Carnot limit vs. efficiency at maximum power (Curzon-Ahlborn)
Finite-time thermodynamics: Real engines operate in finite time, trading efficiency for power. The Curzon-Ahlborn efficiency η_CA = 1 − √(T_c/T_h) maximizes output power for a Novikov-Chambadal-Curzon-Ahlborn endoreversible engine.
Stochastic regime: For microscale engines (molecular motors, colloidal particles), fluctuations matter. The Jarzynski equality ⟨e^{−βW}⟩ = e^{−βΔF} holds exactly; work fluctuations satisfy the Crooks fluctuation theorem P(W)/P(−W) = e^{β(W−ΔF)}.
Power-efficiency tradeoff: Ẇ = κ(T_h − T_c)·η(1−η/η_Carnot) — maximum at η_CA. Efficiency at max power for stochastic engines is bounded: η_CA/2 ≤ η_MP ≤ η_CA.