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The Drude model treats conduction electrons as a classical ideal gas that undergoes collisions with a characteristic relaxation time tau, yielding the DC conductivity sigma = ne2 tau / m and the Hall coefficient R_H = -1/ne. It correctly predicts Ohm's law and the Wiedemann-Franz ratio but fails for the electronic specific heat (predicting 3/2 k_B per electron, far too large). The Sommerfeld model corrects this by applying Fermi-Dirac statistics: at temperature T, only the fraction ~k_BT/E_F of electrons near the Fermi surface are thermally active, giving a specific heat C_el = (pi2/2)(k_BT/E_F)(Nk_B) that is linear in T and much smaller than the classical prediction.
The Drude model (1900) is the simplest theory of electrical conduction: treat the n conduction electrons per unit volume as a classical ideal gas that undergoes random collisions every tau seconds on average, each collision randomizing the electron's velocity. Between collisions, an applied electric field E accelerates each electron, producing a drift velocity v_d = -eE tau/m and a current density j = nev_d = (ne2 tau/m)E. This immediately gives Ohm's law with conductivity sigma = ne2 tau/m. The model also predicts the Hall effect (R_H = -1/ne), AC conductivity (sigma(omega) = sigma_0/(1 - i omega tau)), and a Wiedemann-Franz-like ratio between thermal and electrical conductivity.
The Drude model has two major failures, both rooted in its classical treatment of electron statistics. First, the specific heat: equipartition gives each electron 3k_B/2, predicting a total electronic specific heat of (3/2)nk_B — far larger than what is observed. Experimentally, the electronic specific heat at room temperature is roughly 1% of the classical value. Second, the magnetic susceptibility: classical electrons should exhibit Curie-like paramagnetism proportional to 1/T, but metals show temperature-independent Pauli paramagnetism.
Sommerfeld (1928) resolved both problems by a single change: replacing the Maxwell-Boltzmann distribution with the Fermi-Dirac distribution. At temperature T, the occupation of states follows f(E) = 1/(e(E-E_F)/k_BT + 1). Since E_F is typically 5-10 eV while k_BT at room temperature is only 0.025 eV, the Fermi function is nearly a step function. Only electrons within ~k_BT of E_F can be thermally excited — a fraction k_BT/E_F of the total. This immediately gives a specific heat C_el = gamma T with gamma proportional to g(E_F) (and thus to m*/m), roughly 100 times smaller than the classical prediction at room temperature. Similarly, only Fermi-surface electrons can flip their spin in a magnetic field, giving temperature-independent Pauli paramagnetism chi = mu_B2 g(E_F).
The Sommerfeld model retains the free-electron assumption (no lattice potential, parabolic dispersion) and the phenomenological relaxation time tau. Despite this simplicity, it quantitatively explains the electronic specific heat and magnetic susceptibility of simple metals and provides the correct framework for understanding transport. Its limitations — inability to explain band gaps, the sign of the Hall coefficient in some metals, or the origin of tau itself — are addressed by adding the periodic lattice potential (Bloch's theorem) and the theory of electron-phonon and electron-impurity scattering.