---
title: "Phase-Space Filaments Interactive Lab | Sandbox Physics"
description: "Why does a ripple in a hot plasma fade with no collisions at all?"
canonical_url: "https://sandboxphysics.com/labs/phase-space-filaments/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/phase-space-filaments/"
last_modified: "2026-10-29"
---

FR31 · Where does a plasma wave go?

# Phase-Space Filaments

**Why does a ripple in a hot plasma fade with no collisions at all?** Electrons with different speeds drift apart, so the bunch of charge that made the field is smeared into finer and finer stripes in velocity. The field fades even though nothing is lost from the electrons. How fast it fades depends on the few electrons moving close to the wave speed. This collisionless fading is Landau damping.

Start a small density ripple and watch its electric field die away while the electrons fold into ever finer stripes in velocity. A coarse velocity grid brings the ripple back at the wrong time.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Model assumptions

Electrons move in one dimension through a fixed, uniform ion background, in a periodic box. The program evolves the electron distribution over position and velocity and computes the field from the electron density. Time is measured in inverse plasma frequencies, speed in thermal speeds and length in Debye lengths. Magnetic fields and ion motion are left out. The box holds one wavelength of the ripple. The comparison rate is the complex root of the dielectric function of a Maxwellian, computed separately from the simulation. Collisions are off unless the collision slider in the advanced settings is raised.

Measured damping rate $\gamma_{\mathrm{fit}}$ $-0.154\,\omega_p$

Dielectric-root rate $\gamma_{\mathrm{root}}$ $-0.153\,\omega_p$

Velocity cells per stripe at the end $2\pi/(kt\,\Delta v)$ $3.83$

Grid recurrence time $T_R$ $268\,\omega_p^{-1}$

### How the field fades

simulated field dielectric-root rate (reference) grid recurrence

### Stripes in velocity get finer

measured stripe power free-streaming prediction

Equations, approximations and sources

Electron distribution with fixed ions, in units of the plasma frequency, thermal speed and Debye length:

$$
\frac{\partial f}{\partial t}+v\frac{\partial f}{\partial x}-E\frac{\partial f}{\partial v}=\nu\frac{\partial}{\partial v}\Big[(v-u)f+T\frac{\partial f}{\partial v}\Big],\qquad \frac{\partial E}{\partial x}=1-\int f\,dv
$$

The program splits each step into streaming in position, a kick from the field and, when switched on, a weak-collision step. The comparison rate is the root of the Maxwellian dielectric function:

$$
1-\frac{1}{2k^2}Z'\!\left(\frac{\omega}{\sqrt2\,k}\right)=0
$$

A velocity grid cannot hold stripes finer than one cell, so the free-streaming field comes back at the recurrence time, where the last factor is the cell width:

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

This is a numerical model with one space and one velocity dimension. It omits magnetic fields, ion motion, and the experiment geometry of the original echo measurements. Exported records include every parameter, the model version and the full field history.

- [R. Fitzpatrick, Plasma Physics lecture notes: Landau damping and the plasma dispersion function](https://farside.ph.utexas.edu/teaching/plasma/Plasma/node109.html)
- [Continue through the kinetic plasma course](https://sandboxphysics.com/stories/kinetic-plasma/)

[Read the kinetic plasma course: from phase-space stripes to wave echoes](https://sandboxphysics.com/stories/kinetic-plasma/)

Physics tutorial

## Phase-Space Filaments

**Background** Start a small density ripple and watch its electric field die away while the electrons fold into ever finer stripes in velocity. A coarse velocity grid brings the ripple back at the wrong time.

**Why it matters** A wave in a hot, collisionless plasma loses its field even though no friction acts. The missing field turns into structure in the electron velocities, and that structure can come back as an echo.

Start with the essentials

- **Focus question**: Why does a ripple in a hot plasma fade with no collisions at all?
- **One-sentence intuition**: Each electron keeps its speed, but electrons with different speeds slide apart, so the charge bunch spreads into velocity stripes and its field averages away.

### Core mathematical model

#### Electron Vlasov–Poisson equations in these units

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

Electrons carry charge minus one, so the field pushes them with acceleration minus E. Ions are a fixed background of density one.

#### Free streaming winds the stripes

$$
f_1(x,v,t)=\alpha\,M(v)\cos\bigl(k(x-vt)\bigr),\qquad \Delta v_{\text{stripe}}=\frac{2\pi}{kt}
$$

Without a field, a ripple keeps its height but its velocity stripes get narrower in proportion to one over time.

