Entropy Production & Clausius Inequality

Clausius Inequality: For any cycle, ∮δQ/T ≤ 0, with equality only for reversible (Carnot) cycles. The deficit equals the entropy produced: σ_total = -∮δQ/T ≥ 0.

Entropy Production: σ = J_Q · X_T where J_Q = Q̇ is the heat flux and X_T = (1/T₂ - 1/T₁) is the thermodynamic force (affinity). This is the Onsager linear response regime: σ = L · X².

Irreversibility α: α=0 → Carnot (reversible, zero entropy production). α=1 → maximum irreversibility. Actual efficiency η = η_Carnot · (1-α). The entropy production rate grows quadratically with ΔT for small differences (linear response) and saturates at large ΔT.

Second Law: The arrow of time. The universe's entropy only increases; at best, local decreases are more than offset by the entropy dumped into the environment.