Optimal Control of Heat Engines in Non-equilibrium Statistical Mechanics
Optimal Control of Heat Engines in Non-equilibrium Statistical Mechanics
复制标题
非平衡统计力学中热机的优化控制
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发表时间：
2017
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通讯作者：
Yunlong Huang
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作者：
Yunlong Huang
论文题目：非平衡统计力学中热机的最优控制黄云龙，2017年论文指导人：P. S教授克里希纳普拉萨德省热力发动机是一种循环运行的统计机械系统，它将来自热浴的热量供应转化为机械功。通过改变系统参数来操作热力发动机。由于该非平衡统计力学系统是在有限时间内运行的，因此它是一个耗散系统。在这篇论文中，我们的研究集中在两个热机：一个是随机振荡器，另一个是连接到Nyquist-Johnson电阻（一个随机驱动的电阻-电容电路）的电容。在随机振荡器中，通过改变势阱的刚度，系统可以将热转换为机械功。在阻容电路中，机械功的输出是由于电容器电容的变化。这两个热机是参数控制的。热机参数空间中的路径称为协议。本文第一章在近平衡假设下，利用线性响应理论、涨落理论和随机热力学，考虑了热机参数空间中的逆扩散张量。随机振子的逆扩散张量在由势阱的刚度和热浴的逆温度组成的参数空间中诱导出双曲空间结构。电阻-电容电路的逆扩散张量在由电容器电容和热浴温度的逆组成的参数空间中诱导出欧氏空间结构。热机的平均耗散率由系统参数的切空间上的二次型（具有正定的逆扩散张量）给出。沿着热力发动机的有限时间协议，除了能量耗散之外，还有两个感兴趣的辅助量：一个是热力发动机的提取功，另一个是从浴槽到发动机的总热量供应。这两个量是分析热机效率的基础。在第二章中，结合热机的能量耗散和做功，我们介绍了这两种热机的亚黎曼几何结构。在第三章中，我们在定义了热机效率的概念之后，证明了热机的次黎曼几何中的最优控制问题与热机效率最大化问题的等价性。这样，我们就把几何控制理论引入了非平衡统计力学。特别地，我们阐明了共扼点理论与热机工作回路之间的关系。作为相关的计算，我们在第四章中将双曲空间中的等周问题作为最优控制问题来解决。在前四章理论分析的基础上，本文最后一章采用水平集方法、中点近似法和打靶法对两种热机的最大效率工作回路进行了设计。计算了这些协议的相关效率。
Title of dissertation: Optimal Control of Heat Engines in Non-equilibrium Statistical Mechanics Yunlong Huang, 2017 Dissertation directed by: Professor P. S. Krishnaprasad Dept. of Electrical & Computer Engineering A heat engine is a cyclically operated statistical mechanical system which converts heat supply from a heat bath into mechanical work. The heat engine is operated by varying the system parameter. As it is operated in finite time, this non-equilibrium statistical mechanical system is a dissipative system. In this dissertation, our research focuses on two heat engines: one is a stochastic oscillator and the other is a capacitor connected to a Nyquist-Johnson resistor (a stochastically driven resistor-capacitor circuit). In the stochastic oscillator, by varying the stiffness of the potential well, the system can convert heat to mechanical work. In the resistor-capacitor circuit, the output of mechanical work is due to the change of the capacitance of the capacitor. These two heat engines are parametrically-controlled. A path in the parameter space of a heat engine is termed as a protocol. In the first chapter of this dissertation, under the near-equilibrium assumption, with the help of linear response theory, fluctuation theorem and stochastic thermodynamics, we consider an inverse diffusion tensor in the parameter space of a heat engine. The inverse diffusion tensor of the stochastic oscillator induces a hyperbolic space structure in the parameter space composed of the stiffness of the potential well and the inverse temperature of the heat bath. The inverse diffusion tensor of the resistor-capacitor circuit induces a Euclidean space structure in the parameter space composed of the capacitance of the capacitor and the inverse temperature of the heat bath. The average dissipation rate of a heat engine is given by a quadratic form (with a positive-definite inverse diffusion tensor) on the tangent space of the system parameter. Along a finite-time protocol of a heat engine, besides the energy dissipation, there are two auxiliary quantities of interest: one is the extracted work of the heat engine and the other is the total heat supply from the bath to the engine. These two quantities are fundamental to the analysis of the efficiency of a heat engine. In Chapter 2, combining the energy dissipation and the extracted work of a heat engine, we introduce sub-Riemannian geometry structures underlying both heat engines. In Chapter 3, after defining efficiency of a heat engine, we show the equivalence between an optimal control problem in the sub-Riemannian geometry of the heat engine and the problem of maximizing the efficiency of the heat engine. In this way, we bring geometric control theory to non-equilibrium statistical mechanics. In particular, we explicate the relation between conjugate point theory and the working loops of a heat engine. As a related calculation, we solve the isoperimetric problem in hyperbolic space as an optimal control problem in Chapter 4. Based on the theoretical analysis in the first four chapters, in the final chapter of the dissertation, we adopt level set methods, mid-point approximation and shooting method to design maximum-efficiency working loops of both heat engines. The associated efficiencies of these protocols are computed.