Ratchetta

Ratchetta, a gear-warlock of the Paradox Parlour, wants his ratchet to steal work from random heat, but his wheel will not turn.

The Physics

The ratchet and pawl, analysed by Smoluchowski (1912) and made famous in the Feynman Lectures (Vol. I, ch. 46), is a vane in a gas joined by an axle to a ratchet wheel with a spring-loaded pawl. Molecules hitting the vane seem to push the wheel forward, while the pawl blocks it going back. But the pawl is also at a temperature and also jiggles. When both are at the same temperature, the pawl lifts and lets the wheel slip back as often as the vane pushes it forward.

rate forward ∝ e^{−ε/k_BT₁}, rate back ∝ e^{−ε/k_BT₂} ⇒ T₁ = T₂ gives ⟨W⟩ = 0

• ε: energy needed to lift the pawl over one tooth (J)
• k_B: Boltzmann constant, exact in the 2019 SI
• T₁: temperature of the gas around the vane (K)
• T₂: temperature of the ratchet and pawl (K)
• ⟨W⟩: average work delivered per unit time

With equal temperatures the two rates match, which is what the second law (Kelvin–Planck form) requires: no cyclic device can turn heat from a single reservoir into work. With T₁ ≠ T₂ it does work, as a heat engine. The same idea underlies Brownian motors and models of molecular motors.

Honest caveat: This is the simplified rate picture from Feynman's lectures. Feynman also argued that with two temperatures the device could reach Carnot efficiency; Parrondo and Español (Am. J. Phys. 64, 1125, 1996) showed this is wrong, because the axle couples both reservoirs at once and heat always leaks. Real molecular motors work only because they are driven out of equilibrium, for example by ATP.

On the sticker

Ratchetta holds a glowing amber ratchet gear in a cave where both boxes glow the same temperature colour. Printed text: RATCHETTA with the line T1 = T2 ⇒ W = 0. Pixel-art kiss-cut vinyl sticker, white, in 2″, 3″, 4″ and 6″ sizes.