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Journal of Physical Studies

The Bogoliubov weakly imperfect Bose-gas

Valentin Anatol'evich Zagrebnov

Published Jan 1, 1999• 1 citation
Publisher

Abstract

It is shown that the condensation in the Bogoliubov Weakly Imperfect Bose{Gas (WIBG) may appear in two stages.If the interaction is such that the pressure of the WIBG does not coincide with the pressure of the Perfect Bose{Gas (PBG), then the WIBG may manifest two kinds of condensations: a non{conventional condensation in the zero{mode due to the interaction (the rst stage) and a conventional (generalized of type I) Bose{Einstein condensation in modes next to the zero{mode due to the particle density saturation (the second stage).

DOI
10.30970/jps.03.239

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It is shown that the condensation in the Bogoliubov Weakly Imperfect Bose{Gas (WIBG) may appear in two stages.If the interaction is such that the pressure of the WIBG does not coincide with the pressure of the Perfect Bose{Gas (PBG), then the WIBG may manifest two kinds of condensations: a non{conventional condensation in the zero{mode due to the interaction (the rst stage) and a conventional (generalized of type I) Bose{Einstein condensation in modes next to the zero{mode due to the particle density saturation (the second stage).
We show that the pressure of the Bogoliubov weakly imperfect Bose gas (WIBG) can be calculated exactly in the thermodynamic limit. We point out the sufficient and necessary conditions for it not to equate with the pressure of the ideal Bose gas (IBG). We prove that they differ only in that part of the phase diagram where the WIBG has a Bose condensate. We show that in contast to the conventional Bose condensate (e.g. in the IBG) the condensate in the WIBG is due to an effective attraction between bosons in the zero-mode.
We present an exactly soluble boson model which manifests two kinds of condensation. They occur in two stages: for intermediate densities one has a non-conventional Bose condensation in the lowest mode k = 0, which is due to a diagonal perturbation of the imperfect Bose gas Hamiltonian, whereas for large densities , this condensation coexists with conventional (generalized, non-extensive) Bose-Einstein condensation in non-zero modes condensation, corresponding to the standard mechanism of saturation.
We present a review of some published results concerning evolution of the notion of (conventional ) Bose-Einstein condensation (BEC) predicted in Ideal Bose-Gas (IBG) in 1925. The first step forward was the notion of the conventional generalised BEC à la van den Berg-Lewis-Pulé discovered in the IBG and later in certain models with interactions in 1980s. The next development was a discovery of non-conventional condensation of bosons that occurs only due to interaction. Finally we discuss an example of (non-ideal) the Weakly Imperfect Bose-Gas (WIBG), where the conventional and non-conventional condensates may coexist due to interaction.
The exactness of the Bogoliubov approximation is discussed from the point of view of the condensation properties of the Bose gas. For the imperfect Bose gas, the authors find that the approximation yields the correct condensate density when the thermodynamic limit is taken by isotropic dilation; this is not the case in general for other ways of going to the infinite volume limit. They also prove the existence of Bose-Einstein condensation for a class of Bose gases with weak interactions and an energy gap.
The nature of the Einstein-Bose condensation in an imperfect boson gas is discussed. It is argued that no matter how low the temperature, provided it is not zero, macroscopic second-sound waves should be weakly excited as thermal fluctuations about equilibrium. This implies that the single-particle momentum distribution must be smeared so that the state of zero momentum cannot contain a finite fraction of the particles. A microscopic mechanism which independently leads to a momentum spread is discussed. It is suggested that the actual Einstein-Bose condensation of an imperfect gas takes place in energy rather than momentum but that even in terms of this variable it becomes perfectly sharp only at zero temperature. This hypothesis is formulated analytically in terms of correlation functions of the second-quantized boson field.
We study a non-interacting (perfect) Bose-gas in random external potentials (traps).It is shown that a generalized Bose-Einstein condensation in the random eigenstates manifests if and only if the same occurs in the one-particle kinetic-energy eigenstates, which corresponds to the generalized condensation of the free Bose-gas.Moreover, we prove that the amounts of both condensate densities are equal.This statement is relevant for justification of the Bogoliubov approximation in the theory of disordered boson systems.