Dynamical thermalization of disordered nonlinear lattices

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.PACS numbers: 05.45.-a, 63.50.-x, 63.70.+h The studies of ergodicity and dynamical thermalization in regular nonlinear lattices have a long history initiated by the Fermi-Pasta-Ulam problem [1] but they are still far from being complete (see, e.g., [2] for thermal transport in nonlinear chains and [3] for thermalization in a Bose-Hubbard model). In this letter, we study how the dynamical thermalization appears in nonlinear disordered chains where all linear modes are exponentially localized. Such modes appear due to the Anderson localization, introduced in the context of electron transport in disordered solids [4,5,6] and describing various physical situations like wave propagation in a random medium [7], localization of a Bose-Einstein condensate [8] and quantum chaos [9].Effects of nonlinearity on localization properties have attracted large interest recently. Indeed, nonlinearity naturally appears for localization of a Bose-Einstein condensate, as its evolution is described by the nonlinear Gross-Pitaevskii equation [10]. An interplay of disorder, localization, and nonlinearity is also important in other physical systems like wave propagation in nonlinear disordered media [11,12] and chains of nonlinear oscillators with randomly distributed frequencies [13].The main question here is whether the localization is destroyed by nonlinearity. It has been addressed recently using two physical setups. In refs. [14,15]it was demonstrated that an initially concentrated wavepacket spreads apparently indefinitely, although subdiffusively, in a disordered nonlinear lattice. For a transmission through a nonlinear disordered layer [16,17], chaotic destruction of localization leads to a drastically enhanced transparency.Here we study the thermalization properties of the dynamics of a nonlinear disordered lattice -discrete Anderson nonlinear Schrödinger equation (DANSE). We describe in details the features of the time-evolution of an initially localized excitation towards a statistical equilibrium in a finite lattice (we stress that this evolution is purely deterministic -and the relaxation to equilibrium is due to determinsitc chaos.). Below we argue that a statistically stationary state is characterized by the Gibbs energy equipartition across the linear eigenmodes (Eq. (5)) and call a relaxation to such an equilibrium state thermalization. Because thermalization is due to deterministic chaos, its rate heavily dependes on the statistical properties of the chaos. As is typical for nonlinear Hamiltonian systems, depending on initial conditions one can obtain solutions belonging to a "chaotic sea" or to "regul...

Dynamical thermalization in Bose-Hubbard systems

“…This feature has been noted and used for nonlinear chains with disorder [31,32] and Bose-Einstein condensates, described by the GrossPitaivskii equation, in chaotic two-dimensional billiards [33]. It is interesting to note that in these nonlinear systems [31][32][33] the DTC is still valid but it is induced by a nonlinear mean-field interactions between linear states. (12) where β and µ were chosen such that k n k = N = 8 and k E k n k + N 2 U/L = Eα with Eα/N corresponding to the centers of the abscissa intervals (e.g.…”

Nonlinear Lattice Waves in Random Potentials

“…n = σ /2 + 1 corresponds to n-body interactions, and σ = 1 relates to quadratic Kerr media in nonlinear optics. Mulansky [55] presented numerical simulations of the gDNLS model for a few integer values of σ and single site excitations, and fitted the dependence m 2 (t) ∼ t α with exponents α which depend on σ (see open circle data in left plot in Fig.10). In [56] numerical simulations of the gDNLS model were performed for non integer values of σ on rather short time scales, leaving the characteristics of the asymptotic (t → ∞) evolution of wave packets aside.…”

Energy spreading in strongly nonlinear disordered lattices

“…In this case, due to Anderson localization, linear eigenmodes are exponentially localized and the spectrum is purely discrete [4]. Recent numerical experiments with nonlinear disordered lattices have demonstrated that the initially localized wave packets spread in a very weak, subdiffusive manner [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. A complete theoretical understanding of the subdiffusive behavior has not been presented, but it is mostly agreed that the spreading in these models is induced by weak chaos.…”