- 00-495 V. A. Zagrebnov, J.-B. Bru
- The Bogoliubov model of Weakly Imperfect Bose-Gas
Dec 10, 00
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Abstract. We present a systematic account of known rigorous results about
the Bogoliubov model of Weakly Imperfect Bose Gas (WIBG).
This model is a basis of the celebrated Bogoliubov theory of superfluidity.
The theory is based on two Bogoliubov's ans tze: the first truncates
the full Hamiltonian of the interacting bosons to produce the WIBG,
whereas the second substitutes some operators by complex numbers
(the Bogoliubov approximation). After some historical remarks,
and physical and mathematical motivations of this Bogoliubov
treatment of the WIBG, we turn to revision of the Bogoliubov's ans tze
from the point of view of rigorous quantum statistical mechanics.
Since the exact calculation of the pressure and the behaviour of
the Bose condensate in the WIBG are available, we review these results
stressing the difference between them and the Bogliubov theory.
One of the main feature of the mathematical analysis of the WIBG is that
it takes into account quantum fluctuations ignored by the second Bogoliubov ansatz.
It is these fluctuations which are responsible for indirect attraction between
bosons in the fundamental mode. The latter is the origin of a nonconventional
Bose condensation in this mode, which has a dynamical nature.
A (generalized) conventional Bose-Einstein condensation appears in the WIBG
only in the second stage as a result of the standard mechanism of the total
particle density saturation. It coexists with the nonconventional condensation.
We give also a review of some models related to the WIBG and to the Bogoliubov
theory, where the similar two-stage Bose condensation may take place. They indicate
possibilities to go beyond the Bogoliubov theory and the Hamiltonian for the WIBG.