These notes explain our most-used normalisation for the Vlasov-Poisson system.
\[\begin{aligned}
\omega_{p}^{2} = \dfrac{n_{0} e^{2}}{\eps_{0} m_{e}}
\end{aligned}\]
\[\begin{aligned}
\lambda = \dfrac{v_{th}}{\omega_{p}}
\end{aligned}\]
\[\begin{aligned}
v_{th} = \sqrt{\dfrac{k_{B} T}{m_{e}}} = \lambda \omega_{p}
\end{aligned}\]
\[\begin{aligned}
\tilde{t} \equiv \omega_{p} t
\end{aligned}\]
\[\begin{aligned}
\tilde{x} \equiv \dfrac{x}{\lambda}
\end{aligned}\]
\[\begin{aligned}
\tilde{v} \equiv \dfrac{v}{v_{th}}
\end{aligned}\]
\[\begin{aligned}
\tilde{\partial}_{t} &\equiv \dfrac{\partial_{t}}{\omega_{p}} &
d\tilde{t} &\equiv \omega_{p} dt
\end{aligned}\]
\[\begin{aligned}
\tilde{\partial}_{x} &\equiv \lambda \partial_{x} &
d\tilde{x} &\equiv \dfrac{dx}{\lambda}
\end{aligned}\]
\[\begin{aligned}
\tilde{\partial}_{v} &\equiv v_{th} \partial_{v} &
d\tilde{v} &\equiv \dfrac{dv}{v_{th}}
\end{aligned}\]
\[\begin{aligned}
\left[ \phy \right] = V = \dfrac{\mrm{kg} \, \mrm{m^{2}}}{\mrm{A} \, \mrm{s^{3}}}
\hspace{3em}
\rightarrow
\hspace{3em}
\left[ \dfrac{e}{m_{e}} \, \phy \right] = \dfrac{\mrm{m}^{2}}{\mrm{s^{2}}}
\end{aligned}\]
\[\begin{aligned}
\tilde{\phy} \equiv \dfrac{e}{m_{e}} \dfrac{\phy}{v_{th}^{2}} = \dfrac{e}{m_{e}} \dfrac{\phy}{\lambda^{2} \omega_{p}^{2}}
\end{aligned}\]
\[\begin{aligned}
\phy = \dfrac{m_{e}}{e} \, v_{th}^{2} \, \tilde{\phy} = \dfrac{m_{e}}{e} \, \lambda^{2} \omega_{p}^{2} \, \tilde{\phy}
\end{aligned}\]
\[\begin{aligned}
H_{s} = \dfrac{m_{s}}{2} v^{2} + q_{s} \phy
\end{aligned}\]
\[\begin{aligned}
H_{s} = \dfrac{m_{s}}{2} \, v_{th}^{2} \, \tilde{v}^{2} + \mrm{sign} (q_{s}) \, e \dfrac{m_{e}}{e} v_{th}^{2} \tilde{\phy}
= m_{s} v_{th}^{2} \left[ \dfrac{1}{2} \, \tilde{v}^{2} + \mrm{sign} (q_{s}) \dfrac{m_{e}}{m_{s}} \, \tilde{\phy} \right]
\end{aligned}\]
\[\begin{aligned}
\tilde{H}_{s}
= \dfrac{1}{m_{s} v_{th}^{2}} \, H_{s}
= \dfrac{1}{2} \, \tilde{v}^{2} + \mrm{sign} (q_{s}) \dfrac{m_{e}}{m_{s}} \, \tilde{\phy}
\end{aligned}\]
\[\begin{aligned}
\partial_{t} f_{s} + \{ f_{s}, H_{s} \}
= \partial_{t} f_{s} + \partial_{x} f_{s} \, \partial_{v} H_{s} - \partial_{v} f_{s} \, \partial_{x} H_{s}
= 0
\end{aligned}\]
\[\begin{aligned}
0
