Vlasov Equation

Source lecture(s): pc368_lec04_vlasov, pc368_lec12_landau_damping

Intuition

In a collisionless plasma, particles stream freely through the electromagnetic field. The Vlasov equation tells us how the 6-dimensional phase-space density \(f(\mathbf{r}, \mathbf{v}, t)\) evolves without assuming thermodynamic equilibrium. It is the kinetic theory backbone of plasma physics.

Formal Definition

The Vlasov equation is the collisionless Boltzmann equation:

\[ \frac{\partial f}{\partial t} + \mathbf{v}\cdot\frac{\partial f}{\partial \mathbf{r}} + \frac{q}{m}\bigl(\mathbf{E} + \mathbf{v}\times\mathbf{B}\bigr)\cdot \frac{\partial f}{\partial \mathbf{v}} = 0 \]

It states that \(df/dt = 0\) along particle trajectories in phase space.

Mathematical Formulation

The fields are self-consistently coupled via Maxwell’s equations:

\[ \nabla\cdot\mathbf{E} = \frac{\rho}{\varepsilon_0}, \qquad \nabla\times\mathbf{B} = \mu_0\mathbf{J} + \mu_0\varepsilon_0\frac{\partial \mathbf{E}}{\partial t} \]

with

\[ \rho = \sum_s q_s \int f_s \, d^3v, \qquad \mathbf{J} = \sum_s q_s \int \mathbf{v}\, f_s \, d^3v. \]

Derivation

Start from Liouville’s theorem: in the absence of collisions, phase-space density is conserved along Hamiltonian trajectories. Write the Hamiltonian for a charged particle in EM fields, \(H = \frac{1}{2}m v^2 + q\phi - q\mathbf{A}\cdot\mathbf{v}\), and apply the chain rule:

\[ \frac{df}{dt} = \frac{\partial f}{\partial t} + \dot{\mathbf{r}}\cdot\frac{\partial f}{\partial \mathbf{r}} + \dot{\mathbf{p}}\cdot\frac{\partial f}{\partial \mathbf{p}} = 0. \]

Using Lorentz force \(\dot{\mathbf{v}} = (q/m)(\mathbf{E} + \mathbf{v}\times\mathbf{B})\) produces the Vlasov equation.

Worked Example

Landau damping seed: Consider a one-dimensional electrostatic perturbation with potential \(\phi(x,t) = \phi_0 e^{i(kx - \omega t)}\). Linearize the Vlasov–Poisson system and find the dispersion relation.

Linearize \(f = f_0(v) + f_1\) and \(\phi\). Fourier transforming yields \((-i\omega + ikv) f_1 + (e/m_e)\phi_0 \delta f_0/\delta v = 0\). Solving for \(f_1\), inserting into Poisson, and closing with \(\varepsilon(\omega,k) = 1 + \chi_e = 0\) gives the dielectric function. The solution is obtained by integrating along the Landau contour.

Common Mistakes

Vlasov is not the same as fluid equations. Fluid equations are velocity moments; Vlasov retains the full velocity dependence.
Collisions are ignored completely. Resistivity, Landau damping, and wave particle interactions arise from microphysics beyond the collisionless hypothesis.
Fields are not given a priori. They must be solved self-consistently.

Quiz Questions

Conceptual: Why does \(df/dt = 0\) hold even though particles feel forces?
Computational: Compute the linear Landau damping rate for a Maxwellian plasma with \(T_e = 1\) keV and \(\lambda_D = 10^{-3}\) m.
MCQ: Which Maxwell equation closes the Vlasov–Maxwell system?
A) \(\nabla\cdot\mathbf{B} = 0\)
B) \(\nabla\cdot\mathbf{E} = \rho/\varepsilon_0\)
C) \(\nabla\times\mathbf{E} = -\partial \mathbf{B}/\partial t\)
D) All of the above