Plasma Physics Course

Total: 8 parts covering plasma fundamentals through advanced topics

Part 2, Chapter 1

Vlasov Equation

Collisionless Boltzmann equation for plasma kinetic theory

1.1 Introduction

The Vlasov equation is the fundamental equation of collisionless plasma kinetic theory. In hot, tenuous plasmas where the collision frequency is much smaller than the plasma frequency (Ξ½coll β‰ͺ Ο‰p), binary Coulomb collisions can be neglected on short time scales, and the plasma evolution is governed by the collective electromagnetic fields generated by all particles.

Unlike fluid descriptions (MHD, two-fluid theory), the Vlasov equation retains the full velocity-space information through the distribution function f(r, v, t). This allows us to describe phenomena like:

  • Landau damping: Collisionless wave damping through wave-particle resonance
  • Two-stream instability: Instability from counter-streaming particle populations
  • Particle trapping: Nonlinear trapping in wave potential wells
  • Cyclotron resonances: Interactions at Ο‰ = nΞ©c

πŸ’‘ The Vlasov equation represents the "self-consistent field approximation" where each particle moves in the mean field generated by all other particles, neglecting discrete particle-particle collisions.

1.2 Derivation from Liouville's Theorem

Phase Space Density

We define the distribution function fs(r, v, t) for species s such that:

$$f_s(\mathbf{r}, \mathbf{v}, t) \, d^3r \, d^3v = \text{number of particles of species } s$$
$$\text{in volume } d^3r \text{ about } \mathbf{r} \text{ with velocity in } d^3v \text{ about } \mathbf{v}$$

The phase space density evolves according to the trajectories of individual particles. In the absence of collisions, particles follow deterministic paths in the 6D phase space (r, v):

$$\frac{d\mathbf{r}}{dt} = \mathbf{v}$$
$$\frac{d\mathbf{v}}{dt} = \frac{q_s}{m_s}\left(\mathbf{E} + \mathbf{v} \times \mathbf{B}\right)$$

Liouville's Theorem

Liouville's theorem states that the phase space density is constant along particle trajectories (incompressibility of phase space flow in Hamiltonian systems). The total time derivative is:

$$\frac{df_s}{dt} = \frac{\partial f_s}{\partial t} + \frac{d\mathbf{r}}{dt} \cdot \nabla_{\mathbf{r}} f_s + \frac{d\mathbf{v}}{dt} \cdot \nabla_{\mathbf{v}} f_s = 0$$

Substituting the equations of motion:

$$\frac{\partial f_s}{\partial t} + \mathbf{v} \cdot \nabla_{\mathbf{r}} f_s + \frac{q_s}{m_s}(\mathbf{E} + \mathbf{v} \times \mathbf{B}) \cdot \nabla_{\mathbf{v}} f_s = 0$$

This is the Vlasov equation!

Note that the Lorentz force is perpendicular to v, so (v Γ— B) Β· βˆ‡vf causes rotation in velocity space but doesn't change the magnitude of velocities. Only the electric fieldE can accelerate or decelerate particles.

1.3 Self-Consistent Fields: Vlasov-Maxwell System

The Vlasov equation is coupled to Maxwell's equations through the charge and current densities:

Charge and Current Densities

$$\rho(\mathbf{r}, t) = \sum_s q_s \int f_s(\mathbf{r}, \mathbf{v}, t) \, d^3v$$
$$\mathbf{J}(\mathbf{r}, t) = \sum_s q_s \int \mathbf{v} \, f_s(\mathbf{r}, \mathbf{v}, t) \, d^3v$$

Maxwell's Equations

Gauss's Law

$$\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}$$

No Magnetic Monopoles

$$\nabla \cdot \mathbf{B} = 0$$

Faraday's Law

$$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$$

Ampère-Maxwell Law

$$\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0\epsilon_0 \frac{\partial \mathbf{E}}{\partial t}$$

Closed System: The Vlasov-Maxwell system is a nonlinear integro-differential system. The distribution function f determines ρ and J, which appear in Maxwell's equations to determine E and B, which in turn appear in the Vlasov equation forf. This self-consistency captures collective plasma behavior.

1.4 Linearization: Electrostatic Perturbations

For small-amplitude waves, we linearize around an equilibrium:

$$f_s = f_{s0}(v) + f_{s1}(\mathbf{r}, \mathbf{v}, t), \quad |f_{s1}| \ll f_{s0}$$
$$\mathbf{E} = \mathbf{E}_1(\mathbf{r}, t), \quad \mathbf{B} = \mathbf{B}_0 + \mathbf{B}_1(\mathbf{r}, t)$$

Electrostatic Approximation

For waves with phase velocity much less than c, we can neglect magnetic field perturbations (βˆ‡ Γ— E1 β‰ˆ 0) and write E1 = -βˆ‡Ο†1. The linearized Vlasov equation becomes:

$$\frac{\partial f_{s1}}{\partial t} + \mathbf{v} \cdot \nabla f_{s1} - \frac{q_s}{m_s} \nabla \phi_1 \cdot \frac{\partial f_{s0}}{\partial \mathbf{v}} = 0$$

Fourier-Laplace Analysis

For plane waves ∝ exp(ikΒ·r - iΟ‰t), the linearized Vlasov equation in 1D (k = kx) gives:

$$f_{s1}(k, v, \omega) = \frac{iq_s}{m_s} \frac{k \phi_1(k,\omega)}{\omega - kv} \frac{\partial f_{s0}}{\partial v}$$

The perturbed charge density is:

$$\rho_1 = \sum_s q_s \int f_{s1} \, dv = \sum_s \frac{iq_s^2}{m_s} \phi_1(k,\omega) k \int \frac{\partial f_{s0}/\partial v}{\omega - kv} \, dv$$

Poisson's equation βˆ‡Β²Ο†1 = -ρ1/Ξ΅0 in Fourier space becomes:

$$-k^2 \phi_1 = -\frac{\rho_1}{\epsilon_0}$$

Combining these gives the plasma dispersion relation:

$$1 + \sum_s \frac{\omega_{ps}^2}{k^2} \int \frac{k \, \partial f_{s0}/\partial v}{\omega - kv} \, dv = 0$$

Plasma Dispersion Relation (Ξ΅(k,Ο‰) = 0)

Landau Prescription: The integral has a pole at v = Ο‰/k (resonant particles). Proper treatment requires analytic continuation from Im(Ο‰) > 0 (Landau contour), leading to Landau damping even for real k.

