摘要翻译：
Schmiedl和Seifert（{\It Phys.rev.lett.}{\bf 98},108301（2007））要求外部控制参数的最优协议，以使在有限时间内驱动纳米尺度系统从一个平衡态到另一个平衡态所需的平均功最小化，发现欧拉-拉格朗日方程是非局部相关函数的积分-微分方程。对于两个线性例子，我们展示了如何解析地求解这个积分-微分方程。对于非线性物理系统，我们给出了如何在数值上找到最优协议，证明了可能同时存在几个不同的最优协议，并给出了分别具有一跳、二跳和三跳的最优协议。
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英文标题：
《Computing the optimal protocol for finite-time processes in stochastic
  thermodynamics》
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作者：
Holger Then and Andreas Engel
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最新提交年份：
2008
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分类信息：

一级分类：Physics        物理学
二级分类：Statistical Mechanics        统计力学
分类描述：Phase transitions, thermodynamics, field theory, non-equilibrium phenomena, renormalization group and scaling, integrable models, turbulence
相变，热力学，场论，非平衡现象，重整化群和标度，可积模型，湍流
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英文摘要：
  Asking for the optimal protocol of an external control parameter that minimizes the mean work required to drive a nano-scale system from one equilibrium state to another in finite time, Schmiedl and Seifert ({\it Phys. Rev. Lett.} {\bf 98}, 108301 (2007)) found the Euler-Lagrange equation to be a non-local integro-differential equation of correlation functions. For two linear examples, we show how this integro-differential equation can be solved analytically. For non-linear physical systems we show how the optimal protocol can be found numerically and demonstrate that there may exist several distinct optimal protocols simultaneously, and we present optimal protocols that have one, two, and three jumps, respectively.
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PDF链接：
https://arxiv.org/pdf/710.3297