# Optimal Control of Hydrogen Atom-Like Systems as Thermodynamic Engines in Finite Time

## Abstract

**:**

## 1. Introduction

## 2. Background

#### 2.1. Aspects of the Hydrogen Atom-Like System

_{2}[32], and we could translate our analysis from using $\left(\kappa ,T\right)$ to $\left(p,T\right)$ as control variables, in principle.

#### 2.2. Thermodynamic Cycles for the Hydrogen Atom-Like System

#### 2.3. Statistical Mechanics and Thermodynamics of Cycles

## 3. Optimal Control Criterion for Finite-Time Thermodynamic Cycles of General Statistical Mechanical Systems

#### 3.1. Finite-Time Optimal Control along a General Path in $\left(\kappa ,T\right)$ Space

#### 3.2. Finite-Time Optimal Control along Isothermal Legs

#### 3.3. Finite-Time Optimal Control along Iso-$\kappa $ Legs

#### 3.4. Finite-Time Optimal Control for Adiabatic Legs

#### 3.4.1. General Adiabatic Paths

#### 3.4.2. Special Adiabatic Paths

## 4. Application to the Hydrogen Atom-Like System

#### 4.1. Iso-$\kappa $-Adiabatic Cycle

#### 4.2. Iso-$\kappa $-Isothermal Cycle

_{T}and N

_{K}of the individual legs will be determined from the ratios $\frac{{\tau}^{\left(j\right)}}{{\left(\Delta t\right)}_{T,\kappa}}$, where we again will assume that the relaxation times are constant. Employing Equations (37), (42) and (56)–(58), the total thermodynamic length of the cycle is then

#### 4.3. Isothermal-Adiabatic Cycle

## 5. Approximation of the Hydrogen Atom-Like System as a Two-Level System

#### 5.1. Preliminaries

#### 5.2. Thermodynamic Cycles

## 6. Summary and Discussion

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Sketch of an iso-$\kappa $-isothermal cycle for a hydrogen atom-like system. Branches $\left[1\right]\to \left[2\right]$ and $\left[3\right]\to \left[4\right]$ are iso-$\kappa $-legs and branches $\left[2\right]\to \left[3\right]$ and $\left[4\right]\to \left[1\right]$ are isothermal legs, respectively. The four corners of the cycle are the points $\left[1\right]=\left({\kappa}^{\left[1\right]},{T}^{\left[1\right]}\right)=\left({\kappa}_{in},{T}_{1}\right)$, $\left[2\right]=\left({\kappa}^{\left[2\right]},{T}^{\left[2\right]}\right)=\left({\kappa}_{in},{T}_{2}\right)$, $\left[3\right]=\left({\kappa}^{\left[3\right]},{T}^{\left[3\right]}\right)=\left({\kappa}_{f},{T}_{2}\right)$ and $\left[4\right]=\left({\kappa}^{\left[4\right]},{T}^{\left[4\right]}\right)=\left({\kappa}_{f},{T}_{1}\right)$ in the $\left(\kappa ,T\right)$ plane. Note that for given ${\kappa}_{in}$, ${\kappa}_{f}>{\kappa}_{in}$ and ${T}^{\left[1\right]}={T}_{1}$, there remains only one variable, e.g., ${T}^{\left[2\right]}={T}_{2}$, we are free to choose; all other variables are fixed by the choice of path types. All cycles for which ${T}_{2}>{T}_{1}$ are feasible iso-$\kappa $-isothermal cycles.

**Figure 2.**Sketch of an iso-$\kappa $-adiabatic cycle for a hydrogen atom-like system. Branches $\left[1\right]\to \left[2\right]$ and $\left[3\right]\to \left[4\right]$ are iso-$\kappa $-legs and branches $\left[2\right]\to \left[3\right]$ and $\left[4\right]\to \left[1\right]$ are (special) adiabatic legs, respectively. Note that the adiabatic legs run along straight lines through the origin. The four corners of the cycle are the points $\left[1\right]=\left({\kappa}^{\left[1\right]},{T}^{\left[1\right]}\right)=\left({\kappa}_{in},{T}_{1}\right)$, $\left[2\right]=\left({\kappa}^{\left[2\right]},{T}^{\left[2\right]}\right)=\left({\kappa}_{in},{T}_{2}\right)$, $\left[3\right]=\left({\kappa}^{\left[3\right]},{T}^{\left[3\right]}\right)=\left({\kappa}_{f},{T}_{3}\right)$, and $\left[4\right]=\left({\kappa}^{\left[4\right]},{T}^{\left[4\right]}\right)=\left({\kappa}_{f},{T}_{4}\right)$ in the $\left(\kappa ,T\right)$ plane. Note that for given ${\kappa}_{in}$, ${\kappa}_{f}>{\kappa}_{in}$, and ${T}^{\left[1\right]}={T}_{1}$ there remains only one variable, e.g., ${T}^{\left[2\right]}={T}_{2}$, we are free to choose; all other variables are fixed by the choice of path types. All cycles for which ${T}_{2}>{T}_{1}$ are feasible iso-$\kappa $-adiabatic cycles.

