= Low-temperature energy of gapless bosonic excitations
{title2=$\epsilon(k)\propto k^s\ \Longrightarrow\ U-U_0\propto T^{1+d/s}$}
In $d$ dimensions, isotropic modes with dispersion $\epsilon=A k^s$ have <density of states> proportional to $\epsilon^{d/s-1}$. Their equilibrium number is not conserved, so the excitation <chemical potential> is zero. Integrating $\epsilon D(\epsilon)/(e^{\epsilon/(k_BT)}-1)$ and rescaling by $k_BT$ 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.
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