Landau damping is a very important phenomenon in plasma discovered by prof. Landau theoretically. It can only be described by kinetic theory. It is a very important interaction mechanism between waves and energetic particles in future collisionless fusion plasma.

kinetic equation

The Vlosov equation for electrons in the one dimensional (x axis) plasma is like this:

$\frac{\partial f}{\partial t} + v\frac{\partial f}{\partial x} - \frac{e}{m_e}\frac{\partial f}{\partial v} = 0$

Take a time and space Fourier transformation of the above equation as: $\frac{\partial}{\partial t} \sim -i\omega$ and $\nabla \sim \frac{\partial}{\partial x} = ik$. Also we perform a linearization as: $f(t, x, v) = f_0(v) + f_1(x, t)$. And suppose the distribution function has the format as:

$f = n_{e0}(\frac{m_e}{2\pi k_B T_e})^{1/2}e^{-\frac{m_e v^2}{2k_B T_e}} + f_{e1}e^{i(kx - \omega t)}$

Keeping only the first order term, the linearized equation becomes:

$-i\omega f_1 + v i k f_1 - \frac{e E}{m}\frac{\partial f_0}{\partial v} = 0$

Then we can get the perturbed electron distribution function.

$f_{e1} = \frac{i e E}{m_e}\frac{\partial f_0/\partial v}{(\omega - k v)}$

Notice that term $\omega - kv = 0$ will cause a singular for this function. The value of the perturbation function $f_1$ will reach to infinit once the particles have the speed close to $v = \frac{\omega}{k}$. Particles close to this velocity are also called the resonant particles.
During this derivation we have: $\frac{\partial f_0}{\partial t} = 0, \frac{\partial f_0}{\partial x} = 0, \frac{\partial f_1}{\partial v} = 0$, but $\frac{\partial f_0}{\partial v} \neq 0$.

For the perturbation part of Possion’s equation (also the Gauss’s law in Maxwell’s equation). $\nabla\cdot\vec{E_1} = \frac{1}{\epsilon_0}\rho_{e1}$. The electric charge is:

$\rho_{e1} = \int (-e) f_{e1}dv = \int (-e)\frac{i e E}{m_e}\frac{\partial f_0/\partial v}{(\omega - k v)}dv$

Then the whole equation becomes:
$ik E_1 = -\frac{e}{\epsilon_0}\int \frac{i e E_1}{m_e}\frac{\partial f_0/\partial v}{(\omega - k v)}dv$. Then we can get the function as:

$kE_1 = -\frac{\omega_{pe}^2}{k n_e}E_1\int\frac{\partial f_{e0}/\partial v}{\omega/k - v}dv$

From which we an get the:

$(1+\frac{\omega_{pe}^2}{n_e k^2}\int\frac{1}{n_e}\frac{\partial f_{e0}/ \partial v}{\omega/k - v}dv)E_1 = 0$

If we wish the electric perturbation $E_1$ to have meaningful solution, this equation requires that:

$1+\frac{\omega_{pe}^2}{n_e k^2}\int\frac{1}{n_e}\frac{\partial f_{e0}/ \partial v}{\omega/k - v}dv =\epsilon(\omega, k) = 0$

This should be the kinetic dispersion relation of Langmuire wave. This term $\epsilon(\omega, k)$ is also called the general dielectric function. And the above equation can be simplified as: $\epsilon E_1 = 0$

Cold, warm and hot plasmas

To get the detailed dispersion relation of waves in this expression, we should conduct some simplification to it. First we have the exact expression of:
$\frac{\partial f_0}{\partial v} = $

Landau damping of Langmuir wave