  In this letter, a multi-wave quasi-resonance framework is established to
analyze energy diffusion in classical lattices, uncovering that it is
fundamentally determined by the characteristics of eigenmodes. Namely, based on
the presence and the absence of extended modes, lattices fall into two
universality classes with qualitatively different thermalization behavior. In
particular, we find that while the one with extended modes can be thermalized
under arbitrarily weak perturbations in the thermodynamic limit, the other
class can be thermalized only when perturbations exceed a certain threshold,
revealing for the first time the possibility that a lattice cannot be
thermalized, violating the hypothesis of statistical mechanics. Our study
addresses conclusively the renowned Fermi-Pasta-Ulam-Tsingou problem for large
systems under weak perturbations, underscoring the pivotal roles of both
extended and localized modes in facilitating energy diffusion and
thermalization processes.
