We give an exact solution for the complete distribution of component sizes in random networks with arbitrary degree distributions. The solution tells us the probability that a randomly chosen node belongs to a component of size $s$ for any $s$. We apply our results to networks with the three most commonly studied degree distributions—Poisson, exponential, and power-law—as well as to the calculation of cluster sizes for bond percolation on networks, which correspond to the sizes of outbreaks of epidemic processes on the same networks. For the particular case of the power-law degree distribution, we show that the component size distribution itself follows a power law everywhere below the phase transition at which a giant component forms, but takes an exponential form when a giant component is present.

A recent study [D. Helbing, A. Johansson, and H. Z. Al-Abideen, Phys. Rev. E 75, 046109 (2007)] has revealed a “turbulent” state of pedestrian flows, which is characterized by sudden displacements and causes the falling and trampling of people. However, turbulent crowd motion is not reproduced well by current many-particle models due to their insufficient representation of the local interactions in areas of extreme densities. In this contribution, we extend the repulsive force term of the social force model to reproduce crowd turbulence. We perform numerical simulations of pedestrians moving through a bottleneck area with this model. The transitions from laminar to stop-and-go and turbulent flows are observed. The empirical features characterizing crowd turbulence, such as the structure function and the probability density function of velocity increments, are reproduced well; i.e., they are well compatible with an analysis of video data during the annual Muslim pilgrimage.

We report on a honeycomb hexagon pattern in dielectric barrier discharge. It bifurcates from a square pattern as the applied voltage is increased. A phase diagram of the pattern types as a function of the gas component and applied voltage is presented. The spatial Fourier spectrum of the honeycomb hexagon pattern is a hexagonal superstructure with three wave vectors ${\stackrel{⃑}{k}}_1^h$, ${\stackrel{⃑}{k}}_2^s$ and ${\stackrel{⃑}{k}}_3^s$ at least, demonstrating that the pattern is a superlattice pattern. Measurements of the correlation between discharge filaments indicate that the honeycomb hexagon pattern is an interleaving of three different transient hexagonal sublattices, whose spatial wave vectors are exactly equal to ${\stackrel{⃑}{k}}_1^h$, ${\stackrel{⃑}{k}}_2^s$ and ${\stackrel{⃑}{k}}_3^s$, respectively. These three wave modes fulfill a triad resonance condition ${\stackrel{⃑}{k}}_3^s-{\stackrel{⃑}{k}}_2^s={\stackrel{⃑}{k}}_1^h$.

Steady-state solutions to the advection-diffusion equation for a passive scalar, with a chaotic divergence-free flow, are determined using a discrete-time, finite-difference model. The physical system studied is a density of particles diffusing across a chaotic layer. The impact of the advective structures on the solutions is illustrated, with special attention given to the cantori. It is argued that cantori play an important role in restricting transport and that coordinates adapted to cantori, called ghost coordinates, provide a natural framework about which the dynamics may be organized; for example, the averaged density profile becomes a smoothed devil’s staircase in ghost coordinates.

We study a class of models incorporating threshold-activated coupling on a lattice of chaotic elements, evolving under updating rules incorporating varying degrees of synchronicity. Interestingly, we observe that asynchronous updating, both random and sequential, yields more spatiotemporal order than parallel (synchronous) updating. Further, the order induced by random asynchronous updating is very robust and occurs even for small asynchronicities in the temporal evolution of the local dynamics. So this case study suggests a very different mechanism for inducing regularity in extended systems.

The dynamic behavior of cavitation and gas bubbles under negative pressure has been studied numerically to evaluate the effect of gas bubble injection into a liquid on the suppression of cavitation inception. In our previous studies, it was demonstrated by direct observation that cavitation occurs in liquid mercury when mechanical impacts are imposed, and this will cause cavitation damage in spallation neutron sources, in which liquid mercury is bombarded by a high-power proton beam. In the present paper, we describe numerical investigations of the dynamics of cavitation bubbles in liquid mercury using a multibubble model that takes into account the interaction of a cavitation bubble with preexisting gas bubbles through bubble-radiated pressure waves. The numerical results suggest that, if the mercury includes gas bubbles whose equilibrium radius is much larger than that of the cavitation bubble, the explosive expansion of the cavitation bubble (i.e., cavitation inception) is suppressed by the positive-pressure wave radiated by the injected bubbles, which decreases the magnitude of the negative pressure in the mercury.