#### Damping rate from the dielectric root

$$
1+\frac{1}{k^{2}}\Bigl[1+\zeta Z(\zeta)\Bigr]=0,\qquad \zeta=\frac{\omega}{\sqrt{2}\,k}
$$

Z is the plasma dispersion function. For $k\lambda_D=0.5$ the root is $\omega=1.4157-0.1534\,i$ in units of the plasma frequency.

#### When a velocity grid fakes a return

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

A velocity grid with spacing $\Delta v$ cannot tell a stripe that narrow from no stripe at all, so the density rebuilds itself at this recurrence time.

### Common difficulties

#### Damping without friction

**Typical misconception** The wave fades because electrons lose energy in collisions.

**Better mental model** The run has no collisions. Electrons near the wave speed gain the energy the field loses; the total energy stays fixed.

#### Model scale

**Typical misconception** The grid reproduces a laboratory plasma.

**Better mental model** Electrons move in one dimension through a fixed, uniform ion background, in a periodic box. The program evolves the electron distribution over position and velocity and computes the field from the electron density. Time is measured in inverse plasma frequencies, speed in thermal speeds and length in Debye lengths. Magnetic fields and ion motion are left out. The box holds one wavelength of the ripple. The comparison rate is the complex root of the dielectric function of a Maxwellian, computed separately from the simulation. Collisions are off unless the collision slider in the advanced settings is raised.

### Run the experiment

1. 01

   #### Watch the field fade

   Choose A ripple fades and keep the self-consistent field on.

   **What to observe:** The cyan field shrinks along the gold reference line while the phase-space picture fills with tilted stripes.

2. 02

   #### Remove the field

   Choose Streaming without a field.

   **What to observe:** The density still fades, but faster and with a curved logarithmic history: phase mixing alone gives a Gaussian decay, not one fixed rate.

3. 03

   #### Starve the velocity grid

   Choose A coarse grid brings it back, then compare the stripe chart with the gold line.

   **What to observe:** The measured stripe wave number hits the top of the chart and wraps around; the field returns at the coral recurrence line, a numerical artifact.

Physical conclusion

Landau damping removes the field, not the information: the distribution keeps every stripe until collisions or a too-coarse grid erase it.

## Related experiments and articles

Kinetic plasma · Reading path

### Kinetic plasma from phase-space stripes to wave echoes

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **Why electrons in a plasma ring like a bell**](https://sandboxphysics.com/stories/kinetic-plasma/plasma-oscillations-and-units/)
2. [Lab **Radio Echoes from the Ionosphere** See the plasma frequency reflect radio waves](https://sandboxphysics.com/labs/ionospheric-radio-sounding/)
3. [Article **Follow every electron at once in phase space**](https://sandboxphysics.com/stories/kinetic-plasma/phase-space-and-the-vlasov-equation/)
4. [Article **Where a fading ripple hides, and how a grid brings it back**](https://sandboxphysics.com/stories/kinetic-plasma/phase-mixing-and-recurrence/)
5. [Lab **Plasma Wave Echo** Make an echo with two pulses](https://sandboxphysics.com/labs/plasma-wave-echo/)
6. [Article **Calculating how fast a plasma wave fades**](https://sandboxphysics.com/stories/kinetic-plasma/landau-damping-rate/)
7. [Lab **Landau Damping and Trapping** Follow the energy to resonant electrons](https://sandboxphysics.com/labs/landau-damping-trapping/)
8. [Article **Two pulses fade, then a third signal appears**](https://sandboxphysics.com/stories/kinetic-plasma/plasma-wave-echoes/)
9. [Article **How rare collisions erase a plasma’s memory**](https://sandboxphysics.com/stories/kinetic-plasma/weak-collisions-and-irreversibility/)

Fields & matter · Starter path

## From a field to an electron microscope

Map an electric field and steer a charged particle. Then find the electron's wavelength and use electron beams to see a crystal and a surface.

[Start here → **Make an Electric Field Visible** Move the probe around a dipole and check that the field always points downhill on the potential landscape. Observe · 8 min](https://sandboxphysics.com/labs/electric-field/)

1. [1 Make an Electric Field Visible](https://sandboxphysics.com/labs/electric-field/)
2. [2 A Charge Bending through Fields](https://sandboxphysics.com/labs/charged-particle/)
3. [3 Electron Wavelength & Voltage](https://sandboxphysics.com/labs/electron-wavelength/)
4. [4 Electron Diffraction & Reciprocal Space](https://sandboxphysics.com/labs/electron-diffraction/)
5. [5 TEM Bright Field, Dark Field & SAED](https://sandboxphysics.com/labs/tem-mode-selector/)
6. [6 SEM Secondary-Electron Topography](https://sandboxphysics.com/labs/sem-topography/)

[View the full path →](https://sandboxphysics.com/topics/fields-and-matter/#starter-path)