1.5 Conservation Laws

Particle Conservation

Integrating the Vlasov equation over all velocities gives the continuity equation:

$$\frac{\partial n_s}{\partial t} + \nabla \cdot (n_s \mathbf{u}_s) = 0$$

where ns = ∫ fs d³v and nsus = ∫ vfs d³v

Energy Conservation

Multiplying by (msvΒ²/2) and integrating gives the kinetic energy evolution:

$$\frac{\partial}{\partial t}\left(\sum_s \int \frac{1}{2}m_s v^2 f_s \, d^3v\right) + \nabla \cdot \mathbf{S} = -\mathbf{J} \cdot \mathbf{E}$$

Energy exchange between particles and fields

Entropy-Like Invariant

The Vlasov equation conserves an H-function (analogous to entropy):

$$H = \sum_s \int f_s \ln f_s \, d^3r \, d^3v = \text{constant}$$

This shows the Vlasov equation is time-reversible and non-dissipative, unlike the Boltzmann equation with collisions.

1.6 Physical Interpretation

Wave-Particle Resonance

Particles with velocity v = Ο‰/k (resonant particles) move with the wave phase velocity. They remain in phase with the wave and can exchange energy coherently.

β†’ Landau damping, transit-time damping

Collective Effects

All particles contribute to the self-consistent fields. Even a small perturbation can grow if the distribution function has an unstable feature (e.g., positive slope βˆ‚f/βˆ‚v > 0 somewhere).

β†’ Two-stream instability, bump-on-tail instability

Phase Mixing

Different velocity particles have different orbital periods. Phase space structures become increasingly fine-grained (filamentation), appearing as damping in coarse-grained observables.

β†’ Reversible "effective damping" without collisions

No Thermalization

The Vlasov equation alone cannot drive the distribution toward Maxwellian. Thermalization requires collisions (Fokker-Planck terms) or nonlinear wave-particle interactions (quasilinear diffusion).

β†’ Need Fokker-Planck or quasilinear theory

1.7 Validity and Limitations

Validity Criteria

1. Weak Coupling (Ideal Plasma)

$$\Lambda = n \lambda_D^3 \gg 1$$

Many particles in a Debye sphere. Binary collisions are rare compared to collective interactions.

2. Collisionless Regime

$$\nu_{coll} \ll \omega_p, \quad \nu_{coll} \ll k v_{th}$$

Collision frequency much smaller than plasma oscillation frequency and wave frequency.

3. Non-Relativistic

$$v_{th} \ll c$$

Thermal velocities much less than speed of light. For relativistic plasmas, use relativistic Vlasov equation.

4. Classical Statistics

$$n \lambda_{th}^3 \ll 1$$

Thermal de Broglie wavelength Ξ»th = h/(mvth) much smaller than inter-particle spacing. Quantum effects negligible.

When Vlasov Equation Fails

  • Long time scales (t ≫ 1/Ξ½coll): Collisions cause diffusion in velocity space β†’ Fokker-Planck equation
  • Strong turbulence: Wave-wave interactions, mode coupling β†’ Need nonlinear kinetic theory
  • Strongly magnetized plasmas: Gyrokinetic equation more efficient (averages over fast gyro-motion)
  • Relativistic plasmas: Relativistic Vlasov equation with Ξ³ factor and 4-momentum
  • Quantum plasmas: Wigner function, quantum kinetic equations

1.8 Applications

🌌 Space Plasmas

  • β€’ Solar wind expansion (collisionless)
  • β€’ Magnetospheric physics (wave-particle interactions)
  • β€’ Auroral particle acceleration
  • β€’ Planetary radiation belts

πŸ”₯ Fusion Plasmas

  • β€’ RF wave heating (ICRH, ECRH, LHCD)
  • β€’ Fast particle physics (alpha particles, NBI)
  • β€’ Plasma stability (kinetic instabilities)
  • β€’ Runaway electron generation

πŸ’₯ Laser-Plasma Interactions

  • β€’ Parametric instabilities (SRS, SBS, TPD)
  • β€’ Hot electron generation
  • β€’ Particle acceleration mechanisms
  • β€’ Inertial confinement fusion

⚑ Astrophysical Shocks

  • β€’ Collisionless shock structure
  • β€’ Cosmic ray acceleration (Fermi mechanism)
  • β€’ Pulsar wind termination shocks
  • β€’ Supernova remnant shocks

πŸ“š Key Takeaways

  • βœ“ Vlasov equation describes collisionless plasma evolution in phase space
  • βœ“ Coupled to Maxwell equations through self-consistent ρ and J
  • βœ“ Linearization reveals plasma waves and instabilities (Landau damping, two-stream)
  • βœ“ Wave-particle resonance at v = Ο‰/k is key to kinetic effects
  • βœ“ Valid when Ξ½coll β‰ͺ Ο‰p (hot, tenuous plasmas)
  • βœ“ Foundation for understanding collisionless plasma phenomena in space, fusion, and astrophysics
← Part II OverviewNext: Landau Damping β†’