Dynamical thermalization of Frenkel-Kontorova model in the thermodynamic limit

Dynamic crossover towards energy equipartition in the Fermi-Pasta-Ulam-Tsingou β model with long-range interactions

Energy equipartition can be established in short-range systems after the dynamic process of thermalization. However, energy distribution between different degrees of freedom in systems with long-range interactions is unclear. We study the dynamics of energy relaxation in the Fermi-Pasta-Ulam-Tsingou β model with long-range quartic interactions, which decay as 1/d^{δ} with d being the lattice distance. The dynamic crossover of a mode-energy distribution from localized to equipartitioned with the increase of the power δ is observed. A transition of mode-energy distribution is identified around the value of δ=1, which usually serves as the distinction between strong and weak long-range couplings. We elucidate that the varying frequency overlapping of the mode-energy power spectrum is responsible for this dynamic crossover. Through further calculation of the spectral entropy, the minimum duration of quasistationary states, τ_{QSS}, is found at δ=2, which may provide possible dynamic explanations for the peculiar behavior of heat transport in long-range lattice chains. In addition, the double scaling in τ_{QSS} as a function of energy density is also observed in our long-range lattices. Our results not only contribute to understanding the dynamics of energy relaxation in long-range systems, but also shed light on the longstanding problem of thermalization and low-dimensional heat transport in short-range systems.

Wave-Turbulence Origin of the Instability of Anderson Localization against Many-Body Interactions

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.

Universal scaling of the thermalization time in one-dimensional lattices

We show that, in the thermodynamic limit, a one-dimensional (1D) nonlinear lattice can always be thermalized for arbitrarily small nonlinearity, thus proving the equipartition theorem for a class of systems. Particularly, we find that in the lattices with nearest-neighbor interaction potential V(x)=x^{2}/2+λx^{n}/n with n≥4, the thermalization time, T_{eq}, follows a universal scaling law; i.e., T_{eq}∝λ^{-2}ε^{-(n-2)}, where ε is the energy per particle. Numerical simulations confirm that it is accurate for an even n, while a certain degree of deviation occurs for an odd n, which is attributed to the extra vibration modes excited by the asymmetric interaction potential. This finding suggests that although the symmetry of interactions will not affect the system reaching equipartition eventually, it affects the process toward equipartition. Based on the scaling law found here, a unified formula for the thermalization time of a 1D general nonlinear lattice is obtained.

Dynamical glass in weakly nonintegrable Klein-Gordon chains

Integrable many-body systems are characterized by a complete set of preserved actions. Close to an integrable limit, a nonintegrable perturbation creates a coupling network in action space which can be short or long ranged. We analyze the dynamics of observables which become the conserved actions in the integrable limit. We compute distributions of their finite time averages and obtain the ergodization time scale T_{E} on which these distributions converge to δ distributions. We relate T_{E} to the statistics of fluctuation times of the observables, which acquire fat-tailed distributions with standard deviations σ_{τ}^{+} dominating the means μ_{τ}^{+} and establish that T_{E}∼(σ_{τ}^{+})^{2}/μ_{τ}^{+}. The Lyapunov time T_{Λ} (the inverse of the largest Lyapunov exponent) is then compared to the above time scales. We use a simple Klein-Gordon chain to emulate long- and short-range coupling networks by tuning its energy density. For long-range coupling networks T_{Λ}≈σ_{τ}^{+}, which indicates that the Lyapunov time sets the ergodization time, with chaos quickly diffusing through the coupling network. For short-range coupling networks we observe a dynamical glass, where T_{E} grows dramatically by many orders of magnitude and greatly exceeds the Lyapunov time, which satisfies T_{Λ}≲μ_{τ}^{+}. This effect arises from the formation of highly fragmented inhomogeneous distributions of chaotic groups of actions, separated by growing volumes of nonchaotic regions. These structures persist up to the ergodization time T_{E}.

Intermittent many-body dynamics at equilibrium

The equilibrium value of an observable defines a manifold in the phase space of an ergodic and equipartitioned many-body system. A typical trajectory pierces that manifold infinitely often as time goes to infinity. We use these piercings to measure both the relaxation time of the lowest frequency eigenmode of the Fermi-Pasta-Ulam chain, as well as the fluctuations of the subsequent dynamics in equilibrium. The dynamics in equilibrium is characterized by a power-law distribution of excursion times far off equilibrium, with diverging variance. Long excursions arise from sticky dynamics close to q-breathers localized in normal mode space. Measuring the exponent allows one to predict the transition into nonergodic dynamics. We generalize our method to Klein-Gordon lattices where the sticky dynamics is due to discrete breathers localized in real space.