Boltzmann Transport Equation

Core Idea

The Boltzmann transport equation (BTE) governs the distribution function f(r, k, t) of electrons in a solid subject to external fields and scattering. In steady state, the balance between the driving term (electric and magnetic fields changing k, temperature gradients changing the local equilibrium) and the collision integral (scattering returning the distribution toward equilibrium) determines transport coefficients: electrical conductivity, thermal conductivity, thermoelectric power, and magnetoresistance. The relaxation-time approximation replaces the full collision integral with -(f - f_0)/tau, making the BTE analytically solvable and recovering the Drude formula as a special case.

Explainer

The Drude and Sommerfeld models provide the qualitative picture of metallic transport, but to calculate transport coefficients quantitatively — especially when scattering rates depend on energy, when temperature gradients or magnetic fields are present, or when the Fermi surface is anisotropic — you need the Boltzmann transport equation. The BTE tracks the non-equilibrium distribution function f(r, k, t), which gives the probability of finding an electron at position r with crystal momentum k at time t. In equilibrium, f = f_0 (the Fermi-Dirac distribution). External perturbations drive f away from f_0, and scattering processes push it back.

The BTE in its general form is df/dt + v_k · nabla_r f + (F/hbar) · nabla_k f = I_coll{f}, where v_k = (1/hbar) nabla_k E(k) is the group velocity, F is the external force (electric and magnetic), and I_coll is the collision integral encoding all scattering mechanisms. The relaxation-time approximation simplifies I_coll to -(f - f_0)/tau, asserting that scattering restores equilibrium exponentially with time constant tau. This approximation, while crude, captures the essential physics of most transport phenomena and is analytically tractable.

For electrical conductivity in the relaxation-time approximation, the BTE yields sigma = e² integral tau(k) v_k v_k (-df_0/dE) [d^3k/(2pi)³]. The crucial factor -df_0/dE is a sharply peaked function at E_F (width ~k_BT at low temperature), enforcing that only Fermi-surface electrons contribute. For an isotropic metal this reduces to sigma = (1/3) e² v_F² tau g(E_F), recovering the Drude formula with the correct Sommerfeld modifications. For anisotropic Fermi surfaces, the tensor character of the conductivity emerges naturally from the k-dependent velocity and scattering rate.

The BTE framework extends to all transport phenomena: thermal conductivity (response to a temperature gradient), thermoelectric effects (coupling between heat and charge currents, giving the Seebeck and Peltier coefficients), and magnetotransport (Hall effect, magnetoresistance, de Haas-van Alphen oscillations in the semiclassical regime). The Mott formula for thermopower, the Wiedemann-Franz law, and the Kohler rule for magnetoresistance all emerge as special cases. The BTE remains the workhorse of transport theory in condensed matter, succeeded by the Kubo formula only when quantum coherence effects become important.