Development of a Collisionless Plasma Kinetic Solver and an Investigation of One-Dimensional Plasma Waves and Instabilities #
Shielded potential of a test electron:
\[\phi(r) = \frac{-e}{4 \pi \epsilon_0 r} e ^{- r / \lambda_D}\]where the Debye length is \( \lambda_D = \sqrt{\frac{\epsilon_0 T_e}{ n_e e}} \) . The mean free path between large-angle collisions is estimated as
\[\lambda_{mfp} \sim \frac{\epsilon_0 T_e ^2}{\phi_e n_e \log ( \Lambda)}\] where \(\phi_e = e^2 / 4 \pi \epsilon_0\) are the constants from the Coulomb force law.
Smooth out the discreteness of particles via spatial average over small volumes:
\[\rho \rightarrow \langle \rho_c \rangle \qquad \vec E \rightarrow \langle \vec E \rangle + \delta \vec E\]The mean field \(\langle \vec E \rangle\) is responsible for collective modes of plasma motion. Estimate the collisionality of the plasma by comparing the length scales \(\lambda_{mfp} / \lambda_D\)
\[\frac{\lambda_{mfp}}{\lambda_D} \sim \frac{T_e ^{3/2}}{n_e ^{1/2}}\]Plasma is seen to become collisionless as the temperature becomes high or the plasma becomes more rarified.
Phase space mechanics #
To arrive at a kinetic equation governing the collisionless mechanics, consider the one-dimensional motion of a single particle
\[\dot{x} = v \qquad \dot v = F(x)\]and define the phase space coordinates as \(\vec{\dot r} = \vec F \equiv [ v, F(x) ]\) . The flux vector is similar to the velocity field of a fluid flow. The streamlines of \(\vec F\) are the streamlines which a particle will follow if the flux is constant in time. Phase flow is always analogous to that of an incompressible fluid because the flux divergence is zero:
\[\div [v, F(x)] = \pdv{v}{x} + \pdv{F(x)}{v}\]If the phase fluid density is given by a function \(f(x, v, t)\) where \(t\) is the time parameter, because any instantiation of a particle can not leave the phase plane, the probability density will be conserved. We can write a conservation law:
\[\pdv{f(x, v, t)}{t} = - \div ( f (x, v, t) \vec F)\]and due to the flow’s incompressibility
\[\div (f \vec F) = f ( \div \vec F) + \vec F \cdot \grad f = \vec F \cdot \grad f \\ \rightarrow \pdv{f}{t} = - \left[ v, F(x) \right] \cdot \left[ \pdv{f}{x}, \pdv{f}{v} \right] \\ \rightarrow \pdv{f}{t} + v \pdv{f}{x} + F(x) \pdv{f}{v} = 0\]the probability density \( f \) satisfies a simple advection equation in the phase space.
Vlasov Equation #
Beginning from the probability density kinetic equation, one can arrive at the Boltzmann equation for the ensemble-averaged velocity distribution function \( f(\vec x, \vec v, t) \) of a gas or plasma
\[\pdv{f}{t} + \vec v \cdot \grad_x f + \frac{1}{m} \vec F \cdot \grad_v f = C(f)\]where \( C(f) \) is in general an integral operator representing inter-particle correlations. We arrive at the Vlasov-Poisson system of equations if we entirely neglect correlations and only consider the electrostatic potential. Restricted to a single spatial dimension, it has the form
\[\pdv{f _ \alpha}{t} + v \pdv{f _ \alpha}{x} - \frac{Z _ \alpha}{m _ \alpha} \pdv{\phi}{x} \pdv{f _ \alpha}{v} = 0 \\ \dv{ ^2 \phi}{x^2} = - \frac{1}{\epsilon_0} \sum_\alpha Z_\alpha \int_{-\infty} ^\infty f _ \alpha(x, v, t) \dd v\]The most striking difference between the dynamics of neutral fluids and a collisionless plasma is the appearance of non-equilibrium velocity distributions in plasma as a result of collective behavior. The collision operator enforces local thermodynamic equilibrium, driving the velocity distribution to Maxwellian.
\[f(x, v) = n(x) \sqrt{ \frac{1}{2 \pi v_{th} ^2 (x) }} \exp \left( - \frac{(v - u(x))^2}{2 v_{th}^2 (x)} \right)\]where \[n(x) = \int_{-\infty} ^\infty f(x, v, t) \\ u(x) = \frac{1}{n(x)} \int_{-\infty} ^\infty v f(x, v, t) \dd v \\ v_{th} ^2 (x) = \frac{1}{n(x)} \int_{-\infty} ^\infty (v - u(x))^2 f(x, v, t) \dd v\]
In absence of collisions, there is no driver for the velocity distribution to posess normal statistics. The structure of the distribution function within the phase space is generated through wave-particle resonance, where the waves with phase velocity \( v_{ph} = \omega / k \) resonate with particles traveling at the same velocity. This process is Landau resonance.