Jaffe | last1=Glimm | first1=James | last2=Jaffe | first2=Arthur | title=Quantum physics | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=2nd | isbn=978-0-387-96476-8 | id={{MathSciNet | id = 887102}} | year=1987}} |
*{{Citation | author1-link=James Glimm | author2-link=Arthur Jaffe | last1=Glimm | first1=James | last2=Jaffe | first2=Arthur | title=Quantum physics | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=2nd | isbn=978-0-387-96476-8 | id={{MathSciNet | id = 887102}} | year=1987}} |
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*{{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|pages=685–697|doi=10.1017/S0305004100011191}} |
*{{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|pages=685–697|doi=10.1017/S0305004100011191}} |
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Revision as of 21:50, 8 April 2013
In statistical mechanics, an Ursell function or connected correlation function, is a cumulant of a random variable. It is also called a connected correlation function as it can often be obtained by summing over connected Feynman diagrams (the sum over all Feynman diagrams gives the correlation functions).
If X is a random variable, the moments sn and cumulants (same as the Ursell functions) un are related by the exponential formula:
(where E is the expectation). Formally, the quantities un are functions of the random variable X, in the same way as the expectation:
The Ursell functions for multivariate random variables are defined analogously to the above, and in the same way as multivariate cumulants.[1]
The function was named after Harold Ursell, who introduced it in 1927.
References
- ^ Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi:10.1007/BF01221652, please use {{cite journal}} (if it was published in a bona fide academic journal, otherwise {{cite report}} with
|doi=10.1007/BF01221652instead.
- Glimm, James; Jaffe, Arthur (1987), Quantum physics (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-96476-8, MR887102
- Ursell, H. D. (1927), "The evaluation of Gibbs phase-integral for imperfect gases", Proc. Cambridge Philos. Soc, 23: 685–697, doi:10.1017/S0305004100011191