Low-temperature energy of gapless bosonic excitations ($ϵ(k)\propto k^s⟹U-U_0\propto T^{1+d/s}$)

In $d$ dimensions, isotropic modes with dispersion $ϵ=A{k}^s$ have density of states proportional to ${ϵ}^{d/s-1}$. Their equilibrium number is not conserved, so the excitation chemical potential is zero. Integrating $ϵD(ϵ)/(e^{ϵ/(k_{B}T)}-1)$ and rescaling by $k_{B}T$ gives the displayed energy power and heat capacity proportional to $T^{d/s}$. In three dimensions, linear acoustic phonons give $T^4$ energy and $T^3$ heat capacity; quadratic gapless ferromagnetic magnons give $T^{5/2}$ energy and $T^{3/2}$ heat capacity. A gap changes the asymptotically low-temperature law to activated behaviour. The mode approximation applies below the relevant band cutoff and when interactions between excitations are negligible.

One isotropic three-dimensional bosonic branch with dispersion $ϵ=\alpha p^2$ and zero chemical potential has density of states per volume $\sqrt{ϵ}/(4{\pi }^2{\hslash }^3{\alpha }^{3/2})$. Integrating its Bose-Einstein energy and differentiating temperature gives the title coefficient. Identical independent branches multiply it by their number. Thus a measured $T^{3/2}$ coefficient fixes branch multiplicity divided by ${\alpha }^{3/2}$, not a dispersion coefficient independent of that counting convention. A gap or low-dimensional spectrum changes the asymptotic law.