Abstract
A stronger version of the Bogoliubov inequality is used to derive an upper bound for the anomalous average ¦<o(x)>|s of an interacting nonrelativislic Bose fielda(x) at a finite temperature. This bound is ¦a(x)2|s <pR, whereR satisfies 1 -R = (RT/2T c v/2, withv the dimensionality, andT c the critical temperature in the absence of interactions. The formation of nonzero averages is closely related to the Bose-Einstein condensation and ¦<a(X)>|·2 is often believed to coincide with the mean densitypa of the condensate. We have found nonrigorous arguments supporting the inequality po ⩾ ¦<a(X)>|·2, which parallels the result of Griffiths in the case of spin systems.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
L. J. Rodriguez and H. A. Gersch,Phys. Rev. A 9:2085 (1974).
D. Schiff and L. Verlet,Phys. Rev. 160:208 (1967); M. H. Kalos, D. Levesque, and L. Verlet,Phys. Rev. A 9:2178 (1974).
P. C. Hohenberg,Phys. Rev. 158:383 (1967).
G. V. Chester, M. E. Fisher, and N. D. Mermin,Phys. Rev. 185:760 (1969).
D. Ruelle,Statistical Mechanics (Benjamin, New York, 1969).
D. Ruelle,Comm. Math. Phys. 18:127 (1970).
N. N. Bogoliubov,Phys. Abh. S.U. 6(1):229 (1962).
G. Roepstorff,Comm. Math. Phys. 53:143 (1977).
J. Ginibre,Comm. Math. Phys. 8:26 (1968).
J. C. Garrison and J. Wong,Comm. Math. Phys. 26:1 (1972).
M. Bouziane and Ph. A. Martin,J. Math. Phys. 17:1848 (1976).
N. N. Bogoliubov,Lectures on Quantum Statistics, Vol. 2 (Gordon and Breach, New York, 1970).
R. Haag,Nuovo Cimento 25:287 (1962).
R. Griffiths,Phys. Rev. 152:240 (1966).
K. Hepp and E. H. Lieb,Phys. Rev. A 8:2517 (1973).
F. J. Dyson, E. H. Lieb, and B. Simon, Phase transitions in quantum spin systems with isotropic and non-isotropic interactions, to appear.
G. Roepstorff,Comm. Math. Phys. 46:253 (1976).
G. Lindblad,Comm. Math. Phys. 39:111 (1974).
R. Haag, N. M. Hugenholtz, and M. Winnink,Comm. Math. Phys. 5:215 (1967).
E. Feenberg,Theory of Quantum Fluids (Academic Press, New York, 1969).
R. T. Rockafellar,Convex Analysis (Princeton Univ. Press, 1972).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Roepstorff, G. Bounds for a Bose condensate in dimensions v ⩾ 3. J Stat Phys 18, 191–206 (1978). https://doi.org/10.1007/BF01014310
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01014310