**Figure 3.**Sketch of an adiabatic-isothermal cycle for a hydrogen atom-like system. Branches $\left[1\right]\to \left[2\right]$ and $\left[3\right]\to \left[4\right]$ are (special) adiabatic legs and branches $\left[2\right]\to \left[3\right]$ and $\left[4\right]\to \left[1\right]$ are isothermal legs, respectively. Note that the adiabatic legs run along straight lines through the origin. The four corners of the cycle are the points $\left[1\right]=\left({\kappa}^{\left[1\right]},{T}^{\left[1\right]}\right)=\left({\kappa}_{in},{T}_{1}\right)$, $\left[2\right]=\left({\kappa}^{\left[2\right]},{T}^{\left[2\right]}\right)=\left({\kappa}_{2},{T}_{2}\right)$, $\left[3\right]=\left({\kappa}^{\left[3\right]},{T}^{\left[3\right]}\right)=\left({\kappa}_{f},{T}_{2}\right)$, and $\left[4\right]=\left({\kappa}^{\left[4\right]},{T}^{\left[4\right]}\right)=\left({\kappa}_{4},{T}_{1}\right)$ in the $\left(\kappa ,T\right)$ plane. Note that for given ${\kappa}_{in}$, ${\kappa}_{f}>{\kappa}_{in}$ and ${T}^{\left[1\right]}={T}_{1}$, there remains only one variable, e.g., ${T}^{\left[2\right]}={T}_{2}$, we are free to choose; all other variables are fixed by the choice of path types. However, only cycles for which ${T}_{2}^{max}>{T}_{2}>{T}_{1}$, with ${T}_{2}^{max}=\frac{{\kappa}_{f}}{{\kappa}_{in}}{T}_{1}$, are feasible adiabatic-isothermal cycles for the hydrogen atom-like system.

**Figure 4.**Sketch of an optimal setting of step points along a (special) adiabatic leg moving from corner $\left[3\right]$ to corner $\left[4\right]$, where the excess heat associated with being “off-target” with respect to the ideal adiabatic path is minimized. Note that the density of step points $\left({\kappa}_{i},{T}_{i}\right)$ along the leg increases with temperature, approximately in a square-root fashion. Red lines are the isothermal and blue lines the iso-$\kappa $ sub-pieces, respectively, which connect two points $\left({\kappa}_{i-1},{T}_{i-1}\right)$ and $\left({\kappa}_{i},{T}_{i}\right)$ along the perfect adiabatic path via the virtual intermediary off-target points $\left({\kappa}_{i-1},{T}_{i}\right)$.

**Figure 5.**Sketch of an a-thermal iso-$\kappa $-“adiabatic” cycle for a two-state system comprising the ground state and the first excited state of the hydrogen-like atom. Branches $\left[1\right]\to \left[2\right]$ and $\left[3\right]\to \left[4\right]$ are iso-$\kappa $-legs and branches $\left[2\right]\to \left[3\right]$ and $\left[4\right]\to \left[1\right]$ are “adiabatic” legs, respectively. Note that the adiabatic legs run along straight lines that pass through the origin and through the point $\left(\kappa ,E\right)=\left(1,-\alpha \right)$ for the adiabatic leg in the ground state, and through the origin and through the point $\left(\kappa ,E\right)=\left(1,-\alpha /4\right)$ for the adiabatic leg in the first excited state, respectively. The four corners of the cycle are the points $\left[1\right]=\left({\kappa}^{\left[1\right]},{E}^{\left[1\right]}\right)=\left({\kappa}_{in},{E}_{1}\right)$, $\left[2\right]=\left({\kappa}^{\left[2\right]},{E}^{\left[2\right]}\right)=\left({\kappa}_{in},{E}_{2}\right)$, $\left[3\right]=\left({\kappa}^{\left[3\right]},{E}^{\left[3\right]}\right)=\left({\kappa}_{f},{E}_{3}\right)$, and $\left[4\right]=\left({\kappa}^{\left[4\right]},{E}^{\left[4\right]}\right)=\left({\kappa}_{f},{E}_{4}\right)$ in the $\left(\kappa ,E\right)$ plane. Note that ${E}^{\left[1\right]}={\kappa}_{in}{E}_{1}\left(\kappa =1\right)$, ${E}^{\left[2\right]}={\kappa}_{in}{E}_{2}\left(\kappa =1\right)$, ${E}^{\left[3\right]}={\kappa}_{f}{E}_{2}\left(\kappa =1\right)$, and ${E}^{\left[4\right]}={\kappa}_{f}{E}_{1}\left(\kappa =1\right)$.

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Schön, J.C. Optimal Control of Hydrogen Atom-Like Systems as Thermodynamic Engines in Finite Time. *Entropy* **2020**, *22*, 1066.
https://doi.org/10.3390/e22101066

**AMA Style**

Schön JC. Optimal Control of Hydrogen Atom-Like Systems as Thermodynamic Engines in Finite Time. *Entropy*. 2020; 22(10):1066.
https://doi.org/10.3390/e22101066

**Chicago/Turabian Style**

Schön, Johann Christian. 2020. "Optimal Control of Hydrogen Atom-Like Systems as Thermodynamic Engines in Finite Time" *Entropy* 22, no. 10: 1066.
https://doi.org/10.3390/e22101066