# Casimir and Casimir-Polder Forces in Graphene Systems: Quantum Field Theoretical Description and Thermodynamics

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## Abstract

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## 1. Introduction

## 2. The Lifshitz Theory of the Casimir and Casimir-Polder Forces and Its Generalizations

## 3. Experimental and Thermodynamic Problems of the Lifshitz Theory

## 4. Different Approaches to Theoretical Description of the Electromagnetic Response of Graphene

## 5. Quantum Field Theoretical Description of the Electromagnetic Response of Graphene via the Polarization Tensor

## 6. Low-Temperature Behavior of the Casimir-Polder Free Energy and Entropy for an Atom and a Pristine Graphene Sheet

## 7. Low-Temperature Behavior of the Casimir Free Energy and Entropy for Two Pristine Graphene Sheets

## 8. Low-Temperature Behavior of the Casimir-Polder Free Energy and Entropy for an Atom and a Graphene Sheet Possessing the Energy Gap and Chemical Potential

## 9. Low-Temperature Behavior of the Casimir-Polder Free Energy and Entropy for Two Graphene Sheets Possessing the Energy Gap and Chemical Potential

## 10. Discussion: How Graphene May Point the Way to Resolution of Thermodynamic Problems in the Lifshitz Theory

## 11. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The mean measured gradients of the Casimir force between an Au-coated sphere and an Au-coated plate are shown by crosses as a function of separations. For clarity only each third experimental data point is plotted. The bottom and top lines demonstrate theoretical predictions of the Lifshitz theory obtained with inclusion and neglect of the relaxation of conduction electrons, respectively.

**Figure 2.**The mean measured gradients of the Casimir force between a Ni-coated sphere and a Ni-coated plate are shown by crosses as a function of separations. The top and bottom lines demonstrate theoretical predictions of the Lifshitz theory obtained with inclusion and neglect of the relaxation of conduction electrons, respectively.

**Table 1.**Up to an order of magnitude asymptotic behaviors at arbitrarily low temperature of the thermal corrections to the Casimir-Polder energy (line 2) and of the Casimir-Polder entropy (line 3) under different relationships between the energy gap $\Delta $ and chemical potential $\mu $ (line 1).

$\mathbf{\Delta}=\mathit{\mu}=0$ | $\mathbf{\Delta}>2\mathit{\mu}\u2a7e0$ | $\mathbf{\Delta}=2\mathit{\mu}\ne 0$ | $0\u2a7d\mathbf{\Delta}<2\mathit{\mu}$ | |
---|---|---|---|---|

${\delta}_{T}{\mathcal{F}}_{CP}(a,T)$ | $-\frac{{\alpha}_{0}{\left({k}_{B}T\right)}^{3}}{{v}_{F}^{2}{\hslash}^{2}a}$ | $-\frac{{\alpha}_{0}{\left({k}_{B}T\right)}^{5}}{{\left(\hslash c\right)}^{3}\Delta}$ | $-\frac{{\alpha}_{0}{k}_{B}T}{{a}^{3}}$ | $-\frac{{\alpha}_{0}{\mu}^{2}{\left({k}_{B}T\right)}^{2}}{{\left(\hslash c\right)}^{2}a\sqrt{4{\mu}^{2}-{\Delta}^{2}}}$ |

${S}_{CP}(a,T)$ | $\frac{{\alpha}_{0}{k}_{B}{\left({k}_{B}T\right)}^{2}}{{v}_{F}^{2}{\hslash}^{2}a}$ | $\frac{{\alpha}_{0}{k}_{B}{\left({k}_{B}T\right)}^{4}}{{\left(\hslash c\right)}^{3}\Delta}$ | $\frac{{\alpha}_{0}{k}_{B}}{{a}^{3}}$ | $\frac{{\alpha}_{0}{\mu}^{2}{k}_{B}^{2}T}{{\left(\hslash c\right)}^{2}a\sqrt{4{\mu}^{2}-{\Delta}^{2}}}$ |

**Table 2.**Up to an order of magnitude asymptotic behaviors at arbitrarily low temperature of the thermal corrections to the Casimir energy (line 2) and of the Casimir entropy (line 3) under different relationships between the energy gap $\Delta $ and chemical potential $\mu $ (line 1).

$\mathbf{\Delta}=\mathit{\mu}=0$ | $\mathbf{\Delta}>2\mathit{\mu}\u2a7e0$ | $\mathbf{\Delta}=2\mathit{\mu}\ne 0$ | $0\u2a7d\mathbf{\Delta}<2\mathit{\mu}$ | |
---|---|---|---|---|

${\delta}_{T}{\mathcal{F}}_{C}(a,T)$ | $\frac{{\left({k}_{B}T\right)}^{3}}{{\left(\hslash c\right)}^{2}}\mathrm{ln}\frac{a{k}_{B}T}{\hslash c}$ | $-\frac{{\left({k}_{B}T\right)}^{5}}{{\left(\hslash c\right)}^{2}{\Delta}^{2}}$ | $-\frac{{k}_{B}T}{{a}^{2}}$ | $-\frac{a(4{\mu}^{2}-{\Delta}^{2}){\left({k}_{B}T\right)}^{2}}{{\left(\hslash c\right)}^{3}}$ |

${S}_{C}(a,T)$ | $-{k}_{B}{\left(\frac{{k}_{B}T}{\hslash c}\right)}^{2}\mathrm{ln}\frac{a{k}_{B}T}{\hslash c}$ | $\frac{{k}_{B}{\left({k}_{B}T\right)}^{4}}{{\left(\hslash c\right)}^{2}{\Delta}^{2}}$ | $\frac{{k}_{B}}{{a}^{2}}$ | $\frac{a(4{\mu}^{2}-{\Delta}^{2}){k}_{B}^{2}T}{{\left(\hslash c\right)}^{3}}$ |

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**MDPI and ACS Style**

Klimchitskaya, G.L.; Mostepanenko, V.M.
Casimir and Casimir-Polder Forces in Graphene Systems: Quantum Field Theoretical Description and Thermodynamics. *Universe* **2020**, *6*, 150.
https://doi.org/10.3390/universe6090150

**AMA Style**

Klimchitskaya GL, Mostepanenko VM.
Casimir and Casimir-Polder Forces in Graphene Systems: Quantum Field Theoretical Description and Thermodynamics. *Universe*. 2020; 6(9):150.
https://doi.org/10.3390/universe6090150

**Chicago/Turabian Style**

Klimchitskaya, Galina L., and Vladimir M. Mostepanenko.
2020. "Casimir and Casimir-Polder Forces in Graphene Systems: Quantum Field Theoretical Description and Thermodynamics" *Universe* 6, no. 9: 150.
https://doi.org/10.3390/universe6090150