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Journal of Physics Conference Series

The equation of state of one-dimensional Gross-Pitaevskii equation

Teguh Budi Prayitno

Published Oct 13, 2014• 4 citations
Publisher

Abstract

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.

DOI
10.1088/1742-6596/539/1/012016

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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.
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.
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.
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.
In experiments, Bose-Einstein condensates are prepared by cooling a dilute Bose gas in a trap. After the phase transition has been reached, the trap is switched off and the evolution of the condensate observed. The evolution is macroscopically described by the Gross-Pitaevskii equation. On the microscopic level, the dynamics of Bose gases are described by the NNN-body Schrödinger equation. We review our article [BdS12] in which we construct a class of initial data in Fock space which are energetically close to the ground state and prove that their evolution approximately follows the Gross-Pitaevskii equation. The key idea is to model two-particle correlations with a Bogoliubov transformation.
We study condensation of trapped bosons in the limit when the number of particles tends to infinity. For the noninteracting gas we prove that there is no phase transition in any dimension, but in any dimension at any temperature the system is 100% condensated into the one-particle ground state. In the case of an interacting gas we show that for a family of suitably scaled pair interactions, the Gross-Pitaevskii scaling included, a less-than-100% condensation into a one-particle eigenstate, which may depend on the interaction strength, persists at all temperatures.
Self-consistent excited states of condensates are solutions of the Gross-Pitaevskii (GP) equation and have been amply discussed in the literature and related to experiments. By applying a recently introduced more general mean field which includes the GP one as a special case, we find a new class of self-consistent excited states. In these states macroscopic numbers of bosons reside in different one-particle functions, i.e., the states are fragmented. Still, a single chemical potential is associated with the condensate. A numerical example is presented, illustrating that the energies of the new, fragmented states are much lower than those of the GP excited states, and that they are stable to variations of the particle number and shape of the trap potential.