We develop an efficient numerical scheme to solve accurately the set of nonlinear integral equations derived previously in [A. Saichev and D. Sornette, J. Geophys. Res. 112, B04313 (2007)], which describes the distribution of interevent times in the framework of a general model of earthquake clustering with long memory. Detailed comparisons between the linear and nonlinear versions of the theory and direct synthetic catalogs show that the nonlinear theory provides an excellent fit to the synthetic catalogs, while there are significant biases resulting from the use of the linear approximation. We then address the suggestions proposed by some authors to use the empirical distribution of interevent times to obtain a better determination of the so-called clustering parameter. Our theory and tests against synthetic and empirical catalogs find a rather dramatic lack of power for the distribution of interevent times to distinguish between quite different sets of parameters, casting doubt on the usefulness of this statistic for the specific purpose of identifying the clustering parameter.

We suggest an approach to partial synchronization of chaotic systems with uncertainty. This method contains two steps: (i) transforming the synchronization system into the canonical form by the well-known feedback linearization theory and (ii) finding a control signal to ensure the asymptotic stability of the canonical system. This partial synchronization approach requires very little system information by applying a finite-time convergence technique to estimate uncertainties caused by unknown states, parameters, or structure. We also argue in detail that this partial synchronization method can be extended to parameter identification, (sub)structure estimation, and even phase detection. Several examples are presented to illustrate the partial synchronization approach suggested.

Loose packings of spheres with bidisperse or log-normal distributions are generated by random sequential deposition. Porosity, conductivity, and permeability are determined. The porosities correspond to loose packings, but they follow the usual trends for bidisperse packings. The conductivity and permeability follow power laws as functions of the porosity of the packings. Several other quantities such as the classical Kozeny constant are successfully represented as functions of porosity. Some dimensionless representations gather the numerical data on curves valid for all particle distributions. Finally, comparisons with experimental data are satisfactory.

The interaction of solitons with bond defects in discrete nonlinear Schrödinger (NLS) chains is investigated. A perturbed NLS equation is derived on the basis of a microscopic model. Localized soliton-defect solutions are obtained and their stability is analyzed. Scattering of propagating solitons from bond defects is studied numerically and a variety of scattering patterns is obtained. A phase diagram is constructed showing the corresponding parametric regions.

The study addresses the azimuthal jumping motion of an adsorbed polar molecule in a periodic $n$-well potential under the action of an external alternating electric field. Starting from the perturbation theory of the Pauli equation with respect to the weak field intensity, explicit analytical expressions have been derived for the time dependence of the average dipole moment as well as the frequency dependences of polarizability and the average angular velocity, the three quantities exhibiting conspicuous stochastic resonance. As shown, unidirectional rotation can arise only provided simultaneous modulation of the minima and maxima of the potential by an external alternating field. For a symmetric potential of hindered rotation, the average angular velocity, if calculated by the second-order perturbation theory with respect to the field intensity, has a nonzero value only at $n=2$, i.e., when two azimuthal wells specify a selected axis in the system. Particular consideration is given to the effect caused by the asymmetry of the two-well potential on the dielectric loss spectrum and other Brownian motion parameters. When the asymmetric potential in a system of dipole rotators arises from the average local fields induced by an orientational phase transition, the characteristics concerned show certain peculiarities which enable detection of the phase transition and determination of its parameters.

We derive the generalized Fokker-Planck equation associated with a Langevin equation driven by arbitrary additive white noise. We apply our result to study the distribution of symmetric and asymmetric Lévy flights in an infinitely deep potential well. The fractional Fokker-Planck equation for Lévy flights is derived and solved analytically in the steady state. It is shown that Lévy flights are distributed according to the beta distribution, whose probability density becomes singular at the boundaries of the well. The origin of the preferred concentration of flying objects near the boundaries in nonequilibrium systems is clarified.

An analytic effective medium theory is constructed to study the mean access times for random walks on hybrid disordered structures formed by embedding complex networks into regular lattices, considering transition rates $F$ that are different for steps across lattice bonds from the rates $f$ across network shortcuts. The theory is developed for structures with arbitrary shortcut distributions and applied to a class of partially disordered traversal enhanced networks in which shortcuts of fixed length are distributed randomly with finite probability. Numerical simulations are found to be in excellent agreement with predictions of the effective medium theory on all aspects addressed by the latter. Access times for random walks on these partially disordered structures are compared to those on small-world networks, which on average appear to provide the most effective means of decreasing access times uniformly across the network.

