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"Kami telah membahas fungsi partisi dari kondensasi Bose-Einstein di dalam perangkap parabola yang dinyatakan oleh persamaan Gross-Pitaevskii satu dimensi. Fungsi partisi itu sendiri dirumuskan hanya dengan meninjau semua tingkat-tingkat energi dari osilator kuantum makroskopik yang mirip seperti di dalam mekanika statistika. Solusi-solusi dari tingkat-tingkat energi untuk kasus ini dapat diturunkan dengan mengikuti metode yang menggunakan teori perturbasi bebas waktu. Pada kasus ini, persamaan Gross-Pitaevskii satu dimensi dapat diperlakukan sebagai osilator kuantum makroskopik dengan menerapkan kondisi bahwa faktor nonlinearnya sangat kecil. Selain itu, perumusan analitik untuk energi tingkat dasar dapat diperoleh dengan menggunakan metode tersebut. Namun demikian, tingkat-tingkat eksitasinya tidak diberikan secara eksplisit. Saat ini, kami melanjutkan pekerjaan sebelumnya untuk menurunkan tingkat-tingkat keadaan lainnya supaya dapat merumuskan fungsi partisi. Akan tetapi, kami tidak mendapatkan bentuk analitik dari fungsi partisi karena integral dari suku-suku nonlinear tidak dapat membentuk hubungan rekursif. Akibatnya, tidak hanya fungsi partisi tetapi juga energi bebas Helmholtz dan entropi harus dikaji ulang untuk memeriksa sifat konvergennya.

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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 that applies the time-independent perturbation theory. In this case, 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, the analytical expression for the ground state energy can be obtained by applying the method. However, the higher level states were not explicitly provided. In this research we followed up on the 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. "

"We consider the correction of ground state energy of one-dimensional Gross-Pitaevskii equation by adding a gain-loss term as a time-dependent external potential. The interesting purpose of this term is that it can be used to explain the experimental results especially in the nonlinear fiber optics regarding the pulse propagation and collapse-revival of the condensate in the Bose-Einstein condensation. In the Bose-Einstein condensation itself, the function can represent thatcondensate can interact with the normal atomic cloud. Some analytical solutions have been obtained by choosing anansatz solution of the wave function and its solution can be dark or bright soliton. Since the Gross-Pitaevskii equation can be treated as a macroscopic quantum oscillator, we can use time-dependent perturbation theory as in ordinaryquantum mechanics to find the ground state energy correction if we assume other terms to be very small. In addition, time-dependent potential allows a transition from one energy level to others. In this case, we expand the solution of nonstationary one-dimensional wave function as a linear superposition of harmonic oscillator normalized eigen functions. To get the recursive formulas, we suggest an option to formulate the coefficients after inserting the initial condition which must be satisfied such as in quantum mechanics. "