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arXiv 1903.08115cond-mat.stat-mechmath.PR
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停时熵产生的积分涨落关系

Integral Fluctuation Relations for Entropy Production at Stopping Times

Izaak Neri, Edgar Roldán, Simone Pigolotti, Frank Jülicher

备注：
37 pages, 6 figures
发表信息：
J. Stat. Mech. (2019) 104006
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摘要：

停止时间$T$是随机过程的轨迹第一次满足特定标准的时间。本文利用鞅理论导出了定态物理系统在随机停时$T$的随机熵产生$S_{\rm tot}$的积分涨落关系$\langle e^{-S_{\rm tot}(T)}\rangle=1$.这种涨落关系隐含着$\langle S_{\rm tot}(T)\rangle\geq 0$定律，它指出，即使在停止时间停止一个随机过程，也不可能平均减少熵，我们称之为停止时间热力学第二定律。这个定律意味着当系统在随机时间停止时，系统可以从其环境中提取的平均热量和功的界限。此外，积分涨落关系意味着熵产生的某些涨落是普适的或受普适函数的约束。这些普适性质是通过选择适当的停止时间从积分涨落关系中得到的：例如，当$T$是熵产生的首次通过时间时，我们得到了熵产生负记录统计量的一个界。我们说明这些结果的简单模型的非平衡系统描述的朗之万方程，并揭示了两个有趣的现象。首先，我们证明了等温介观系统可以从他们的环境中提取平均热量时，停止在一个巧妙选择的时刻和第二定律在停止时间提供了一个约束的平均提取的热量。其次，我们证明了一个自治的随机热机，如费曼棘轮，在停止时间的平均效率可以大于卡诺效率和热力学第二定律在停止时间提供了一个约束的平均效率在停止时间。

英文摘要：

A stopping time $T$ is the first time when a trajectory of a stochastic process satisfies a specific criterion. In this paper, we use martingale theory to derive the integral fluctuation relation $\langle e^{-S_{\rm tot}(T)}\rangle=1$ for the stochastic entropy production $S_{\rm tot}$ in a stationary physical system at stochastic stopping times $T$. This fluctuation relation implies the law $\langle S_{\rm tot}(T)\rangle\geq 0$, which states that it is not possible to reduce entropy on average, even by stopping a stochastic process at a stopping time, and which we call the second law of thermodynamics at stopping times. This law implies bounds on the average amount of heat and work a system can extract from its environment when stopped at a random time. Furthermore, the integral fluctuation relation implies that certain fluctuations of entropy production are universal or are bounded by universal functions. These universal properties descend from the integral fluctuation relation by selecting appropriate stopping times: for example, when $T$ is a first-passage time for entropy production, then we obtain a bound on the statistics of negative records of entropy production. We illustrate these results on simple models of nonequilibrium systems described by Langevin equations and reveal two interesting phenomena. First, we demonstrate that isothermal mesoscopic systems can extract on average heat from their environment when stopped at a cleverly chosen moment and the second law at stopping times provides a bound on the average extracted heat. Second, we demonstrate that the average efficiency at stopping times of an autonomous stochastic heat engines, such as Feymann's ratchet, can be larger than the Carnot efficiency and the second law of thermodynamics at stopping times provides a bound on the average efficiency at stopping times.

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