Chaotic diffusion weighed on the standard map

Does deterministic chaos produce diffusion — does momentum in the kicked rotor's chaotic sea random-walk, and if so is the diffusion coefficient the quasilinear K²/4 of independent kicks, or does the determinism leave a measurable fingerprint?

Units: (momentum)²/step — Rechester–White D_RW(K=20) = (K²/4)·[1 − 2J₂(20)(1 − J₂(20))], J₂(20) = −0.16034135; 37.2% above quasilinear

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The finding

Chaotic diffusion weighed on the standard map: momentum in the kicked rotor's chaotic sea random-walks, but the kicks are NOT independent — raw iteration of p' = p + K sin θ (no diffusion law, no Bessel function in the generator) returns D(K=20) = 136.27 ± 0.15 vs the Rechester–White correlation-corrected 137.21 (rel 6.8e-3), tracks the full J₂ oscillation across K = 8.4–20 (slope 0.97, r 0.990, every large deviation's sign correct: suppressed to 0.65·K²/4 at K = 10, enhanced to 1.37·K²/4 at K = 20), and falsifies the quasilinear rival D ≡ K²/4 in BOTH directions — while the same machinery with the angle genuinely re-randomized each kick recovers K²/4 to 0.4%, so the oscillation is carried entirely by the deterministic angular correlations; KAM tori block transport at K = 0.5 (D/D_ql ≤ 5e-4) and the K = 6.5 accelerator mode goes superdiffusive (D/D_ql = 132 and climbing)

Method

Generator: the raw Chirikov standard map p' = p + K sin θ, θ' = θ + p' (mod 2π), ensembles of 10⁵–1.5×10⁵ seeded orbits started at p₀ = 0, θ₀ uniform; D̂ = (⟨Δp²⟩(500) − ⟨Δp²⟩(50))/(2·450) (two-time difference kills the correlation intercept), diffusive linearity checked by the window ratio D[275→500]/D[50→275]. No K²/4, no Bessel function, and no diffusion law anywhere in the generator — the Rechester–White prediction is computed only in the scorer (J₂ by Simpson quadrature). Channels: (1) scalar D at K = 20, 6 seeds; (2) K-sweep over 10 values spanning three periods of the J₂ oscillation (accelerator windows excluded); (3) random-phase control — identical machinery, θ redrawn uniformly each kick; (4) KAM regime K = 0.5; (5) accelerator regime K = 6.5; (6) verbatim mirror of the module's on-screen K = 5, M = 2400 measurement. Gates A–I, all seeded and deterministic, ~63 s.

The law it recovers

D(K) = (K²/4)·[1 − 2J₂(K)(1 − J₂(K))] — the quasilinear K²/4 dressed by the 2-step angular correlation whose Fourier weight is J₂(K). Sweep: measured D/D_ql − 1 regressed on the RW prediction gives slope 0.968 through the origin, Pearson r 0.990, mean |residual| 0.030 over 10 K values; the sign of every deviation with |prediction| ≥ 0.1 is correct, and the oscillation's phase is pinned by the J₂ zeros (measured D/D_ql ≈ 1 at K = 8.4, 11.6, 18 — all within 0.05 of unity). The residuals (+0.09 at K = 9.2, +0.07 at K = 14) are the known truncation of the second-order Fourier-path expansion at moderate K, disclosed and budgeted, not noise (per-point SE ≈ 0.004).

Measurements, controls & cross-checks

Recovered se

0.147

Rival

Name

quasilinear / independent-kick theory: D ≡ K²/4 for all K in the chaotic sea (no oscillation)

Falsified

D/D_ql = 0.651 at K = 10 and 1.369 at K = 20 — wrong in both directions by 35–37%, hundreds of SE. The decisive control: the SAME estimator on the SAME kick sequence with θ genuinely re-randomized each step returns D/D_ql = 0.9963 at both K = 10 and 20 (|dev| ≤ 0.004) — so independent kicks really do give K²/4, and the deviation of the deterministic map is carried entirely by the angular correlations chaos fails to destroy in one step.

Additional regimes

K = 0.5 < K_c = 0.9716: D/D_ql = 4.9e-4 — KAM tori block transport entirely, quasilinear is infinitely wrong below criticality. K = 6.5 (period-1 accelerator window): D/D_ql = 132 with window ratio 2.20 — superdiffusion, no finite D exists; the estimator detects the anomaly instead of averaging it away.

Module systematic

Mirror K5 M2400

• 6.212
• 6.061
• 6.292
• 6.016

Mirror mean

6.145

Oracle grade K5

D = 6.297 ± 0.017 → net correlation correction +0.8% relative to K²/4 = 6.25

Note

the module's on-screen K = 5 'D ≈ K²/4 to about a percent' is numerically right but was framed as 'quasilinear is best here' — in fact K = 5 sits near J₂'s first zero (5.136) where the RW correction happens to nearly vanish; RW second order alone predicts −8.9% at K = 5 and misses, a disclosed truncation failure of the second-order formula at moderate K (the oracle therefore gates the scalar at K = 20 where truncation is <1%). Module docstring and panel updated to disclose the J₂ story; module M = 2400 statistics give ±0.2 on-screen scatter, consistent with the oracle mirror (|mirror − oracle-grade| = 0.15).

