Normalisation for the Vlasov-Poisson System

These notes explain our most-used normalisation for the Vlasov-Poisson system.

Reference Quantities

• plasma frequency

\[\begin{aligned} \omega_{p}^{2} = \dfrac{n_{0} e^{2}}{\eps_{0} m_{e}} \end{aligned}\]

• Debye length

\[\begin{aligned} \lambda = \dfrac{v_{th}}{\omega_{p}} \end{aligned}\]

• thermal velocity

\[\begin{aligned} v_{th} = \sqrt{\dfrac{k_{B} T}{m_{e}}} = \lambda \omega_{p} \end{aligned}\]

Coordinates

• time

\[\begin{aligned} \tilde{t} \equiv \omega_{p} t \end{aligned}\]

• space

\[\begin{aligned} \tilde{x} \equiv \dfrac{x}{\lambda} \end{aligned}\]

• velocity

\[\begin{aligned} \tilde{v} \equiv \dfrac{v}{v_{th}} \end{aligned}\]

Derivatives and Differentials

• time

\[\begin{aligned} \tilde{\partial}_{t} &\equiv \dfrac{\partial_{t}}{\omega_{p}} & d\tilde{t} &\equiv \omega_{p} dt \end{aligned}\]

• space

\[\begin{aligned} \tilde{\partial}_{x} &\equiv \lambda \partial_{x} & d\tilde{x} &\equiv \dfrac{dx}{\lambda} \end{aligned}\]

• velocity

\[\begin{aligned} \tilde{\partial}_{v} &\equiv v_{th} \partial_{v} & d\tilde{v} &\equiv \dfrac{dv}{v_{th}} \end{aligned}\]

Electrostatic Potential

• units

\[\begin{aligned} \left[ \phy \right] = V = \dfrac{\mrm{kg} \, \mrm{m^{2}}}{\mrm{A} \, \mrm{s^{3}}} \hspace{3em} \rightarrow \hspace{3em} \left[ \dfrac{e}{m_{e}} \, \phy \right] = \dfrac{\mrm{m}^{2}}{\mrm{s^{2}}} \end{aligned}\]

• normalisation

\[\begin{aligned} \tilde{\phy} \equiv \dfrac{e}{m_{e}} \dfrac{\phy}{v_{th}^{2}} = \dfrac{e}{m_{e}} \dfrac{\phy}{\lambda^{2} \omega_{p}^{2}} \end{aligned}\]

• inverse

\[\begin{aligned} \phy = \dfrac{m_{e}}{e} \, v_{th}^{2} \, \tilde{\phy} = \dfrac{m_{e}}{e} \, \lambda^{2} \omega_{p}^{2} \, \tilde{\phy} \end{aligned}\]

Hamiltonian

• charged particle in electrostatic potential

\[\begin{aligned} H_{s} = \dfrac{m_{s}}{2} v^{2} + q_{s} \phy \end{aligned}\]

• replace quantities with normalised versions

\[\begin{aligned} H_{s} = \dfrac{m_{s}}{2} \, v_{th}^{2} \, \tilde{v}^{2} + \mrm{sign} (q_{s}) \, e \dfrac{m_{e}}{e} v_{th}^{2} \tilde{\phy} = m_{s} v_{th}^{2} \left[ \dfrac{1}{2} \, \tilde{v}^{2} + \mrm{sign} (q_{s}) \dfrac{m_{e}}{m_{s}} \, \tilde{\phy} \right] \end{aligned}\]

• normalisation

\[\begin{aligned} \tilde{H}_{s} = \dfrac{1}{m_{s} v_{th}^{2}} \, H_{s} = \dfrac{1}{2} \, \tilde{v}^{2} + \mrm{sign} (q_{s}) \dfrac{m_{e}}{m_{s}} \, \tilde{\phy} \end{aligned}\]

Vlasov Equation

• Vlasov equation in Poisson bracket notation

\[\begin{aligned} \partial_{t} f_{s} + \{ f_{s}, H_{s} \} = \partial_{t} f_{s} + \partial_{x} f_{s} \, \partial_{v} H_{s} - \partial_{v} f_{s} \, \partial_{x} H_{s} = 0 \end{aligned}\]

