Sharp Bounds on the Mean Efficiency of a Fluctuating Machine
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 $η=W/(W+T_0S)$, which lies in $[0,1]$ pointwise for non-negative dissipation, and ask what the energy budget alone determines about its mean. With $α=T_0\langle S\rangle/W$ and $σ^2$ the relative variance of the dissipation, $1/(1+α)<\langleη\rangle\leσ^2/(1+σ^2)+1/\{(1+σ^2)[1+α(1+σ^2)]\}$, 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 $σ^2/(1+σ^2)$ 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 $T_0S/W$, 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 $α^{-1}e^{1/α}E_1(1/α)$. 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 $q/(1+αb)+(1-q)/(1+αc)$, $q=σ^2/[σ^2+(1-b)^2]$, $c=1+σ^2/(1-b)$, 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.