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arXiv:1901.04245 · New Journal of Physics

Universal law of thermalization for one-dimensional perturbed Toda lattices

Weicheng Fu, Yong Zhang, Hong Zhao

Published Mar 20, 2019• 24 citations
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Abstract

The Toda lattice is a nonlinear but integrable system. Here we study the thermalization problem in one-dimensional, perturbed Toda lattices in the thermodynamic limit. We show that the thermalization time, T eq , follows a universal law; i.e. T eq ∼ −2 , where the perturbation strength , , characterizes the nonlinear perturbations added to the Toda potential. This universal law applies generally to weak nonlinear lattices due to their equivalence to perturbed Toda systems.

DOI
10.1088/1367-2630/ab115a

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The Toda lattice is a nonlinear but integrable system. Here we study the thermalization problem in one-dimensional, perturbed Toda lattices in the thermodynamic limit. We show that the thermalization time, T eq , follows a universal law; i.e. T eq ∼ −2 , where the perturbation strength , , characterizes the nonlinear perturbations added to the Toda potential. This universal law applies generally to weak nonlinear lattices due to their equivalence to perturbed Toda systems.
Nonintegrability is a necessary condition for the thermalization of a generic Hamiltonian system. In practice, the integrability can be broken in various ways. As illustrating examples, we numerically studied the thermalization behaviors of two types of one-dimensional (1D) diatomic chains in the thermodynamic limit. One chain was the diatomic Toda chain whose nonintegrability was introduced by unequal masses. The other chain was the diatomic Fermi-Pasta-Ulam-Tsingou-β chain whose nonintegrability was introduced by quartic nonlinear interaction. We found that these two different methods of destroying the integrability led to qualitatively different routes to thermalization in the near-integrable region, but the thermalization time, T_{eq}, followed the same scaling law; T_{eq} was inversely proportional to the square of the perturbation strength. This law also agreed with the existing results of 1D monatomic lattices. All these results imply that there is a universal scaling law of thermalization that is independent of the method of breaking integrability.
Understanding how systems achieve thermalization is a fundamental task in statistical physics. This Letter presents both analytical and numerical evidence showing that thermalization can be universally achieved in sufficiently large two- and three-dimensional lattices via weak nonlinear interactions. Thermalization time follows a universal scaling law unaffected by lattice structures, types of interaction potentials, or whether the lattice is ordered or not. Moreover, this study highlights the critical impact of dimensionality and degeneracy on thermalization dynamics.
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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−2T_{\mathrm{eq}}\propto g^{-2}Teq​∝g−2, where ggg 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 ggg decreases, and thermalization becomes controlled by higher-order processes. Numerical studies reveal successive regimes Teq∝g−γT_{\mathrm{eq}}\propto g^{-γ}Teq​∝g−γ with γ=2,4,6γ=2,4,6γ=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.
Whether and how a system approaches equilibrium is a central issue in nonequilibrium statistical physics, underpinning our understanding of both thermalization and heat transport. Bogoliubov's three-stage (initial, kinetic, and hydrodynamic) evolution hypothesis provides a qualitative framework, but quantitative advances have mostly been restricted to near-integrable systems such as dilute gases. In this work, we investigate the relaxation dynamics of a one-dimensional diatomic hard-point gas and present a phase diagram that comprehensively characterizes relaxation behavior across the full parameter space, from near-integrable to far-from-integrable regimes. We first analyze thermalization (local energy relaxation in nonequilibrium states) and identify three universal dynamical regimes: (1) In the near-integrable regime, kinetic processes dominate (initial + kinetic phase), local energy relaxation decays exponentially, and the thermalization time τ scales with perturbation strength δ as τ ∝ δ − 2 . (2) In the far-from-integrable regime, hydrodynamic effects dominate (initial + hydrodynamic phase), local energy relaxation follows power-law decay, and thermalization time scales linearly with system size N . (3) In the intermediate regime, Bogoliubov phase emerges, marked by the transition from kinetic to hydrodynamic relaxation. The phase diagram further reveals that hydrodynamic behavior can emerge even in small systems when the system is sufficiently far from the integrable regime, challenging the conventional view that such effects are only significant in large systems. In the thermodynamic limit, the system's relaxation behavior depends on the order in which the limits ( N → ∞ or δ → 0 ) are taken. We then turn to heat transport (decay of heat-current fluctuations in equilibrium) and demonstrate that its behavior is fully consistent with the thermalization process, leading to the first unified theoretical description of thermalization and heat transport. Our results establish a comprehensive view of relaxation in classical many-body systems and suggest a route for studying similar dynamics in quantum systems, highlighting universal scaling laws and the interplay of kinetic and hydrodynamic processes across integrability regimes.
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.