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On Gibbs quantum and classical particle systems with three-body forces

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Ukrainian Mathematical Journal Aims and scope

Abstract

For equilibrium quantum and classical systems of particles interacting via ternary and pair (nonpositive) infinite-range potentials, a low activity convergent cluster expansion for their grand canonical reduced density matrices and correlation functions is constructed in the thermodynamic limit.

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References

  1. D. Ruelle, Statistical Mechanics. Rigorous Results, Benjamin (1969).

  2. D. Ruelle, “Superstable interactions in classical statistical mechanics,” Commun. Math. Phys., 18, 127–150 (1970).

    Article  MathSciNet  Google Scholar 

  3. D. Ya. Petrina, V. I. Gerasimenko, and P. V. Malyshev, Mathematical Foundations of Classical Statistical Mechanics, Gordon and Breach (1989).

  4. R. Esposito, F. Nicolo, and M. Pulvirenti, “Super-stable interactions in quantum statistical mechanics: Maxwell-Boltzmann statistics,” Ann. Ins. H. Poincaré, 36, No. 2, 127–158 (1982).

    MathSciNet  Google Scholar 

  5. R. Gielerak and V. Zagrebnov, “Analyticity and independence on the classical boundary conditions of the infinite volume thermal KMS states for a class of continuous systems. I. The Maxwell-Boltzmann statistics case,” Helv. Phys. Acta, 64, 1226–1246 (1991).

    MathSciNet  Google Scholar 

  6. J. Ginibre, “Reduced density matrices of quantum gases. I. Limit of infinite volume,” J. Math. Phys., 6, No. 2, 238–251 (1965).

    Article  MathSciNet  Google Scholar 

  7. D. Ya. Petrina, Mathematical Foundations of Quantum Statistical Mechanics. Continuous Systems, Kluwer, Dordrecht (1995).

    Google Scholar 

  8. Hui-Hsuing Kuo, “Gaussian measures in Banach spaces,” in: Lecture Notes in Mathematics, Vol. 463, Springer, Berlin (1975).

    Google Scholar 

  9. I. I. Gikhman and A. V. Skorokhod, Theory of Random Processes [in Russian], Vol. 1, Nauka, Moscow (1971).

    Google Scholar 

  10. M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. 2, Academic Press, New York (1975).

    MATH  Google Scholar 

  11. H. Moraale, “The Kirkwood-Salsburg equation and the virial expansion for many-body potentials,” Phys. Lett. A, 59, No. 1, 9–10 (1976).

    Article  Google Scholar 

  12. W. Greenberg, “Thermodynamic states of classical systems,” Commun. Math. Phys., 22, 259–268 (1971).

    Article  Google Scholar 

  13. V. I. Skrypnik, “On generalized Gibbs-type solutions of the diffusion Bogolyubov-Strel’tsova hierarchy,” Teor. Mat. Fiz., 58, No. 3, 398–420 (1984).

    Google Scholar 

  14. W. I. Skrypnik, “Correlation functions of infinite system of interacting Brownian particles; local in time evolution close to equilibrium,” J. Stat. Phys., 35, No. 5/6, 587–602 (1985).

    MathSciNet  Google Scholar 

  15. V. I. Skrypnik, “Mean-field limit in a generalized Gibbs system and an equivalent system of interacting Brownian particles,” Teor. Mat. Fiz., 76, No. 1, 100–117 (1988).

    MathSciNet  Google Scholar 

  16. V. I. Skrypnik, “Sine-Gordon transformations in nonequilibrium systems of Brownian particles,” Ukr. Mat. Zh., 49, No. 10, 1404–1421 (1997).

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics, Ukrainian Academy of Sciences, Kyiv

    V. I. Skrypnik

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  1. V. I. Skrypnik
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 7, pp. 976–996, July, 2006.

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Skrypnik, V.I. On Gibbs quantum and classical particle systems with three-body forces. Ukr Math J 58, 1106–1128 (2006). https://doi.org/10.1007/s11253-006-0123-4

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  • DOI: https://doi.org/10.1007/s11253-006-0123-4

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