Normalization

The reduced equilibrium distribution function $F_0$ satisfies the normalization condition $\int_{- \infty}^{\infty} F_0 (v_z) d v_z = n_0$, where $n_0$ the number density of electrons. Denote the thermal velocity of the equilibrium distribution by $v_t$. Then Eq. (70) can be written

$$\frac{\partial \hat{F}_1}{\partial t} = - i k v_z \hat{F}_1 - \fr... ...c{\partial F_0}{\partial v_z} \right) \int_{- \infty}^{\infty} \hat{F}_1 d v_z,$$ (72)

which can be further written

$$\frac{\partial \hat{F}_1}{\partial t} = - i k v_t \overline{v}_z ... ...F_0}{\partial v_z} \right) \int_{- \infty}^{\infty} \hat{F}_1 d \overline{v}_z,$$ (73)

where $\overline{v}_z = v_z / v_t$, $\omega_p = \sqrt{n_0 q^2 / (\varepsilon_0 m)}$ is the electron plasma frequency. Define $\overline{t} = t \omega_p$, then Eq. (73) is written

$$\frac{\partial \hat{F}_1}{\partial \overline{t}} = - \frac{i k v_... ...F_0}{\partial v_z} \right) \int_{- \infty}^{\infty} \hat{F}_1 d \overline{v}_z,$$ (74)

Define $a = i k v_t / \omega_p$, then the above equation is written

$$\frac{\partial \hat{F}_1}{\partial \overline{t}} = - a \overline{... ...F_0}{\partial v_z} \right) \int_{- \infty}^{\infty} \hat{F}_1 d \overline{v}_z,$$ (75)