Isotropic, rotating, and stratified turbulent flows are analyzed using a scale- and direction-dependent flatness. The anisotropy of the spatial fluctuations of the energy distribution can hereby be quantified for different length scales. This measure allows one to distinguish between longitudinal and transversal intermittency as well as between horizontal and vertical intermittency. The difference between longitudinal and transversal intermittency is argued to be related to the incompressiblity constraint. A large difference between horizontal and vertical intermittency for stratified turbulence can be explained by an energy depletion of the horizontal plane in Fourier space.

We present a constant-pressure molecular dynamics simulation study of the phase behavior of binary (50:50) Gay-Berne liquid crystal mixtures consisting of elongated particles with different lengths $\left(L_A>L_B\right)$ and equal diameters. We focus on systems at dense liquid-state conditions. Considering three mixtures characterized by different values of $L_{A\left(B\right)}$ and different length ratios $q=L_B∕L_A<1$, we find complex fluid-fluid phase behavior resulting from the interplay between nematic, smectic-$A$-type, or smectic-$B$-type orientational ordering, on the one hand, and demixing into two phases of different composition (fractionation), on the other hand. The driving “forces” of demixing transitions are the temperature and the length ratio. Indeed, in the system characterized by the largest value of $q$ $\left(q=0.86\right)$ orientational order occurs already in mixed states, whereas full fractionation is found at $q=0.71$. The two resulting states are either of type smectic-$B$-nematic (intermediate temperatures) or smectic-$B$-smectic-$B$ (low temperatures). In the intermediate case $q=0.80$ we observe a stepwise ordering and demixing behavior on cooling the system from high temperatures. Moreover, our results show that the stability range of (partially) nematic structures in mixtures of sufficiently small $q$ can be significantly larger than in the pure counterparts, in qualitative agreement with experimental observations.

We present a theoretical analysis of the phase behavior of colloid-polymer mixtures which applies to all polymer/colloid size ratios $q$. It accounts for the crossover from a constant length scale $R$ (radius of gyration) in the colloid limit (small $q$) to the concentration-dependent correlation length $\xi$ in the protein limit $(q>1)$. We obtain predictions that fully agree with observations and simulations. In the protein limit the colloid concentrations $\eta$ along the binodals become independent of $q$ and the polymer concentrations $\phi$ scale as $q^{1∕\gamma }$, where $\gamma =0.77$ is the scaling exponent in $\xi \sim {\phi }^{-\gamma }$: phase diagrams plotted as $\phi q^{-1∕\gamma }$ vs $\eta$ are then independent of $q$. The liquid window in the protein limit is narrow.

Vesicles under shear flow exhibit various dynamics: tank treading $\left(\mathit{TT}\right)$, tumbling $\left(\mathit{TB}\right)$, and vacillating breathing $\left(\mathit{VB}\right)$. The $\mathit{VB}$ mode consists in a motion where the long axis of the vesicle oscillates about the flow direction, while the shape undergoes a breathing dynamics. We extend here the original small deformation theory [C. Misbah, Phys. Rev. Lett. 96, 028104 (2006)] to the next order in a consistent manner. The consistent higher order theory reveals a direct bifurcation from $\mathit{TT}$ to $\mathit{TB}$ if $C_a\equiv \tau \dot{\gamma }$ is small enough—typically below 0.5, but this value is sensitive to the available excess area from a sphere ($\tau =\text{vesicle}$ relaxation time towards equilibrium shape, $\dot{\gamma }=\text{shear}$ rate). At larger $C_a$ the $\mathit{TB}$ is preceded by the $\mathit{VB}$ mode. For $C_a⪢1$ we recover the leading order original calculation, where the $\mathit{VB}$ mode coexists with $\mathit{TB}$. The consistent calculation reveals several quantitative discrepancies with recent works, and points to new features. We briefly analyze rheology and find that the effective viscosity exhibits a minimum in the vicinity of the $\mathit{TT}\text{−}\mathit{TB}$ and $\mathit{TT}\text{−}\mathit{VB}$ bifurcation points. At small $C_a$ the minimum corresponds to a cusp singularity and is at the $\mathit{TT}\text{−}\mathit{TB}$ threshold, while at high enough $C_a$ the cusp is smeared out, and is located in the vicinity of the $\mathit{VB}$ mode but in the $\mathit{TT}$ regime.

