In 1955, Fermi, Pasta, Ulam, and Tsingou simulated a chain of nonlinear oscillators expecting to see energy spread uniformly across all modes — thermalization. Instead, the energy returned nearly to its initial state, cycling through a small number of modes without reaching equilibrium. The FPUT paradox: simple nonlinearity doesn't guarantee thermalization.
Seventy years later (arXiv:2603.23347), the picture is richer but the paradox persists in modified form. Thermalization does occur, but on timescales that depend sensitively on the energy and the type of nonlinearity. Below a critical energy, the FPUT recurrences dominate and the system is effectively integrable. Above it, energy spreads across modes and equipartition is eventually reached.
The universal patterns are the interesting part. The transition between integrable and thermalizing behavior, the scaling of thermalization time with energy, and the role of resonances between modes follow common patterns across different types of nonlinearity. The details of the lattice matter less than the energy regime and the mode structure.
The through-claim: the FPUT paradox was never about whether nonlinear systems thermalize — they do, eventually. It was about the existence of a long intermediate regime where the system behaves as though it's integrable despite being nonlinear. Seventy years of study have characterized this intermediate regime rather than resolving it. The paradox became the subject.