Abstract
In this paper we give a precise mathematical formulation of the relation between Bose condensation and long cycles and prove its validity for the perturbed mean field model of a Bose gas. We decompose the total density ρ=ρshort+ρlong into the number density of particles belonging to cycles of finite length (ρshort) and to infinitely long cycles (ρlong) in the thermodynamic limit. For this model we prove that when there is Bose condensation, ρlong is different from zero and identical to the condensate density. This is achieved through an application of the theory of large deviations. We discuss the possible equivalence of ρlong≠ 0 with off-diagonal long range order and winding paths that occur in the path integral representation of the Bose gas
Similar content being viewed by others
References
R. P. Feynman, Atomic theory of the λ transition in Helium, Phys. Rev. 91:1291, (1953); R. P. Feynman, Statistical Mechanics, Chap. 11 (Benjamin, 1974).
O. Penrose L. Onsager (1956) ArticleTitleBose–Einstein condensation and Liquid Helium Phys. Rev. 104 576 Occurrence Handle10.1103/PhysRev.104.576 Occurrence Handle1956PhRv..104..576P
D. Chandler P.G. Wolynes (1981) ArticleTitleExploiting the isomorphism between quantum theory and classical statistical mechanics of polyatomic fluids J. Chem. Phys. 74 4078 Occurrence Handle10.1063/1.441588 Occurrence Handle1981JChPh..74.4078C
R.P. Sear J.A. Cuesta (2001) ArticleTitleWhat do emulsification failure and Bose–Einstein condensation have in common? Europhys. Lett. 55 451 Occurrence Handle10.1209/epl/i2001-00436-6
A.M.J. Schakel (2001) ArticleTitlePercolation, Bose–Einstein condensation and string proliferation Phys. Rev. E 63 126115 Occurrence Handle10.1103/PhysRevE.63.026115 Occurrence Handle2001PhRvE..63B6115S
A. Sütö, Percolation transition in the Bose gas, J. Phys. A: Math. Gen. 26:4689 (1993); Percolation transition in the Bose gas: II, J. Phys. A: Math. Gen. 35:6995 (2002).
M. Berg Particlevan den T.C. Dorlas J.T. Lewis J.V. Pulé (1999) ArticleTitleA perturbed mean field model of an interacting boson gas and the Large Deviation Principle Commun. Math. Phys. 127 1
J.Ginibre, in Statistical Mechanics and Quantum Field Theory, C. DeWitt and R. Stora, eds. (Gordon and Breach, Les Houches, 1971), p. 327.
F. Cornu (1996) ArticleTitleCorrelations in quantum plasmas Phys. Rev. E 53 4562 Occurrence Handle1996PhRvE..53.4562C Occurrence Handle97c:82043
D.M. Ceperley (1995) ArticleTitlePath integrals in the theory of condensed Helium Rev. Mod. Phys. 67 279 Occurrence Handle10.1103/RevModPhys.67.279 Occurrence Handle1995RvMP...67..279C
Ph.A. Martin (2003) ArticleTitleQuantum Mayer graphs: application to Bose and Coulomb gases Acta Phys. Pol. B 34 3629 Occurrence Handle2003AcPPB..34.3629M
Dorlas T.C., Lewis J.T., V Pulé J. (1991). Condensation in Some Perturbed Meanfield Models of a Bose Gas, Helv. Phys. Acta 1200
T.C. Dorlas T.C. Lewis J. V Pulé (1992) ArticleTitleCondensation in a variational problem on the space of measures Arch. Rational Mech. Anal. 118 245 Occurrence Handle10.1007/BF00387897 Occurrence Handle93g:49033
R.B. Griffiths (1964) ArticleTitleA proof that the free energy of a spin system is extensive J. Math. Phys. 5 1215 Occurrence Handle10.1063/1.1704228
L. Landau and E. Lifshitz, Statistical Mechanics (Pergamon Press, 1976).
R.S. Ellis J. Gough J.V. Pulé (1993) ArticleTitleThe Large Deviation Principle for measures with random weights Rev. Math. Phys. 5 659 Occurrence Handle10.1142/S0129055X93000206 Occurrence Handle94j:60051
S.R.S. Varadhan (1989) ArticleTitleAsymptotic probabilities and differential equations Commun. Pure Appl. Math. 19 261 Occurrence Handle34 #3083
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Dorlas, T.C., Martin, P.A. & Pule, J.V. Long Cycles in a Perturbed Mean Field Model of a Boson Gas. J Stat Phys 121, 433–461 (2005). https://doi.org/10.1007/s10955-005-7582-0
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s10955-005-7582-0