Integrable systems have as many conserved quantities as degrees of freedom. Energy, momentum, and an infinite tower of higher conservation laws constrain the dynamics so completely that thermalization — the loss of memory of initial conditions — cannot occur. Break the integrability and thermalization proceeds. But how quickly depends on how the integrability is broken.
Weak integrability breaking perturbations produce quasi-conserved quantities — quantities that are almost but not exactly conserved (arXiv:2603.11712). The quasi-conservation slows thermalization dramatically. The system thermalizes eventually, but on timescales far longer than the dynamical timescale, because the nearly-conserved quantities constrain the accessible phase space almost as effectively as exact conservation would.
The alpha-FPUT model — the celebrated Fermi-Pasta-Ulam-Tsingou chain with cubic nonlinearity — is proven to be a genuine weak integrability breaking perturbation of the harmonic oscillator chain. The cubic nonlinearity is not an arbitrary perturbation; it is structurally special in that it generates an infinite hierarchy of quasi-conserved quantities. This explains the FPUT recurrence: the system appeared to return to its initial state because the quasi-conservation laws restricted it to a submanifold of phase space that repeatedly passed near the starting configuration.
Any cubic, translationally invariant perturbation of the harmonic chain is a weak integrability breaking perturbation. The result is not specific to the FPUT Hamiltonian but to the symmetry class. Cubic perturbations of any harmonic system produce anomalous thermalization because they all generate quasi-conservation hierarchies.