Quantum statistical systems in $D$-dimensional space using a fractional derivative

We investigate the thermodynamic properties of some quantum statistical systems with a fractional Hamiltonian in $D$-dimensional space. We calculate the partition function of the system of $N$ fractional quantum oscillators and the thermodynamic quantities associated with it. We consider the thermal and critical properties of both Bose and Fermi gases in the context of the fractional energy and described by a fractional derivative.