We analyze the effects caused by an anomalous single-particle dispersion relation suggested in several quantum-gravity models, upon the thermodynamics of a Bose-Einstein condensate trapped in a generic 3-dimensional power-law potential. We prove that the shift in the condensation temperature, caused by a deformed dispersion relation, described as a non-trivial function of the number of particles and the shape associated to the corresponding trap, could provide bounds for the parameters associated to such deformation. Additionally, we calculate the fluctuations in the number of particles as a criterium of thermodynamic stability for these systems. We show that the apparent instability caused by the anomalous fluctuations in the thermodynamic limit can be suppressed considering the lowest energy associated to the system in question.
We analyze the effects caused by an anomalous single-particle dispersion relation suggested in several quantum-gravity models, upon the thermodynamics of a Bose-Einstein condensate trapped in a generic 3-dimensional power-law potential. We prove that the shift in the condensation temperature, caused by a deformed dispersion relation, described as a non-trivial function of the number of particles and the shape associated to the corresponding trap, could provide bounds for the parameters associated to such deformation. Additionally, we calculate the fluctuations in the number of particles as a criterium of thermodynamic stability for these systems. We show that the apparent instability caused by the anomalous fluctuations in the thermodynamic limit can be suppressed considering the lowest energy associated to the system in question.
Bose-Einstein condensation (BEC) of an ideal gas is investigated, beyond the thermodynamic limit, for a finite number N of particles trapped in a generic three-dimensional power-law potential. We derive an analytical expression for the condensation temperature Tc in terms of a power series in ${x}_{0}={\ensuremath{\varepsilon}}_{0}/{k}_{B}{T}_{c}$, where ${\ensuremath{\varepsilon}}_{0}$ denotes the zero-point energy of the trapping potential. This expression, which applies in Cartesian, cylindrical, and spherical power-law traps, is given analytically at infinite order. It is also given numerically for specific potential shapes as an expansion in powers of x0 up to the second order. We show that, for a harmonic trap, the well-known first-order shift of the critical temperature $\ensuremath{\Delta}{T}_{c}/{T}_{c}\ensuremath{\propto}{N}^{\ensuremath{-}1/3}$ is inaccurate when $N\ensuremath{\leqslant}{10}^{5}$, the next order (proportional to ${N}^{\ensuremath{-}1/2}$) being significant. We also show that finite-size effects on the condensation temperature cancel out in a cubic trapping potential, e.g., $V(r)\ensuremath{\propto}{r}^{3}$. Finally, we show that in a generic power-law potential of higher order, e.g., $V(r)\ensuremath{\propto}{r}^{\ensuremath{\alpha}}$ with $\ensuremath{\alpha}>3$, the shift of the critical temperature becomes positive. This effect provides a large increase of Tc for relatively small atom numbers. For instance, an increase of about $+$40% is expected with 104 atoms in a $V(r)\ensuremath{\propto}{r}^{12}$ trapping potential.
We analyze the consequences caused by a deformed dispersion relation, suggested in several quantum gravity models, upon a bosonic gas. Concerning the ground state of the Bogoliubov space of this system, we deduce the corrections in the pressure, the speed of sound, and the corresponding healing length. Indeed, we prove that the corrections in the relevant thermodynamic properties associated with the ground state, defines a non trivial function of the density of particles and the deformation parameters, allowing us to constrain, in principle, the form of the modified energy-momentum dispersion relation.
A generalization of the semiclassical approximation is suggested allowing for an essential extension of its region of applicability. In particular, it becomes possible to describe Bose-Einstein condensation of a trapped gas in low-dimensional traps and in traps of low confining dimensions, for which the standard semiclassical approximation is not applicable. The result of the modified approach is shown to coincide with purely quantum-mechanical calculations for harmonic traps, including the one-dimensional harmonic trap. The advantage of the semiclassical approximation is in its simplicity and generality. Power-law potentials of arbitrary powers are considered. The effective thermodynamic limit is defined for any confining dimension. The behavior of the specific heat, isothermal compressibility, and density fluctuations is analyzed, with an emphasis on low confining dimensions, where the usual semiclassical method fails. The peculiarities of the thermodynamic characteristics in the effective thermodynamic limit are discussed.
We analytically calculate the statistics of Bose-Einstein condensate (BEC) fluctuations in an interacting gas trapped in a three-dimensional cubic or rectangular box with the Dirichlet, fused or periodic boundary conditions within the mean-field Bogoliubov and Thomas-Fermi approximations. We study a mesoscopic system of a finite number of trapped particles and its thermodynamic limit. We find that the BEC fluctuations, first, are anomalously large and non-Gaussian and, second, depend on the trap's form and boundary conditions. Remarkably, these effects persist with increasing interparticle interaction and even in the thermodynamic limit-only the mean BEC occupation, not BEC fluctuations, becomes independent on the trap's form and boundary conditions.
In the present work, we analyze the corrections caused by an anomalous dispersion relation, suggested in several quantum gravity models, upon the speed of sound in a weakly interacting Bose-Einstein condensate, trapped in a potential of the form . We show that the corresponding ground-state energy and consequently, the associated speed of sound, present corrections with respect to the usual case, which may be used to explore the sensitivity to Planck-scale effects on these relevant properties associated with the condensate. Indeed, we stress that this type of macroscopic bodies may be more sensitive, under certain conditions, to Planck-scale manifestations than its constituents. In addition, we prove that the inclusion of a trapping potential, together with many-body contributions, improves the sensitivity to Planck-scale signals, compared to the homogeneous system.
We study thermodynamic properties of a one-dimensional ideal Bose gas trapped by a steep potential of an exponential type U(q)=U0[e(2q/a)b−1]. Fugacity, energy, and heat capacity of such a system are calculated for various combinations of the potential parameters as well for several values of the number of particles N. Both the thermodynamic limit and finite N are considered. Estimations for the single-particle spectrum asymptotics are obtained in the analytical form involving the Lambert W function. In the thermodynamic limit, the Bose–Einstein condensation is predicted for 0 < b < 2. We associate such behavior with an effective temperature-dependent space dimensionality arising due to the influence of the external potential of the analyzed type.