Onsager Reciprocal Relations — Irreversible Thermodynamics

Phenomenological Matrix L

—

Force X₁ (∇T⁻¹):—

Force X₂ (∇μ):—

Flux J₁ (heat):—

Flux J₂ (matter):—

Entropy prod. σ:—

Seebeck coeff α:—

L₁₁ (κ heat)50

L₁₂=L₂₁ (cross)15

L₂₂ (σ electric)40

ΔT (thermal force)30

Δμ (chem force)-10

Onsager (1931): near equilibrium, thermodynamic fluxes are linear in forces: J_i = Σ_j L_ij X_j. The reciprocal relations L_ij = L_ji follow from microscopic reversibility (detailed balance). For coupled heat/matter transport: J_heat = L₁₁ X₁ + L₁₂ X₂, J_matter = L₂₁ X₁ + L₂₂ X₂. Seebeck effect: open-circuit voltage from temperature gradient. Peltier effect: heat pumping by electric current. Positive semi-definiteness of L ensures σ = Σ J_i X_i ≥ 0.