The Bogoliubov model of weakly imperfect Bose gas

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, although the physical phenomenon is, of course, more complicated than the model. The theory is based on two Bogoliubovʹs ansätze: the first truncates the full Hamiltonian of the interacting bosons to produce the WIBG, whereas the second substitutes some operators by c-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 Bogoliubovʹs ansätze 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 features 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 a similar two-stage Bose condensation may take place. They indicate possibilities to go beyond the Bogoliubov theory and the Hamiltonian for the WIBG.