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MAKARA of Science Series

The Partition Function of the Bose-Einstein Condensation in Parabolic Trap

Teguh Budi Prayitno, Sinta Latifa

Published Nov 27, 2012• 1 citation
Publisher

Abstract

Normal 0 false false false IN X-NONE X-NONE MicrosoftInternetExplorer4 We have discussed the partition function of the Bose-Einstein condensation in parabolic trap associated to the one-dimensional Gross-Pitaevskii equation. The partition function itself is constructed by considering all the energy levels of the macroscopic quantum oscillator which is similar to statistical mechanics. The solutions of the energy levels for this case can be derived by pursuing the method proposed by Kivshar et al. that applies the time-independent perturbation theory. In their paper, they demonstrated that the one-dimensional Gross Pitaevskii equation can be treated as the one-dimensional macroscopic quantum oscillator on condition that the nonlinearity is very small. Moreover, they found the analytical expression for the ground state energy by applying the method. However, the higher level states were not explicitly provided. In this research we followed up on their former work to derive explicitly the other states in order to formulate the partition function. However, we did not find the closed form of the partition function since the results of nonlinear term integral could not form the recursion relation. As a consequence, not only should the partition function but also the Helmholtz free energy and entropy should be reevaluated to check their convergences. Normal 0 false false false IN X-NONE X-NONE MicrosoftInternetExplorer4 Keywords: Gross-Pitaevskii equation, partition function, quantum oscillator, thermodinamical properties.

