# Magnetic Engine for the Single-Particle Landau Problem

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Partition Function for the Single-Particle Landau Problem

## 3. Thermodynamics and Magnetic Engine

#### 3.1. The First Law of Thermodynamics: A Microscopic Approach

#### 3.2. Magnetic Engine

- Deducting the relation between the magnetic field and the temperature along an isoentropic trajectory solving the differential equation of first order given by$$dS(B,T)={\left(\frac{\partial S}{\partial B}\right)}_{T}dB+{\left(\frac{\partial S}{\partial T}\right)}_{B}dT=0,$$$$\frac{dB}{dT}=-\frac{{C}_{B}}{T{\left(\frac{\partial S}{\partial B}\right)}_{T}},$$
- The other possibility is connecting the value for the entropy in two isoentropic trajectories in the form$$\begin{array}{c}S({B}_{1},{T}_{1})=S({B}_{2},{T}_{2})\hfill \\ S({B}_{2},{T}_{3})=S({B}_{1},{T}_{4}),\hfill \end{array}$$$$r=\frac{{l}_{{B}_{1}}}{{l}_{{B}_{2}}},$$

#### Magnetic Engine for the Landau Problem

## 4. Results and Discussion

^{−1}, which means an active area of $A\propto {10}^{-7}$ m

^{2}, by using the fact that the universal flux quantum has an order of ${\mathsf{\Phi}}_{0}\propto {10}^{-15}$ Wb. In the left panel of Figure 2, we plot the solution for the case $S(T,B)=S(10,1)$, and in the right panel we plot the solution for the case $S(T,B,{10}^{8})=S(10,1,{10}^{8})$. The contrast is evident: in the simple scenario an increase in the magnetic field implies an increase in the temperature. However, for the case with degeneracy, the rise in the magnetic field leads to a decrease in the temperature. The explanation of this fact lies in the behavior of the entropy at low temperatures, because of $S{(T,B,\lambda )}_{T\to 0}\sim {k}_{B}ln\left(g\right)$, where g is directly proportional to B. This is discussed in Figure 3 where we show the entropy behavior in these two different scenarios. In the non-degenerate case, when we increase the magnetic field, the function $S\left(T,B\right)$ intersects the starting value of the entropy always in a higher value than the initial one, reflected in the left frame of Figure 3. This explains the linearity that we obtain in a plot B vs. T for the left panel of Figure 2. The opposite occurs for the degenerate case, the function ${S}_{L}\left(T,B,{10}^{8}\right)$, which intersects the starting value of the entropy always in a lower value than the initial one, as we see in the right frame of Figure 3. From this same figure, we can conclude that the entropy function for the degenerate case collapses to approximately the same value for higher temperature for different values of the magnetic field. Therefore, since the magnetic field is the external parameter that makes the engine work, we have a region of temperature and magnetic field where it is valid to carry out this cycle.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

**Figure A1.**A parametric solution of the differential equation along the adiabatic trajectories for the Landau case. The dotted line represents the exact solution and the dot-dashed line the asymptotic case for $u<<1$. We can clearly see the constant value 0.5 for the solution in the case of $u>>1$ from the dotted line in the figure. The solid line represents the proposal curve given by Equation (A13) showing a good fit for the problem under study.

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**Figure 1.**Pictorial description for the novel-magnetic engine represented as an entropy versus a magnetic field diagram.

**Figure 2.**Behavior of the magnetic field versus the temperature for the case without the degeneracy factor (

**a**) and the case with the degeneracy factor $\frac{\mathsf{\Phi}\left(B\right)}{2{\mathsf{\Phi}}_{0}}$ (

**b**). We select the factor $\frac{A}{2{\mathsf{\Phi}}_{0}}\propto {10}^{8}$ T

^{−1}for this example.

**Figure 3.**The isoentropic trajectories behavior for the two cases under discussion. In (

**a**) we plot the non-degenerate case $S(T,B)=S(10,1)$ and in (

**b**) the degenerate case ${S}_{L}(T,B,{10}^{8})={S}_{L}(10,1,{10}^{8})$.

**Figure 4.**Total work (

**a**) and input heat (

**b**) versus the r parameter along the cycle for the case with degeneracy (dot dashed line) and without degeneracy (dashed line).

**Figure 5.**Efficiency for different cases of interest. For this case, the dotted red line corresponds to the value of Carnot cycle for a machine operating between the two temperatures ${T}_{1}=4$ K and ${T}_{3}=10$ K.

**Figure 7.**Magnetization along the first iso-magnetic trajectory as a function of T in the range of 4 K to 10 K. We selected the different values for ${B}_{2}$ that we found from numerical calculations.

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Peña, F.J.; González, A.; Nunez, A.S.; Orellana, P.A.; Rojas, R.G.; Vargas, P. Magnetic Engine for the Single-Particle Landau Problem. *Entropy* **2017**, *19*, 639.
https://doi.org/10.3390/e19120639

**AMA Style**

Peña FJ, González A, Nunez AS, Orellana PA, Rojas RG, Vargas P. Magnetic Engine for the Single-Particle Landau Problem. *Entropy*. 2017; 19(12):639.
https://doi.org/10.3390/e19120639

**Chicago/Turabian Style**

Peña, Francisco J., Alejandro González, Alvaro S. Nunez, Pedro A. Orellana, René G. Rojas, and Patricio Vargas. 2017. "Magnetic Engine for the Single-Particle Landau Problem" *Entropy* 19, no. 12: 639.
https://doi.org/10.3390/e19120639