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The Vlasov–Poisson System

Source: PHY653B Ch. 2

Intuition

Refuse to close the hierarchy and this is what you are left with: the full distribution function evolving in phase space, coupled to the field it generates. It is the most honest description of a collisionless plasma, and the most expensive.

The system

\[\frac{\partial f}{\partial t} + v\frac{\partial f}{\partial x} - E\frac{\partial f}{\partial v} = 0, \qquad \frac{\partial E}{\partial x} = 1 - \int f\,dv\]

in the normalised units of the course (\(\omega_{pe} = 1\), \(\lambda_D = 1\), electrons on a fixed ion background). Two statements: \(f\) is constant along particle trajectories, and those trajectories are set by the field the particles themselves make.

Characteristics — and why it is not "just advection"

The Vlasov equation says \(df/dt = 0\) along

\[\frac{dx}{dt} = v,\qquad \frac{dv}{dt} = -E(x,t)\]

so \(f\) is merely transported in phase space, never created or destroyed. This is why the semi-Lagrangian scheme works: each split step is an exact shift.

The nonlinearity is entirely in the coupling. \(E\) depends on \(f\) through Poisson, and \(f\) depends on \(E\) through its characteristics. Freeze \(E\) and the problem is linear advection; let it respond and you get the whole of kinetic plasma physics — Landau damping, two-stream growth, trapping, phase-space holes.

What you gain

  • Noise-free. Unlike PIC, there is no statistical sampling error. Small effects — a damping rate of \(-0.0126\) at \(k\lambda_D = 0.3\) — are measurable without drowning in \(1/\sqrt{N}\) noise.
  • The whole distribution. Tails, beams, and the resonant region near \(v = \omega/k\) are resolved as well as the bulk. In PIC the tail is where you have fewest markers, and the tail is usually what matters.
  • Smooth diagnostics. Fitting a growth rate to a clean exponential is a very different experience from fitting one to a noisy one.

What you pay

  • Dimensionality. 1-D in space and velocity is a 2-D grid. Full 3-D + 3-V is six dimensions: a modest \(128^6\) is \(4\times10^{12}\) cells. This is why the method lives in reduced geometry.
  • Velocity truncation. The domain must be cut at some \(v_{\max}\) (typically \(5\)–\(6\,v_{th}\)), and particles that reach the edge are simply lost. Fine for a Maxwellian, dangerous for anything with a hot tail or an accelerating population.
  • Recurrence. Filamentation is unbounded in the continuum, but a grid has a finest resolvable \(\Delta v\). The scheme therefore has an artefact you can predict exactly — and cannot remove.

Common mistakes

  • Forgetting to subtract the mean charge. The periodic Poisson equation has no solution unless the net charge is zero. If quasi-neutrality is not enforced, the field solve either fails or silently returns garbage in the \(k = 0\) mode.
  • Setting \(v_{\max}\) too small. Truncating at \(3v_{th}\) removes exactly the resonant particles that carry Landau damping, and the measured rate comes out wrong.
  • Assuming noise-free means error-free. Vlasov codes trade statistical error for discretisation error — numerical diffusion from interpolation, and recurrence. Different errors, not fewer.

Knowledge graph position

Prerequisites: Vlasov equation, closure problem, collision operators. Leads to: semi-Lagrangian solver, gyrokinetics.

Quiz

Q1 (conceptual). Where is the nonlinearity in Vlasov–Poisson?

Answer

Entirely in the coupling: \(E\) is a functional of \(f\) via Poisson, and \(f\) is transported along characteristics set by \(E\). The Vlasov equation itself is linear in \(f\) for a given field — which is exactly what makes operator splitting into exact shifts possible.

Q2 (computational). Estimate the memory for a 3-D + 3-V grid at \(128\) points per dimension in double precision.

Answer

\(128^6 = 4.4\times10^{12}\) cells \(\times\) 8 bytes \(\approx 3.5\times10^{13}\) bytes \(= 35\) TB for a single copy of \(f\) — and a time integrator typically needs several. This is the dimensionality wall that motivates PIC and gyrokinetics.

Q3 (MCQ). Compared with PIC, a Vlasov grid solver primarily trades:

  • (a) accuracy for speed
  • (b) statistical noise for dimensionality cost and grid artefacts
  • (c) stability for accuracy
  • (d) nothing — it is strictly better
Answer

(b). No sampling noise, but the cost scales as a power of dimension and you inherit interpolation diffusion plus recurrence. Neither method dominates; they fail differently.