Exec certificate

MEASURED BY EXECUTING THE SHIPPED MODULE (gates J–N, added this run; module file untouched — sha-pinned b84ec8b7…). The shipped ChaoticDiffusionModule.ts, type-stripped and executed headless, is bit-identical to the oracle's gate-I mirror on all 4 mirror seeds (Object.is on D̂) — the on-screen 'D measured' IS the oracle mirror, proven by execution rather than prose. 1500-call lockstep at fl(1/120): the Float32Array θ/p gas + 28,800-float render buffers bit-exact against an op-order replica at EVERY call, with 7 recollapse bursts at pinned calls [206,427,645,852,1068,1284,1494] and a 14,400-draw stream fingerprint (the live stream is a 1800-draw init θ-fill plus 1800-draw bursts whose TIMING rides the f32 chaos — lockstep IS the stream proof). fixedUpdate ignores dt entirely (a dt=999 twin is bit-identical): exactly one map step per engine call, no accumulator regime at all. HUD byte-equal at all 250 every-6th writes (1 distinct — D̂ is init-frozen from the derived mulberry32(seed^0x85ebca6b) f64 stream, fully decoupled from the f32 gas), chart byte-equal to the replica template; the measurement slice is token-free of dQL/'/ 4'/6.25/J₂/Bessel/137 (the on-screen K²/4 = 6.25 is display-only); the estimator is proven live on synthetic inputs (LS(6n)/2 = 3 and LS(100+4n)/2 = 2 exactly — intercept absorbed, integer sums exact in f64; pRms(all-2) = 2); the recollapse threshold is STRICT > at exactly 52 by hand-probe (θ=0 kicks are exactly zero, pRms(52) = 52, no burst; 53 bursts with θ redrawn from stream position 1800).

Screen bias measured

The original finding called mirror-vs-oracle-grade 'consistent' (±0.2 on-screen scatter) — an ESTIMATE, and executing deeper corrects it (#97): at 48 seeds the screen protocol reads D̂ = 6.2015 ± 0.0237 (sd 0.164) vs oracle-grade 6.2971 ± 0.0171 → bias −0.096 = −1.5%, z = −3.3 — a REAL finite-protocol bias, not seed scatter. Mechanism measured, not assumed: the module starts p₀ ~ U[0,2π) (the oracle-grade estimator starts p₀ = 0); a paired p₀-ensemble twin at the same two-time estimator and depth gives p₀=0 minus p₀-uniform = +0.117 ± 0.055, and the gap persists through the 500→2500 window, fading only by n ≈ 2500–5000 — a slow initial-ensemble correlation transient at K = 5 (the reference's disclosed sticky-shoulder region K = 5–7.6), while the estimator form is innocent: late-only LS (n ≥ 180) and two-time twins agree with the module's full-NS LS to z ≈ 0. Both readings sit inside gate I's ±0.45 mirror window; the HUD's 'D measured' is an honest finite-protocol measurement, now priced at −1.5% with its mechanism attributed.

What it reduces to

Chirikov's quasilinear diffusion (Phys. Rep. 52, 263 (1979)) with the Rechester–White correlation correction (Phys. Rev. Lett. 44, 1586 (1980); Rechester–Rosenbluth–White, Phys. Rev. A 23, 2664 (1981)): deterministic chaos produces genuine momentum diffusion whose coefficient oscillates around K²/4 with the Bessel weight J₂(K) of the surviving 2-step angular correlation. Non-circular because the generator contains only the bare map — the recovered ⟨Δp²⟩ growth is linear (window ratios 1.00 ± 0.015), the J₂ oscillation emerges from iteration and is matched against a quadrature computed only in the scorer, and the same machinery run on genuinely independent kicks reproduces the rival's K²/4 instead. The 0.68% deficit at K = 20 and the +0.09/+0.07 sweep residuals at K = 9.2/14 are consistent with the known higher-order Fourier-path terms the second-order RW formula drops (~(2/πK)^{3/2}); anomalous regimes (KAM blocking, accelerator superdiffusion) are gated as regimes where NO diffusion coefficient exists, matching Meiss, Rev. Mod. Phys. 64, 795 (1992).

Confidence & reproduction

Confidence

high

Validation

derisk-pass

Re-run the check

npm run derisk -- chaosdiff (scripts/chaosdiff-derisk.mjs — 14 gates: the 9 physics gates A–I (untouched this run: scalar D(K=20), worst seed, diffusive linearity, J₂-law sweep, quasilinear rival falsified both directions, random-phase control, KAM block, accelerator anomaly, module mirror) + the honest-module certificate J–N (sha-pin + mechanical type-strip + rng-site census + answer-free measurement-slice token scan + stream-position probes; executed init bit== f64/f32 replica with D̂ === gate-I mirror by Object.is; 1500-call bit-exact lockstep with 7 pinned recollapse bursts and a 14,400-draw stream fingerprint; dt-ignore and strict-52-threshold hand-probes; HUD/chart byte-equality with synthetic-input estimator proofs; 48-seed exec closure pricing the screen protocol at −1.5% ± 0.5% vs oracle-grade with the p₀-ensemble mechanism twin), deterministic, ~67 s; tamper self-tests: cert sha ⇒ only J fails; measurement-kick ×1.001 ⇒ K+M+N fail with the 9 physics gates, the lockstep and the recovery untouched; known_value → 130 ⇒ A+B only, recovered value byte-unchanged, restored by hand.)

Oracle

scripts/oracles/chaosdiff.reference.json

Sources

B. V. Chirikov, Phys. Rep. 52, 263 (1979); A. B. Rechester & R. B. White, Phys. Rev. Lett. 44, 1586 (1980); A. B. Rechester, M. N. Rosenbluth & R. B. White, Phys. Rev. A 23, 2664 (1981); J. D. Meiss, Rev. Mod. Phys. 64, 795 (1992)