Sharp Bounds on the Mean Efficiency of a Fluctuating Machine

Abstract

The efficiency of a machine at the scale of thermal fluctuations is random, and the conventional ratio $-W/Q_h$ has no moment of any order: its input heat fluctuates through zero. We work instead with the exergetic ratio $\eta =W/(W+T_{0}S)$, which lies in $[0,1]$ pointwise for non-negative dissipation, and ask what the energy budget alone determines about its mean. With $\alpha =T_0⟨S⟩/W$ and ${\sigma }^2$ the relative variance of the dissipation, $1/(1+\alpha )<⟨\eta ⟩\le {\sigma }^2/(1+{\sigma }^2)+1/\{(1+{\sigma }^2)[1+\alpha (1+{\sigma }^2)]\}$, both ends sharp, with no distributional assumption. The floor is Jensen's inequality: fluctuating dissipation raises the mean efficiency above its deterministic value, and the mean alone gives nothing more. The ceiling is attained by an intermittently reversible law, dissipating nothing in a fraction ${\sigma }^2/(1+{\sigma }^2)$ of realisations, a prediction testable on trajectories. A third moment lifts the floor. Fixed delivered work is not required: when it too fluctuates, the bounds hold with the moments taken on $T_{0}S/W$, and the thermodynamic uncertainty relation on the work current converts the ceiling into a precision-efficiency frontier, whose zero-variance member is the known bound on a motor's ratio-of-means efficiency, shown here to be unsafe for the mean of the fluctuating ratio. Inside the interval lies the maximum-entropy benchmark ${\alpha }^{-1}{e}^{1/\alpha }{E}_1(1/\alpha )$. Finally the bounds are worked out for a motor with futile cycles, observed until a fixed number of steps is delivered. There the dissipation cannot fall below the reversible cost of that work, and this floor $b$ sharpens the ceiling to $q/(1+\alpha b)+(1-q)/(1+\alpha c)$, $q={\sigma }^2/[{\sigma }^2+(1-b{)}^2]$, $c=1+{\sigma }^2/(1-b)$, removing 40-67 per cent of the width. It is saturated when slips are rare: the extremal law is an operating regime, not an idealisation.

v3: adds Section 9, a molecular motor with futile cycles, and Proposition 16, an upper bound under a known dissipation floor. 37 pages, 5 figures. Code: doi:10.5281/zenodo.22812101