&= \omega_{p} \tilde{\partial}_{t} f_{s} + \dfrac{1}{\lambda v_{th}} \left[ \tilde{\partial}_{x} f_{s} \, \tilde{\partial}_{v} H_{s} - \tilde{\partial}_{v} f_{s} \, \tilde{\partial}_{x} H_{s} \right] \\
&= \omega_{p} \tilde{\partial}_{t} f_{s} + \dfrac{v_{th}}{\lambda v_{th}^{2}} \, m_{s} v_{th}^{2} \left[ \tilde{\partial}_{x} f_{s} \, \tilde{\partial}_{v} \tilde{H}_{s} - \tilde{\partial}_{v} f_{s} \, \tilde{\partial}_{x} \tilde{H}_{s} \right] \\
&= \omega_{p} \tilde{\partial}_{t} f_{s} + \omega_{p} \dfrac{m_{s}}{m_{e}} \left[ \tilde{\partial}_{x} f_{s} \, \tilde{\partial}_{v} \tilde{H}_{s} - \tilde{\partial}_{v} f_{s} \, \tilde{\partial}_{x} \tilde{H}_{s} \right]
\end{aligned}\]
\[\begin{aligned}
\tilde{\partial}_{t} f_{s} + \dfrac{m_{s}}{m_{e}} \left[ \tilde{\partial}_{x} f_{s} \, \tilde{\partial}_{v} \tilde{H}_{s} - \tilde{\partial}_{v} f_{s} \, \tilde{\partial}_{x} \tilde{H}_{s} \right] = 0
\end{aligned}\]
\[\begin{aligned}
- \eps_{0} \, \Delta \phy = \sum \limits_{s} q_{s} n_{s}
\end{aligned}\]
\[\begin{aligned}
n_{s} &= \int f_{s} \, dv , &
n_{0} &= \dfrac{1}{L_{x}} \int f_{e} \, dx \, dv = \bracket{n_{e}}
\end{aligned}\]
\[\begin{aligned}
- \eps_{0} \, \Delta \phy = q_{i} n_{i} + q_{e} n_{e} = q_{i} \bracket{n_{e}} + q_{e} \int f \, dv
\end{aligned}\]
\[\begin{aligned}
- \eps_{0} \, \dfrac{1}{\lambda^{2}} \tilde{\partial}_{x}^{2} \phy
&= q_{e} \, \big( n_{e} - \bracket{n_{e}} \big)
%&= - \dfrac{q_{e} v_{th} \lambda}{\tilde{L}_{x} \lambda} \int f \, d\tilde{x} \, d\tilde{v} + q_{e} v_{th} \int f \, d\tilde{v}
\\
\eps_{0} \, \dfrac{\lambda^{2}}{\lambda^{2}} \dfrac{m_{e}}{e} \omega_{p}^{2} \, \tilde{\partial}_{x}^{2} \tilde{\phy} &= e n_{0} - e n_{e}
\\
\eps_{0} \, \dfrac{m_{e}}{e^{2}} \dfrac{n_{0} e^{2}}{\eps_{0} m_{e}} \, \tilde{\partial}_{x}^{2} \tilde{\phy} &= n_{0} - n_{e}
\\
\tilde{\partial}_{x}^{2} \tilde{\phy} &= 1 - \dfrac{n_{e}}{n_{0}}
\end{aligned}\]
\[\begin{aligned}
\tilde{\partial}_{t} \tilde{f} + \left[ \tilde{\partial}_{x} \tilde{f} \, \tilde{\partial}_{v} \tilde{H} - \tilde{\partial}_{v} \tilde{f} \, \tilde{\partial}_{x} \tilde{H} \right] &= 0, &
\tilde{\partial}_{x}^{2} \tilde{\phy} &= 1 - \tilde{n}_{e} , &
\tilde{H} &= \dfrac{1}{2} \, \tilde{v}^{2} + \tilde{\phy}
\end{aligned}\]