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.
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.
The Fermi-Pasta-Ulam-Tsingou problem, i.e., the problem of energy equipartition among normal modes in a weakly nonlinear lattice, is here studied in two types of two-dimensional (2D) lattices, more precisely in lattices with square cell and triangular cell. We apply the wave-turbulence approach to describe the dynamics and find multi-wave resonances play a major role in the transfer of energy among the normal modes. We show that, in general, the thermalization time in 2D systems is inversely proportional to the squared perturbation strength in the thermodynamic limit. Numerical simulations confirm that the results are consistent with the theoretical prediction no matter systems are translation-invariant or not. It leads to the conclusion that such systems can always be thermalized by arbitrarily weak many-body interactions. Moreover, the validity for disordered lattices implies that the localized states are unstable.
The Fermi--Pasta--Ulam--Tsingou (FPUT) problem provides a paradigmatic framework for understanding thermalization in weakly nonlinear many-body Hamiltonian systems. This focused review summarizes major developments in near-integrable dynamics, wave resonances, and phonon kinetic theory, emphasizing recent results on thermalization-time scaling in nonlinear lattices. A coherent picture emerges in the thermodynamic limit based on the eigenmode properties of an appropriate integrable reference system. If the reference system has extended eigenmodes and the leading resonant or quasi-resonant processes form a sufficiently connected network, the thermalization time follows Teq∝g−2, where g measures the effective deviation from integrability. This scaling is found broadly in ordered and weakly disordered lattices and is robust to dimensionality, interaction potential, integrability-breaking mechanism, and multimode initial conditions. Identifying the correct integrable reference is essential; in some one-dimensional FPUT-type lattices with cubic interactions, the nearby Toda lattice, rather than the harmonic chain, provides the proper reference. If all reference eigenmodes are localized, spatial-overlap constraints progressively fragment low-order resonance networks as g decreases, and thermalization becomes controlled by higher-order processes. Numerical studies reveal successive regimes Teq∝g−γ with γ=2,4,6, together with weak system-size dependence. Whether this hierarchy persists asymptotically and whether a finite thermalization threshold exists remain open questions. We also discuss finite-size effects, strongly nonintegrable dynamics, heat transport, localization, and higher-dimensional lattices.
Multi-phonon resonance conditions underpin kinetic theories of phonon transport and lattice thermalization. We show that exact resonance matching, nonzero interaction coefficients, and network connectivity do not guarantee persistent energy diffusion. Symmetry-enforced balance relations drive exact-resonant collision currents to nonthermal zero-flux states, producing kinetic arrest from individual resonant sets to connected networks. Complete energy spreading is sustained by quasi-resonances. The thermodynamic and weak-nonlinearity limits do not commute: the leading kinetic behavior is recovered in the former, whereas at fixed finite size the thermalization time diverges through higher-order crossovers as the nonlinearity vanishes. Exact-resonance existence and connectivity are therefore kinematic, not sufficient dynamical, criteria for phonon energy diffusion.
Whether Anderson localization is robust against many-body interactions and, closely related, whether a disordered many-body system can be thermalized are long outstanding issues. In this Letter, we address these issues with the wave-turbulence theory. We show that, in general, the thermalization time in one-dimensional disordered lattice systems is inversely proportional to the squared interaction strength in the thermodynamic limit. It leads to the conclusion that such systems can always be thermalized by arbitrarily weak many-body interactions and thus the localized states are unstable.
In their seminal work, Fermi, Pasta, Ulam and Tsingou explored the connection between statistical mechanics and dynamical properties, such as chaos and ergodicity. Even today, seventy years later, the topic is not fully understood: while most results of statistical mechanics require the ergodic hypothesis to be rigorously proved, there are many indications that these predictions, both in and out of equilibrium, hold even in the absence of a rigorous form of ergodicity. Motivated by the above considerations, in this work we reconsider the point of view that the relevant ingredients for the validity of statistical mechanics are the large number of degrees of freedom and the choice of extensive observables, while the details of the dynamics do not play an essential role. This is the idea behind Khinchin's famous proof of the typicality of macroscopic observables at equilibrium. We extend this perspective to the context of non equilibrium, by investigating the thermalization properties of both harmonic (integrable) and nonharmonic (chaotic) oscillator chains initially prepared in out-of-equilibrium conditions. In integrable systems, thermalization occurs, or not, depending on the observable. In the chaotic regime, instead, thermalization is reached by any observable, although the relaxation timescale might be larger than the observation time.
We show how to systematically construct weak integrability breaking perturbations (WIBs) for classical integrable models on the lattice. These perturbations, which allow quasi-conserved quantities, have mostly been explored in quantum systems, where they are expected to delay the onset of thermalization and diffusive transport to timescales far exceeding those predicted by Fermi's golden rule. However, accessing such long-time dynamics in quantum models is computationally challenging. Classical integrable lattice models offer a complementary setting for probing transport and long-time dynamics under WIBs. In this work, we specialize our general framework to construct several families of WIBs for the Ishimori model, the Toda chain, and the Harmonic Oscillator Chain (HOC). Such constructions can help quantify how WIBs contribute to anomalous transport and serve as a benchmark for thermalization studies in perturbed integrable models. An important example is the Fermi-Pasta-Ulam-Tsingou (FPUT) model: Starting from the HOC, we show that the cubic nonlinearity (the alpha-FPUT interaction) is a genuine WIB perturbation. Using the integrals of motion (IoMs) of the Toda lattice, we explicitly construct corrections to the entire hierarchy of the HOC IoMs, thereby obtaining an infinite tower of quasi-conserved quantities for the alpha-FPUT chain. We further identify the corresponding adiabatic gauge potential (AGP) as a nontrivial trilocal generator in real space, and show that, more generally, any cubic, translationally invariant, momentum-conserving perturbation of the HOC admits such a generator and is therefore a WIB. Together with our transport and AGP-variance studies, our results provide a unified classical framework for weak integrability breaking and for diagnosing anomalous thermalization and transport in nearly integrable Hamiltonian lattice systems.