波动机械平均效率的尖锐界
Sharp Bounds on the Mean Efficiency of a Fluctuating Machine
浏览论文内容
中文总结 AI 辅助
针对热涨落机械的效率均值，本文推导了基于能量预算的尖锐上下界，下界由Jensen不等式给出，上界对应间歇可逆极值，并建立了精度-效率前沿。
中文摘要 AI 辅助
在热涨落尺度下，机械的效率是一个随机变量，通常不具有任何阶的矩：分母中的输入热$-W/Q_h$会穿过零点而波动。我们转而使用火用比$\eta=W/(W+T_0S)$，其分母仅当分子为零时才为零，因此对于非负耗散，它逐点位于$[0,1]$内，且所有阶的矩均存在。我们问：仅由能量预算能确定其均值的什么信息？设$\alpha=T_0\langle S\rangle/W$为单位有用功的平均耗散，$\sigma^2$为耗散的相对方差，则$1/(1+\alpha)<\langle\eta\rangle\le\sigma^2/(1+\sigma^2)+1/[(1+\sigma^2)(1+\alpha(1+\sigma^2))]$，两端均为尖锐界，且无需任何分布假设。下界即Jensen不等式：波动的耗散总是将平均效率提高到其确定性值之上。上界由一种间歇可逆律达到，该律在$\sigma^2/(1+\sigma^2)$比例的实现中完全不耗散；它是矩问题的极值而非实际可实现的机械，但一个稳定性结果将其转化为预测：在界附近测得的设备必须间歇运行，这仅凭轨迹记录即可检验。三阶矩完全封闭了区间。当输出的功也波动时，这些界在比值$T_0S/W$的矩下成立，热力学不确定性关系随后将上限转化为精度-效率前沿：输出更可重复的机械具有严格更低的效率上限。在零耗散方差时，这退化为已知的分子马达均值比效率的界，将其识别为家族中的一员，并表明它对于波动比值的均值是不安全的。区间内部是最大熵基准$\alpha^{-1}e^{1/\alpha}E_1(1/\alpha)$。
英文摘要
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.
发表机构
- USMBA（穆莱伊斯梅尔大学）
机构由 AI 辅助整理，请以论文原文为准。
