The Chaplygin sleigh is a rigid body on a plane constrained to move only in the direction its blade points — like a sled that can't slide sideways. The constraint is nonholonomic: it restricts velocities, not positions. At zero temperature, the mechanics are straightforward. At finite temperature, the naive approach — adding thermal noise and friction to the constrained equations of motion — produces a perpetual motion machine.
Jagla, Bloch, and Rojo show why. The naive Langevin equation satisfies the fluctuation-dissipation relation only for the unconstrained degrees of freedom. The constraint eliminates sideways motion, but the thermal bath doesn't know about the constraint — it kicks the sleigh in all directions, including the forbidden one. The constraint converts forbidden sideways momentum into allowed forward motion, extracting net work from thermal fluctuations without any temperature gradient. This violates the second law.
The resolution: model the nonholonomic constraint as the limit of strong viscous friction at the contact point. Before taking the limit, the sleigh can slide sideways, but the viscous force resists it. The fluctuation-dissipation theorem then requires stochastic forces at the contact proportional to the viscous drag — additional noise sources that the naive model omits. In the rigid-constraint limit, these contact forces persist and exactly cancel the spurious work extraction.
The phantom engine — the apparent ability to extract work from a single thermal reservoir — appears whenever an idealized constraint is coupled to a thermal bath without accounting for the constraint's physical mechanism. The constraint is not a mathematical boundary condition; it's a limiting case of a physical interaction, and the interaction has fluctuations that the limit must preserve.