# Why do the $C_v$ of gapless systems have a power law behaviour?

The functional dependence of the heat capacity $$C_v$$ of systems with gapless excitations (e.g., lattice with acoustic phonons, Heisenberg ferromagnet with spin waves etc) is like a power law $$C_v\sim T^n.\tag{1}$$ But for systems with gapped energy levels this is not true. For example, in the Einstein's model a crystal the heat capacity $$C_v\neq T^n.$$ For an even simpler example, consider a three-level system with non-degenerate energy levels $$\varepsilon_1, \varepsilon_2, \varepsilon_3$$. The heat capacity is $$C_v=\frac{\partial}{\partial T}U=\frac{\partial}{\partial T}\Big[\frac{(\varepsilon_1 e^{-\varepsilon_1/kT}+\varepsilon_2 e^{-\varepsilon_2/kT}+\varepsilon_3 e^{-\varepsilon_3/kT})}{( e^{-\varepsilon_1/kT}+e^{-\varepsilon_2/kT}+ e^{-\varepsilon_3/kT})}\Big]\tag{2}$$ which clearly shows that $$C_v$$ is cannot be a power law; not even at low temperatures.

I am looking for two answers here. One mathematical and the other physical.

Question 1 How is it that $$C_v$$, which in general, is a complicated function of $$T$$ [of the form $$(2)$$ or its generalization] smoothly converge to a power law $$(1)$$ in the limit of gapless levels?

Question 2 Can we interpret it physically i.e., what is so special about gapless systems? What is going on?