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Journal of Statistical Mechanics Theory and Experiment

Exact thermodynamic theory of an ideal boson gas in a one-dimensional harmonic trap

Ze Cheng

Published Nov 13, 2015• 6 citations
Publisher

Abstract

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.

DOI
10.1088/1742-5468/2015/11/p11003

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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.
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.
One can derive an analytic result for the issue of Bose–Einstein condensation (BEC) in anisotropic 2D harmonic traps. We find that the number of uncondensed bosons is represented by an analytic function, which includes a series expansion of q-digamma functions in mathematics. One can utilize this analytic result to evaluate various thermodynamic functions of ideal bosons in 2D anisotropic harmonic traps. The first major discovery is that the internal energy of a finite number of ideal bosons is a monotonically increasing function of anisotropy parameter p. The second major discovery is that, when p≥0.5, the changing with temperature of the heat capacity of a finite number of ideal bosons possesses the maximum value, which happens at critical temperature Tc. The third major discovery is that, when 0.1≤p<0.5, the changing with temperature of the heat capacity of a finite number of ideal bosons possesses an inflection point, but when p<0.1, the inflection point disappears. The fourth major discovery is that, in the thermodynamic limit, at Tc and when p≥0.5, the heat capacity at constant number reveals a cusp singularity, which resembles the λ-transition of liquid helium-4. The fifth major discovery is that, in comparison to 2D isotropic harmonic traps (p=1), the singular peak of the specific heat becomes very gentle when p is lowered.
Motivated by quantum statistical mechanics, we propose an accurate analytical solution to the problem of Bose–Einstein condensation (BEC) of ideal bosons in a two-dimensional anisotropic harmonic trap. The study reveals that the number of noncondensed bosons is characterized by an analytical function, which relates to a series expansion of q-digamma functions in mathematics. The q-digamma function is a function of temperature, boson number, and anisotropic parameter. The analytical solution describes fully the experimental results of the BEC of ideal bosons in a two-dimensional anisotropic harmonic trap. We derive the analytical expressions of the critical temperature and the condensate fraction in the thermodynamic limit. The first main conclusion is that for a fixed temperature and boson number, there is a critical anisotropic parameter, which is the precise onset of BEC in this harmonically trapped two-dimensional system. The second main conclusion is that the critical temperature in a two-dimensional anisotropic harmonic trap is larger than that in a two-dimensional isotropic harmonic trap.
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.
Abstract Within the framework of quantum statistical mechanics, 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. The results of numerical calculation of the analytical solution agree completely with the foregoing experimental results in the BEC of harmonically trapped, two-dimensional, and ideal atoms. The analytical expressions of the critical temperature and the condensate fraction are derived in the thermodynamic limit. It is found that the two-dimensional critical temperature is smaller than the one-dimensional critical temperature by an order of magnitude or more.
Abstract We have proposed an exact analytical solution to the problem of Bose–Einstein condensation (BEC) of harmonically trapped, one-dimensional, and ideal atoms. It is found that the number of atoms in vapor is characterized by an analytical function, which involves a q -digamma function in mathematics. We employ the q -digamma function to calculate the spatial density n ( z;T , N ) of ideal Bose atoms in a one-dimensional harmonic trap. The first main finding in this paper is that when Bose atoms are in the normal state, the density profile exhibits Friedel oscillations. The second main finding is that when Bose atoms are in the BEC state, the density profile exhibits a sharp peak with extremely narrow width. The third main finding is that the central peak of the spatial density is a monotonically increasing function of the number of atoms N but is a monotonically decreasing function of temperature T .