Make a chain of springs slightly nonlinear, put all the energy in one mode — does it spread out and thermalise (equipartition), as statistical mechanics assumes?
▶ Launch the interactive simulationIntegrate the 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, fixed ends) with energy-conserving velocity Verlet, starting all the energy in mode 1. Project onto normal modes each step and watch the per-mode energy fractions E_k(t)/E_tot over ~two recurrence periods.
the FPUT recurrence (1955): energy does NOT equipartition. It flows into a handful of low modes — mode 1 drains to ~6% — and then almost entirely RETURNS to mode 1 (~98%), the chain nearly retracing its initial state, periodically. The energy never reaches the equipartition line E_tot/N_modes. This non-thermalisation in a nonlinear system seeded soliton theory, KAM, and the modern study of integrability — the surprise that the obvious assumption of statistical mechanics simply failed.