How rare collisions erase a plasma’s memory

Collisions blur the distribution in velocity, and a blur wipes out fine stripes much faster than coarse ones. The stripes that carry an echo grow finer the longer the delay, so a collision rate thousands of times smaller than the plasma frequency can still erase the echo while leaving ordinary Landau damping untouched.

Sandbox Physics · 2026-10-29

Derivation

A collision model that keeps the conservation laws

The simulation uses a model collision operator of drag plus diffusion in velocity, introduced by Dougherty in 1964. At each position it pulls electrons toward the local mean velocity $u$ and spreads them with the local temperature $T$ , both computed from the same column of the distribution. That choice conserves the number of electrons, their momentum and their kinetic energy.

The operator also never lowers the entropy $-\int f\ln f\,dv$ , and the only distribution it leaves unchanged is a Maxwellian with the local $u$ and $T$ . In a test with collisions only, the entropy rises at every step and ends within $6\times10^{-10}$ of the Maxwellian value with the same energy.

C[f]=\nu\,\frac{\partial}{\partial v}\Big[(v-u)\,f+T\,\frac{\partial f}{\partial v}\Big]

Fine stripes go first

A stripe pattern $e^{i\kappa v}$ under diffusion with coefficient $\nu T$ decays at the rate $\nu T\kappa^2$ . A smooth Maxwellian has $\kappa$ of order one; the stripes from chapter 3 have $\kappa=kt$ , which reaches $20$ at $t=40$ for $k=0.5$ . The stripes are erased $400$ times faster than the smooth part.

Integrating that rate along the stripe’s history gives the total loss. For a stripe whose wave number grows as $kt$ from zero, the accumulated exponent after time $t$ is $\nu k^2t^3/3$ : it grows with the cube of time.

\frac{d}{dt}\ln|\hat f(\kappa)|=-\nu T\kappa^2,\qquad \int_0^t \nu (ks)^2\,ds=\frac{\nu k^2t^3}{3}

Worked example

Worked example: estimate the echo loss

The echo-producing stripes tighten as $k_1t$ until the second pulse, then loosen at the rate $k_2-k_1$ until they vanish at $\tau^{\prime}$ . Adding the two pieces, with $T=1$ , gives the estimate below. For modes 1 then 2, $k_1=0.5$ , $k_2=1$ , $\tau=20$ , the bracket is $1333$ . With $\nu=0.002$ the estimate predicts that the echo shrinks by $e^{-2.67}\approx0.07$ .

The simulation gives a ratio of $0.18$ , an exponent of $1.69$ . The free-streaming estimate leaves out the drag half of the collision operator and the self-consistent field, and in our runs it overestimates the exponent by roughly $20\%$ to $60\%$ , more for stronger suppression; at $\tau=30$ the measured exponents are $0.38$ , $0.74$ and $1.40$ for $\nu=10^{-4}$ , $2\times10^{-4}$ and $4\times10^{-4}$ , against estimates $0.45$ , $0.90$ and $1.80$ . The cubic growth with delay is the robust part: a collision rate that barely touches a single wave removes most of a long-delay echo.

\ln\frac{E_{\mathrm{echo}}(\nu)}{E_{\mathrm{echo}}(0)}\approx-\nu\left[\frac{k_1^2\tau^3}{3}+\frac{(k_1\tau)^3}{3(k_2-k_1)}\right]=-\frac{\nu\,k_1^2k_2\,\tau^3}{3(k_2-k_1)}

Where irreversibility enters

Without collisions, the Vlasov equation is reversible and the echo proves it: information hidden in the stripes comes back. With collisions, the finest stripes are converted into a smooth change of the distribution, and the entropy rises. No later pulse can recover them.

The 1967 paper noted that this sensitivity could be used to measure weak collision rates, because Coulomb collisions act through exactly this velocity-space diffusion, enhanced for a stripe of wave number $kv\tau$ by a factor of order $(kv\tau)^2$ . Su and Oberman published a theory of collisional echo damping in 1968; comparing the simulation with their result is left as a further check.

Predict, change, explain

Predict how the echo changes if you keep the collision rate fixed and lengthen the delay from 20 to 30.

Set the collision slider to 0.001 in the echo Lab, then compare delays of 20 and 30. Repeat with collisions off.

Use the cubic dependence on delay to explain why the longer delay suffers far more, even though each pulse’s own damping barely changes.

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. By what factor does the estimated collision exponent grow when the delay increases from 20 to 30?

Read an explained answer

It scales as $\tau^3$ , so by $(30/20)^3\approx3.4$ .

2. Why does a collision rate of $0.002$ barely change the Landau damping of a single $k=0.5$ wave?

Read an explained answer

Landau damping is set by the smooth slope of the distribution near the wave speed. Collisions change that slope at a rate of order $\nu$ , about $0.002$ , far slower than the damping rate $0.153$ .

3. Does the collision operator conserve the fine-grained entropy?

Read an explained answer

No. It never decreases it and increases it whenever the distribution is not a local Maxwellian. That is the difference from the Vlasov equation of chapter 2.

Sources to read alongside

R. W. Gould, T. M. O’Neil and J. H. Malmberg, Plasma Wave Echo, Physical Review Letters 19, 219 (1967)

Read in full from the Caltech repository copy. Figure 1 sketches the two pulses and the echo; Equations (4) and (5) give the echo time and envelope. Our reproduction checks the echo time, the echo mode and the envelope rates in a one-dimensional simulation.

C. H. Su and C. Oberman, Collisional Damping of a Plasma Echo, Physical Review Letters 20, 427 (1968)

Theory of how weak collisions reduce an echo. Only the bibliographic record and abstract were checked; the collision estimate in this course is derived separately.

J. P. Dougherty, Model Fokker-Planck Equation for a Plasma and Its Solution, Physics of Fluids 7, 1788 (1964)

Origin of the drag-plus-diffusion collision model used in the simulation. Only the bibliographic record was checked; the conservation properties used here are verified numerically.

A collision model that keeps the conservation laws

Fine stripes go first

Worked example: estimate the echo loss

Where irreversibility enters

Predict, change, explain

Try explaining it yourself

1. By what factor does the estimated collision exponent grow when the delay increases from 20 to 30?

2. Why does a collision rate of 0.0020.0020.002 barely change the Landau damping of a single k=0.5k=0.5k=0.5 wave?

3. Does the collision operator conserve the fine-grained entropy?

Sources to read alongside

2. Why does a collision rate of $0.002$ barely change the Landau damping of a single $k=0.5$ wave?