PHYSICAL REVIEW E 8, 56 9 Dynaical theralization of disordered nonlinear lattices Mario Mulansky, Karsten Ahnert, Arkady Pikovsky,, and Dia L. Shepelyansky, Departent of Physics and Astronoy, Potsda University, Karl-Liebknecht-Straße, D-76 Potsda-Gol, Gerany Laboratoire de Physique Théorique (IRSAMC), Université de Toulouse UPS, F-6 Toulouse, France LPT (IRSAMC), CNRS, F-6 Toulouse, France Received March 9; revised anuscript received October 9; published Noveber 9 We study nuerically how the energy spreads over a finite disordered nonlinear one-diensional lattice, where all linear odes are exponentially localized by disorder. We establish eergence of dynaical theralization characterized as an ergodic chaotic dynaical state with a Gibbs distribution over the odes. Our results show that the fraction of theralizing odes is finite and grows with the nonlinearity strength. DOI:./PhysRevE.8.56 PACS nuber s : 5.5. a, 6.5. x, 6.7. h The studies of ergodicity and dynaical theralization in regular nonlinear lattices have a long history initiated by the Feri-Pasta-Ula proble but they are still far fro being coplete see, e.g., for theral transport in nonlinear chains and for theralization in a Bose-Hubbard odel. In this paper, we study how the dynaical theralization appears in nonlinear disordered chains where all linear odes are exponentially localized. Such odes appear due to the Anderson localization introduced in the context of electron transport in disordered solids 6 and describing various physical situations such as wave propagation in a rando ediu 7, localization of a Bose-Einstein condensate 8, and quantu 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. An interplay of disorder, localization, and nonlinearity is also iportant in other physical systes such as wave propagation in nonlinear disordered edia, and chains of nonlinear oscillators with randoly distributed frequencies. The ain question here is whether the localization is destroyed by nonlinearity. It has been addressed recently using two physical setups. In Refs.,5 it was deonstrated that an initially concentrated wave packet spreads apparently indefinitely, although subdiffusively, in a disordered nonlinear lattice. For a transission through a nonlinear disordered layer 6,7, chaotic destruction of localization leads to a drastically enhanced transparency. Here we study the theralization properties of the dynaics of a nonlinear disordered lattice discrete Anderson nonlinear Schrödinger equation DANSE. We describe in details the features of the tie evolution of an initially localized excitation toward a statistical equilibriu in a finite lattice. We stress that this evolution is purely deterinistic and that the relaxation to equilibriu is due to deterinistic chaos. Below we argue that a statistically stationary state is characterized by the Gibbs energy equipartition across the linear eigenodes Eq. 5 and call a relaxation to such an equilibriu state theralization. Because theralization is due to deterinistic chaos, its rate heavily depends on the statistical properties of the chaos. As is typical for nonlinear Hailtonian systes, depending on initial conditions one can obtain solutions belonging to a chaotic sea or to regular islands. Moreover, one can expect the forer to theralize while the latter do not lead to theralization. We nuerically find nontheralizing odes and characterize their dependence on the nonlinearity and the lattice length. We stress here that our analysis heavily relies on nuerical siulations as analytic ethods appear to be hardly applicable for disordered nonlinear systes. In nuerics, a difference between theralizing and nontheralizing states as well as between chaotic and nonchaotic states is liited by the axial integration tie: it ight happen that the states which do not theralize up to soe tie will theralize in the future. There is no way to overcoe this liitation in a siple way because of a possibility for slow processes such as Arnold diffusion, characteristic tie of which lies far beyond any coputationally accessibility. Nevertheless, perforing an analysis based on large but finite tie scales, we can, on one hand, ake predictions for experients, and on the other hand, obtain a coarse-grained description of the dynaics. Accordingly, the results below should be understood as valid for available integration ties, without a straightforward extrapolation for asyptotically large ties. We describe a nonlinear disordered ediu by the DANSE odel: i n = E n n + n n + n+ + n, t where characterizes nonlinearity and the on-site energies E n or frequencies are independent rando variables distributed uniforly in the range W/ E n W/ they are shifted in such a way that E= corresponds to the central energy of the band. We consider a finite lattice n N with periodic boundary conditions. Then DANSE is a classical dynaical syste with the Hailton function H = n E n n + n n + n n + n. It describes recent experients with nonlinear photonic lattices cf. Eq. in, where one follows, along a transversally disordered finite nonlinear crystal, the evolution of a single-site or a single-ode initial state. This corresponds to the setup of our theralization proble. Thus, the properties below can be observed experientally as theralization of photons provided the crystal is long enough. In the context 59-755/9/8 5 /56 5 56-9 The Aerican Physical Society
