The Fermi-Pasta-Ulam-Tsingou problem after 70 years: Universal laws of thermalization in lattice systems

TL;DR

This paper reviews 70 years of research on the Fermi-Pasta-Ulam-Tsingou problem, revealing universal laws of thermalization in lattice systems, classifying systems into two types based on mode localization and their thermalization behavior.

Contribution

It provides a comprehensive classification of lattice systems into two universal classes and elucidates the mechanisms governing their thermalization, including effects of disorder and localization.

Findings

Class 1 systems thermalize with time scaling as g^{-2}.

Class 2 systems' thermalization time is system-size independent.

Disorder can accelerate or suppress thermalization depending on the class.

Abstract

Over the past decade, substantial progress has been made in clarifying a central question of the Fermi-Pasta-Ulam-Tsingou problem: whether weakly nonlinear lattice systems thermalize and, if so, through what mechanisms. The current understanding is as follows. (a) Classical lattice systems fall into two universal classes. In the first, the Hamiltonian has extended normal modes. For sufficiently large systems, the thermalization time scales as $T_{\rm eq}\sim g^{-\gamma}$ with $\gamma=2$, where $g$ denotes the effective nonlinear strength, i.e., the perturbation strength or degree of non-integrability. Thus, in the thermodynamic limit, these systems inevitably thermalize. Typical examples include common one-, two-, and three-dimensional lattice models. In the second class, all normal modes are localized. Here the relaxation time is essentially independent of system size. Although one may still formally write $T_{\rm eq}\sim g^{-\gamma}$, the exponent $\gamma$ diverges as $g\to0$, implying that arbitrarily weak nonlinear perturbations cannot induce thermalization. For sufficiently small $g$, such systems may therefore be viewed, in a theoretical sense, as thermal insulators. (b) In systems of the first class, disorder does not obstruct thermalization. Rather, by breaking translational symmetry and relaxing wave-vector resonance constraints, it increases the number of quasi-resonant processes and can therefore accelerate thermalization. (c) In systems of the second class, when both on-site potentials and disorder are present, all normal modes become localized in sufficiently large systems, suppressing thermalization. The perturbative framework underlying these conclusions will also be presented systematically, with particular emphasis on the thermalization criterion based on resonance-network connectivity, an approach rooted in weak wave turbulence theory.