We investigate through computational simulations with a pore network model the formation of patterns caused by erosion-deposition mechanisms. In this model, the geometry of the pore space changes dynamically as a consequence of the coupling between the fluid flow and the movement of particles due to local drag forces. Our results for this irreversible process show that the model is able to reproduce typical natural patterns caused by well-known erosion processes. Moreover, we observe that, within a certain range of porosity values, the grains form clusters that are tilted with respect to the horizontal with a characteristic angle. We compare our results to recent experiments for granular material in flowing water and show that they present a satisfactory agreement.

We study the solid mechanical properties of several thixotropic suspensions as a function of the shear stress history applied during their flow stoppage and their aging in their solid state. We show that their elastic modulus and yield stress depend strongly on the shear stress applied during their solid-liquid transition (i.e., during flow stoppage) while applying the same stress only before or only after this transition may induce only second-order effects: there is negligible dependence of the mechanical properties on the preshear history and on the shear stress applied at rest. We also found that the suspensions age with a structuration rate that hardly depends on the stress history. We propose a physical sketch based on the freezing of a microstructure whose anisotropy depends on the stress applied during the liquid-solid transition to explain why the mechanical properties depend strongly on this stress. This sketch points out the role of the internal forces in the colloidal suspensions’ behavior. We finally discuss briefly the macroscopic consequences of this phenomenon and show the importance of using a controlled-stress rheometer.

Population stratification is a problem encountered in several areas of natural science, engineering, and public health. We tackle this problem by mapping a population and its element attributes onto a hypergraph, a natural extension of the concept of graph or network to encode associations among any number of elements. On this hypergraph, we construct a statistical model reflecting our intuition about how the element attributes can emerge from a postulated population structure. Finally, we introduce the concept of stratification representativeness as a mean to identify the simplest stratification already containing most of the information about the population structure. We demonstrate the power of this framework stratifying an animal and a human population based on phenotypic and genotypic properties, respectively.

We numerically study the dynamics of a discrete spring-block model introduced by Olami, Feder, and Christensen (OFC) to mimic earthquakes and investigate to what extent this simple model is able to reproduce the observed spatiotemporal clustering of seismicity. Following a recently proposed method to characterize such clustering by networks of recurrent events [J. Davidsen, P. Grassberger, and M. Paczuski, Geophys. Res. Lett. 33, L11304 (2006)], we find that for synthetic catalogs generated by the OFC model these networks have many nontrivial statistical properties. This includes characteristic degree distributions, very similar to what has been observed for real seismicity. There are, however, also significant differences between the OFC model and earthquake catalogs, indicating that this simple model is insufficient to account for certain aspects of the spatiotemporal clustering of seismicity.

We consider the delayed feedback control method for stabilization of unstable rotating waves near a fold bifurcation. Theoretical analysis of a generic model and numerical bifurcation analysis of the rate-equations model demonstrate that such orbits can always be stabilized by a proper choice of control parameters. Our paper confirms the recently discovered invalidity of the so-called “odd-number limitation” of delayed feedback control. Previous results have been restricted to the vicinity of a subcritical Hopf bifurcation. We now refute such a limitation for rotating waves near a fold bifurcation. We include an application to all-optical realization of the control in three-section semiconductor lasers.

Tracer particles on the surface of a turbulent flow have a very intermittent distribution. This preferential concentration effect is studied in a two-dimensional synthetic compressible flow, both in the inertial (self-similar) and in the dissipative (smooth) range of scales, as a function of the compressibility $\mathcal{C}$. The second moment of the concentration coarse grained over a scale $\mathit{r}$, $⟨n_r^2⟩$, behaves as a power law in both the inertial and the dissipative ranges of scale, with two different exponents. The shapes of the probability distribution functions of the coarse-grained density ${\mathit{n}}_{\mathit{r}}$ vary as a function of scale $\mathit{r}$ and of compressibility $\mathcal{C}$ through the combination $\mathcal{C}/r^{\kappa }$ $\left(\kappa \simeq 0.5\right)$, corresponding to the compressibility, coarse grained over a domain of scale $\mathit{r}$, averaged over Lagrangian trajectories.

