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 $-W/Q_h$ 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.