The effect of non-local derivative on Bose-Einstein condensation

In this paper, we study the effect of non-local derivative on Bose-Einstein condensation. Firstly, we consider the Caputo-Fabrizio derivative of fractional order \alpha to derive the eigenvalues of non-local Schr\"odinger equation for a free particle in a 3D box. Afterwards, we consider 3D Bose-Einstein condensation of an ideal gas with the obtained energy spectrum. Interestingly, in this approach the critical temperatures Tc of condensation for 1 < \alpha < 2 are greater than the standard one. Furthermore, the condensation in 2D is shown to be possible. Second and for comparison, we presented, on the basis of a spectrum established by N. Laskin, the critical transition temperature as a function of the fractional parameter {\alpha} for a system of free bosons governed by an Hamiltonian with power law on the moment (H~p\alpha). In this case, we have demonstrated that the transition temperature is greater than the standard one. By comparing the two transition temperatures (relative to Caputo-Fabrizio and to Laskin), we have found for fixed \alpha and the density \rho that the transition temperature relative to Caputo-fabrizio is greater than relative to Laskin.