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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D. Ruelle,Statistical Mechanics, New York (1969).
D. Ruelle,Rev. Mod. Phys.,36, 580 (1964).
G. I. Kalmykov,Teor. Mat. Fiz.,84, 279 (1990).
G. I. Kalmykov,Diskretnaya Matematika,4, No. 2, 66 (1992).
G. I. Kalmykov,Teor. Mat. Fiz.,97, 452 (1993).
G. M. Fikhtengol'ts,Course of Differential and Integral Calculus, Vol. 2 [in Russian], Nauka, Moscow (1969).
A. Isihara,Statistical Physics, Academic Press, New York (1971).
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
- Correlation Function
- Connected Graph
- Canonical Ensemble
- Convergent Series
- Grand Canonical Ensemble