One-dimensional Vlasov-Poisson equations

In this section, we consider the self-consistent-field theory of Landau damping. The Vlasov equation is written

$$\frac{\partial f}{\partial t} +\mathbf{v} \cdot \nabla f + \frac{q}{m} (\mathbf{E}+\mathbf{v} \times \mathbf{B}) \cdot \nabla_v f = 0,$$ (58)

where $f$ is the electron distribution function, $q$ and $m$ are the charge and mass of electrons, respectively. The linearized version of the above equation is written

$$\frac{\partial f_1}{\partial t} +\mathbf{v} \cdot \nabla f_1 + \f... ...\frac{q}{m} (\mathbf{E}_1 +\mathbf{v} \times \mathbf{B}_1) \cdot \nabla_v f_0 .$$ (59)

We consider the case of $\mathbf{B}_0 = 0$ and $\mathbf{E}_0 = 0$. Further consider only the electrostatic case, i.e. $\partial \mathbf{B}_1 / \partial t = 0$, i.e., there is not magnetic fluctuation. Then it follows from Faraday's law that the perturbed electric field can be written as $\mathbf{E}_1 = - \nabla \Phi$. Then the linearized Vlasov equation (59) is written

$$\frac{\partial f_1}{\partial t} +\mathbf{v} \cdot \nabla f_1 = - \frac{q}{m} \mathbf{E}_1 \cdot \nabla_v f_0 .$$ (60)

Consider the one-dimensional case where $f_1$, and $\Phi$ are both independent of $x$ and $y$ coordinates, then the above equation is written

$$\frac{\partial f_1}{\partial t} + v_z \frac{\partial f_1}{\partial z} = - \frac{q}{m} E_1 \frac{\partial f_0}{\partial v_z} .$$ (61)

Integrating both sides of Eq. (61) over $v_x$ and $v_y$, we obtain

$$\frac{\partial F_1}{\partial t} + v_z \frac{\partial F_1}{\partial z} = - \frac{q}{m} E_1 \frac{\partial F_0}{\partial v_z},$$ (62)

where

$$F_0 = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} f_0 d v_x d v_y,$$ (63)

and

$$F_1 = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} f_1 d v_x d v_y,$$ (64)

which are called the reduced distribution functions. Poisson's equation is written

$$\frac{\partial E}{\partial z} = \frac{1}{\varepsilon_0} \left( n_i q_i + q \int_{- \infty}^{\infty} f d^3 v \right),$$ (65)

where $n_i$ and $q_i$ are the number density and charge of ions, respectively. In equilibrium the number density of electrons and ions are equal to each other. Assuming the number density of the massive ions remain unchanged, Poisson's equation for the perturbed quantities is written

$$\frac{\partial E_1}{\partial z} = \frac{1}{\varepsilon_0} q \int_{- \infty}^{\infty} f_1 d^3 v,$$ (66)

In terms of the reduced distribution function, equation (66) is written

$$\frac{\partial E_1}{\partial z} = \frac{1}{\varepsilon_0} q \int_{- \infty}^{\infty} F_1 d v_z .$$ (67)

Equations (62) and (67) governs the time evolution of $F_1$ and $E_1$. Consider the case that $F_0$ is spatially uniform, then all the coefficients of Eqs. (62) and (67) are independent of the spatial coordinate. In this case, different spatial Fourier harmonics are decoupled. Therefore, we can consider the case that there are only one Fourier harmonics $e^{i k z}$ in both $F_1$ and $E_1$, i.e., $F_1$ and $E_1$ are written, respectively, as

$$F_1 = \hat{F}_1 (t, v_z) e^{i k z},$$ (68)

$$E_1 = \hat{E}_1 (t) e^{i k z} .$$ (69)

Note that $\hat{F}_1$ and $\hat{E}_1$ in Eqs (68) and (69) are usually complex-valued (to allow arbitrary phases angle in $z$). Then using Eq. (67) in Eq. (62) yields

$$\frac{\partial \hat{F}_1}{\partial t} = - i k v_z \hat{F}_1 - \fr... ...rtial F_0}{\partial v_z} \frac{1}{i k} \int_{- \infty}^{\infty} \hat{F}_1 d v_z$$ (70)

Given an equilibrium distribution function $F_0 (v_z)$ and an initial perturbation $\hat{F}_1 (0, v_z)$, equation (70) can be solved analytically by using the Laplace transformation. Here, to avoid the fancy mathematics involved in the Laplace transformation, I solve Eq. (70) by a direct numerical method.

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$$\frac{\partial F_1}{\partial t} + v_z \frac{\partial F_1}{\partial z} = - \frac{q}{m} E_1 \frac{\partial F_0}{\partial v_z}$$ (71)

$$F_1 (t) - F_1 (t_1) = - \frac{q}{m} \int_{t_1}^t \frac{\partial F_0}{\partial v_z} E_1 d t'$$

Using $E_1 = \hat{E}_1 e^{- i \omega t' + i k z'}$, the above equation is written

$$F_1 (t) - F_1 (t_1) = - \frac{q}{m} \int_{t_1}^t \frac{\partial F_0}{\partial v_z} \hat{E}_1 e^{- i \omega t' + i k z'} d t'$$

$$F_1 (t) - F_1 (t_1) = - \frac{q}{m} \int_{t_1}^t \frac{\partial F_0}{\partial v_z} \hat{E}_1 e^{- i \omega t' + i k (z - v (t - t'))} d t'$$

$$F_1 (t) - F_1 (t_1) = - \frac{q}{m} e^{i k z - i k v t} \frac{\partial F_0}{\partial v_z} \hat{E}_1 \int_{t_1}^t e^{- i (\omega - k v) t'} d t'$$

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