Sharp Bounds on the Mean Efficiency of a Fluctuating Machine

RESUMO

The efficiency of a machine at the scale of thermal fluctuations is a random variable that conventionally has no moments of any order: the input heat in the denominator of $-W/Q_h$ fluctuates through zero. We work instead with the exergetic ratio $\eta=W/(W+T_0S)$, whose denominator vanishes only with its numerator, so that for non-negative dissipation it lies in $[0,1]$ pointwise and every moment exists, and we ask what the energy budget alone determines about its mean. With $\alpha=T_0\langle S\rangle/W$ the mean dissipation per unit useful work and $\sigma^2$ the relative variance of the dissipation, $1/(1+\alpha)<\langle\eta\rangle\le\sigma^2/(1+\sigma^2)+1/[(1+\sigma^2)(1+\alpha(1+\sigma^2))]$, with both ends sharp and no distributional assumption. The lower bound is Jensen's inequality: fluctuating dissipation always raises the mean efficiency above its deterministic value. The upper bound is attained by an intermittently reversible law, which dissipates nothing at all in a fraction $\sigma^2/(1+\sigma^2)$ of realisations; it is a moment-problem extremal rather than a realised machine, but a stability result turns it into a prediction: a device measured near the bound must operate intermittently, testable against the trajectory record alone. A third moment closes the bracket entirely. When the delivered work also fluctuates, the bounds hold with the moments taken on the ratio $T_0S/W$, and the thermodynamic uncertainty relation then converts the ceiling into a precision-efficiency frontier: a machine with more reproducible output has a strictly lower efficiency ceiling. At zero dissipation variance this reduces to the known bound on a molecular motor's ratio-of-means efficiency, identifying it as one member of a family and showing it unsafe for the mean of the fluctuating ratio. Inside the interval lies the maximum-entropy benchmark $\alpha^{-1}e^{1/\alpha}E_1(1/\alpha)$.