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The large deviation principle and some models of an interacting boson gas. (English) Zbl 0679.76124
Summary: This is a study of the equilibrium thermodynamics of the Huang-Yang- Luttinger model of a boson gas [K. Huang, C. N. Yang and J. M. Luttinger, Phys. Rev. 105, 776-784 (1957)] with a hard-sphere repulsion using large deviation methods; we contrast its properties with those of the mean field model. We prove the existence of the grand canonical pressure in the thermodynamic limit and derive two alternative expressions for the pressure as a function of the chemical potential. We prove the existence of condensate for values of the chemical potential above a critical value and verify a prediction of Thouless that there is a jump in the density of condensate at the critical value. We show also that, at fixed mean density, the density of condensate is an increasing function of the strength of the repulsive interaction. In an appendix, we give proofs of the large deviation results used in the body of the paper.

MSC:
76Y05 Quantum hydrodynamics and relativistic hydrodynamics
82B10 Quantum equilibrium statistical mechanics (general)
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[1] London, F.: On the Bose-Einstein condensation. Phys. Rev.54, 947-954 (1938) · Zbl 0019.42703 · doi:10.1103/PhysRev.54.947
[2] London, F.: Superfluids, Vol. II. New York: Wiley 1954
[3] Huang, K., Yang, C.N., Luttinger, J.M.: Imperfect bose gas with hard-sphere interactions. Phys. Rev.105, 776-784 (1957) · Zbl 0077.21001 · doi:10.1103/PhysRev.105.776
[4] Thouless, D.J.: The quantum-mechanics of many-body systems. New York: Academic Press 1961 · Zbl 0103.23502
[5] Lewis, J.T.: Why do bosons condense? In: Statistical mechanics and field theory: mathematical aspects proceedings, Groningen 1985. Dorlas, T.C., Hugenholtz, N.M., Winnink, M. (eds.). Lecture Notes in Physics, Vol. 257. Berlin, Heidelberg, New York: Springer 1986
[6] Varadhan, S.R.S.: Asymptotic probabilities and differential equations. Commun. Pure Appl. Math.19, 261-286 (1966) · Zbl 0147.15503 · doi:10.1002/cpa.3160190303
[7] Berg, M. van den, Lewis, J.T., Pulé, J.V.: A general theory of Bose-Einstein condensation. Helv. Phys. Acta59, 1271-1288 (1986)
[8] Berg, M. van den: On boson condensation into an infinite number of low-lying levels. J. Math. Phys.23, 1159-1161 (1982) · doi:10.1063/1.525445
[9] Berg, M. van den, Lewis, J.T.: On the free Boson gas in a weak external potential. Commun. Math. Phys.81, 475-494 (1981) · doi:10.1007/BF01208269
[10] Berg, M. van den, Lewis, J.T., Smedt, P. de: Condensation in the imperfect Boson gas. J. Stat. Phys.37, 697-707 (1984) · doi:10.1007/BF01010502
[11] Hove, L. van: Math. Revs.18, 836 (1957)
[12] Critchley, R.H.: Approximate equilibrium states for two models of an interacting Boson gas. J. Math. Phys.21, 359-363 (1980) · doi:10.1063/1.524423
[13] Ellis, R.: Entropy, large deviations and statistical mechanics. Berlin, Heidelberg, New York: Springer 1985 · Zbl 0566.60097
[14] Hepp, K., Lieb, E.H.: Equilibrium statistical mechanics of matter interacting with the quantized radiation field. Phys. Rev. A8, 2517-2525 (1973)
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