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Gibbs paradox of some thermodynamic properties in one-dimensional Gross-Pitaevskii equation

Teguh Budi Prayitno, R. Syariati

Published Jun 7, 2019• 0 citations
Publisher

Abstract

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.

DOI
10.1201/9780429461903-10

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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.
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.
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.
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.
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.
We consider the statistical mechanics of systems of particles satisfying the qqq-commutation relations recently proposed by Greenberg and others. We show that although the commutation relations approach Bose (Fermi) relations for $q\ensuremath{\rightarrow}1$ ($q\ensuremath{\rightarrow}\ensuremath{-}1$), the partition functions of free gases are independent of qqq in the range $\ensuremath{-}1<q<1$. The partition functions exhibit Gibbs' paradox in the same way as a classical gas without a correction factor 1N!\frac{1}{N}!N1​! for the statistical weight of the NNN-particle phase space; i.e., the statistical mechanics does not describe a material for which entropy, free energy, and particle number are extensive thermodynamical quantities.
Abstract Chapter 6 introduces quantum-mechanical ensemble theory by proving the asymptotic equivalence of the quantum-mechanical, microcanonical ensemble average with the quantum grand canonical ensemble average for many-particle systems, based on the method of Darwin and Fowler. The procedures involved identify the grand partition function, entropy and other statistical thermodynamic variables, including the grand potential, Helmholtz free energy, thermodynamic potential, Gibbs free energy, Enthalpy and their relations in accordance with the fundamental laws of thermodynamics. Accompanying saddle-point integrations define temperature (inverse thermal energy) and chemical potential (Fermi energy). The concomitant emergence of quantum statistical mechanics and Bose–Einstein and Fermi–Dirac distribution functions are discussed in detail (including Bose condensation). The magnetic moment is derived from the Helmholtz free energy and is expressed in terms of a one-particle retarded Green’s function with an imaginary time argument related to inverse thermal energy. This is employed in a discussion of diamagnetism and the de Haas-van Alphen effect.