MULANSKY et al. of any-particle quantu systes, Eq. is used as an effective ean-field Hailtonian of interacting bosons. For = all eigenstates are exponentially localized with the localization length l 96W for weak disorder at the center of the energy band 6. Below we ainly focus on the case of oderate disorder W=, for which l 6 at the center of the band and l.5 at E=. For each particular realization of disorder a set of eigenenergies and of corresponding eigenodes n can be found. In this eigenode representation n = C n the Hailtonian reads H = C + V knji C k C n C j C i, knji with C = and V l / are the transition atrix eleents 8. This representation is ostly suitable to characterize the spreading of the field over the lattice, since in this basis the transitions take place only due to nonlinearity. Also, the nonlinear correction to the energy is sall /l for one excited ode. To study the dynaical theralization in a lattice, we perfored direct nuerical siulation of DANSE using ainly two ethods: the unitary Crank-Nicholson operator splitting schee with step t=. as described in 5 and an eighth-order Runge-Kutta integration with step t=.; in both cases the total energy and the noralization have been preserved with high accuracy and both integration schees gave siilar results for all lattice lengths N used. Such a restriction of the accuracy check to the conserved quantities is suitable for chaotic systes. A coparison with other nuerical ethods for DANSE 9 goes beyond the scope of this paper and will be perfored in a longer publication. We started with two types of localized initial states: A one site seeded, i.e., n = n,j and B one ode initially excited, i.e., C =,k. For different realizations of disorder, we seeded different possible sites/odes and followed the evolution of the field solving Eq. up to ties in selected runs 8. The level of spreading is characterized by the entropy of the ode distribution, S = ln, = C, where overline eans tie averaging. For one excited ode S= while S=ln N for a unifor distribution over all odes in a lattice of length N. To give an ipression of a tie evolution of the theralization process we show in Fig. several representative tie dependencies of entropy. One can see that for the setup B soe odes reain localized during the coplete integration tie cf., while others after soe transient tie evolve to a state with large entropy. For setup A, the entropy grows in all cases with a tendency to saturation soe states see to saturate at about S ln N, while others reain at values definitely saller than ln N up to the axial integration tie. Especially the results fro B indicate a strong energy dependence of the spreading behavior, which is studied in this work. In our discussion below we focus therefore on the setup B as the ostly nontrivial one. To derive an approxiate expression for the statistically stationary distribution, we ention that it should satisfy PHYSICAL REVIEW E 8, 56 9 6 8 tie FIG.. Color online Tie evolution of entropy S Eq. in DANSE with N= and = for a particular realization of disorder and different initial states: bold black curves with arkers single-ode initial states B with energies E =.,.76,.9,.6,.5 curves fro top to botto at t = 8, two botto cases are very close, solid red/gray curves single-site initial states A; ten randoly chosen states. The dashed line shows the level S=ln. The tie averaging has been perfored over doubling tie intervals between successive arkers. = and E=, where, in view of the discussion above, we have neglected the nonlinear contribution to the energy. Then the condition of axial entropy leads, after a standard calculation, to a Gibbs distribution: = Z exp /T, Z = exp /T. 5 Here T is an effective teperature of the syste: it has no eaning as a physical teperature but serves as a paraeter characterizing the distribution; it is a function of the total energy E of the state and of the realization of disorder. The entropy and the energy are related to each other via usual expressions, e.g., : E = T ln Z/ T, S = E/T +lnz. 6 This value of entropy yields the axial possible equipartition for the given initial energy, and the values of Fig. obtained via a nuerical siulation of the disordered nonlinear lattice should be copared with these values fro the Gibbs distribution. Because we have anyhow neglected the effects of nonlinearity in the theoretical value of the entropy, we adopt other siplifications: approxiate the density of states of the disordered syste as a constant in an interval and consider the energy eigenvalues in a particular realization of disorder as independent rando variables distributed according to this density. Furtherore, we assue the variations of the partition su to be sall and use ln Z ln Z, where brackets denote averaging over disorder realizations. In this siplest approxiation we obtain ln Z ln N + ln sinh /T ln /T. 7 At W= we have see Figs. and below and this theory gives the dependence S E within a few percent accuracy copared to S averaged over disorder within Gibbs coputations with exact nuerical values. This justifies, for the paraeters used, the approxiation above. We note that T=+,, correspond to E=,,+, respectively as in the standard two-level proble, see related discussion in. 56-