DOI
10.7454/mss.v16i2.1401

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Cambridge University Press eBooks•Nov 29, 2001•2 citations
Normal 0 false false false IN X-NONE X-NONE MicrosoftInternetExplorer4 We have discussed the partition function of the Bose-Einstein condensation in parabolic trap associated to the one-dimensional Gross-Pitaevskii equation. The partition function itself is constructed by considering all the energy levels of the macroscopic quantum oscillator which is similar to statistical mechanics. The solutions of the energy levels for this case can be derived by pursuing the method proposed by Kivshar et al. that applies the time-independent perturbation theory. In their paper, they demonstrated that the one-dimensional Gross Pitaevskii equation can be treated as the one-dimensional macroscopic quantum oscillator on condition that the nonlinearity is very small. Moreover, they found the analytical expression for the ground state energy by applying the method. However, the higher level states were not explicitly provided. In this research we followed up on their former work to derive explicitly the other states in order to formulate the partition function. However, we did not find the closed form of the partition function since the results of nonlinear term integral could not form the recursion relation. As a consequence, not only should the partition function but also the Helmholtz free energy and entropy should be reevaluated to check their convergences. Normal 0 false false false IN X-NONE X-NONE MicrosoftInternetExplorer4 Keywords: Gross-Pitaevskii equation, partition function, quantum oscillator, thermodinamical properties.
We have constructed the equation of state of one-dimensional Gross-Pitaevskii equation by benefiting from the formulation of a single macroscopic particle partition function and then modeling N undistinguishable macroscopic particles in canonical ensemble. This macroscopic particle, in which two or more atoms include, is literally a condensate that can be observed as an assembly in the system. In this case, we suppose all condensates are confined in the anisotropic parabolic trap and interaction between two condensates can be ignored by applying semi infinite cigar-shaped trap. It is also shown that the equation of state is indeed an ideal gas with the new thermodynamic interpretation of volume and pressure. Moreover, Even though our resulting partition function was portrayed as a sum of exponential functions, we prove that the series is indeed convergent.
We have studied the ground state energy of one-dimensional macroscopic quantum oscillator by adding gravity term as a time-independent external potential. The macroscopic quantum oscillator itself is based on discussion about Gross-Pitaevskii equation (GPE) describing Bose-Einstein condensate in anisotropic parabolic trap. That gravitational potential displaces the symmetry axis of Bose-Einstein condensation (BEC) with respect to symmetry axis of magnetic field that creates trapping potential of the Bose-Einstein condensate (BECs) in research on atomic lasers. Since the Gross-Pitaevskii equation can be treated as macroscopic quantum oscillator, we use time-independent perturbation theory as in ordinary quantum mechanics to find the ground state energy correction by assuming that other terms are very small. We also formulate the appropriate coefficients of expansion in the solution of one-dimensional GPE by considering the characteristics of harmonic oscillator eigenfunctions and find that the existence of the gravity can only be included if the higher coefficients are considered. Keywords: Gravitational potential, Gross-Pitaevskii, Quantum Oscillator.
A single condensate partition function in the canonical ensemble of the one-dimensional Gross-Pitaevskii equation has been used to construct the partition function of N noninteracting condensates. In this paper, we employed the appropriate transformation in order to make the partition function of a single condensate to be an extensive quantity by introducing a harmonic volume as the inverse cube of the average geometric trapping frequency. We found that some expressions of the thermodynamic quantities must be corrected to keep the consistency of the thermodynamic expressions based on the Gibbs paradox in the statistical mechanics. This only can be done if a set of condensates is considered as the indistinguishable macroscopic quantum oscillators with a weak nonlinearity instead of the distinguishable ones.
Abstract By using the one-dimensional canonical partition function, we modelled an ideal gas-like form of a set of Bose-Einstein condensates confined by a three-dimensional anisotropic parabolic trap. The model itself was constructed by taking the eigenenergies of the one-dimensional Gross-Pitaevskii equation in the longitudinal direction and enabling the harmonic volume as the inverse cube of average geometric trapping frequency to substitute the real volume. In this paper, we showed that the condensates form an ideal gas represented by its equation of state and have similar mature to Einstein’s solid-like model with the corrections in both the low and high temperatures in the internal energy formulations.
Abstract Considering a set of condensates as N indistinguishable noninteracting quantum oscillators and replacing an available volume with a harmonic volume, we derived an ideal gas model by applying the canonical partition function via the thermodynamic relation. In this case, we take a longitudinal part of the three-dimensional Gross-Pitaevskii equation in which the effect of gravitation of earth works. Since we used the time-independent perturbation theory, to include gravity effect, we must put the higher coefficient in the formulation of each energy state whose value is determined by the separated amplitude formula in each state. It will also be shown that even though the equation of state represents an ideal gas, the appropriate heat capacity approaches Einstein’s solid-like model if the certain conditions are employed.
The time-dependent behaviour of Bose–Einstein condensed clouds, such as collective modes and the expansion of a cloud when released from a trap, is an important source of information about the physical nature of the condensate. In addition, the spectrum of elementary excitations of the condensate is an essential ingredient in calculations of thermodynamic properties. In this chapter we treat the dynamics of a condensate at zero temperature starting from a time-dependent generalization of the Gross–Pitaevskii equation used in Chapter 6 to describe static properties. From this equation one may derive equations very similar to those of classical hydrodynamics, which we shall use to calculate properties of collective modes. We begin in Sec. 7.1 by describing the time-dependent Gross–Pitaevskii equation and deriving the hydrodynamic equations. We then use the hydrodynamic equations to determine the excitation spectrum of a homogeneous Bose gas (Sec. 7.2). Subsequently, we consider modes in trapped clouds (Sec. 7.3) within the hydrodynamic approach, and also describe the method of collective coordinates and the related variational method. In Sec. 7.4 we consider surface modes of oscillation, which resemble gravity waves on a liquid surface. The variational approach is used in Sec. 7.5 to treat the free expansion of a condensate upon release from a trap. Finally, in Sec. 7.6 we discuss solitons, which are exact one-dimensional solutions of the time-dependent Gross-Pitaevskii equation. General formulation In the previous chapter we saw that the equilibrium structure of the condensate is described by a time-independent Schrödinger equation with a nonlinear contribution to the potential to take into account interactions between particles.