Scaling and Thermodynamics of a Trapped Bose-Condensed Gas

We investigate the thermodynamics of a Bose gas interacting with repulsive forces and confined in a harmonic trap. We show that the relevant parameters of the system (temperature, number $N$ of atoms, harmonic oscillator length, deformation of the trap, $s$-wave scattering length) fix its thermodynamic behavior in the large $N$ limit through two dimensionless scaling parameters. These are the reduced temperature $t=T/T_c^0$ and the ratio $\eta$ between the $T=0\mathrm{}$ value of the chemical potential, evaluated in the Thomas-Fermi limit, and the critical temperature $T_c^0$ of the noninteracting model. The scaling functions relative to the condensate fraction, energy, chemical potential, and moment of inertia are calculated within the Popov approximation.