# Kinetics of Degenerate Electron–Positron Plasmas

^{1}

^{2}

^{3}

^{4}

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## Abstract

**:**

## 1. Introduction

## 2. Conditions for Formation of Relativistic Degenerate Plasma

#### 2.1. Degenerate Plasmas in Compact Astrophysical Objects

#### 2.2. Strong Electromagnetic Fields in Astrophysical Sources

#### 2.3. Pair Creation in Ultraintense Lasers

#### 2.4. Fermion Critical Density

## 3. Relativistic Kinetic Equations

#### 3.1. Derivation of Relativistic Kinetic Equations from Quantum Theory

#### 3.2. Collision Integrals in the Relativistic Kinetic Equation

#### 3.3. Binary Interactions

#### 3.4. Triple Interactions

#### 3.5. Kinetic versus Thermal Equilibrium

## 4. Bose–Einstein Condensation of Photons in Relativistic Plasma

## 5. Thermalization of Superdegenerate Plasma

## 6. Phase Space Evolution of Pairs Created in Strong Electric Fields

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Notes

1 | http://www.xfel.eu (accessed on 8 September 2022). |

2 | https://eli-laser.eu/ (accessed on 8 September 2022). |

3 | https://xcels.ipfran.ru/img/site-XCELS.pdf (accessed on 8 September 2022). |

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**Figure 1.**Number density-energy density diagram of a photon–electron–positron plasma. Green curve corresponds to thermal equilibrium state. Black curve shows the transition from nondegenerate $D>1$ to degenerate $D<1$ plasma, where D is defined by Equation (2). Red curve corresponds to fully degenerate pair state defined by Equation (3). Vertical line on the left corresponds to the transition from nonrelativistic to relativistic pair plasma ($\theta =0.3$). Vertical line on the right corresponds to relativistic pair plasma with $\theta =1$.

**Figure 2.**Thermal average occupation numbers (top) and thermal spectral energy density (bottom) of pairs as function of their kinetic energy for selected temperatures: $\theta =0.5$ (blue), $\theta =10$ (orange), $\theta =100$ (green). The limiting spectral density for pairs according to Pauli principle is shown in red.

**Figure 3.**Time evolution of energy density (top) and particle number density (bottom) of photons (blue), electrons/positrons (orange), all together (green) in nonrelativistic case. Black line represents the final equilibrium quantity. Final equilibrium temperature is $\theta ={k}_{B}T/{m}_{e}{c}^{2}\simeq 0.1$.

**Figure 4.**Top: The spectral energy density (dots) with the associated Planck fit (solid) for selected time moments from left to right: ${10}^{-15},\phantom{\rule{3.33333pt}{0ex}}{10}^{-11},\phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ s, in nonrelativistic case. Bottom: emission and absorption coefficients for photons (binary reactions: emission (blue) and absorption (cyan); triple reactions: emission (purple) and absorption (red). The left panel represents initial distribution of photons; the middle one shows photon condensation, while the right one corresponds to the final state.

**Figure 5.**The time evolution of the spectral index of the power law distribution of photons below the peak, starting at the moment when it is first established.

**Figure 6.**Time evolution of energy density (top) and particle number density (bottom) for relativistic photon-electron plasma with degenerate initial pair state (solid) and nondegenerate initial electron state (dashed). Blue: photons, orange: electrons, red: positrons and green: total. Final equilibrium temperature is ${\theta}_{f\phantom{\rule{-0.166667em}{0ex}}in}=1.9$.

**Figure 8.**Phase space distributions of electrons (left column) and photons (right column) for the initial condition ${E}_{0}=100\phantom{\rule{0.166667em}{0ex}}{E}_{c}$. Top: $2.3\times {10}^{2}\phantom{\rule{0.166667em}{0ex}}{t}_{c}$, middle: $2.3\times {10}^{2}\phantom{\rule{0.166667em}{0ex}}{t}_{c}$, bottom: $4.6\times {10}^{6}\phantom{\rule{0.166667em}{0ex}}{t}_{c}$.

Binary Processes | Triple Processes |
---|---|

Møller, Bhabha | Bremsstrahlung |

${e}^{\pm}{{e}^{\pm}}^{\prime}\phantom{\rule{3.33333pt}{0ex}}\leftrightarrow \phantom{\rule{3.33333pt}{0ex}}{{e}^{\pm}}^{\u2033}$${{e}^{\pm}}^{\u2034}$ | ${e}^{\pm}{{e}^{\pm}}^{\prime}\phantom{\rule{3.33333pt}{0ex}}\leftrightarrow \phantom{\rule{3.33333pt}{0ex}}{{e}^{\pm}}^{\u2033}{{e}^{\pm}}^{\u2034}\gamma $ |

${e}^{\pm}{e}^{\mp}\phantom{\rule{3.33333pt}{0ex}}\leftrightarrow \phantom{\rule{3.33333pt}{0ex}}{{e}^{\pm}}^{\prime}$${{e}^{\mp}}^{\prime}$ | ${e}^{\pm}{e}^{\mp}\phantom{\rule{3.33333pt}{0ex}}\leftrightarrow \phantom{\rule{3.33333pt}{0ex}}{{e}^{\pm}}^{\prime}{{e}^{\mp}}^{\prime}\gamma $ |

Single Compton | Double Compton |

${e}^{\pm}\gamma \phantom{\rule{3.33333pt}{0ex}}\leftrightarrow \phantom{\rule{3.33333pt}{0ex}}{e}^{\pm}{\gamma}^{\prime}$ | ${e}^{\pm}\gamma \phantom{\rule{3.33333pt}{0ex}}\leftrightarrow \phantom{\rule{3.33333pt}{0ex}}{{e}^{\pm}}^{\prime}{\gamma}^{\prime}{\gamma}^{\u2033}$ |

Pair production and annihilation | Radiative pair production, triplet production and three photon annihilation |

$\gamma {\gamma}^{\prime}\phantom{\rule{3.33333pt}{0ex}}\leftrightarrow \phantom{\rule{3.33333pt}{0ex}}{e}^{\pm}{e}^{\mp}$ | $\gamma {\gamma}^{\prime}$$\phantom{\rule{3.33333pt}{0ex}}\leftrightarrow \phantom{\rule{3.33333pt}{0ex}}{e}^{\pm}{e}^{\mp}$${\gamma}^{\u2033}$ |

${e}^{\pm}\gamma $$\phantom{\rule{3.33333pt}{0ex}}\leftrightarrow \phantom{\rule{3.33333pt}{0ex}}{{e}^{\pm}}^{\prime}{e}^{\mp}{{e}^{\pm}}^{\u2033}$ | |

${e}^{\pm}{e}^{\mp}\phantom{\rule{3.33333pt}{0ex}}\leftrightarrow \phantom{\rule{3.33333pt}{0ex}}\gamma {\gamma}^{\prime}$${\gamma}^{\u2033}$ |

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Vereshchagin, G.; Prakapenia, M.
Kinetics of Degenerate Electron–Positron Plasmas. *Universe* **2022**, *8*, 473.
https://doi.org/10.3390/universe8090473

**AMA Style**

Vereshchagin G, Prakapenia M.
Kinetics of Degenerate Electron–Positron Plasmas. *Universe*. 2022; 8(9):473.
https://doi.org/10.3390/universe8090473

**Chicago/Turabian Style**

Vereshchagin, Gregory, and Mikalai Prakapenia.
2022. "Kinetics of Degenerate Electron–Positron Plasmas" *Universe* 8, no. 9: 473.
https://doi.org/10.3390/universe8090473