DYNAMICAL THERMALIZATION OF DISORDERED 6 6 We copare in Fig. Gibbs distribution 5 with the results of direct nuerical siulations of DANSE using N d disorder realizations. Here we present the values averaged over tie and over different realization of disorder in dependence of the nuber of the initially seeded ode. The odes have been ordered according to their energy, so that = corresponds to the axial energy. One can see a good correspondence between the nuerics and the siple theory 5 in the sense that states at the band edges reain localized, while states in the center generally spread. However, there is one crucial discrepancy: the peaks on the diagonal = indicate that there are cases when there is no theralization within our siulation tie and the energy reains in the initially seeded ode. To characterize theralized and nontheralized cases quantitatively, we copare in Fig. nuerical values for S E according to Eq. with the theoretical Gibbs coputation given by Eqs. 5 7. Clearly, the Gibbs theory gives a satisfactory global description of nuerical data. The nontheralized odes in this presentation are those at the botto of the graph; these states are absent for the setup A where initial sites are seeded. Again, as discussed above, nontheralized should be understood as nontheralized within the integration tie. 6 6 FIG.. Color online Left: tie and disorder averaged probability in ode for initial state in ode. Right: theoretical values according to Gibbs distribution 5. Here N =, =, N d =5. - -..8 Lyapunov exponent FIG.. Color online Left panel: final entropies after an evolution during tie interval 7 averaged over a tie interval of 6. The states evolving fro initial odes in the iddle of the band see text are arked with black circles, while those at the edges of the band are arked by the red gray pluses. The curve shows approxiate theory 7. Right panel: Lyapunov exponents averaged over a tie interval 6 vs entropy for the sae sets with the sae arkers. Here N=, =, N d =7. Note that the states with S and E have nearly zero Lyapunov exponent although hardly visible in the right panel because overlapped by the red/gray pluses..8.6.. PHYSICAL REVIEW E 8, 56 9 It appears appropriate to discuss the dynaics of the odes in the iddle of the energy band and at the edges separately. For the odes in the iddle of the band, the axial entropy according to Eq. 6 is close to ln N, and one clearly distinguishes the theralized odes and those that reain localized, as those reaching the axial entropy and those reaining at the level S, correspondingly. Theralization is associated with the chaotic dynaics of the DANSE lattice. To deonstrate this, we calculated the largest Lyapunov exponents shown in Fig. right panel. All odes with S, i.e., those that do not theralize, have nearly vanishing, while for the theralized states S the positive values of clearly indicate chaos. The above distinction between theralized and nontheralized states is less evident for odes at the band edges shown by red gray pluses in Fig.. Here already the theoretical value of entropy given by Eqs. 5 7 is rather sall. Hence, the spreading can go over a few available odes only. Nevertheless, also here one can see fro Fig. a clear correlation between the entropy and the Lyapunov exponent. Moreover, in several cases the Lyapunov exponent at the edge of the spectru is definitely larger than in the iddle. This happens because the energy spreads over a sall nuber of odes; hence, the effective nonlinearity is larger due to larger aplitudes of each ode, and therefore chaos is stronger. Above, we did not account for a spatial organization of the ode structure. The latter is less iportant for the odes in the iddle of the band, where one can always expect to find neighbors with a close energy. Contrary to this, for the energies at the edges the issue of spatial distance becoes essential. Indeed, since here the theralization is possible only over a few odes, it is iportant whether these odes are spatially separated or not. For linear eigenodes and the natural easure of this separation is the coupling atrix eleent V according to Eq.. It is exponentially sall for spatially separated odes due to their localization. One can expect that a ode at the edge of the spectru will be theralized only if the coupling V to other few odes in the lattice with a close energy is large, which is a rather rare event. Finally, we discuss how the theralization properties depend on the nonlinearity constant. In Fig. we show the dependence S E for different nonlinearities. For =.5 a large portion of the initial states reains nontheralized, while for = all states are theralized at least in the sense that their entropy is close to the axial possible one; as discussed above this is a good criterion in the iddle of the band. To deterine how the fraction of theralized states depends on nonlinearity we use the following procedure. For the initial odes in the iddle of the band i.e., for E we classified those that reach ore than the half of the axial entropy i.e., the level ln N / as theralized and those that reain below this level as nontheralized. The fraction f th of the theralized odes, shown in Fig. 5, onotonously increases with. At fixed the nuerical data indicate saturation of f th at large N, but ore detailed checks at larger sizes and longer ties are required. For exaple, recent results on self-induced transparency of a disordered 56-
