Harmonic-trap phase-space density ($\mathcal{D}=N(\hslash \bar{\omega }/k_{B}T{)}^3$)

For a classical ideal gas in a three-dimensional harmonic trap, the peak phase-space density is $\mathcal{D}=N(\hslash \bar{\omega }/k_{B}T{)}^3$. At fixed trap frequencies, atom loss accompanied by $T\propto N^{\alpha }$ increases $\mathcal{D}$ only for $\alpha >1/3$.