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Physical review. E

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

Jian Wang, Ai-chen Li

Published Jul 25, 2022• 6 citations

Abstract

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.

DOI: 10.1103/physreve.106.014135

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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.

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

T_{\mathrm{eq}}\propto 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

T_{\mathrm{eq}}\propto 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.

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.

Abstract We investigate the equilibration process of the strongly coupled quartic Fermi–Pasta–Ulam–Tsingou model by adding Langevin baths to the ends of the chain. The time evolution of the system is investigated by means of extensive numerical simulations and shown to match the results expected from equilibrium statistical mechanics in the time-asymptotic limit. Upon increasing the nonlinear coupling, the thermalization of the energy spectrum displays an increasing asymmetry in favor of small-scale, high-frequency modes, which relax significantly faster than the large-scale, low-frequency ones. The global equilibration time is found to scale linearly with system size and shown to exhibit a power-law decay with the strength of the nonlinearity and temperature. Nonlinear interaction adds to energy distribution among modes, thus speeding up the thermalization process.

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Abstract We present a numerical study of dynamical correlations (structure factors) of the long-range generalization of the Fermi–Pasta–Ulam oscillator chain, where the strength of the interaction between two lattice sites decays as a power of the inverse of their distance. The structure factors at finite energy density display distinct peaks, corresponding to long-wavelength propagating modes, whose dispersion relation is compatible with the predictions of the linear theory. We demonstrate that dynamical scaling holds, with a dynamical exponent z that depends weakly on in the range . The lineshapes have a non-trivial functional form and appear somehow independent of . Within the accessible time and size ranges, we also find that the short-range limit is hardly attained even for relatively large values of .

We study the statistics and short-time dynamics of the classical and the quantum Fermi-Pasta-Ulam chain in the thermal equilibrium. We analyze the distributions of single-particle configurations by integrating out the rest of the system. At low temperatures, we observe a systematic increase in the mobility of the chain when transitioning from classical to quantum mechanics due to zero-point energy effects. We analyze the consequences of quantum dispersion on the dynamics at short times of configurational correlation functions.