A theoretical model has to stand the test against the real world to be of any practical use. The first step is to identify parameters in the model estimated from experimental data. In many applications where renewal point data are available, models of first-hitting times of underlying diffusion processes arise. Despite the seemingly simplicity of the model, the problem of how to estimate parameters of the underlying stochastic process has resisted solution. The few attempts have either been unreliable, difficult to implement, or only valid in subsets of the relevant parameter space. Here we present an estimation method that overcomes these difficulties, is computationally easy and fast to implement, and also works surprisingly well on small data sets. The method is illustrated on simulated and experimental data. Two common neuronal models—the Ornstein-Uhlenbeck and Feller models—are investigated.

Vertical shaking of a mixture of small and large beads can lead to segregation where the large beads either accumulate at the top of the sample, the so-called Brazil nut effect (BNE), or at the bottom, the reverse Brazil nut effect (RBNE). Here we demonstrate experimentally a sharp transition from the RBNE to the BNE when the particle coefficient of friction increases due to aging of the particles. This result can be explained by the two competing mechanisms of buoyancy and sidewall-driven convection, where the latter is assumed to grow in strength with increasing friction.

In a joint experimental and modeling approach we demonstrate chaos synchronization imposed by a delayed shared feedback coupling between two nonlinear electro-optic oscillators. Robust identical synchronization is obtained for both symmetric and strongly asymmetric timing of the mutual coupling, offering great potential for applications such as chaos-based communications. We further demonstrate antisynchronization as well as generalized synchronization with vanishing linear correlation, by detuning the nonlinearity in one of the oscillators.

We present a Monte Carlo algorithm that facilitates efficient parallel tempering simulations of the density of states $g\left(E\right)$. We show that the algorithm eliminates the supercritical slowing down in the case of the $Q=20$ and $Q=256$ Potts models in two dimensions, typical examples for systems with extreme first-order phase transitions. As recently predicted, and shown here, the microcanonical heat capacity along the calorimetric curve has negative values for finite systems.

A dynamical theory of geophysical precipitation pattern formation is presented and applied to irreversible calcium carbonate (travertine) deposition. Specific systems studied here are the terraces and domes observed at geothermal hot springs, such as those at Yellowstone National Park, and speleothems, particularly stalactites and stalagmites. The theory couples the precipitation front dynamics with shallow water flow, including corrections for turbulent drag and curvature effects. In the absence of capillarity and with a laminar flow profile, the theory predicts a one-parameter family of steady state solutions to the moving boundary problem describing the precipitation front. These shapes match the measured shapes near the vent at the top of observed travertine domes well. Closer to the base of the dome, the solutions deviate from observations and circular symmetry is broken by a fluting pattern, which we show is associated with capillary forces causing thin film break-up. We relate our model to that recently proposed for stalactite growth, and calculate the linear stability spectrum of both travertine domes and stalactites. Lastly, we apply the theory to the problem of precipitation pattern formation arising from turbulent flow down an inclined plane and identify a linear instability that underlies scale-invariant travertine terrace formation at geothermal hot springs.

We develop the semiclassical method of complex trajectories in application to chaotic dynamical tunneling. First, we suggest a systematic numerical technique for obtaining complex tunneling trajectories by the gradual deformation of the classical ones. This provides a natural classification of the tunneling solutions. Second, we present a heuristic procedure for sorting out the least suppressed trajectory. As an illustration, we apply our technique to the process of chaotic tunneling in a quantum mechanical model with two degrees of freedom. Our analysis reveals rich dynamics of the system. At the classical level, there exists an infinite set of unstable solutions forming a fractal structure. This structure is inherited by the complex tunneling paths and plays a central role in the semiclassical study. The process we consider exhibits the phenomenon of optimal tunneling: the suppression exponent of the tunneling probability has a local minimum at a certain energy which is thus (locally) the optimal energy for tunneling. We test the proposed method by comparison of the semiclassical results with the results of the exact quantum computations and find a good agreement.

