Energy refuses to thermalise

Make a chain of springs slightly nonlinear and put all the energy in one mode: does the energy spread out and thermalise (equipartition — the equal sharing over all modes that statistical mechanics assumes), or does it do something else?

Units: dimensionless — the equipartition prediction for the mode-1 energy fraction, 1/(N−1) with N−1 = 31 modes; the value mode 1 would relax to IF the chain thermalised. This is the prediction the experiment FALSIFIES.

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The finding

Energy refuses to thermalise: the Fermi–Pasta–Ulam–Tsingou recurrence — mode-1 energy in a weakly nonlinear chain returns to 97.8% ± 0.2% at ≈146.5 fundamental periods over 24 seeds (30× the equipartition share), with the effective mode number pinned at ≈5.6 of 31, falsifying the naive equipartition hypothesis of statistical mechanics (Fermi–Pasta–Ulam–Tsingou 1955)

Method

Integrate the fixed-end FPUT-α chain ẍ_j = (x_{j+1}+x_{j−1}−2x_j) + α[(x_{j+1}−x_j)²−(x_j−x_{j−1})²] (N=32, α=0.25) with energy-conserving velocity Verlet from an initial mode-1 profile x_j = A·sin(jπ/N), and project onto the linear normal modes each sample to read the per-mode energies E_k = ½Q̇_k² + ½ω_k²Q_k² (ω_k = 2·sin(kπ/2N)). Two emergent observables are measured per run: the recurrence fraction (the fraction of the total energy that RETURNS to mode 1 after the initial drain) and the effective number of excited modes n_eff = exp(−Σ p_k ln p_k), p_k = E_k/E_tot. Sweep 24 seeds (mode-1 IC perturbed by 1% white noise) for uncertainty. The equipartition prediction it is tested against — mode-1 share → 1/(N−1) = 0.0323, n_eff → 31 — is loaded ONLY to score the falsification and never touches the dynamics. Non-circular on four fronts: (i) velocity Verlet conserves the true Hamiltonian to < 1e-3 so the recurrence is dynamical, not drift; (ii) a linear control (α=0) shows mode 1 is an exact eigenmode with zero transfer (n_eff = 1), so the energy flow is genuinely the nonlinearity, not a projection leak; (iii) a stronger-nonlinearity gauge (α=0.7, A=2.5) makes n_eff climb 2.8× toward equipartition, proving n_eff tracks thermalization and isn't pinned low by construction; (iv) the fundamental frequency ω_1 is recovered BLIND from the linear chain's mode-1 zero-crossings, matching 2·sin(π/2N) to 0.000%.

The law it recovers

FPUT-α chain: ẍ_j = (x_{j+1}+x_{j−1}−2x_j) + α[(x_{j+1}−x_j)²−(x_j−x_{j−1})²]; mode energies E_k = ½Q̇_k² + ½ω_k²Q_k², ω_k = 2·sin(kπ/2N); fundamental period T1 = 2π/ω_1 ≈ 64.03.

Measurements, controls & cross-checks

Recovered recurrence fraction

0.978

Recovered recurrence uncertainty

0.0018

Worst seed recurrence

0.9745

Recurrence over equipartition

30.3

Recurrence time T1

146.5

Recurrence time T1 uncertainty

0.2

Drain min fraction

0.087

N eff max

5.62

N eff at equipartition

31

Energy drift max

4.2400e-5

Linear control

Name

linear chain α=0 — springs made exactly harmonic

Mode1 frac min

1

N eff max

1

Note

With no nonlinearity mode 1 is an exact normal mode: the energy placed there stays there forever (a phonon never decays), fraction 1.000, n_eff = 1.000. This proves the transfer seen in the FPUT case is driven by the α-nonlinearity, not a leak in the mode projection.

Thermalization gauge

Name

stronger-nonlinearity gauge α=0.7, A=2.5 — pushed toward the stochasticity threshold

N eff max

15.55

Ratio over base

2.77

Note

Raising the energy/nonlinearity spreads the energy over many more modes (n_eff climbs from ≈5.6 toward the equipartition 31), 2.8× the base — so n_eff faithfully gauges thermalization and the base localization at ≈5.6 is a real physical fact. Above Izrailev–Chirikov's energy threshold the same chain does head for equipartition.

Dispersion crosscheck

Omega1 recovered

0.098135

Omega1 expected

0.098135

Rel error

0

Note

ω_1 recovered blind from the linear chain's mode-1 zero-crossings reproduces the analytic 2·sin(π/2N) to 0.000%, confirming the normal-mode machinery used to read the energies is correct (the frequency is never fed into the measurement).

