Optimal probing scale for current fluctuations in a Brownian gyrator
Abstract
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√(D/γ), the thermal radius of the trap, and for strong driving r*=1.5563√(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.
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BibTeX
@misc{farih2026optimal,
title = {{Optimal probing scale for current fluctuations in a Brownian gyrator}},
author = {Badr Farih},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24748},
eprint = {2609.24748},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24748},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Optimal probing scale for current fluctuations in a Brownian gyrator
AU - Badr Farih
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24748
UR - https://arxiv.org/abs/2609.24748
AB - 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√(D/γ), the thermal radius of the trap, and for strong driving r^*=1.5563√(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.
ER -