Investigation of the Bose-Einstein condensation based on fractality using fractional mathematics

Although atomic Bose gases are investigated in the dilute gas regime, the physical properties of the Bose-Einstein condensates are affected by interparticle interactions and the fractal nature of the space where the Bose systems are evolving. Theoretical predictions of the traditional Bose-Einstein thermostatistics do not account for the deviations from the experimental results, which are related to internal energy, specific heat, transition temperature, etc. On the other hand, in this study, fractional calculus is introduced where effects of the fractality of space are taken into account. Meanwhile, the order of the fractional derivative a is handled as a measure of the fractality of space. In this content, some improvements which take into account the effects of the fractal nature of the system are made in the standard physical results of the Bose-Einstein condensation phenomena. As an example, for the dilute atomic gas Li-7, the measured transition temperature of Bose-Einstein condensation could be obtained for the value of alpha approximate to 0.976, and one could predict that the interparticle interactions have a weak attractive nature consistent with experiment (Bradley et al 1995 Phys. Rev. Lett. 75 1687). Thus, a fractional mathematical theory is established in coherence with experimental results of Bose-Einstein condensation.