Blind directions of linear response and the limits of finite-perturbation bounds on efficiency fluctuations
Abstract
Thermodynamic uncertainty relations bound the fluctuations of time-antisymmetric currents. Stochastic efficiency is not one: a ratio of two odd quantities, it is even under path reversal, and for the conventional ratio −W/Qh no moment exists. We work with the exergetic ratio η=W/(W+T0S), whose moments exist, and ask how much of its variance a finite perturbation of the dynamics can certify when linear response cannot. The question is specific to nonlinear observables: for every current, and every observable linear in transition counts and residence times, the multiparameter Cramér-Rao bound over rate and site perturbations already equals the variance. For η it does not. The Cramér-Rao bound vanishes on a hyperplane of perturbation directions, while the Barankin multi-point bound, applied to tilted path measures, stays positive there and recovers 75% of the variance of η in a three-state motor model where every linear-response bound is zero. The Gram matrix of tilted Markov-jump path measures is a Feynman-Kac matrix exponential whose potential is the Hellinger integrand of the jump intensities, and the cost of a perturbation is its χ2 divergence, which no dissipation bound controls. The bound captures 1.25-1.57 times the best linear-response bound, and the gain persists on networks of up to sixteen states. For an Ornstein-Uhlenbeck process the construction is exactly solvable: the optimal tilt scales as T−1/2 and three test points nearly reach the supremum. Two limits are established. The gain is a short-window effect that disappears as the counts become Gaussian, and the bound cannot be turned into inference: coarse measurements bound χ2 only from below. The construction is a computational instrument for a model in hand, not an uncertainty relation.
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BibTeX
@misc{farih2026blind,
title = {{Blind directions of linear response and the limits of finite-perturbation bounds on efficiency fluctuations}},
author = {Badr Farih},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.37940},
eprint = {2609.37940},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37940},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Blind directions of linear response and the limits of finite-perturbation bounds on efficiency fluctuations
AU - Badr Farih
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2609.37940
UR - https://arxiv.org/abs/2609.37940
AB - Thermodynamic uncertainty relations bound the fluctuations of time-antisymmetric currents. Stochastic efficiency is not one: a ratio of two odd quantities, it is even under path reversal, and for the conventional ratio −W/Q_h no moment exists. We work with the exergetic ratio η=W/(W+T_0S), whose moments exist, and ask how much of its variance a finite perturbation of the dynamics can certify when linear response cannot. The question is specific to nonlinear observables: for every current, and every observable linear in transition counts and residence times, the multiparameter Cramér-Rao bound over rate and site perturbations already equals the variance. For η it does not. The Cramér-Rao bound vanishes on a hyperplane of perturbation directions, while the Barankin multi-point bound, applied to tilted path measures, stays positive there and recovers 75% of the variance of η in a three-state motor model where every linear-response bound is zero. The Gram matrix of tilted Markov-jump path measures is a Feynman-Kac matrix exponential whose potential is the Hellinger integrand of the jump intensities, and the cost of a perturbation is its χ^2 divergence, which no dissipation bound controls. The bound captures 1.25-1.57 times the best linear-response bound, and the gain persists on networks of up to sixteen states. For an Ornstein-Uhlenbeck process the construction is exactly solvable: the optimal tilt scales as T^−1/2 and three test points nearly reach the supremum. Two limits are established. The gain is a short-window effect that disappears as the counts become Gaussian, and the bound cannot be turned into inference: coarse measurements bound χ^2 only from below. The construction is a computational instrument for a model in hand, not an uncertainty relation.
ER -