Linear Landau damping

In the early phase of the simulation

$$t \ll \sqrt{2 \pi} / \omega_b,$$ (8)

where $\omega_b$ is the bounce angular frequency of particles in the trough of the wave, the trapped particles effects can be neglected. This phase can be considered as the linear phase where the linear Landau damping theory (discussed later in this note) is valid. The bounce angular frequency of particles in the trough of the wave is given by[1]

$$\omega_b = \sqrt{\frac{k q E_0}{m}} .$$ (9)

Using this, the condition (8) is written as

$$\frac{t}{T} \ll \sqrt{\frac{m \omega^2}{2 \pi k q E}} .$$ (10)

This condition reduces to $t / T \ll 1$ for the case plotted in Fig. 2, where we see that the total kinetic energy of the particles increases monotonously with time during this period. From the data in Fig. 2, the temporal change rate of the total kinetic energy is estimated as $d \overline{E}_k / d \overline{t} = 0.08$. Next, I compare this result with those given by the analytic formula (39) (given later in this note), which is written

$$\frac{d E_k}{d t} = - \frac{\pi \omega}{\vert k\vert k} \frac{(q E)^2}{2 m} \left[ \frac{d f (v_0)}{d v_0} \right]_{v_0 = \omega / k} .$$

Using Eq. (7) and $v_t / v_p = 1$, the above expression is written

$$\frac{d E_k}{d t} = \pi \frac{(q E)^2}{2 m \omega} \frac{1}{\sqrt{2 \pi}} \exp \left( - \frac{1}{2} \right)$$

Multiplying by $T$ and then dividing by $m v_t^2 / 2$, the above expression is written

$$\frac{d \overline{E}_k}{d \overline{t}} = 4 \pi^2 \frac{(k q E)^2}{m^2 \omega^4} \frac{1}{2 \sqrt{2 \pi}} \exp \left( - \frac{1}{2} \right)$$

Using $2 \pi k q E / m \omega ^2 = 1$, the above expression is written as

$$\frac{1}{2 \sqrt{2 \pi}} \exp \left( - \frac{1}{2} \right) = 0.12$$

The result given by the analytic formula is slightly different from that of the simulation (0.12 vs 0.08). Considering the various approximations used in deriving the analytic formula, the two results can be considered to be in agreement with each other.