Where a fading ripple hides, and how a grid brings it back

The density ripple fades because the distribution folds into stripes across velocity, and those stripes get finer at a steady rate. A computer stores the stripes on a velocity grid. When they become finer than one grid cell, the grid misreads them as coarse stripes, and the ripple reappears at a time set only by the grid.

Sandbox Physics · 2026-10-29

Derivation

The stripes tighten at a fixed rate

Write the ripple as the complex mode $e^{ikx}$ , keeping in mind that the physical ripple is its real part. Under free streaming, $e^{ikx}$ becomes $e^{ik(x-vt)}=e^{ikx}e^{-iktv}$ . At a fixed position, the distribution now oscillates in velocity with wave number $\kappa = kt$ : the stripes tighten in proportion to time.

Density integrates over velocity. Integrating a smooth Maxwellian against a fast oscillation gives almost zero, and the faster the oscillation, the closer to zero. That is phase mixing. With the field on, the electrons also push on each other, but the stripes keep tightening at very nearly the same rate.

e^{ik(x-vt)}=e^{ikx}\,e^{-i\kappa v},\qquad \kappa(t)=kt

A grid can only hold stripes so fine

A velocity grid with spacing $\Delta v$ samples $e^{-i\kappa v}$ only at $v_j=j\Delta v$ . When $\kappa\Delta v$ reaches $2\pi$ , every sample equals its value at $\kappa=0$ . The grid then holds the same numbers as the original smooth ripple, the velocity integral no longer cancels, and the field returns.

The return time is the recurrence time. It depends on the wave number and the velocity spacing, and on nothing physical. Because it arrives on a schedule set by the grid, refining the velocity grid is the clean test: a recurrence moves later, while a physical signal stays put.

k\,T_R\,\Delta v=2\pi\qquad\Longrightarrow\qquad T_R=\frac{2\pi}{k\,\Delta v}

Worked example

Worked example: two grids, one ripple

The phase-space Lab spans velocities from $-6$ to $6$ . With $256$ cells, $\Delta v=12/256\approx0.047$ , and a $k=0.5$ ripple recurs at $T_R=2\pi/(0.5\times0.047)\approx268$ , far beyond the 70-unit run. At $t=40$ one stripe spans $2\pi/(kt)\approx0.31$ in velocity, still about seven cells. The grid resolves it.

With $32$ cells, $\Delta v=0.375$ and $T_R\approx33.5$ . Run the coarse-grid preset and the field climbs back near $t=33.5$ , exactly where the dashed recurrence line sits on the history chart. The damping rate measured before the recurrence is still $-0.1536$ , the same as on the fine grid, because the early fit window ends before the grid fails.

Three things that can look like a returning signal

A returning field can be physics, a grid recurrence or a reflection from the edge of the velocity box. The Lab’s velocity box is periodic, so a stripe that leaves at the top re-enters at the bottom; that does no harm while the distribution is negligible near the limits, which is why the Maxwellian runs use limits of $\pm6$ and the beam runs $\pm9$ .

Each suspect has its own test. Doubling the number of velocity cells doubles a recurrence time. Widening the velocity box at fixed spacing changes a boundary effect. A physical signal survives both. The project benchmarks apply both tests to the echo of chapter 6: with $128$ , $256$ and $512$ cells, and with limits of $6$ and $12$ , the echo arrives at the same time.

An experiment to think through

Predict, change, explain

Predict the recurrence time for a $k=0.5$ ripple on a 64-cell velocity grid.
Choose 64 velocity cells in the phase-space Lab and watch for the field to return. Then try 128.
Use the stripe spectrogram to explain the return: the measured stripe power follows the gold line until the stripes reach the grid limit.

Watch a ripple fold into velocity stripes Make an echo with two pulses

The three phase-space Labs solve a one-dimensional electron model in plasma units. Each Lab exports its parameters, model version and field history.

Try explaining it yourself

1. What recurrence time do you predict for $k=0.5$ and 64 velocity cells over $[-6,6]$ ?

Read an explained answer

$\Delta v=12/64=0.1875$ , so $T_R=2\pi/(0.5\times0.1875)\approx67$ . That falls just inside the 70-unit run.

2. Why does the recurrence happen at a single time instead of slowly?

Read an explained answer

Every velocity sample returns to its starting phase at the same moment, because $kT_R v_j=2\pi j$ for all grid points. The cancellation that hid the ripple undoes itself everywhere at once.

3. A colleague reports a revived field at $t=67$ on a 64-cell grid and calls it an echo. What one test settles it?

Read an explained answer

Rerun with 128 cells. A recurrence moves to $t\approx134$ ; a physical revival stays at $t=67$ .

Sources to read alongside

R. Fitzpatrick, Plasma Physics lecture notes, University of Texas at Austin: Landau damping, resonant particles and the plasma dispersion function
Three linked pages derive the linear response of a Maxwellian plasma, the role of electrons near the wave speed, and the plasma dispersion function used for our dielectric roots.
C. Z. Cheng and G. Knorr, The integration of the Vlasov equation in configuration space, Journal of Computational Physics 22, 330 (1976)
The usual reference for splitting a Vlasov step into position and velocity shifts, the approach our solver takes. Only the bibliographic record was checked; the solver is tested against exact limits and conservation laws instead.

The stripes tighten at a fixed rate

A grid can only hold stripes so fine

Worked example: two grids, one ripple

Three things that can look like a returning signal

Predict, change, explain

Try explaining it yourself

1. What recurrence time do you predict for k=0.5k=0.5k=0.5 and 64 velocity cells over [−6,6][-6,6][−6,6]?

2. Why does the recurrence happen at a single time instead of slowly?

3. A colleague reports a revived field at t=67t=67t=67 on a 64-cell grid and calls it an echo. What one test settles it?

Sources to read alongside

1. What recurrence time do you predict for $k=0.5$ and 64 velocity cells over $[-6,6]$ ?

3. A colleague reports a revived field at $t=67$ on a 64-cell grid and calls it an echo. What one test settles it?