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Low Temperature Physics

Ideal Bose gas in steep one-dimensional traps

Andrij A. Rovenchak, Yuri Krynytskyi

Published Jan 1, 2022• 2 citations
Publisher

Abstract

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.

DOI
10.1063/10.0008959

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Abstract We have proposed an analytical solution to the problem of Bose–Einstein condensation (BEC) of harmonically trapped, two-dimensional, and ideal atoms. It is found that the number of atoms in vapor is characterized by an analytical function, which involves a series of q -digamma functions in mathematics. We employ the analytical solution to calculate the internal energy E , the entropy S , the Helmholtz free energy F , and the heat capacity at constant number C N of ideal Bose atoms in a two-dimensional isotropic harmonic trap. The first main finding in this paper is that the variation with temperature of the internal energy of a finite number of ideal Bose atoms has an inflection point, which occurs at the transition temperature T c . The second main finding is that the internal energy of a finite number of ideal Bose atoms has a classic limit E = 2 N k B T as T ≫ T c . The third main finding is that the variation with temperature of the heat capacity of a finite number of ideal Bose atoms has a maximal value, which occurs at T c . The fourth main finding is that the heat capacity of a finite number of ideal Bose atoms has a classic limit C N / k B = 2 N as T ≫ T c . The fifth main finding is that in the thermodynamic limit, at critical temperature T c the heat capacity at constant number shows a cusp singularity, which is analogous to the λ -transition of liquid helium. Consequently, the transition between normal and condensed states is a second-order phase transition.
Thermodynamic properties of ideal Bose gas trapped in an external generic power law potential are investigated systematically from the grand thermodynamic potential in ddd dimensional space. The most general conditions for Bose-Einstein condensate and the discontinuous conditions of heat capacity at the critical temperature in presence of generic power law potential are presented in this manuscript. The dependence of the physical quantities on external potential, particle characteristics and space dimensionality are discussed. The more general results obtained in this paper presents an unified illustration of Bose-Einstein condensation of ideal Bose systems as they reduces to the expressions and conclusions available in the literature with appropiate choice of power law exponent.
We employ the q-digamma function to calculate the internal energy E, the entropy S, the Helmholtz free energy F, and the heat capacity at constant number of ideal Bose atoms in a one-dimensional harmonic trap. The first main finding in this paper is that in a one-dimensional harmonic trap and when the temperature approaches zero, the entropy of a finite number of ideal Bose atoms is nonzero but the entropy of an infinite number of ideal Bose atoms is zero. The second main finding is that the heat capacity at constant number has a classic limit as . In the thermodynamic limit, we find that at critical temperature the heat capacity at constant number is discontinuous. Hence, at there is a jump in the heat capacity and this jump is calculated. Consequently, the transition between normal and condensed states is a second-order phase transition.
We analyze the thermodynamic limit—modeled as the open-trap limit of an isotropic harmonic potential—of an ideal, non-relativistic Bose gas with a special emphasis on the phenomenon of Bose–Einstein condensation. This is accomplished by the use of an asymptotic expansion of the grand potential, which is derived by ζ-regularization techniques. Herewith, we can show that the singularity structure of this expansion is directly interwoven with the phase structure of the system: In the non-condensation phase, the expansion has a form that resembles usual heat kernel expansions. By this, thermodynamic observables are directly calculable. In contrast, the expansion exhibits a singularity of infinite order above a critical density, and a renormalization of the chemical potential is needed to ensure well-defined thermodynamic observables. Furthermore, the renormalization procedure forces the system to exhibit condensation. In addition, we show that characteristic features of the thermodynamic limit, such as the critical density or the internal energy, are entirely encoded in the coefficients of the asymptotic expansion.
Within the framework of quantum statistical mechanics, we have proposed an analytical solution to the problem of Bose–Einstein condensation (BEC) in harmonically trapped, one-dimensional and ideal atoms. It is found that the number of atoms of vapor is characterized by an analytical function, which involves a q -digamma function in mathematics. The results of the numerical calculation of the analytical solution agree completely with the previous experimental results in the BEC of harmonically trapped, one-dimensional and ideal atoms. The analytical expressions of the critical temperature and the condensate fraction are derived in the thermodynamic limit.
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.