Normalization scheme

It is convenient to normalize time with respect to $\omega_p^{-1}$, where

$$\omega_p^2 = \frac{n_0\,e^2}{\epsilon_0\,m_e}$$ (294)

is the so-called plasma frequency: i.e., the typical frequency of electrostatic electron oscillations. Likewise it is convenient to normalize length with respect to the so-called Debye length:

$$\lambda_D = \frac{v_{th}}{\omega_p},$$

which is the length-scale above which the electrons exhibit collective (i.e., plasma-like) effects, instead of acting like individual particles.

Our normalized equations take the form:

$$\frac{dx_i}{dt} = v_i,$$ (295)

$$\frac{dv_i}{dt} = - E(x_i),$$ (296)

$$E(x) = - \frac{d\phi(x)}{dx},$$ (297)

$$\frac{d^2\phi(x)}{dx^2} = \frac{n(x)}{n_0}-1.$$ (298)

whereas our initial distribution function becomes

$$f(x,v) = \frac{n_0}{2}\left\{\frac{1}{\sqrt{2\,\pi}} \,{\rm ... .../2}+ \frac{1}{\sqrt{2\,\pi}}\,{\rm e}^{-(v+v_b)^2/2} \right\}.$$ (299)

Note that $v_{th}=1$ in normalized units.

Let us solve the above system of equations in the domain $0\leq x\leq L$. Furthermore, for the sake of simplicity, let us adopt periodic boundary conditions: i.e., let us identify the left and right boundaries of our solution domain. It follows that $n(0)=n(L)$, $\phi(0)=\phi(L)$, and $E(0) = E(L)$. Moreover, any electron which crosses the right boundary of the solution domain must reappear at the left boundary with the same velocity, and vice versa.