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

TL;DR

This paper investigates how non-local derivatives, specifically the Caputo-Fabrizio derivative, influence Bose-Einstein condensation, revealing increased critical temperatures and the possibility of 2D condensation, with comparisons to Laskin's fractional spectrum approach.

Contribution

It introduces a novel analysis of Bose-Einstein condensation using non-local derivatives, showing enhanced critical temperatures and extending the phenomenon to 2D systems.

Findings

Critical temperatures for condensation are higher with non-local derivatives.

Condensation in 2D becomes possible under this approach.

Caputo-Fabrizio derivative yields higher transition temperatures than Laskin's spectrum.

Abstract

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.