In 1912 Smoluchowski imagined a microscopic ratchet and pawl — a toothed wheel that could only turn one way, jostled by the random thermal kicks of a gas. Surely it would slowly wind itself up, making useful work out of pure noise. In 1963 Feynman dismantled the dream in Lectures I·46: the pawl is in the bath too. At equal temperature the thermal jiggle lifts the pawl exactly as often as the gas pushes the wheel — it twitches, but it does not turn. Heat one end (gas) hotter than the other (pawl) and it becomes a real engine — bounded by the very same Carnot wall every machine here obeys.
A tiny wheel with asymmetric teeth, a spring-loaded pawl that blocks the backward step. Random gas kicks can
only ratchet it forward… can't they? ⟨ω⟩ > 0 from nothing but noise would be a perpetual-motion
machine of the second kind.
The pawl is just as small and just as warm. At Θ_pawl = Θ_gas it lifts on its own thermal
fluctuations exactly often enough to let the wheel slip back. Forward and backward hops detail-balance:
net rotation is 0 for any tooth shape.
Watch the trail ring around the wheel: at ΔΘ = 0 the forward and backward twitches paint the wheel equally both ways — it rocks around its datum line and never leaves. Cool the pawl and the forward tint wins; the wheel ratchets. Load it, and it does work against the torque — but the efficiency stays pinned under the Carnot ceiling, the same wall the Carnot bench proves. Make the teeth symmetric, or pull the pawl, and a real ΔΘ rectifies nothing. You cannot win, and you cannot break even.
The demon and the pawl are the same machine in two languages · one fact wearing two coats. Carnot: the ceiling. · Demon: the receipt. · Ratchet: the ceiling, with teeth.
η = 1 − T_c/T_h is proven the wall no cycle can cross.
This bench imports that very carnotEfficiency() to bound the loaded ratchet — never redrawn,
never beaten.Four claims, one ledger, re-run live in the self-test badge above: (1) the net drift is
zero at Θ_pawl = Θ_gas within the measured ±KSIG·σ error bars, for any tooth shape — the
heart; (2) sign(⟨ω⟩) flips with sign(ΔΘ) and vanishes as ΔΘ → 0;
(3) the loaded efficiency η = W/Q_h never crosses the carnotEfficiency() imported
from the Carnot bench; and (4) a symmetric wheel and a removed pawl each rectify nothing. The page's core is
the byte-twin of core.mjs — the Node test re-extracts it and proves the page === the module === the
Carnot sibling.