Abstract:
The bosonic atoms used in present day experiments on Bose-Einstein condensation are made up of fermionic electrons and nucleons. In this Letter we demonstrate how the Pauli exclusion principle for these constituents puts an upper limit on the Bose-Einstein-condensed fraction. Detailed numerical results are presented for hydrogen atoms in a cubic volume and for excitons in semiconductors and semiconductor bilayer systems. The resulting condensate depletion scales differently from what one expects for bosons with a repulsive hard-core interaction. At high densities, Pauli exclusion results in significantly more condensate depletion. These results also shed a new light on the low condensed fraction in liquid helium II.

Abstract:
We study the perfect Bose gas in random external potentials and show that there is generalized Bose-Einstein condensation in the random eigenstates 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. Our method is based on the derivation of an explicit formula for the occupation measure in the one-body kinetic-energy eigenstates which describes the repartition of particles among these non-random states. This technique can be adapted to re-examine the properties of the perfect Bose gas in the presence of weak (scaled) non-random potentials, for which we establish similar results.

Abstract:
In an equilibrium three-dimensional (3D) disordered condensate, it's well established that disorder can generate an amount of normal fluid equaling to $\frac{4}{3}$ of the condensate depletion. The concept that the superfluid is more volatile to the existence of disorder than the condensate is crucial to the understanding of Bose glass phase. In this Letter, we show that, by bringing a weakly disordered 3D condensate to nonequilibrium regime via a quantum quench in the interaction, disorder can destroy superfluid significantly more, leading to a steady state in which the normal fluid density far exceeds $\frac{4}{3}$ of the condensate depletion. This suggests a possibility of engineering Bose Glass in the dynamic regime. As both the condensate density and superfluid density are measurable quantities, our results allow an experimental demonstration of the dramatized interplay between the disorder and interaction in the nonequilibrium scenario.

Abstract:
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.

Abstract:
We study Bose-Einstein Condensation (BEC) in the Infinite-Range Hopping Bose-Hubbard model for repulsive on-site particle interaction in presence of ergodic random one-site potentials with different distributions. We show that the model is exactly soluble even if the on-site interaction is random. But in contrast to the non-random case, we observe here new phenomena: instead of enhancement of BEC for perfect bosons, for constant on-site repulsion and discrete distributions of the single-site potential there is suppression of BEC at some fractional densities. We show that this suppression appears with increasing disorder. On the other hand, the BEC suppression at integer densities may disappear, if disorder increases. For a continuous distribution we prove that the BEC critical temperature decreases for small on-site repulsion while the BEC is suppressed at integer values of density for large repulsion. Again, the threshold for this repulsion gets higher, when disorder increases.

Abstract:
We investigate the field-induced insulator-to-superfluid transition of bosonic quasiparticles in $S=1/2$ weakly-coupled dimer antiferromagnets. In presence of realistic disorder due to site dilution of the magnetic lattice, we show that the system displays an extended Bose-glass phase characterized by the localization of the hard-core quasiparticles.

Abstract:
We present a theoretical study of the effects of the next-nearest-neighbor (NNN) hopping ($t_2$) on the properties of non-interacting bosons in optical lattices in the presence of an Aubry-Andr\'{e} quasi-disorder. First we investigate, employing exact diagonalization, the effects of $t_2$ on the localization properties of a single boson. The localization is monitored using an entanglement measure as well as with inverse participation ratio. We find that the sign of $t_2$ has a significant influence on the localization effects. We also provide analytical results in support of the trends found in the localization behavior. Further, we extend these results including the effects of a harmonic potential which obtains in experiments. Next, we study the effects of $t_2$ on Bose-Einstein condensation. We find that, a positive $t_2$ strongly enhances the low temperature thermal depletion of the condensate while a negative $t_2$ reduces it. It is also found that, for a fixed temperature, increasing the quasi-disorder strength reduces the condensate fraction in the extended regime while enhancing it in the localized regime. We also investigate the effects of boundary conditions and that of the phase of the AA potential on the condensate. These are found to have significant effects on the condensate fraction in the localization transition region.

Abstract:
We study the stationary state of a simple exclusion process on a ring which was recently introduced by Arndt {\it et al} [J. Phys. A {\bf 31} (1998) L45;cond-mat/9809123]. This model exhibits spatial condensation of particles. It has been argued that the model has a phase transition from a ``mixed phase'' to a ``disordered phase''. However, in this paper exact calculations are presented which, we believe, show that in the framework of a grand canonical ensemble there is no such phase transition. An analysis of the fluctuations in the particle density strongly suggests that the same result also holds for the canonical ensemble.

Abstract:
We introduce an irreversible discrete multiplicative process that undergoes Bose-Einstein condensation as a generic model of competition. New players with different abilities successively join the game and compete for limited resources. A player's future gain is proportional to its ability and its current gain. The theory provides three principles for this type of competition: competitive exclusion, punctuated equilibria, and a critical condition for the distribution of the players' abilities necessary for the dominance and the evolution. We apply this theory to genetics, ecology and economy.