Optimal probing scale for current fluctuations in a Brownian gyrator
A driven colloidal rotor sustains a circulating current whose fluctuations one would like to bound. Any such bound rests on how strongly the current responds to a perturbation and on how much extra dissipation that perturbation costs, so there is a well-posed question of where to push: the response per unit Onsager-Machlup cost depends on the radius at which the probe acts, and is maximised at a definite one. We answer this for the Brownian gyrator and a quadrupolar shear under a Gaussian envelope, a probe chosen so that linear response is exactly blind to it at every observation window, so that the entire signal is second order. The problem separates: the cost is radial and does not see the circulation, while the response lives in the $m=3$ angular sector, where the resolvent reduces to Kummer's equation with $b=4$ and the susceptibility is a hypergeometric function of the envelope width. Maximising the ratio gives the optimal probing radius in closed form. It is set by whichever of the system's two clocks is faster: for weak driving $r^*=1.1264\sqrt{D/γ}$, the thermal radius of the trap, and for strong driving $r^*=1.5563\sqrt{D/Ω}$, the distance diffused in one radian of rotation, with the trap stiffness dropping out entirely. The crossover is at $Ω=γ$. Both limits, and the prefactors, are confirmed against direct simulation.