We study the nature of electronic states in a tight-binding one-dimensional model with the on-site energies exhibiting long-range correlated disorder and nonrandom hopping amplitudes. The site energies describe the trace of a fractional Brownian motion with a specified spectral density S͑k͒~1͞k a . Using a renormalization group technique, we show that for long-range correlated energy sequences with persistent increments (a . 2) the Lyapunov coefficient (inverse localization length) vanishes within a finite range of energy values revealing the presence of an Anderson-like metal-insulator transition.[S0031-9007(98)07502-4] PACS numbers: 73.20.Jc, 05.40. + j, 72.15.Rn The simplest theoretical model containing the basic ingredients for studying the nature of one-electron states in disordered systems was introduced by Anderson [1] which considered one-electron moving in a lattice endowed by a random potential and allowed to hop only to nearestneighbor sites. Anderson pointed out that if the disorder is very strong the wave function may become exponentially localized with a characteristic localization length l c . Further, scaling arguments applied to noninteracting electron systems in the presence of uncorrelated disorder have indicated that all one-electron states are exponentially localized in one and two dimensions for any amount of disorder, with a true metal-insulator transition taking place only in 3D on which one-electron states may remain extended for weak disorder [2,3].The scaling prediction of the absence of extended states in one dimension agrees with a series of analytical results which show that all wave functions must have an exponentially decaying envelope whenever the potential assumes random values uncorrelated from site to site [4]. In recent years, there has been a growing interest in the study of the role played by correlations in the nature of the one-electron states of low-dimensional disordered systems. The reason for that is based on the fact that a series of one-dimensional versions of the Anderson model has been shown to exhibit a breakdown of Anderson's localization induced by internal correlations on the disorder distribution [5][6][7][8][9][10][11][12][13]. Most of these models consider on-site energies e n distributed in such a way that the impurity always appears in finite segments of fixed size. Extended states arise from resonant modes which present no backscattering through these finite structures. Such states form a discrete set of energy values. Therefore, these models do not present a true disorder induced metal-insulator transition in the thermodynamic limit which is characterized by the presence of mobility edges separating extended and localized energy eigenstates. Also, chains with correlated offdiagonal interactions [5,9,11] have been reported to display delocalized states. More recently, thermally annealed disordered chains with the on-site energies correlated as ͗e i e j ͘~e 2ji2jj͞j have also been investigated [14]. It has been shown that the localization length monoton...

Power-law sensitivity to initial conditions at the edge of chaos provides a natural relation between the scaling properties of the dynamics attractor and its degree of nonextensivity as prescribed in the generalized statistics recently introduced by one of us (C.T.) and characterized by the entropic index q. We show that general scaling arguments imply that 1/(1 − q) = 1/α min − 1/α max , where α min and α max are the extremes of the multifractal singularity spectrum f (α) of the attractor. This relation is numerically checked to hold in standard one-dimensional dissipative maps. The above result sheds light on a longstanding puzzle concerning the relation between the entropic index q and the underlying microscopic dynamics. 05.45.+b; 05.20.-y; 05.90.+m Typeset using REVT E X

An important application involving two-species reaction-diffusion systems relates to the problem of finding the best statistical strategy for optimizing the encounter rate between organisms. We investigate the general problem of how the encounter rate depends on whether organisms move in Lévy or Brownian random walks. By simulating a limiting generalized searcher-target model (e.g., predator-prey, mating partner, pollinator-flower), we find that Lévy walks confer a significant advantage for increasing encounter rates when the searcher is larger or moves rapidly relative to the target, and when the target density is low.

Power-law sensitivity to initial conditions, characterizing the behaviour of dynamical systems at their critical points (where the standard Liapunov exponent vanishes), is studied in connection with the family of nonlinear 1D logistic-like maps xt+1 = 1 − a |xt| z , (z > 1; 0 < a ≤ 2; t = 0, 1, 2, . . .).The main ingredient of our approach is the generalized deviation law lim ∆x(0)→01−q (equal to e λ 1 t for q = 1, and proportional, for large t, to t 1 1−q for q = 1; q ∈ R is the entropic index appearing in the recently introduced nonextensive generalized statistics). The relation between the parameter q and the fractal dimension d f of the onset-to-chaos attractor is revealed: q appears to monotonically decrease from 1 (Boltzmann-Gibbs, extensive, limit) to −∞ when d f varies from 1 (nonfractal, ergodic-like, limit) to zero. 05.45.+b; 05.20.-y; 05.90.+m

