Classical equipartition predicts that each atom in a three-dimensional solid contributes approximately $3k_B$ to the constant-volume heat capacity, giving the Dulong-Petit result $$C_V\approx3Nk_B.$$ This works at high temperature but fails dramatically as $T\to0$, where measured heat capacities vanish.

The failure is quantum mechanical. Lattice vibrations have quantized energies and low-temperature thermal energy cannot excite high-frequency modes freely.

In the Einstein model, all $3N$ vibrational modes are assigned one frequency $\omega_E$. The heat capacity becomes $$C_V=3Nk_B\left(\frac{\Theta_E}{T}\right)^2 \frac{e^{\Theta_E/T}}{(e^{\Theta_E/T}-1)^2},$$ where $\Theta_E=\hbar\omega_E/k_B$. It approaches $3Nk_B$ at high temperature and falls exponentially at low temperature.

The Debye model improves the low-frequency physics by treating acoustic modes as a continuum with approximately linear dispersion up to a cutoff chosen to give $3N$ modes. At low temperature it predicts $$C_V\propto T^3,$$ in good agreement with many insulating crystalline solids.

The contrast illustrates a general principle: heat capacity measures how the accessible energy spectrum opens as temperature rises. Quantum statistics and the density of vibrational states determine which degrees of freedom can actually store thermal energy.