# Thermodynamic Explanation of Landau Damping by Reduction to Hydrodynamics

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## Abstract

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## 1. Introduction

## 2. Ehrenfest Reduction

#### 2.1. General Formulation

#### 2.2. Hamiltonian Version of Ehrenfest Reduction

#### 2.2.1. Hamiltonian Structure of the Vlasov Equation

#### 2.2.2. Formal Solution of Hamiltonian Evolution

#### 2.2.3. Projection of the Poisson Bracket

#### 2.2.4. Comparing the Solutions on Different Levels

#### 2.2.5. The Special Case of Constant Poisson Bivector

#### 2.2.6. Canonical Hamiltonian System

## 3. From Vlasov to Mechanical Equilibrium

#### 3.1. Projection

#### 3.2. Construction of the Reduced Evolution

#### 3.3. Features of the Reduced Evolution

#### 3.3.1. Conservation of Total Energy

#### 3.3.2. Dissipativity of Reduced Evolution

#### 3.3.3. Homogeneous Equilibrium Solution

#### 3.3.4. Some Qualitative Insight into Macroscopic Evolution Equations: Linearization.

## 4. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Hamiltonian interpretation of the Ehrenfest reduction. Step 1: The exact more detailed evolution equations are first solved formally to obtain their solution at time $t+\tau $. Step 2: This solution, $\mathbf{x}(t+\tau )$, is then projected to the less detailed level to obtain $\mathbf{\pi}(\mathbf{x}(t+\tau ))$. Step ${1}^{\prime}$: Alternative route is to first project $\mathbf{x}(t)$ to $\mathbf{y}(t)$. Step ${2}^{\prime}$: The less detailed evolution equation (generated by the projection of the Poisson bracket) then takes $\mathbf{y}(t)$ and gives $\mathbf{y}(t+\tau )$. Step ${2}^{\u2033}$: We have thus $\mathbf{\pi}(\mathbf{x}(t+\tau ))$ and $\mathbf{y}(t+\tau )$, which should ideally be equal, but they are typically not. The value $\mathbf{\pi}(\mathbf{x}(t+\tau ))$ is of course more precise because it is constructed from the detailed evolution equations. To make the value $\mathbf{y}(t+\tau )$ more precise, the less detailed evolution equations are altered by adding the difference between the self-regularized detailed and less-detailed equations. Such equations for $\mathbf{y}$ are then the reduced Ehrenfest equations.

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Pavelka, M.; Klika, V.; Grmela, M.
Thermodynamic Explanation of Landau Damping by Reduction to Hydrodynamics. *Entropy* **2018**, *20*, 457.
https://doi.org/10.3390/e20060457

**AMA Style**

Pavelka M, Klika V, Grmela M.
Thermodynamic Explanation of Landau Damping by Reduction to Hydrodynamics. *Entropy*. 2018; 20(6):457.
https://doi.org/10.3390/e20060457

**Chicago/Turabian Style**

Pavelka, Michal, Václav Klika, and Miroslav Grmela.
2018. "Thermodynamic Explanation of Landau Damping by Reduction to Hydrodynamics" *Entropy* 20, no. 6: 457.
https://doi.org/10.3390/e20060457