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Vlasov Equation

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

Source lecture(s): pc368_lec04_vlasov

Physical Meaning

The Vlasov equation describes the evolution of the phase-space distribution function \(f(\mathbf{r}, \mathbf{v}, t)\) in a collisionless plasma. It states that the density of representative points in phase space is conserved along Hamiltonian trajectories.

Variable Definitions

Symbol Definition SI Units
\(f\) Phase-space density s\(^3\) m\(^{-6}\)
\(\mathbf{r}, \mathbf{v}\) Position and velocity vectors m, m s\(^{-1}\)
\(q, m\) Particle charge and mass C, kg
\(\mathbf{E}, \mathbf{B}\) Self-consistent EM fields V m\(^{-1}\), T
\(t\) Time s

Assumptions

  • Collisionless: No binary Coulomb collisions.
  • Kinetic: Full 6-D phase space; no fluid-closure assumption.
  • Self-consistent: Fields obey Maxwell’s equations with \(\rho = \int q f d^3v\), \(\mathbf{J} = \int q\mathbf{v} f d^3v\).

Derivation

Start from Liouville’s theorem: in Hamiltonian dynamics with Hamiltonian \(H\), phase-space density is conserved:

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

The equations of motion are:

\[\dot{\mathbf{r}} = \frac{\partial H}{\partial \mathbf{p}} = \frac{\mathbf{p}}{m}, \qquad \dot{\mathbf{p}} = -\frac{\partial H}{\partial \mathbf{r}} = q\bigl(\mathbf{E} + \dot{\mathbf{r}}\times\mathbf{B}\bigr)\]

Since \(\dot{\mathbf{r}} = \mathbf{v}\), substituting yields:

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

Applications

  • Linear wave theory: Langmuir, ion acoustic, and Alfvén waves.
  • Collisionless damping: Landau damping, two-stream instability.
  • Turbulence: Gyrokinetic and kinetic simulations.

Connections to Other Equations