Systems of noninteracting bosons trapped by linear potentials V(r) = αr, where r = |r|, are studied in one and three dimensions. For the latter problem, an interpolation formula is suggested for energy levels between n, ℓ ∼ 10 and the WKB limit. Thermodynamic functions are calculated for N = 10, … ,104 particles using discrete energy spectrum. The specific heat and fugacity are compared to the results of the quasiclassical approach, in which the external potential effectively increases the space dimensionality. As expected, the comparison demonstrates that the thermodynamic functions obtained using the discrete spectra rapidly approach the quasiclassical ones in a space with the effectively tripled space dimensionality as N increases.Systems of noninteracting bosons trapped by linear potentials V(r) = αr, where r = |r|, are studied in one and three dimensions. For the latter problem, an interpolation formula is suggested for energy levels between n, ℓ ∼ 10 and the WKB limit. Thermodynamic functions are calculated for N = 10, … ,104 particles using discrete energy spectrum. The specific heat and fugacity are compared to the results of the quasiclassical approach, in which the external potential effectively increases the space dimensionality. As expected, the comparison demonstrates that the thermodynamic functions obtained using the discrete spectra rapidly approach the quasiclassical ones in a space with the effectively tripled space dimensionality as N increases.

Low Temperature PhysicsJul 1, 20211 citation

Similarities and differences between Bose and Fermi gases

Raj K. Pathria

The purported ``equivalence'' between noninteracting Bose and Fermi gases in two dimensions is critically examined. Staying away from the thermodynamic limit, an exact calculation reveals some subtle, but significant, differences between the properties of the two systems that arise characteristically from the statistics obeyed by them. At the same time, certain physical quantities show differences that are not so subtle, in that they remain present even when the thermodynamic limit is taken. The role played by the spatial dimensionality $d$ is also elucidated and it turns out that there is nothing fundamentally special about dimensionality 2; if the single-particle energy spectrum operating in the systems is of the form $\ensuremath{\varepsilon}\ensuremath{\propto}{p}^{s},$ then the equivalence in question arises when and only when $d=s.$ The purported ``equivalence'' between noninteracting Bose and Fermi gases in two dimensions is critically examined. Staying away from the thermodynamic limit, an exact calculation reveals some subtle, but significant, differences between the properties of the two systems that arise characteristically from the statistics obeyed by them. At the same time, certain physical quantities show differences that are not so subtle, in that they remain present even when the thermodynamic limit is taken. The role played by the spatial dimensionality $d$ is also elucidated and it turns out that there is nothing fundamentally special about dimensionality 2; if the single-particle energy spectrum operating in the systems is of the form $\ensuremath{\varepsilon}\ensuremath{\propto}{p}^{s},$ then the equivalence in question arises when and only when $d=s.$

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topicsMar 1, 199825 citations

Thermodynamics of a one-dimensional lattice Bose gas

N. M. Bogoliubov

Theoretical and Mathematical PhysicsJun 1, 19861 citation

Ideal Bose gas in steep one-dimensional traps

Andrij A. Rovenchak, Yuri Krynytskyi

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.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.

Low Temperature PhysicsJan 1, 20222 citations

Thermodynamic properties of the low-dimensional perfect Bose gas

Gaetano Busiello, L. De Cesare, Dimo I. Uzunov

Physica A Statistical Mechanics and its ApplicationsAug 1, 19856 citations

d

is also elucidated and it turns out that there is nothing fundamentally special about dimensionality 2; if the single-particle energy spectrum operating in the systems is of the form $\ensuremath{\varepsilon}\ensuremath{\propto}{p}^{s},$ then the equivalence in question arises when and only when

d=s.