Pressure plays a vital role in changing the transport properties of matter. To understand this phenomenon at a microscopic level, we here focus on a more fundamental problem, i.e., how pressure affects the thermalization properties of solids. As illustrating examples, we study the thermalization behavior of the monatomic chain and the mass-disordered chain of Fermi-Pasta-Ulam-Tsingou-β under different strains in the thermodynamic limit. It is found that the pressure-induced change in integrability results in qualitatively different thermalization processes for the two kinds of chains. However, for both cases, the thermalization time follows the same law-it is inversely proportional to the square of the nonintegrability strength. This result suggests that pressure can significantly change the integrability of a system, which provides a new perspective for understanding the pressure-dependent thermal transport behavior.
Pressure plays a vital role in changing the transport properties of matter. To understand this phenomenon at a microscopic level, we here focus on a more fundamental problem, i.e., how pressure affects the thermalization properties of solids. As illustrating examples, we study the thermalization behavior of the monatomic chain and the mass-disordered chain of Fermi-Pasta-Ulam-Tsingou-β under different strains in the thermodynamic limit. It is found that the pressure-induced change in integrability results in qualitatively different thermalization processes for the two kinds of chains. However, for both cases, the thermalization time follows the same law-it is inversely proportional to the square of the nonintegrability strength. This result suggests that pressure can significantly change the integrability of a system, which provides a new perspective for understanding the pressure-dependent thermal transport behavior.
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.
Abstract We study the effects of two kinds of weak disorders on the thermalization of the Fermi–Pasta–Ulam–Tsingou model by extensive numerical simulations. The disorders are introduced to the mass of atom or coefficient of the quadratic term of potential energy. The initial energy is distributed equally among some lowest frequency modes. We find that the energy transports to high-frequency modes with time t and eventually approaches energy equipartition faster with either weak disorder than that in the homogeneous case. That means weak disorders accelerate the process of thermalization. We further study the effects of two kinds of disorders on the scaling law of equipartition time T eq. We find that T eq satisfies the following scaling law: T eq ∼ (ɛ)−a (|α|)−b for different disorder strengths in the thermodynamic limit. It is found that the exponent a ≈ 1.0 while b depends on the strength of disorder, which are different from b = 2a in the homogeneous case.
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.
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.
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.
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.