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

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.