Bose condensation with power-law dispersion ($ϵ=A{p}^{\alpha },0<\alpha <d$)

For a homogeneous ideal gas in $d$ dimensions with $A,\alpha >0$, the density of states scales as ${ϵ}^{d/\alpha -1}$. At limiting chemical potential zero, the Bose-Einstein distribution is proportional to $1/ϵ$ near zero energy. The excited-particle integral is finite exactly when $d/\alpha >1$. Thus positive-temperature condensation at fixed nonzero density is possible precisely for $0<\alpha <d$. A nonconserved particle number, as for ordinary equilibrium photons, removes the fixed-number saturation mechanism even when the integral is finite.