Abstract
We consider a new way of going to the infinite volume thermodynamic limit for a finite density quantum system and apply it to the case of an ideal Boson gas. We describe two procedures for calculating the particle density in the thermodynamic limit, one local and one global, and show that they give different values for the density. Further calculations show that this discrepancy is caused by lack of macroscopic translation invariance of the system, which is not apparent at the microscopic level. We calculate the limiting value of the expectation function of the Weyl operators both above and below the critical density for Bose-Einstein condensation, and show that the condensate has paradoxical properties of a similar type to those recently discovered for the rotating Boson gas.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Araki, H., Woods, E.J.: Representations of the canonical commutation relations describing a non-relativistic infinite free Bose gas. J. Math. Phys.4, 637–662 (1963).
Cannon, J.T.: Infinite volume limits of the canonical free Bose gas states on the Weyl algebra. Commun. math. Phys.29, 89–104 (1973).
Davies, E.B.: Quantum stochastic processes. Commun. math. Phys.15, 277–304 (1969).
Davies, E.B.: The thermodynamic limit for an imperfect Boson gas. Commun. math. Phys.28, 69–86 (1972).
Davies, E.B.: A generalisation of Kaplansky's theorem. J. London Math. Soc.4, 435–436 (1972).
Davies, E.B.: Properties of the Green's functions of some Schrödinger operators. To appear, in J. London Math. Soc.
Garrod, C., Simmons, C.: Rigorous statistical mechanics for non-uniform systems. J. Math. Phys.13, 1168–1176 (1972).
Gates, D.J., Penrose, O.: The van der Waals limit for classical systems. I. A variational principle. Commun. math. Phys.15, 255–276 (1969).
Huang, K.: Statistical mechanics. London-New York: J. Wiley and Sons 1967.
Kac, M.: Private manuscript.
Kato, T.: Perturbation theory of linear operators. Berlin-Heidelberg-New York: Springer 1966.
Lewis, J.T., Pulè, J.: The equilibrium states of the free Boson gas. To appear.
Lewis, J.T., Pulè, J.: The free Boson gas in a rotating bucket. To appear.
Marchioro, C., Presutti, E.: Thermodynamics of particle systems in the presence of external macroscopic fields. I. Classical case. Commun. math. Phys.27, 146–154 (1972).
Messiah, A.: Quantum mechanics Vol. 1. Amsterdam: North-Holland Publ. Co. 1965.
Millard, K.: A statistical mechanical approach to the problem of a fluid in an external field. J. Math. Phys.13, 222–228 (1972).
Pulè, J.: D. Phil. thesis Oxford University 1972.
Ruelle, D.: Statistical mechanics, rigorous results. New York: W. A. Benjamin Inc. 1969.
Schechter, M.: Spectra of partial differential operators. Amsterdam North-Holland Publ. Co. 1971.
Titchmarsh, E. C.: Eigenfunction expansions, Part 1. Oxford Univ. Press, 2nd edition 1962.
Titchmarsh, E. C.: Eigenfunction expansions, Part 2. Oxford Univ. Press, 1958.
Watson, G.N.: Notes on generating functions of polynomials. J. London Math. Soc.8, 194–199 (1933).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Davies, E.B. The ideal Boson gas in an external scalar potential. Commun.Math. Phys. 30, 229–247 (1973). https://doi.org/10.1007/BF01837360
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01837360