Title: The effect of non-local derivative on Bose-Einstein condensation
Author(s):

	F.E. Bouzenna (Physics Department and LEVRES laboratory, Faculty of Exact Sciences, University of El Oued, 39000, Algeria),
	M.T. Meftah (Physics Department and LRPPS laboratory, Faculty of Mathematics and Matter Sciences, University Kasdi Merbah, Ouargla 30000, Algeria),
	M. Difallah (Physics Department and LABTHOP laboratory, Faculty of Exact Sciences, University of El Oued, 39000, Algeria)

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 α to derive the eigenvalues of non-local Schrödinger 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 T_c of condensation for 1 < α < 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 α for a system of free bosons governed by an Hamiltonian with power law on the moment (H~p^α). 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 α and the density ρ that the transition temperature relative to Caputo-fabrizio is greater than relative to Laskin.

Key words: phase transition, critical phenomena

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