Using large system sizes with extensive statistical sampling, we analyze the scaling properties of crack roughness and damage profiles in the three-dimensional random fuse model. The analysis of damage profiles indicates that damage accumulates in a diffusive manner up to the peak load, and localization sets in abruptly at the peak load, starting from a uniform damage landscape. The global crack width scales as $W\sim L^{0.5}$ and is consistent with the scaling of localization length $\xi \sim L^{0.5}$ used in the data collapse of damage profiles in the postpeak regime. This consistency between the global crack roughness exponent and the postpeak damage profile localization length supports the idea that the postpeak damage profile is predominantly due to the localization produced by the catastrophic failure, which at the same time results in the formation of the final crack. Finally, the crack width distributions can be collapsed for different system sizes and follow a log-normal distribution.

We study phase entrainment of Kuramoto oscillators under different conditions on the interaction range and the natural frequencies. In the first part the oscillators are entrained by a pacemaker acting like an impurity or a defect. We analytically derive the entrainment frequency for arbitrary interaction range and the entrainment threshold for all-to-all couplings. For intermediate couplings our numerical results show a reentrance of the synchronization transition as a function of the coupling range. The origin of this reentrance can be traced back to the normalization of the coupling strength. In the second part we consider a system of oscillators with an initial gradient in their natural frequencies, extended over a one-dimensional chain or a two-dimensional lattice. Here it is the oscillator with the highest natural frequency that becomes the pacemaker of the ensemble, sending out circular waves in oscillator-phase space. No asymmetric coupling between the oscillators is needed for this dynamical induction of the pacemaker property nor need it be distinguished by a gap in the natural frequency.

The nonperturbative non-Gaussian characteristics of diffusive motion are examined in the framework of the large deviation statistical theory, where simple extended mapping models showing chaotic diffusion are taken as an example. Furthermore, by rigorously solving the large deviation statistical quantities, it is found that the same type of anomalous, complex control parameter dependence as that for the diffusion coefficient reported by Klages and Dorfman is also observed in the large deviation statistical quantities such as the weighted average, the generalized diffusion coefficient, and the generalized power spectrum densities.

In this paper the problem of signal propagation in networks of excitatory elements is studied. It is found that the geometry of signal transmission paths depends crucially on how an excitatory element responds to a stimulus. Two types of responses are defined: fast and slow. In the slow response case the signal transmission paths are in the same universality class as optimal paths in the limit of strong disorder. The signal transmission paths formed in the fast response case constitute possibly a new universality class.

We present an experimental study of the compaction dynamics for two-dimensional anisotropic granular systems. The compaction dynamics of rods is measured at three different scales: (i) the macroscopic scale through the packing fraction $\rho$, (ii) the mesoscopic scale through both fractions of aligned grains ${\varphi }_a$ and ideally ordered grains ${\varphi }_{io}$, and (iii) the microscopic scale through both rotational and translational grain mobilities ${\mu }_{r,t}$. At the macroscopic scale, we have observed two stages during the compaction process, suggesting different characteristic time scales for grain relaxation. At the mesoscopic scale, we have observed the formation and the growth of domains made of aligned grains during the first stage of compaction. At the late stage, these domains of aligned grains are sheared to form ideally ordered domains. From a microscopic point of view, measurements reveal that the beginning of the compaction process is essentially related to translational motions of the grains. The grain rotations drive mainly the process during the late stages of compaction.

We present molecular dynamics simulations of the glass transition in a chemically realistic model of 1,4-polybutadiene (PBD). Around $40\mathrm{K}$ above the calorimetric glass transition of this polymer the simulations reveal a well-developed two-stage relaxation of all correlation functions. We have analyzed the time-scale separation between vibrational degrees of freedom (subpicosecond dynamics) and the $\alpha$ relaxation behavior (nanosecond to microsecond dynamics) using the predictions of mode-coupling theory (MCT). Our value for the mode-coupling critical temperature $T_c$ agrees perfectly with prior experimental estimates for PBD. The predictions of MCT for the scaling behavior of the so-called $\beta$ relaxation, the plateau regime separating vibrational dynamics and the $\alpha$ relaxation, are well fulfilled. Furthermore, we are able to derive a consistent set of MCT exponents, completely characterizing the scaling behavior of relaxation processes in the vicinity of $T_c$. For the temperature dependence of the $\alpha$ relaxation we find deviations from MCT predictions which we trace to the influence of torsional barriers on the atomic motions.

We study analytically the emergence of spontaneous collective motion within large bidimensional groups of self-propelled particles with noisy local interactions, a schematic model for assemblies of biological organisms. As a central result, we derive from the individual dynamics the hydrodynamic equations for the density and velocity fields, thus giving a microscopic foundation to the phenomenological equations used in previous approaches. A homogeneous spontaneous motion emerges below a transition line in the noise-density plane. Yet, this state is shown to be unstable against spatial perturbations, suggesting that more complicated structures should eventually appear.