MULANSKY et al. PHYSICAL REVIEW E 8, 56 9 (a) (b).8 (a) - - - - fth, fb.6.. (b) FIG.. Color online Dependence of entropy S on energy E as in Fig. but for N=6, N d =8, and two values of nonlinearity: a =.5; b =. Averaging has been perfored over the tie interval 6 after an initial evolution during tie 6 ; for sall still longer ties are needed to reach theralized state with axial S at given E. The curves are the sae theoretical estiates as in Fig.. nonlinear layer 7 show decrease in critical with lattice size for N. The properties of theralization described above can be incorporated in a general fraework of nonlinear dynaics as follows. One can expect, based on general Kologorov- Arnold-Moser arguents, that for sall nonlinearity regular nonergodic regies predoinate, while for large values of stable solutions are destroyed and a chaotic ergodic state establishes in the lattice. While it is hard to characterize this transition via a general analysis of the dynaics in a highdiensional phase space, it is possible to follow the evolution, as nonlinearity increases, of special resonant odes that ste fro linear ones. Looking for solutions of Eq. in the for n t = n e i t, we arrive at a nonlinear eigenvalue proble n =E n n + n + n + n+ which, of course, at = yields linear frequencies and odes. Starting fro these odes, we followed these solutions to larger nonlinearities using a nuerical continuation and in this way obtained nonlinear resonant odes breathers cf.,. Worth noting, these odes change in the regions where the field is large, while in the tails they follow linear solutions in accordance with. Moreover, we perfored nuerical stability analysis of these breathers and found that they bifurcate at soe critical value of nonlinearity c. The values of c for an enseble of realizations of rando potentials are shown in Fig. 5 b. Additionally, we show in Fig. 5 a a cuulative distribution of c for the sae range of eigenenergies n that is used for the other curves plotted. First of all, note the siilar global behavior of f th and f b which akes us believe that the bifurcations of stable resonant odes are indeed the echanis of the dependence of theralization. However, the curves do not coincide because c is defined as the value of the first bifurcation, which ay not iediately lead to chaos but ay be the first one in a series of transitions to ore irregularity. Strictly speaking, f b should be an upper bound for f th, which is seen in Fig. 5 a. The increase in f th fro t= 6 to 7 shows that it has not saturated yet, but the saturation curve ust lie below f b. ɛ - -.5.5 FIG. 5. Color online a Fraction of theralized after tie 6 odes f th fro the iddle of the band as a function of nonlinearity for N=6 circles, bold line, and 6 pluses. Diaonds show data for t= 7 and N=. The dotted line shows the fraction of the bifurcated breathers f b according to panel b. Panel b : the bifurcation values c for different odes vs their linear energies for N=. To all odes with c we have attributed c =; this set looks like two vertical lines at = on panel b. Rearkably, we have found that the breathers at the edges of the band, i.e., for n, are extreely stable: ost of the reain stable up to large values of 5. This corresponds to the nuerical observation of the strong suppression of the theralization for these odes. We ephasize here that because of the nonlinearity of the syste the superposition principle does not hold. This eans that to observe a stable breather ode one has to prepare initial conditions ostly close to this solution which is achieved here by choosing the initial conditions as a pure linear eigenode case B above. When one initially seeds one site, as in case A or uses other initial conditions not close to a breather, then this initial condition does not produce a breather because the latter typically does not survive nonlinear interaction with other coponents of the solution. If, for exaple, one starts with an excitation of two odes which are both stable at soe value of, one ight still see fast theralization because a superposition of two breathers is not a breather. Our ain conclusion is that the axially theralized state in a disordered nonlinear lattice Eq., which eerges as a result of chaotic dynaics, is described by the Gibbs distribution over the linear odes, with soe effective teperature depending on the initial excitation. Not all odes lead to theralization; soe fraction of the reains localized, but this fraction decreases with nonlinearity. We found that this can be explained by the disappearance via bifurcations as the nonlinearity increases of stable resonant odes breathers steing fro linear eigenstates. Further studies are still required to establish the properties of this theralization in dependence on the nonlinearity strength, disorder, and lattice size. β 56-
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