A Brownian gyrator converts a temperature gradient into rotation. Two degrees of freedom — say the x and y positions of a trapped nanoparticle — are coupled to heat baths at different temperatures. The anisotropy of the thermal noise drives a systematic rotational current. It is a nanoscale heat engine that produces torque from temperature difference.
Most theoretical treatments assume the overdamped limit — negligible inertia, where velocity follows force instantaneously. Real nanoparticles in vacuum have finite mass. As the surrounding gas pressure drops, damping decreases, and inertia matters. The question: does inertia help or hurt?
Neither, exactly. The rotational dynamics and energetics are optimized at critical damping — the boundary between overdamped and underdamped regimes. Below critical damping (underdamped), the particle oscillates, and the oscillations waste energy that could have driven rotation. Above critical damping (overdamped), the sluggish response limits how fast the particle can track the temperature-driven current. At critical damping, the system converts thermal gradient into rotation most efficiently.
The paradox is that spatial signatures of rotation — the visible circulatory current in the particle's trajectory — weaken as damping decreases. The gyrator looks less rotational but performs better. The thing that makes it work is not the same as the thing you would measure by watching it. Performance is hidden in the energetics, not displayed in the trajectory.
This is a clean case of optimization at a boundary. Neither the overdamped nor underdamped limit is optimal. The best engine lives exactly where the two regimes meet — at critical damping, where the system transitions between qualitatively different dynamics. The optimal operating point is the phase boundary itself.
(arXiv:2603.18818)