Sub-Riemannian geometry and finite time thermodynamics Part 1: The stochastic oscillator
Sub-Riemannian geometry and finite time thermodynamics Part 1: The stochastic oscillator
复制标题
亚黎曼几何和有限时间热力学第 1 部分：随机振荡器
DOI：
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发表时间：
2020
期刊：
影响因子：
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通讯作者：
P. Krishnaprasad
中科院分区：
文献类型：
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作者：
Yunlong Huang;P. Krishnaprasad
亚黎曼几何领域在过去40年里蓬勃发展，因为应用科学(在机器人等领域)中出现的问题与关于空间性质的纯数学性质的问题之间存在着强烈的相互作用。控制理论的方法，如由向量场的李括号确定的可控性、与最优控制的最大值原理有关的哈密尔顿方程、哈密尔顿-雅各比-贝尔曼方程等，都被发现是回答这类问题的基本工具。在这篇文章中，我们发现了亚黎曼几何在攻击非平衡统计力学中感兴趣的问题方面的一个有用的作用：如何为受环境波动影响的微纳系统(热机)的运行创建规则，以便能够在运行的一个周期内做一些有用的事情，如将热转化为功？我们利用几何最优控制理论来产生这样的规则，这些规则被选为效率最高的规则。这是通过具体处理一个模型问题--随机振荡器来实现的。对我们的工作至关重要的是时间尺度的分离，物理学家使用的时间尺度非常有效，并在线性响应制度中是合理的。
The field of sub-Riemannian geometry has flourished in the past four decades through the strong interactions between problems arising in applied science (in areas such as robotics) and questions of a pure mathematical character about the nature of space. Methods of control theory, such as controllability properties determined by Lie brackets of vector fields, the Hamilton equations associated to the Maximum Principle of optimal control, Hamilton-Jacobi-Bellman equation etc. have all been found to be basic tools for answering such questions. In this paper, we find a useful role for the vantage point of sub-Riemannian geometry in attacking a problem of interest in non-equilibrium statistical mechanics: how does one create rules for operation of micro- and nano-scale systems (heat engines) that are subject to fluctuations from the surroundings, so as to be able to do useful things such as converting heat into work over a cycle of operation? We exploit geometric optimal control theory to produce such rules selected for maximal efficiency. This is done by working concretely with a model problem, the stochastic oscillator. Essential to our work is a separation of time scales used with great efficacy by physicists and justified in the linear response regime.