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Expansion of the correlation functions of the grand canonical ensemble in powers of the activity

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Abstract

A study is made of the grand canonical ensemble of single-component systems of particles in a region Λ. A new representation of the Ursell functions is given. In it an Ursell function is represented as a sum of products of Mayer and Boltzmann functions over the subset of connected graphs labeled by trees. Such a representation greatly reduces the complexity of the structure of these functions. A new definition of all-round tending of the region Λ to infinity is given. The relationship between this definition and the well-known definition of tending of the set Λ to infinity in the sense of Fisher is demonstrated in examples. It is shown that in the case of all-round tending of the set Λ to infinity a term-by-term passage to the limit can be made in the series in Ruelle's representation of the correlation functions as a finite sum of finite products of convergent series. The domain of convergence of the obtained expansions is discussed. As examples, the expansions of the single-particle and binary correlation functions are obtained.

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Additional information

All-Union Correspondence Institute of the Food Industry. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 101, No. 1, pp. 94–109, October, 1994.

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Kalmykov, G.I. Expansion of the correlation functions of the grand canonical ensemble in powers of the activity. Theor Math Phys 101, 1224–1234 (1994). https://doi.org/10.1007/BF01079260

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Keywords

  • Correlation Function
  • Connected Graph
  • Canonical Ensemble
  • Convergent Series
  • Grand Canonical Ensemble