Screen reconciliation

Url

?world=fput

Hud

Min f1

0.056727618

Recur frac

0.9815066

Recur t

9374.05

Recur t over T1

146.41072

Energy drift

3.6078e-5

Displayed

drains to 6%, returns to 98% · recurrence at ≈ 146 periods · energy drift 3.6e-5

Protocol note

The module's HUD is fully DETERMINISTIC (no RNG, no seed): _runTrace integrates the EXACT noiseless mode-1 IC x_j = A·sin(jπ/N) over trace_time 20000 (~2 recurrences; the oracle ensemble uses 12000 with 1%-IC-noise seeds) and the four static HUD values are read off that trace. The derisk transcribes _runTrace verbatim, pins the five trace floats bit-exact (strict ===) plus the four HUD strings, and the live browser HUD was read back headlessly (vite preview + headless Chrome) and matches all four pinned strings verbatim.

Ensemble reconciliation

Recur frac noiseless minus mean

0.0035

Recur frac sigmas above

1.9

Recur time sigmas

0.6

Note

A-priori DIRECTIONAL gates: IC noise degrades the FPUT recurrence (an imperfect mode-1 profile seeds incoherent background that never rephases), so the noiseless HUD return 0.9815 must sit at or above the perturbed 24-seed mean 0.9780 ± 0.0018, within 5 cloud-σ (observed +1.9σ — the noiseless run is the ensemble's upper envelope, as predicted); the recurrence time 146.41 T1 lands 0.6σ inside the ensemble's 146.5 ± 0.21 T1.

Module systematics

Drain depth ic sensitivity

The HUD's 'drains to 6%' (noiseless min_f1 = 0.0567) is NOT the ensemble's drain (mean 0.0873): drain depth is the IC-noise-sensitive observable — a coherent mode-1 IC transfers more of its energy out at mid-cycle, so the noiseless run drains DEEPER (gated directionally a-priori, min_f1 ≤ ensemble mean). The recurrence fraction and time, by contrast, are noise-robust; the on-screen drain percentage should be read as the exact-IC value, not an ensemble property.

Second recurrence

The module's 2-recurrence window contains a second return of 0.9415 at 2.006× the first recurrence time (pinned bit-exact; gated ≥ 0.9 at 1.9–2.1×): the recurrence is near-periodic with a ~4.0% per-cycle degradation, the visible fact in the module's chart. The HUD's 'returns to 98%' is the FIRST (strongest) return — the max-picker over the full window lands on it because the second is weaker.

Window note

HUD trace_time 20000 vs oracle ensemble trace_time 12000: the ensemble window holds exactly one recurrence, so ensemble and HUD measure the same first-return observable; the longer HUD window adds only the (weaker) second return.

Reference integrity gate

known_value must equal the derived 1/(N−1) for the simulated chain (< 1e-15) — repairs the previously-vacuous known-value tamper test: the falsification-margin gates have deliberate headroom (10×), so a tampered reference now fails on self-consistency instead of slipping under the margin.

What it reduces to

The 1955 Fermi–Pasta–Ulam–Tsingou computer experiment (Los Alamos LA-1940), the birth of nonlinear dynamics as an experimental science. It VALIDATES a landmark result rather than deriving a constant: seeding all the energy in mode 1 of a weakly nonlinear chain, the lab shows the energy does NOT equipartition — instead it drains into a handful of low modes (effective mode number n_eff ≈ 5.6 of 31, never approaching the 31 of a thermalised chain) and then RECURS, returning to 97.8% ± 0.2% of the total in mode 1 at ≈146.5 fundamental periods, robust across 24 perturbed seeds (recurrence fraction std 0.0018). That is ~30× the equipartition prediction 1/(N−1) = 0.032 the chain was expected to relax to — the naive ergodic hypothesis of statistical mechanics simply fails on these timescales. The result is made non-circular four ways: energy is conserved to 4×10⁻⁵ (the recurrence is dynamical, not numerical drift); a linear α=0 control shows zero mode transfer (n_eff = 1) so the flow is genuinely the nonlinearity; a stronger-nonlinearity gauge drives n_eff 2.8× higher toward equipartition, proving the localization metric is not pinned low; and ω_1 is recovered blind from the linear dispersion to 0.000%. It does NOT derive a fundamental constant or claim eternal non-thermalisation — above the Izrailev–Chirikov energy threshold the chain does thermalise; it demonstrates that below it a nonlinear system near-integrably recurs. This near-integrability is the same fact Zabusky & Kruskal explained in 1965 by mapping the chain to the KdV equation (coining 'soliton') — the continuum sequel is the lab's ?world=kdv, and the linear limit is ?world=phonons.

Confidence & reproduction

Confidence

high

Validation

derisk-pass

Re-run the check

npm run derisk -- fput (scripts/fput-derisk.mjs)

Oracle

scripts/oracles/fput.reference.json

Sources

E. Fermi, J. Pasta, S. Ulam & M. Tsingou, 'Studies of Nonlinear Problems', Los Alamos report LA-1940 (1955), reprinted in Collected Papers of Enrico Fermi, Vol. II, p. 978. N. J. Zabusky & M. D. Kruskal, Phys. Rev. Lett. 15, 240 (1965) — the KdV/soliton explanation. F. M. Izrailev & B. V. Chirikov, Sov. Phys. Dokl. 11, 30 (1966) — the energy (stochasticity) threshold for thermalisation. Classic α-model: N=32, α=0.25, all energy in mode 1.