Compared to single-pressure models, multipressure multiphase flow models require additional closure relations to determine the individual pressures of the different phases. These relations are often taken to be evolution equations for the volume fractions. We present a rigorous theoretical framework for constructing such equations for compressible multiphase mixtures in terms of submodels for the relative volumetric expansion rates $\mathrm{\Delta }{E}_i$ of the phases. These quantities are essentially the rates at which the phases dynamically expand or contract in response to pressure differences, and represent the general tendency of the volume fractions to relax toward values that produce local pressure equilibrium. We present a simple provisional model of this type in which $\mathrm{\Delta }{E}_i$ is proportional to pressure differences divided by the time required for sound waves to traverse an appropriate characteristic length. It is shown that the resulting approach to pressure equilibrium is monotonic rather than oscillatory, and occurs instantaneously in the incompressible limit.

The results of numerical study of mass-transfer processes in quasi-two- and three-dimensional nonideal dissipative systems are presented. Simulations were performed for different types of model pair potentials of intergrain interaction that are various combinations of power-law and exponential functions. The calculations were performed in a wide range of parameters typical for laboratory dusty plasma experiments. It was shown that the dynamics of grains in liquidlike systems for short observation times is close to the evolution of thermal oscillations in the crystal lattice.

The results of the experimental study of mass-transfer processes are presented for dust systems, forming in a laboratory plasma of a radio-frequency capacitive discharge. The validity of the Langevin and Green-Kubo equations for the description of the dynamics of dusty grains in laboratory plasma is verified. A method for simultaneous determination of dusty plasma parameters, such as the kinetic temperature of the grains, their friction coefficient, and characteristic oscillation frequency, is suggested. The coupling parameter of the system under study and the minimal values of the grain charges are estimated. The parameters of the dusty subsystem obtained (diffusion coefficients, pair correlation functions, charges, and friction coefficients of the grains) are compared with the existing theoretical and numerical data.

The aim of the paper is to show that splitting of a waveguide leads to fission of bulk solitons in solids. We study the dynamics of a longitudinal bulk solitary wave in a delaminated, symmetric layered elastic bar. First, we consider a two-layered bar and assume that there is a perfect interface when $x<0$ and complete debonding (splitting) when $x>0$, where the axis $Ox$ is directed along the bar. We derive the so-called doubly dispersive equation (DDE) for a long nonlinear longitudinal bulk wave propagating in an elastic bar of rectangular cross section. We formulate the problem for a delaminated two-layered bar in terms of the DDE with piecewise constant coefficients, subject to continuity of longitudinal displacement and normal stress across the “jump” at $x=0$. We find the weakly nonlinear solution to the problem and consider the case of an incident solitary wave. The solution describes both the reflected and transmitted waves in the far field, as well as the diffraction in the near field (in the vicinity of the jump). We generalize the solution to the case of a symmetric $n$-layered bar. We show that delamination can lead to the fission of an incident solitary wave, and obtain explicit formulas for the number, amplitudes, velocities, and positions of the secondary solitary waves propagating in each layer of the split waveguide. We establish that generally there is a higher-order reflected wave even when the leading order reflected wave is absent.

If a stochastic system during some periods of its evolution can be divided into noninteracting parts, the kinetics of each part can be simulated independently. We show that this can be used in the development of efficient Monte Carlo algorithms. As an illustrative example, the simulation of irreversible growth of extended one-dimensional islands is considered. The approach allowed us to simulate the systems characterized by parameters superior to those used in previous simulations.

We derive the macroscopic Fourier’s Law of heat conduction from the exact gain-loss time convolutionless quantum master equation under three assumptions for the interaction kernel. To second order in the interaction, we show that the first two assumptions are natural results of the long time limit. The third assumption can be satisfied by a family of interactions consisting of an exchange effect. The pure exchange model directly leads to energy diffusion in a weakly coupled spin-$1∕2$ chain.

We present a theoretical study of the recently observed dynamical regimes of paramagnetic colloidal particles externally driven above a regular lattice of magnetic bubbles [P. Tierno, T. H. Johansen, and T. M. Fischer, Phys. Rev. Lett. 99, 038303 (2007)]. An external precessing magnetic field alters the potential generated by the surface of the film in such a way to either drive the particle circularly around one bubble, ballistically through the array, or in triangular orbits on the interstitial regions between the bubbles. In the ballistic regime, we observe different trajectories performed by the particles phase locked with the external driving. Superdiffusive motion, which was experimentally found bridging the localized and delocalized dynamics, emerge only by introducing a certain degree of randomness into the bubbles size distribution.