Bose-Einstein Condensation of q-Deformed Bosons Harmonically Trapped on Sierpiński Carpet and Menger Sponge

Bose–Einstein condensation, as a fifth state of matter, can only occur under certain conditions. One of those conditions is the spatial dimensions confining the bosonic systems. We investigated Bose–Einstein condensation for a finite number of harmonically trapped bosons on fractal structures. The investigation involves two approaches; one belongs to standard Bose–Einstein statistics, and the other belongs to the theory of q-deformed bosons. The properties of Bose–Einstein condensates in the two approaches are computed by performing the sum over the energy states. From these two approaches, we attempt to gain insight into the possibility of using q-numbers to assign fractal dimensions via Bose–Einstein condensation. In this endeavor, the bosons are considered ideal to emphasize that the parameter q only represents the fractal dimension of the structures confining the bosons. The results reveal that a condensate of q-deformed bosons with q=0.74 is adequate to represent a condensate of standard bosons on a Sierpiński carpet. The results also reveal that a condensate of q-deformed bosons with q=0.33 is adequate to represent a condensate of standard bosons on a Menger sponge. We also suggest an expression for using the parameter q to measure the interaction between bosons harmonically trapped on fractal structures, which may also help to study the effect of porosity or fractal dimension on the interaction between bosons.