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Quasilinear diffusion and particle trapping

Contrast incoherent resonant diffusion with coherent wave trapping and calculate the amplitude-dependent scale at which linear resonance dynamics can fail.

Before you begin
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The question for today

What you’ll understand

  • State the weak-fluctuation and decorrelation assumptions of a quasilinear diffusion model.
  • Calculate bounce frequency and separatrix velocity width with a declared amplitude convention.
  • Distinguish a self-consistent wave–particle energy budget from externally prescribed diffusion.

Growing or damped plasma waves alter the velocity distribution that originally determined their response. Two useful reductions describe different ways this feedback can occur. A weak, sufficiently decorrelated spectrum can produce a slow diffusion in velocity space, while a coherent wave can trap particles in a moving potential well. Neither mechanism follows simply from the word nonlinear. Their relevant timescales, spectrum, and amplitude determine which approximation is justified and what saturation claims can be made.

Averaging retains slow changes in the distribution

Split the distribution and fields into an average and fluctuations. The average kinetic equation contains a correlation between the fluctuating force and fluctuating distribution. Quasilinear theory evaluates that correlation using a leading response while retaining its accumulated effect on the average. For a homogeneous one-dimensional electrostatic illustration, the result can be expressed as a velocity diffusion whose coefficient is concentrated near wave–particle resonances. The coefficient depends on a chosen spectrum and correlation model.

The usual random-phase, weak-spectrum interpretation requires that individual coherent kicks not trap particles for many bounce periods before decorrelation. Small fluctuation amplitude alone does not guarantee this if a monochromatic wave remains coherent for a long time. Broader kinetic quasilinear formulations exist, but every closure must state what correlations it neglects. A diffusion equation is therefore a model for an averaged distribution, not a claim that each collisionless particle literally performs the same stochastic trajectory in a specified deterministic wave.

∂tF‾=∂v[D(v,t)∂vF‾],Jv=−D∂vF‾,D≥0\partial_t\overline F=\partial_v\left[D(v,t)\partial_v\overline F\right],\qquad J_v=-D\partial_v\overline F,\qquad D\ge0
F̄ is a reduced density per velocity and Jv its velocity-space flux. D has units (m/s)²/s. This schematic homogeneous resonant-diffusion model assumes a justified averaging and decorrelation closure; it is not a general nonlinear Vlasov solver.↔ Scroll wide equations horizontally.

A coherent potential has a separatrix

Consider one sinusoidal potential with constant amplitude and phase velocity. In its moving frame, the nonrelativistic particle Hamiltonian is kinetic energy relative to that phase velocity plus charge times potential. Near a stable potential minimum, expanding the cosine gives harmonic bounce motion. The bounce frequency scales as the square root of wave amplitude. The energy difference between well bottom and separatrix is twice the potential-energy amplitude, giving a precise factor in the maximum trapped velocity excursion.

Declare whether a quoted trapping width is the half-width about phase velocity or the full span. The full span is twice the half-width. Also distinguish the small-oscillation bounce frequency at the bottom from the longer period near the separatrix. A slowly varying self-consistent wave can trap and release particles as its separatrix changes, but the constant-wave Hamiltonian does not predict that evolution by itself. Initial phases and wave coherence remain essential information.

ωb=k∣qϕ0∣m=∣q∣kE0m,Δvhalf=2∣qϕ0∣m,E0=k∣ϕ0∣\omega_b=k\sqrt{\frac{|q\phi_0|}{m}}=\sqrt{\frac{|q|kE_0}{m}},\qquad \Delta v_{\mathrm{half}}=2\sqrt{\frac{|q\phi_0|}{m}},\qquad E_0=k|\phi_0|
φ₀ is the peak potential amplitude in volts, E₀ the peak electric-field amplitude in V/m, k > 0 in m⁻¹, and ωb in s⁻¹. The half-width is evaluated at the well bottom for the separatrix of a single coherent sinusoid; full width is twice this value.↔ Scroll wide equations horizontally.

Saturation and conservation require coupled evolution

If the bounce frequency becomes comparable to a linear growth or damping rate, trapping can modify the distribution before the linear field prediction completes many e-folds. If bounce frequency times correlation time is much smaller than one, a particle samples only a small fraction of a bounce before the force decorrelates, supporting a diffusion interpretation under additional weak-spectrum assumptions. These comparisons are useful diagnostics rather than exact universal thresholds for every spectrum or initial condition.

A positive diffusion coefficient tends to flatten resonant gradients where flux can redistribute particles. With zero boundary flux, particle number is conserved, but particle energy need not be conserved if the wave spectrum is externally held fixed. A closed wave–particle calculation must evolve wave energy consistently with the particle exchange. A plateau can reduce drive, yet it does not determine a unique final amplitude or guarantee Maxwellian equilibrium. Coherent structures, spectral transfer, and collisions can each change the long-time outcome.

