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

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 $T_{\rm eq}\sim g^{-γ}$ with $γ=2$, 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 $T_{\rm eq}\sim g^{-γ}$, the exponent $γ$ 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.

Figures (8)

• Figure 1: Example where the harmonic Hamiltonian is chosen as the integrable reference system, $H=H_0+\sum_{j}\frac{1}{d}|x_{j+1}-x_j|^d$, with $H_0=\sum_{j}\left[\frac{1}{2}p_j^2+\frac{1}{2}(x_{j+1}-x_j)^2\right]$. (Reprinted from Fu:2019R).
• Figure 2: Example where the nonlinear integrable Toda Hamiltonian is chosen as the reference system, $H=H_0+\sum_{j}\frac{1}{n}(x_{j+1}-x_j)^n$, with $H_0=\sum_{j}\left[\frac{1}{2}p_j^2+V_{\rm T}(\alpha,x_{j+1}-x_j)\right]$, where $V_{\rm T}(\alpha,x)=\frac{e^{2\alpha x}-2\alpha x-1}{4\alpha^2}$ is the Toda potential. (Reprinted from Fu:2019).
• Figure 3: Thermalization time as a function of mass inhomogeneity in diatomic lattice systems. (a) Integrability of the nonlinear Toda Hamiltonian is broken by introducing unequal masses, leading to a perturbation whose strength is characterized by the mass difference. The thermalization time follows the inverse-square scaling with the perturbation strength. (b) Results for the diatomic FPUT-$\beta$ chain shown for comparison. In this case the relevant integrable reference system is the harmonic Hamiltonian with the same mass configuration, and therefore the mass difference does not change the effective perturbation strength. (Reprinted from PhysRevE.100.052102)
• Figure 4: Thermalization scaling in 1D disordered lattices. (a) $T_{\text{eq}}$ vs $\varepsilon$ for polynomial potentials of order $n=3,4,5$ (from bottom to top). Data converge to the predicted scaling $T_{\text{eq}}\propto\varepsilon^{1-n}$, confirming the universal law. (b) For the disordered Lennard-Jones potential, the scaling follows $T_{\text{eq}}\propto\varepsilon^{-1}$. (Reprinted from Wang:2020)
• Figure 5: Scaling of thermalization time $T_{\text{eq}}$ with energy density $\varepsilon$ in high-dimensional nonlinear lattices. (a) Ordered and disordered hexagonal (2D) and face‑centered cubic (3D) lattices. (b) Finite‑size effects in the simple cubic lattice. (c) Finite‑size effects in the square lattice. (d) Influence of mass disorder on thermalization in the square lattice. (Reprinted from from Wang:2024)