In a recent paper, Houchmandzadeh et al. [Phys. Rev. E 72, 061920 (2005)] introduce a correlated bigradient model in order to explain the robust scaling of the boundary of hunchback (hb) expression in the early Drosophila embryo. In particular, they stress that recent experiments by Lucchetta et al. [Nature (London) 434, 1134 (2005)], where embryos whose anterior and posterior halves develop at different temperatures still show excellent precision in the hb boundary, are in good agreement with such a model. We would like to show here that this conclusion is unwarranted. This is because the experiments of Lucchetta et al. were done at different temperatures from those studied in the model. Since in other temperature combinations the model does not produce precise boundaries and there are no systematic trends in these deviations, a comparison to the experiment is not possible. Furthermore, we would like to point out that any correlated bigradient model should also take into account the fluctuations of the bicoid profile within an embryo. When forming correlated bigradients of experimental profiles from an online library, we observe that these intraembryo variationss destroy robustness of the hb boundary even in the wild-type situation.

A fixed-mass multifractal (FMA) analysis was used to investigate natural river networks and braided channels. In particular, while the study of natural river networks was performed with fixed-size algorithms (FSAs) in the past, the analysis of natural braided channels was not pursued before to our knowledge. Results showed the multifractal and non-plane-filling nature of all the digitalized data sets. Analysis of the digitalization step (constant or not) was performed and showed that it does not exert a strong influence on the assessed values of the Lipschitz-Hölder exponents and the support dimensions, even if a constant step permits better reconstruction of the right sides of the spectra, for negative moment orders of probabilities. The FMA approach presented two improvements with respect to the FSA one, in terms of oscillations of the scaling curves for negative moment orders of probabilities and of error bars. A more precise assessment of the multifractal spectra is of great importance in the development of multifractal models for the simulation of flood hydrographs.

We investigate the nature of written human language within the framework of complex network theory. In particular, we analyze the topology of Orwell’s 1984 focusing on the local properties of the network, such as the properties of the nearest neighbors and the clustering coefficient. We find a composite power law behavior for both the average nearest neighbor’s degree and average clustering coefficient as a function of the vertex degree. This implies the existence of different functional classes of vertices. Furthermore, we find that the second order vertex correlations are an essential component of the network architecture. To model our empirical results we extend a previously introduced model for language due to Dorogovtsev and Mendes. We propose an accelerated growing network model that contains three growth mechanisms: linear preferential attachment, local preferential attachment, and the random growth of a predetermined small finite subset of initial vertices. We find that with these elementary stochastic rules we are able to produce a network showing syntacticlike structures.

An approach to modeling knowledge acquisition in terms of walks along complex networks is described. Each subset of knowledge is represented as a node, and relations between such knowledge are expressed as edges. Two types of edges are considered, corresponding to free and conditional transitions. The latter case implies that a node can only be reached after visiting previously a set of nodes (the required conditions). The process of knowledge acquisition can then be simulated by considering the number of nodes visited as a single agent moves along the network, starting from its lowest layer. It is shown that hierarchical networks—i.e., networks composed of successive interconnected layers—are related to compositions of the prerequisite relationships between the nodes. In order to avoid deadlocks—i.e., unreachable nodes—the subnetwork in each layer is assumed to be a connected component. Several configurations of such hierarchical knowledge networks are simulated and the performance of the moving agent quantified in terms of the percentage of visited nodes after each movement. The Barabási-Albert and random models are considered for the layer and interconnecting subnetworks. Although all subnetworks in each realization have the same number of nodes, several interconnectivities, defined by the average node degree of the interconnection networks, have been considered. Two visiting strategies are investigated: random choice among the existing edges and preferential choice to so far untracked edges. A series of interesting results are obtained, including the identification of a series of plateaus of knowledge stagnation in the case of the preferential movement strategy in the presence of conditional edges.

We extend the merging model for undirected networks by Kim et al. [Eur. Phys. J. B 43, 369 (2004)] to directed networks and investigate the emerging scale-free networks. Two versions of the directed merging model, friendly and hostile merging, give rise to two distinct network types. We uncover that some nontrivial features of these two network types resemble two levels of a certain randomization/nonspecificity in the link reshuffling during network evolution. Furthermore, the same features show up, respectively, in metabolic networks and transcriptional networks. We introduce measures that single out the distinguishing features between the two prototype networks, as well as point out features that are beyond the prototypes.