We study the stability of a constant volume of fluid spreading down an incline. In contrast to the commonly considered flow characterized by constant fluid flux, in the present problem the base flow is time dependent. We present a method to carry out consistently linear stability analysis, based on simultaneously solving the time evolution of the base flow and of the perturbations. The analysis is performed numerically by using a finite-difference method supplemented with an integral method developed here. The computations show that, after a short transient stage, imposed perturbations travel with the same velocity as the leading contact line. The spectral analysis of the modes evolution shows that their growth rates are, in general, time dependent. The wavelength of maximum amplitude, ${\lambda }_{max}$, decreases with time until it reaches an asymptotic value which is in good agreement with experimental results. We also explore the dependence of ${\lambda }_{max}$ on the cross sectional fluid area $A$, and on the inclination angle $\alpha$ of the substrate. For considered small $A$’s, corresponding to small Bond numbers, we find that the dependence of ${\lambda }_{max}$ on $A$ is in good agreement with experimental data. This dependence differs significantly from the one observed for the films characterized by much larger $A$’s and Bond numbers. We also predict the dependence of ${\lambda }_{max}$ on the inclination angle $\alpha$.

We study non-Gaussian probability density functions (PDF’s) of multiplicative log-normal models in which the multiplication of Gaussian and log-normally distributed random variables is considered. To describe the PDF of the velocity difference between two points in fully developed turbulent flows, the non-Gaussian PDF model was originally introduced by Castaing et al. [Physica D 46, 177 (1990)]. In practical applications, an experimental PDF is approximated with Castaing’s model by tuning a single non-Gaussian parameter, which corresponds to the logarithmic variance of the log-normally distributed variable in the model. In this paper, we propose an estimator of the non-Gaussian parameter based on the $q$th order absolute moments. To test the estimator, we introduce two types of stochastic processes within the framework of the multiplicative log-normal model. One is a sequence of independent and identically distributed random variables. The other is a log-normal cascade-type multiplicative process. By analyzing the numerically generated time series, we demonstrate that the estimator can reliably determine the theoretical value of the non-Gaussian parameter. Scale dependence of the non-Gaussian parameter in multiplicative log-normal models is also studied, both analytically and numerically. As an application of the estimator, we demonstrate that non-Gaussian PDF’s observed in the S&P500 index fluctuations are well described by the multiplicative log-normal model.

Multiple topplings and grain redistribution are two essential features in sandpile dynamics. A renormalization group (RG) approach incorporating these features is investigated. The full enumeration of all relaxations involving such an approach is difficult. Instead, we developed an efficient procedure to sample the relaxations. We applied this RG scheme to a square lattice and a triangular lattice. As shown by the fixed point analysis on a square lattice, the effect of multiple topplings leads the resultant height probabilities towards the exact solution while the effect of grain redistribution does not.

We describe a general method for the multiscale analysis of stochastic lattice models. Beginning with a lattice Langevin formulation of site fluctuations, we derive stochastic partial differential equations by regularizing the transition rules of the model. Subsequent coarse graining is accomplished by calculating renormalization-group (RG) trajectories from initial conditions determined by the regularized atomistic models. The RG trajectories correspond to hierarchies of continuum equations describing lattice models over expanding length and time scales. These continuum equations retain a quantitative connection over different scales, as well as to the underlying atomistic dynamics. This provides a systematic method for the derivation of continuum equations from the transition rules of lattice models for any length and time scales. As an illustration we consider the one-dimensional (1D) Wolf-Villain (WV) model [Europhys. Lett. 13, 389 (1990)]. The RG analysis of this model, which we develop in detail, is generic and can be applied to a wide range of conservative lattice models. The RG trajectory of the 1D WV model shows a complex crossover sequence of linear and nonlinear stochastic differential equations, which is in excellent agreement with kinetic Monte Carlo simulations of this model. We conclude by discussing possible applications of the multiscale method described here to other nonequilibrium systems.