Normalization scheme

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Normalization scheme

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

[画像:\begin{displaymath} \omega_p^2 = \frac{n_0,円e^2}{\epsilon_0,円m_e} \end{displaymath}] (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:

[画像:\begin{displaymath} \lambda_D = \frac{v_{th}}{\omega_p}, \end{displaymath}]

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:

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

$\displaystyle \frac{dv_i}{dt}$ $\textstyle =$ [画像:$\displaystyle - E(x_i),$] (296)

[画像:$\displaystyle E(x)$] $\textstyle =$ [画像:$\displaystyle - \frac{d\phi(x)}{dx},$] (297)

[画像:$\displaystyle \frac{d^2\phi(x)}{dx^2}$] $\textstyle =$ [画像:$\displaystyle \frac{n(x)}{n_0}-1.$] (298)

whereas our initial distribution function becomes

[画像:\begin{displaymath} f(x,v) = \frac{n_0}{2}\left\{\frac{1}{\sqrt{2,円\pi}} ,円{\rm ... .../2}+ \frac{1}{\sqrt{2,円\pi}},円{\rm e}^{-(v+v_b)^2/2} \right\}. \end{displaymath}] (299)

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

Let us solve the above system of equations in the domain 0ドル\leq x\leq L$. Furthermore, for the sake of simplicity, let us adopt periodic boundary conditions: i.e., let us identify the left and right boundaries of our solution domain. It follows that [画像:$n(0)=n(L)$], [画像:$\phi(0)=\phi(L)$], and [画像:$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.

next up previous
Next: Solution of electron equations Up: Particle-in-cell codes Previous: Introduction

Richard Fitzpatrick 2006年03月29日