Particle simulation of wave particle interactions

Consider a longitudinal wave given by

$$\mathbf{E}= \hat{\mathbf{z}} E \cos (k z - \omega t) .$$ (1)

The equations of motion of a test particle in the wave field are given by

$$m \frac{d v}{d t} = q E \cos (k z - \omega t),$$ (2)

and

$$\frac{d z}{d t} = v,$$ (3)

where $v \equiv \mathbf{v} \cdot \hat{\mathbf{z}}$ is the $z$ component of the velocity of the particle.

Normalize $z$ by the wavelength $\lambda$, $t$ by the wave period $T$, $v$ by the phase velocity $v_p$, i.e.,

$$\overline{z} = \frac{z}{\lambda}, \overline{t} = \frac{t}{T}, \overline{v} = \frac{v}{v_p},$$ (4)

where $\lambda = 2 \pi / k$, $T = 2 \pi / \omega$, $v_p = \omega / k$. Using the normalized quantities, Eqs. (2) and (3) are written, respectively, as

$$\frac{d \overline{v}}{d \overline{t}} = \frac{2 \pi k q E}{m \omega^2} \cos [2 \pi ( \overline{z} - \overline{t})],$$ (5)

and

$$\frac{d \overline{z}}{d \overline{t}} = \overline{v} .$$ (6)

The initial distribution function of particles $f_0 (z, v)$ is taken to be uniform in space and Maxwellian in velocity,

$$f_0 (z, v) = f_m (v) = \frac{1}{v_t \sqrt{2 \pi}} \exp \left( - \frac{v^2}{2 v_t^2} \right),$$ (7)

which satisfies the normalization condition $\int_{- \infty}^{\infty} f_0 (v) d v = 1$. In my particle simulation code (/home/yj/project_new/pic_code), $4 \times 10^5$ particles are initially loaded random in $z$ and Maxwellian in $v$. Then the motion equations of every particle are followed numerically to obtain the location and velocity at later time. In the numerical code, when a particle leaves from the region $0 \leqslant \overline{x} \leqslant 1$, it is shifted by one wavelength to return to this region. This shift does not influence the force on the particle and it simulates the situation of infinite length in $z$ direction, where when a particle leave the region $0 \leqslant \overline{x} \leqslant 1$ from the right boundary, a particle of the same velocity will enter the region from the left boundary, and vice versa.

The velocity distribution at later time is obtained by counting the number of particles in each velocity interval. Figure 1a compares the velocity distribution function at $\overline {t} = 0$ and $\overline {t} = 10$, which shows that the distribution is flatted in the resonant region $v / v_p = 1$, which suggests that the total kinetic energy of particles may be increased. Figure 2 plots the temporal evolution of the total kinetic energy of the particles, which confirms that the kinetic energy is increased by the wave. The conservation of energy tell us that the increased kinetic energy of particles must be drawn from the wave, i.e., the wave encounters damping.

Figure 1: Comparison of the velocity distribution function (spatially averaged) at various time, which shows that the distribution is distorted in the resonant region ( $v / v_p \approx 1$). Other parameters: $v_t / v_p = 1$, $2 \pi k q E / m \omega ^2 = 1$.

Figure: Temporal evolution of the total kinetic energy of the particles, where $\overline{E}_k = \sum_{i = 1}^{i = N} \frac{1}{2} m v^2_i / \left( N \frac{1}{2} m v_t^2 \right)$. Other parameters: $v_t / v_p = 1$, $2 \pi k q E / m \omega ^2 = 1$.

Although the above simulation is performed by holding the wave amplitude constant, it takes into account all the nonlinear physics of the particle motion in the wave field. Therefore this is a nonlinear simulation.