Statistical Mechanics and Thermodynamic Properties of Bose-Einstein Condensation in Fractal Media

The phenomenon of Bose–Einstein condensation (BEC), which was predicted by Einstein in 1925 and experimentally realized in 1995, has been the subject of intensive research in the last ecades. On the theoretical side, several approaches have been formulated. One of the important concerns of these approaches has been the conditions under which this phenomenon is realized. Among the known factors affecting this realization is the dimensionality of the bosons' confining medium. The main conclusions of
previous studies are that BEC can occur in 3D and in 2D under a wide range of conditions. Among these conditions is the inhomogeneity of the Bose systems. But, to occur in 1D more stringent conditions are required, among them is the need for the treatment of a finite number of particles. Recently, there has been increased interest in BEC occurrence in media with fractional dimensions for two main reasons. The first is the experimental findings asserting the fractality of bosons' confining media. The second reason is the emergence of fractal geometry as a well–founded research discipline; whereas this emergence was indeed contemporary to the aforementioned experimental findings. However, the formulation of fractal models for the BEC phenomenon is still in its early stages. The present work is mainly devoted to the formulation of such a model and also to the investigation of its thermodynamic behavior through symbolic computation by using the MATHEMATICA® software package as a computational environment. The model formulated in the present work assumes that a finite number of ideal bosons are harmonically trapped in a fractal medium. It also assumes that the applicable statistical mechanics ensemble is the grand canonical. The fractality of the confining medium has been introduced in the formulation by two distinct methods. The first is by adopting an idea due to Rovenchack; which assumes that the degeneracy
factors of the energy levels can be extended to fractal dimensions by converting the factorial functions appearing in the expressions for the degeneracy factors into gamma functions. The second method is to use the well–established nonextensive Tsallis statistics; where the index of nonextensivity is related to the fractal dimension. It is important to mention here that both methods reduce to the standard case when dealing with the integer dimensions
(1D, 2D and 3D). To test the proposed MATHEMATICA® symbolic computational framework, computations of Bose–Einstein condensates for integer 1D, 2D and 3D were first carried out on the basis of the previously mentioned assumptions. The tests confirm the robustness of the computational scheme and the results obtained agree with previous ones.
Due to the success of the tests, computations on the basis of the two models were carried out for bosons harmonically trapped in fractal media which are embedded in 2D and in 3D dimensions. In general, it is found that the condensation temperature in the model based on Tsallis thermostatistics is lower than that obtained on the basis of Boltzmann–Gibbs thermostatistics. It is found that this result agrees with Salasnich result and observations by and other workers in the field. In conclusion, the models presented in this thesis and the proposed symbolic computational scheme can be successfully used to treat the BEC phenomenon in fractal media and permit possible extension.