• replace derivatives and quantities with normalised versions

\[\begin{aligned} 0 &= \omega_{p} \tilde{\partial}_{t} f_{s} + \dfrac{1}{\lambda v_{th}} \left[ \tilde{\partial}_{x} f_{s} \, \tilde{\partial}_{v} H_{s} - \tilde{\partial}_{v} f_{s} \, \tilde{\partial}_{x} H_{s} \right] \\ &= \omega_{p} \tilde{\partial}_{t} f_{s} + \dfrac{v_{th}}{\lambda v_{th}^{2}} \, m_{s} v_{th}^{2} \left[ \tilde{\partial}_{x} f_{s} \, \tilde{\partial}_{v} \tilde{H}_{s} - \tilde{\partial}_{v} f_{s} \, \tilde{\partial}_{x} \tilde{H}_{s} \right] \\ &= \omega_{p} \tilde{\partial}_{t} f_{s} + \omega_{p} \dfrac{m_{s}}{m_{e}} \left[ \tilde{\partial}_{x} f_{s} \, \tilde{\partial}_{v} \tilde{H}_{s} - \tilde{\partial}_{v} f_{s} \, \tilde{\partial}_{x} \tilde{H}_{s} \right] \end{aligned}\]

• normalised equation:

\[\begin{aligned} \tilde{\partial}_{t} f_{s} + \dfrac{m_{s}}{m_{e}} \left[ \tilde{\partial}_{x} f_{s} \, \tilde{\partial}_{v} \tilde{H}_{s} - \tilde{\partial}_{v} f_{s} \, \tilde{\partial}_{x} \tilde{H}_{s} \right] = 0 \end{aligned}\]

Poisson Equation

• Poisson equation

\[\begin{aligned} - \eps_{0} \, \Delta \phy = \sum \limits_{s} q_{s} n_{s} \end{aligned}\]

• densities and average density

\[\begin{aligned} n_{s} &= \int f_{s} \, dv , & n_{0} &= \dfrac{1}{L_{x}} \int f_{e} \, dx \, dv = \bracket{n_{e}} \end{aligned}\]

• Poisson equation for one species (electrons with neutralising ion brackground \(n_{i} = \bracket{n_{e}}\))

\[\begin{aligned} - \eps_{0} \, \Delta \phy = q_{i} n_{i} + q_{e} n_{e} = q_{i} \bracket{n_{e}} + q_{e} \int f \, dv \end{aligned}\]

• replace derivatives and quantities with normalised versions (\(q_{i} = - q_{e} = - e\))

\[\begin{aligned} - \eps_{0} \, \dfrac{1}{\lambda^{2}} \tilde{\partial}_{x}^{2} \phy &= q_{e} \, \big( n_{e} - \bracket{n_{e}} \big) %&= - \dfrac{q_{e} v_{th} \lambda}{\tilde{L}_{x} \lambda} \int f \, d\tilde{x} \, d\tilde{v} + q_{e} v_{th} \int f \, d\tilde{v} \\ \eps_{0} \, \dfrac{\lambda^{2}}{\lambda^{2}} \dfrac{m_{e}}{e} \omega_{p}^{2} \, \tilde{\partial}_{x}^{2} \tilde{\phy} &= e n_{0} - e n_{e} \\ \eps_{0} \, \dfrac{m_{e}}{e^{2}} \dfrac{n_{0} e^{2}}{\eps_{0} m_{e}} \, \tilde{\partial}_{x}^{2} \tilde{\phy} &= n_{0} - n_{e} \\ \tilde{\partial}_{x}^{2} \tilde{\phy} &= 1 - \dfrac{n_{e}}{n_{0}} \end{aligned}\]

Distribution Function

• distribution function \(\tilde{f}\) is normalised such that \(\bracket{\tilde{n}_{e}} = \tilde{n}_{0} = 1\)

• Vlasov-Poisson system for electrons and fixed ions

\[\begin{aligned} \tilde{\partial}_{t} \tilde{f} + \left[ \tilde{\partial}_{x} \tilde{f} \, \tilde{\partial}_{v} \tilde{H} - \tilde{\partial}_{v} \tilde{f} \, \tilde{\partial}_{x} \tilde{H} \right] &= 0, & \tilde{\partial}_{x}^{2} \tilde{\phy} &= 1 - \tilde{n}_{e} , & \tilde{H} &= \dfrac{1}{2} \, \tilde{v}^{2} + \tilde{\phy} \end{aligned}\]