Optimal Control of Heat Engines in Non-equilibrium Statistical Mechanics

Abstract

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 thermo- dynamics, 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 chap- ter 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.