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How PIC Lies

Source: PHY653B Ch. 3

Intuition

PIC is the workhorse of kinetic plasma simulation and it is wrong in specific, catalogued, predictable ways. Knowing the catalogue is the difference between using PIC and being used by it. Every item below produces output that looks entirely plausible.

1. Statistical noise

Markers sample \(f\), so every moment carries Monte-Carlo error:

\[\frac{\delta n}{n} \sim \frac{1}{\sqrt{N_{\rm ppc}}}\]

With 100 markers per cell that is 10% density noise — which appears in the field, pushes the markers, and never averages away because it is dynamically coupled. Consequences:

  • Growth rates measured from noise. A spectrum built from a noisy field has power at every \(k\). Fitting an exponential to a mode that never actually grew is easy and common — see growth-rate fitting.
  • Small damping rates are unmeasurable. \(\gamma = -0.0126\) at \(k\lambda_D = 0.3\) requires the signal to stay above the noise floor for many damping times. A Vlasov solver measures it easily; PIC needs enormous \(N\).

2. Numerical heating

Even a correct PIC code heats. Finite-grid effects couple short-wavelength noise into particle energy, and if \(\Delta x > \lambda_D\) the heating is catastrophic: the plasma warms until \(\lambda_D\) grows to match \(\Delta x\), then stops. The simulation has self-consistently satisfied the resolution requirement by destroying your temperature, and nothing crashes.

\[\Delta x \lesssim \lambda_D, \qquad \omega_{pe}\Delta t \lesssim 0.2\]

Always plot total energy against time. A slow upward creep is the signature.

3. The self-force

Mismatched deposit and interpolate shape functions let a marker push itself. Momentum conservation fails and energy is injected. Matched shapes make it cancel exactly.

4. Two discretisations, two failure modes

Chapter 3's comparison, and a genuinely useful piece of judgement:

Energy-conserving PIC Momentum-conserving PIC
Conserves total energy (to round-off) total momentum exactly
Fails by violating momentum; finite-grid instability suppressed numerical heating if under-resolved
Self-force absent by construction absent only with matched shapes
Use when long runs where energy drift would dominate wave/instability physics where momentum matters

Neither is "correct". They make opposite compromises, and the right choice depends on which conservation law your diagnostic depends on. A code that conserves the quantity you are measuring is not thereby validated — it may be conserving it by construction while getting the distribution wrong.

5. The tail is where you have no markers

A Maxwellian sampled with uniform weights puts almost all markers in the bulk. But the physics that matters — resonant particles, runaways, fusion-relevant fast ions — lives in the tail, where the sampling is worst. Standard remedies are variable weighting and delta-f, both of which are importance sampling under other names.

How to keep yourself honest

  • Plot total energy every run. Non-negotiable.
  • Run the same case at 2× and 4× the marker count. Anything that changes is noise, not physics.
  • Check \(\Delta x/\lambda_D\) and \(\omega_{pe}\Delta t\) before believing anything.
  • Compare against a noise-free solver on a reduced problem where both are affordable — this is exactly what the Vlasov benchmark is for.

Common mistakes

  • Treating a converged-looking run as converged. Convergence in PIC means convergence in \(N_{\rm ppc}\) and grid and time step.
  • Reporting a growth rate without a noise floor. State the noise level and show the mode rose above it.
  • Assuming energy conservation validates the run. See the table: some schemes conserve it by construction.

Knowledge graph position

Prerequisites: PIC shape functions, normalisation & resolution. Leads to: delta-f methods, implicit and energy-conserving PIC, defensible kinetic results.

Quiz

Q1 (conceptual). Why is numerical grid heating so dangerous compared with an outright instability?

Answer

An instability announces itself — the run blows up. Grid heating is smooth, stable and self-limiting: the plasma warms until \(\lambda_D\) matches \(\Delta x\) and then sits there, producing a completely plausible temperature history that happens to be wrong.

Q2 (computational). A PIC run uses 25 markers per cell. What is the rough density noise, and how many markers would be needed to halve it?

Answer

\(1/\sqrt{25} = 20\%\). Halving requires quadrupling the markers to 100 per cell — the \(1/\sqrt{N}\) scaling is why noise reduction is so expensive and why variance-reduction methods are worth the complexity.

Q3 (MCQ). An energy-conserving PIC scheme is preferable when:

  • (a) momentum conservation is the key diagnostic
  • (b) the run is long and energy drift would otherwise dominate the result
  • (c) you want to eliminate statistical noise
  • (d) the plasma is unmagnetised
Answer

(b). It trades exact momentum conservation for exact energy conservation and suppression of the finite-grid instability — the right trade for long runs, the wrong one when your physics depends on momentum balance.