de Moura and Lyra Reply: In the previous Comment [1], it is pointed out that the one-dimensional Anderson model with long-range correlated disordered potentials having fluctuations increasing with the system size does not exhibit extended states. The authors are correct on this specific point and are led to the conclusion that our finding of an Anderson transition for long-range correlated energy landscapes is "not valid." This conclusion does not follow for the reasons discussed below.Without any normalization, the disorder width v and hence the band width DE diverge in the thermodynamic limit. Therefore, all one-particle eigenstates become localized even in higher dimensions, thus preventing the occurrence of an Anderson transition. When comparing systems with distinct sizes, the authors in [1] are actually comparing systems with distinct disorder and band widths which shall not, in principle, be expected to have similar physical properties.In order to avoid the divergence of the disorder width and to allow for nontrivial thermodynamic limit, we normalized the local potentials in such a way that keeps v and DE finite and size independent [2]. The normalization of the physical parameters is a traditional procedure in model systems with a divergent relevant energy scale. This is done, for example, in the tight-binding model in infinite d dimensions, in the spherical d-vector model, and in longrange interacting spin systems.In our Letter [2], we used a size dependent normalization factor which is equivalent to dividing the local potentials by L H . This procedure imposes that the disorder and bandwidths are kept fixed for distinct chain sizes. With this normalization, the 1D Anderson model does exhibit a phase with eigenstates delocalized all over the chain with mobility edges separating localized and delocalized states, provided that the spectral density of the potential decays faster than S͑k͒~1͞k 2 . For long-range correlated off-diagonal disorder, the transition can take place for potentials with weaker correlations, i.e., spectral densities decaying faster than 1͞k [3].Our finding has been recently supported by the work of Izrailev and Krokhin [4]. Using a second order perturbation theory, they show the existence of mobility edges

For a family of logistic-like maps, we investigate the rate of convergence to the critical attractor when an ensemble of initial conditions is uniformly spread over the entire phase space. We found that the phase space volume occupied by the ensemble W (t) depicts a power-law decay with log-periodic oscillations reflecting the multifractal character of the critical attractor. We explore the parametric dependence of the power-law exponent and the amplitude of the log-periodic oscillations with the attractor's fractal dimension governed by the inflexion of the map near its extremal point. Further, we investigate the temporal evolution of W (t) for the circle map whose critical attractor is dense. In this case, we found W (t) to exhibit a rich pattern with a slow logarithmic decay of the lower bounds. These results are discussed in the context of non-extensive Tsallis entropies.

Axis labels appear incorrectly in Fig. 2. The axis labels should appear exchanged as in the correct Fig. 2 shown below. The main results reported in our Letter are not affected by this error, but some secondary conclusions are incorrect. Figure 2(a) is approximately symmetrical with respect to the r v line; therefore, the error in the figure ends up not being important. However, in the case of Fig. 2(b), there is no such symmetry, and some conclusions require modification. Specifically, for Lévy targets it is v rather than r that determines whether Lévy searches confer advantages relative to Brownian searches.We thank E. P. Raposo and M. G. E. da Luz for helpful comments. FIG. 2. Gray-scale graph showing as a function of r and v for (a) Browning targets and (b) Lévy targets. Each set of 4 graphs corresponds to increasing system size L 25, L 50, L 75, and L 100 from top-left to bottom-right.

We investigate the magnetocaloric effect in a diamond chain model on which competing interactions result from the local quantum hopping of interstitial S =1/ 2 spins which are intercalated between nodal Ising spins. The model is exactly solvable by using exact diagonalization and the decoration-iteration mapping onto the one-dimensional Ising model with effective parameters depending on the temperature and the external magnetic field. We analyze the thermodynamic behavior of the effective parameters in light of the ground-state ordering and the level crossing of the low-lying excited states. Further, we investigate the magnetocaloric effect on this spin chain model by computing isoentropy curves in the temperature versus external field parameter space, as well as the adiabatic cooling rate. We show that the adiabatic cooling rate exhibits a pronounced valley-peak structure in the vicinity of the critical fields associated with zero-temperature phase transitions.