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arXiv:0903.2191 · Physical Review E

Dynamical thermalization of disordered nonlinear lattices

Mario Mulansky, Karsten Ahnert, ARKADY S. PIKOVSKY, Dima L. Shepelyansky

Published Nov 24, 2009• 74 citations
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Abstract

We study numerically how the energy spreads over a finite disordered nonlinear one-dimensional lattice, where all linear modes are exponentially localized by disorder. We establish emergence of dynamical thermalization characterized as an ergodic chaotic dynamical state with a Gibbs distribution over the modes. Our results show that the fraction of thermalizing modes is finite and grows with the nonlinearity strength.

DOI
10.1103/physreve.80.056212

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We study numerically how the energy spreads over a finite disordered nonlinear one-dimensional lattice, where all linear modes are exponentially localized by disorder. We establish emergence of dynamical thermalization characterized as an ergodic chaotic dynamical state with a Gibbs distribution over the modes. Our results show that the fraction of thermalizing modes is finite and grows with the nonlinearity strength.
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We study numerically time evolution in classical lattices with weak or moderate nonlinearity which leads to interactions between linear modes. Our results show that in a certain strength range a moderate nonlinearity generates a dynamical thermalization process which drives the system to the quantum Gibbs distribution of probabilities, or average oscillation amplitudes. The effective dynamical temperature of the lattice varies from large positive to large negative values depending on the energy of the initially excited modes. This quantum Gibbs distribution is drastically different from the usually expected energy equipartition over linear modes corresponding to a regime of classical thermalization. Possible experimental observations of this dynamical thermalization are discussed for cold atoms in optical lattices, nonlinear photonic lattices and optical fiber arrays.
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The interplay of fluctuations, ergodicity, and disorder in many-body interacting systems has been striking attention for half a century, pivoted on two celebrated phenomena: Anderson localization predicted in disordered media, and Fermi-Pasta-Ulam-Tsingou (FPUT) recurrence observed in a nonlinear system. The destruction of Anderson localization by nonlinearity and the recovery of ergodicity after long enough computational times lead to more questions. This thesis is devoted to contributing to the insight of the nonlinear system dynamics in and out of equilibrium. Focusing mainly on the GP lattice, we investigated elementary fluctuations close to zero temperature, localization properties, the chaotic subdiffusive regimes, and the non-equipartition of energy in non-Gibbs regime. Initially, we probe equilibrium dynamics in the ordered GP lattice and report a weakly non-ergodic dynamics, and an ergodic part in the non-Gibbs phase that implies the Gibbs distribution should be modified. Next, we include disorder in GP lattice, and build analytical expressions for the thermodynamic properties of the ground state, and identify a Lifshits glass regime where disorder dominates over the interactions. In the opposite strong interaction regime, we investigate the elementary excitations above the ground state and found a dramatic increase of the localization length of Bogoliubov modes (BM) with increasing particle density. Finally, we study non-equilibrium dynamics with disordered GP lattice by performing novel energy and norm density resolved wave packet spreading. In particular, we observed strong chaos spreading over several decades, and identified a Lifshits phase which shows a significant slowing down of sub-diffusive spreading.
How a closed system thermalizes, especially in the absence of global conservation laws but in the presence of disorder and interactions, is one of the central questions in nonequilibrium statistical mechanics. We explore this for a disordered, periodically driven Ising chain. Our numerical results reveal inhomogeneous thermalization leading to a distribution of thermalization timescales within a single disordered sample, which we encode via a distribution of effective local temperatures. Using this, we find an excellent collapse without any fitting parameters of the local relaxation dynamics for the entire range of disorder values in the ergodic regime when adapting the disorder-averaged diagonal entanglement entropy as internal ``time'' of the system. This approach evidences a remarkably uniform parametrization of the dynamical many-body evolution of local temperature within the otherwise highly heterogeneous ergodic regime, independent of the strength of the disorder.
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.