We present an asymptotic analysis of time-delayed feedback control of steady states for large delay time. By scaling arguments, and a detailed comparison with exact solutions, we establish the parameter ranges for successful stabilization of an unstable fixed point of focus type. Insight into the control mechanism is gained by analyzing the eigenvalue spectrum, which consists of a pseudocontinuous spectrum and up to two strongly unstable eigenvalues. Although the standard control scheme generally fails for large delay, we find that if the uncontrolled system is sufficiently close to its instability threshold, control does work even for relatively large delay times.

We study two methods for constructing a nonuniform embedding for multivariate data. A nonuniform embedding is a state space reconstruction which is more flexible than the common delay coordinates with fixed delays since it contains variable delays. Using these methods, we can extract causal relationships among many variables in a more suitable way. We demonstrate that the proposed methods can give more precise predictions and simpler models than some previous methods.

The amplitude ratio of the susceptibility (or second size moment) for two-dimensional percolation is calculated by two series methods and also by Monte Carlo simulation. The first series method is an approach based upon integrating approximations to the scaling function. The second series method directly uses low- and high-density series expansions of the susceptibility, going to unprecedented orders for both bond and site percolation on the square lattice. Putting all methods together we find a consistent value ${\Gamma }^{-}∕{\Gamma }^{+}=162.5\pm 2$, a significant improvement over previous results that placed the value of this ratio variously in the range of 14 to 220.

A broad class of $(N+1)$-species ratio-dependent predator-prey stochastic models, which consist of one predator population and $N$ prey populations, is considered. The effect of a fluctuating environment on the carrying capacities of prey populations is taken into account as colored noise. In the framework of the mean-field theory, approximate self-consistency equations for prey-populations mean density and for predator-population density are derived (to the first order in the noise variance). In some cases, the mean field exhibits Hopf bifurcations as a function of noise correlation time. The corresponding transitions are found to be reentrant, e.g., the periodic orbit appears above a critical value of the noise correlation time, but disappears again at a higher value of the noise correlation time. The nonmonotonous dependence of the critical control parameter on the noise correlation time is found, and the conditions for the occurrence of Hopf bifurcations are presented. Our results provide a possible scenario for environmental-fluctuations-induced transitions between the oscillatory regime and equilibrium state of population sizes observed in nature.

Tumor growth has a number of features in common with a physical process known as molecular beam epitaxy. Both growth processes are characterized by the constraint of growth development to the body border, and surface diffusion of cells and particles at the growing edge. However, tumor growth implies an approximate spherical symmetry that makes necessary a geometrical treatment of the growth equations. The basic model was introduced in a former paper [C. Escudero, Phys. Rev. E 73, 020902(R) (2006)], and in the present work we extend our analysis and try to shed light on the possible geometrical principles that drive tumor growth. We present two-dimensional models that reproduce the experimental observations, and analyze the unexplored three-dimensional case, for which interesting conclusions on tumor growth are derived.

In this paper a continuous probabilistic approach formulated using Kolmogorov-Fokker-Planck forward and backward models is applied to Hubbell’s zero-sum neutral theory for species dynamics in local community. Using this technique the probability density of species abundance, distribution of the first passage time to extinction or fixation and probability of extinction are defined. The resulting values for the distribution of the first passage time to extinction are verified by the simulation study of Hubbell’s zero-sum neutral model for the local community. Based on the sensitivity analysis for the continuous probabilistic models, the realistic classification of local communities subject to their diversity and species dynamics is proposed with respect to the immigration probability, the species metacommunity relative abundance, and the size of local community.

We have applied a variational algorithm based on projected entangled pair states (PEPS) to a two dimensional frustrated spin system, the spin-1/2 antiferromagnetic Heisenberg model on the Shastry-Sutherland lattice. We use the class of PEPS with internal tensor dimension $D=2$, the first step beyond product states ($D=1$ PEPS). We have found that the $D=2$ variational PEPS algorithm is able to capture the physics in both the valence-bond crystal and the Néel ordered state. Also the spin textures giving rise to the magnetization plateaus seen in experiments on ${\mathrm{SrCu}}_2({\mathrm{BO}}_3{)}_2$ are well reproduced. This shows that PEPS with the smallest nontrivial internal dimension, $D=2$, can provide valuable insights into frustrated spin systems.

It is well known that the entropy of the microcanonical ensemble cannot be calculated as the Legendre transform of the canonical free energy when the entropy is nonconcave. To circumvent this problem, a generalization of the canonical ensemble that allows for the calculation of nonconcave entropies was recently proposed. Here, we study the mean-field Curie-Weiss-Potts spin model and show, by direct calculations, that the nonconcave entropy of this model can be obtained by using a specific instance of the generalized canonical ensemble known as the Gaussian ensemble.