Phonon DOS g(ω) counts vibrational modes per frequency interval. The Debye model assumes linear dispersion ω=v_s·k, giving g(ω)∝ω² up to the Debye cutoff ω_D — accurate at low frequencies. Real crystals have van Hove singularities at Brillouin zone boundaries where ∇_k ω=0, causing peaks in the DOS. The Einstein model treats optical modes as a single frequency — a delta peak. Heat capacity C_V = ∫ g(ω)(ℏω/k_BT)² · e^(ℏω/k_BT)/(e^(ℏω/k_BT)−1)² dω → T³ at low T (Debye), constant at high T (Dulong-Petit: 3Nk_B).