Timescale Hierarchy #
What happens to a perturbation in a plasma comprised of \( N \) particles, each with independent \( \vec x \) and \( \vec v \)? The timescales that define the statistical equilibria define some different regimes.
On the timescale of the plasma frequency (\( 1 / \omega_p \)), Coulomb potentials are shielded within the region of a Debye sphere. This is the fastest timescale that’s relevant to plasma dynamics, because it is the speed at which electrons can move.
The velocity distribution relaxes to a local Maxwellian (local thermal equilibrium) on a collisional timescale \( \nu^{-1} = \frac{1}{\omega_p} \frac{\Lambda}{\ln \Lambda} \gg \frac{1}{\omega_p} \), where \( \Lambda \) is the plasma parameter \( \Lambda = n \lambda_D ^3 \gg 1 \)
Macroscopic force balance is established on a timescale \( \nu ^{-1} \sim L / v_{th} \) where \( L \) is a characteristic length of the plasma and \( v_{th} \) is the electron thermal speed. This determines the thermal motion of the electrons.
Hydrodynamic diffusion happens on a much longer timescale, over which the electrons and ions reach a Maxwellian distribution. If magnetized, this time is like \( \frac{1}{\nu} (L / \rho)^2 \), and if the plasma is unmagnetized the timescale is \( \frac{1}{\nu} (L / \lambda_{eff})^2 \).
Timescales and regimes: #
\( t = 0 \), arbitrary initial distribution in space and velocity, determined by the Klimontovich equation
\( t > \frac{1}{\omega_p} \): Debye shielding established in plasma. Kinetic theory holds with the Vlasov equation
\( t > \frac{1}{\nu} \): Distribution functions approach local Maxwellian. Fokker-Planck equation
\( t \gg \frac{1}{\nu} \): Distribution functions approach global Maxwellian. Fully into fluid theory. This can be very long time for a hot plasma, usually scales with \( T_e ^{3/2} \). But as it turns out, fluid assumptions hold pretty well for a lot of laboratory plasmas for other reasons.
Kinetic Theory #
Kinetic theory is concerned with determining the evolution of a distribution \( f(t, \vec x, \vec v) \) for each species.
If \( f \) is a Maxwellian, the temperature specifies the velocity distribution
\[f(t, \vec x, \vec v) \rightarrow f(t, \vec x)\] \[f_m (v_x) = \frac{1}{\sqrt{2 \pi} \sqrt{T_{e}/m}} e^{- \frac{v_x ^2}{2 (T_{ev}/m)}}\]The temperature \( T_{e} \) determines the spread of the distribution.
Kinetic equations #
The distribution function \( f(\vec x, \vec v, t) \) needs to satisfy the Boltzmann equation for conservation in phase space. This is just a statement of conservation of particles
Boltzmann Equation #
Boltzmann Equation
\[\dv{}{t} f(\vec x, \vec v, t) = \pdv{f}{t} + \vec v \cdot \grad f + \frac{\vec F}{m} \cdot \pdv{f}{\vec v} = \left( \dv{f}{t} \right)_C\]
where \( \vec F / m \) is simply the time change of velocity itself
\[\frac{\vec f}{m} = \pdv{\vec v}{t}\]and the collision operator \( \left( \dv{f}{t} \right)_C \) determines particle-particle interactions. For collisions with neutral atoms, we can use the Krook collision term:
\[\left( \dv{f}{t} \right)_C = \frac{f_n - f}{\tau}\]where \( f_n \) is a distribution for neutral particles and \( f \) is the distribution for the species of interest.
For Coulomb collisions, we use the Fokker-Planck collision operator:
\[\left( \dv{f}{t} \right)_C = - \pdv{}{\vec v} \cdot ( f \langle \Delta \vec v \rangle ) \frac{1}{2} \pdv{^2}{\vec v \partial \vec v} \cdot (f \langle \Delta \vec v \Delta \vec v \rangle)\]Coulomb collisions are generally smoother and longer-range.
BBGKY Hierarchy #
A microscopic description of the plasma is the most exact description. With \( > 10^{20} \) particles in a typical volume, we need to expand the microscopic model up to longer scales. This is basically a question of when we can ignore individual particle-to-particle interactions.
At length scales \( \geq \lambda_D \), individual particle effects are shielded, and the collective effects of particles caught within the Debye sphere dominates the dynamics.
The collision term can be modeled as the ensemble average of interactions between may particles. We define the plasma parameter \( \Lambda \) as
\[\Lambda ^{-1} = \frac{1}{n \lambda_D ^3} \ll 1\]We can classify equations for 2-body interactions (\( g_{\alpha \beta} \)), 3-body interactions (\( h_{\alpha \beta \gamma} \)), etc. and order them as
\[\begin{aligned} \uparrow & \qquad f_\alpha \sim O(1) \\ \uparrow & \qquad g_{\alpha \beta} \sim O(\Lambda ^{-1}) \\ \uparrow & \qquad h_{\alpha \beta \gamma} \sim P(\Lambda ^{-2}) \end{aligned}\] \[\dv{f_\alpha}{t} = \sum_\beta \frac{q_\alpha q_\beta}{m_\alpha} \int \dd x' \dv{}{\vec x} \frac{1}{|\vec x - \vec x'|} \cdot \pdv{g_{\alpha \beta}(t, \vec x, \vec x')}{\vec v}\]with \( g_{\alpha \beta} \) having its own evolution equation with \( h_{\alpha \beta \gamma} \) terms. We close the system with \( h_{\alpha \beta \gamma} \rightarrow 0 \).
Vlasov Equations #
If we drop the \( g_{\alpha \beta} \) terms, we ignore binary interactions. We substitute electromagnetic forces for \( \vec F \) to get:
\[\dv{f}{t} = \pdv{f}{t} + \vec v \cdot \grad f + \frac{q}{m} (\vec E + \vec v \cross \vec B) \cdot \pdv{f}{\vec v} = 0\]This is the collision-neglected Boltzmann equation. The rationale for ignoring 2-particle interactions is
\[g_{\alpha \beta} \sim \frac{f_\alpha}{\Lambda}\] and as \( \Lambda \rightarrow \infty \), \( g_{\alpha \beta} \rightarrow 0 \)
This assumption is valid either when density is very high, or \( \lambda_D = \sqrt{\epsilon_0 T_e}{e^2 n_e} \) has to very large, which happens at very high temperatures. Physically, if kinetic energy of interactions dominates the potential energy change, then the plasma behaves like a perfect gas.
The full set of “Vlasov equations” includes the Maxwell equations to complete the picture and determine the interaction between the electromagnetic fields and their sources
\[\rho = \sum_j q_j \int \dd ^3 v f_j\] \[\vec j = \sum_j q_j \int \dd ^3 v \vec v f_j\]
What does setting the correlation term \( \left( \dv{f}{t} \right)_C \) to zero? The left-hand-side of the Boltzmann equation is just the total derivative of the distribution function in phase space, so we have just assumed that \( f \) is constant in phase space.