Entropy Production & Irreversible Thermodynamics

Onsager reciprocal relations · Linear response · σ = J·X ≥ 0
20 K
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0.10
σ = L₁₁X₁² + (L₁₂+L₂₁)X₁X₂ + L₂₂X₂² ≥ 0 (Onsager: L₁₂=L₂₁)
Linear Irreversible Thermodynamics: Near equilibrium, fluxes J_i are linear in thermodynamic forces X_i (affinities): J_i = Σ_j L_ij X_j. The Onsager-Casimir theorem (1931) states L_ij = L_ji — remarkable symmetry derived from microscopic time-reversal invariance.

Entropy Production Rate: σ = Σ_i J_i X_i = Σ_ij L_ij X_i X_j ≥ 0. The matrix L must be positive semidefinite. This constrains cross-coefficients: L₁₂² ≤ L₁₁L₂₂.

Coupled transport: Heat flux can drive particle flux (Soret effect) and vice versa (Dufour effect). Here X₁ = ΔT/T² (thermal force), X₂ = −Δμ/T (chemical force). Thermoelectric efficiency is bounded by the figure of merit ZT.

Left panel: Flux-force diagram in (J₁,J₂) space. Right panel: Entropy production rate σ as function of forces.