ωbτc≪1(decorrelation diagnostic),dσv2dt=2D(constant-D illustration)\omega_b\tau_c\ll1\quad\text{(decorrelation diagnostic)},\qquad \frac{d\sigma_v^2}{dt}=2D\quad\text{(constant-D illustration)}
τc is a force-correlation time in seconds. The variance formula assumes constant D on the full velocity line, decaying boundary tails, and finite initial moments. It is an illustrative diffusion identity, not a universal quasilinear coefficient or a coherent-trapping law.↔ Scroll wide equations horizontally.

Work it through

Calculate the trapped interval

A species has |q|/m = 10⁸ C/kg in a single coherent wave with k = 10 m⁻¹ and peak E₀ = 0.10 V/m. Find peak potential amplitude, small-oscillation bounce frequency, and full separatrix velocity width.

  1. The potential amplitude is ∣ϕ0∣=E0/k=0.10/10=0.010 V|\phi_0|=E_0/k=0.10/10=0.010\,\mathrm V. Thus ∣qϕ0∣/m=106 m2/s2|q\phi_0|/m=10^6\,\mathrm{m^2/s^2}.

  2. The bounce frequency is ωb=k∣qϕ0∣/m=10(1000)=104 s−1\omega_b=k\sqrt{|q\phi_0|/m}=10(1000)=10^4\,\mathrm{s^{-1}}. The central bounce period is 2π/ωb.

  3. The velocity half-width is 2 × 1000 = 2000 m/s, so the full span is 4000 m/s about phase velocity. This width assumes a coherent fixed-amplitude wave.

The result

The potential amplitude is 0.010 V, bounce frequency is 10⁴ s⁻¹, and full separatrix velocity width is 4.0 km/s.

The living notebook

Explore the model

Interactive lab

Interactive lab · oscillator

Only the bottom of a potential well is harmonic

Reset and set BOTH Damping ratio ζ to 0 and Natural frequency ω₀ to 1; predict an undamped unit-amplitude oscillation. Then Reset and set BOTH ζ to 0 and ω₀ to 2; predict half the period. The lab is a linear positive-stiffness oscillator starting at x = 1 with zero velocity. It illustrates a local harmonic bounce only and has no sinusoidal separatrix, trapping width, diffusion, or self-consistent wave evolution.

Only the bottom of a potential well is harmonicUnit mass, x(0) = 1 and v(0) = 0. The analytic solution includes underdamped, critically damped and overdamped cases; time is measured in seconds. Current readings: Displacement x(t): 1; Velocity v(t): 0; Regime: underdamped. All parameters are adjustable with the labeled controls below.02.557.510−1.15−0.5700.571.15Time · sDisplacementunderdamped
Displacement x(t)Current time
Displacement x(t)
1
Velocity v(t)
0
Regime
underdamped
t = 0 s

Unit mass, x(0) = 1 and v(0) = 0. The analytic solution includes underdamped, critically damped and overdamped cases; time is measured in seconds. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forA harmonic approximation describes motion near a stable well bottom and does not reproduce the full trapped-orbit geometry.

Pause & reason

Check your understanding

A single coherent sinusoid has a trapping half-width Δvhalf. What is the full velocity span enclosed by the separatrix at the well bottom?

Your turn

Try it yourself

An illustrative constant velocity diffusivity is D = 10⁶ (m/s)²/s on the full velocity line. A distribution initially has variance 4 × 10⁶ m²/s² and finite decaying tails. Find its variance after 3 s and state whether its mean changes.

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

Integrate the constant-D equation against v and v², using vanishing boundary terms.

Compare with the solution

The variance increase is 2Dt=2(106)(3)=6×106 m2/s22Dt=2(10^6)(3)=6\times10^6\,\mathrm{m^2/s^2}, so the final variance is 107 m2/s210^7\,\mathrm{m^2/s^2}. The first moment remains constant because the integrated first-moment diffusion term is a vanishing boundary term. The standard deviation is 107≃3162 m/s\sqrt{10^7}\simeq3162\,\mathrm{m/s}. This prescribed diffusion increases particle kinetic energy; a self-consistent closed wave model would need to account for the energy source.

Keep the idea

Take it with you

  • Quasilinear diffusion represents a justified average of weak resonant correlations.
  • Coherent trapping introduces an amplitude-dependent bounce scale and a separatrix.
  • A distribution plateau and an early growth rate do not uniquely determine nonlinear saturation.

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