Question

Find relation between moments and cumulants

Answer 1

Cumulant expansionl ) is one of powerful tools in statistical mechanics. For example, although the original Mayer and Mayer's derivation2 ) of cluster expansion involved some complicated combinatorics, Kubol} presented a simplified and straightforward derivation on the basis of cumulant expansion. In doing so, a crucial point is to write cumulants in terms of moments. In the case of cluster expansion, it turns out that the cumulant is equal to the moment in the limit V --> 00 where V is the volume of the system under consideration. An inverse procedure, i.e., to express moments in terms of cumulants is also possible as was discussed by Mehta.3 ) The purpose of this paper is to show that the above procedure is useful to study high temperature expansion for n=O system. In § 2, for the convenience of the reader, we will briefly discuss a procedure to express moments in terms of cumulants. We will apply in § 3 the results to n=O system and show that high temperature expansion studied in previous papers4 ),5) is interpreted from a diagrammatic point of view. The problem of self-avoiding walk is also discussed.


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