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?


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