Wave 28: Debye Shielding, Bayesian Updating & Fabry–Pérot

Platform Stats

Wave 28 at a Glance

Debye Shielding

In a plasma, free electrons cluster around positive test charges and exponentially suppress the electrostatic field beyond the Debye length:

λ_D = √(ε₀ k_B T / n e²)     φ(r) = Ze/(4πε₀r) · exp(−r/λ_D)

The simulation offers three complementary views:

• Potential Profile — log-scale overlay of screened Yukawa vs. bare Coulomb 1/r, with a λ_D marker.
• Shielding Cloud — 2D ImageData colour-map of φ(r) across a cross-section. Warm amber → high potential, dark blue → fully screened.
• Debye Sphere — schematic showing the λ_D radius, electron dot population (capped at 120 for performance), and a validity check N_D ≫ 1.

Controls expose log₁₀ n (m⁻³), temperature T in eV, and charge Z. The info bar reports λ_D in SI units (pm → µm depending on density), N_D, ω_p, and the coupling parameter Γ = e²/(4πε₀ a k_BT) where a = n^−1/3.

Bayesian Updating

The Beta-Binomial model is the textbook conjugate pair for proportion inference. Choose a prior Beta(α, β) — encoding α pseudo-heads and β pseudo-tails observed before the experiment — then flip a coin whose true bias p is set by a slider (hidden from the “Bayesian”). After k heads in n flips the posterior is analytic:

Posterior = Beta(α + k,  β + n − k)

The canvas plots both distributions continuously, with an amber line for the true p and a teal 95% credible interval computed via the regularised incomplete Beta function (Lentz continued-fraction, 80-iteration bisection for quantiles). Four flip speeds and a 500-entry coloured flip-log make it easy to see convergence at a glance.

Fabry–Pérot Interferometer

A Fabry–Pérot etalon is a pair of parallel, partially reflective mirrors separated by distance d. Light reflected back and forth interferes constructively only when the round-trip phase δ = 4πnd·cos(θ)/λ is a multiple of 2π, producing sharp resonance peaks described by the Airy function:

T(λ) = 1 / (1 + F · sin²(δ/2))    F = 4R/(1−R)²

The simulation visualises the transmission spectrum, the Free Spectral Range FSR = λ²/(2nd), and the finesse ℱ = π√R/(1−R) that quantifies how sharp the peaks are. Sliders allow sweeping mirror reflectivity R, cavity length d, refractive index n, and incidence angle θ to directly observe how each affects the resonance structure.

Engineering Notes

Debye shielding: SI-aware value formatting

Plasma density spans from laboratory discharges (~10²⁰ m⁻³) to stellar interiors (~10³² m⁻³), so the Debye length swings from micrometres to femtometres. Rather than printing raw scientific notation, a lightweight fmtSI(v, unit) routine maps the SI prefix (f, p, n, µ, m, k, M, …) automatically by computing floor(log10(v)/3). A seeded Mulberry32 pseudo-random generator ensures the electron dots in the Debye-sphere view are reproducible across slider changes.

Bayesian updating: incomplete Beta via Lentz continued fraction

Computing 95% credible intervals requires the inverse regularised incomplete Beta function I_x(a, b). The forward CDF uses the Lentz modified continued fraction (200 iterations, convergence threshold 10⁻¹²), with the standard symmetry identity I_x(a,b) = 1 − I_1−x(b,a) to select the more numerically stable branch. The quantile is solved by bisection over I_x (80 iterations, converges to 8 significant figures on the unit interval). The logGamma evaluator uses the 7-term Lanczos approximation, supporting α, β down to 0.5 without overflow.

Fabry–Pérot: smooth finesse rendering across all R

For high reflectivity (R → 0.99) the Airy peaks become extremely narrow (ℱ → 300). Rendering them faithfully requires at least ~20 sample points per peak. The simulation adaptively increases the wavelength resolution near peaks by pre-sampling at coarse resolution, locating the minima of (1 − T), then inserting refined sub-samples between adjacent minima. This keeps total evaluations under 2000 even at maximum finesse, well within a single animation frame.

What’s Next — Wave 29 Preview

Planned topics for Wave 29 include:

• Taylor Series Visualiser — interactive animation of Nth-order polynomial approximation to sin, cos, exp, ln, with error band
• Hodgkin–Huxley Neuron — conductance-based action potential, gating variables m/h/n, Euler ODE integration
• Ising Model Phase Transition — 2D Metropolis–Hastings Monte Carlo, spontaneous magnetisation, critical temperature T_c = 2J/k_B ln(1+√2)

All Wave 29 simulations will ship with EN + UK pages on launch day.