We propose a mean-field method to calculate approximately the spacing distribution functions $p^{\left(n\right)}\left(s\right)$ in one-dimensional classical many-particle systems. We compare our method with two other commonly used methods, the independent interval approximation and the extended Wigner surmise. In our mean-field approach, $p^{\left(n\right)}\left(s\right)$ is calculated from a set of Langevin equations, which are decoupled by using a mean-field approximation. We find that in spite of its simplicity, the mean-field approximation provides good results in several systems. We offer many examples illustrating that the three previously mentioned methods give a reasonable description of the statistical behavior of the system. The physical interpretation of each method is also discussed.

We examine the evolution of an Ising ferromagnet endowed with zero-temperature single spin-flip dynamics. A large droplet of one phase in the sea of the opposite phase eventually disappears. An interesting behavior occurs in the intermediate regime when the droplet is still very large compared to the lattice spacing, but already very small compared to the initial size. In this regime the shape of the droplet is essentially deterministic (fluctuations are negligible in comparison with characteristic size). In two dimensions the shape is also universal, that is, independent of the initial shape. We analytically determine the limiting shape of the Ising droplet on the square lattice. When the initial state is a semi-infinite stripe of one phase in the sea of the opposite phase, it evolves into a finger which translates along its axis. We determine the limiting shape and the velocity of the Ising finger on the square lattice. An analog of the Ising finger on the cubic lattice is the translating Ising soliton. We show that far away from the tip, the cross-section of the Ising soliton coincides with the limiting shape of the two-dimensional Ising droplet and we determine a relation between the cross-section area, the distance from the tip, and the velocity of the soliton.

We report numerical results on the velocity statistics of topological defects during the dynamics of phase ordering and nonrelaxational evolution assisted by an external shear flow. We propose a numerically efficient tracking method for finding the position and velocity of defects and apply it to vortices in a uniform field and dislocations in anisotropic stripe patterns. During relaxational dynamics, the distribution function of the velocity fluctuations is characterized by a dynamical scaling with a scaling function that has a robust algebraic tail with an inverse cube power law. This is characteristic of defects of codimension 2, e.g., point defects in two dimensions and filaments in three dimensions, regardless of whether the motion is isotropic (as for vortices) or highly anisotropic (as for dislocations). However, the anisotropic dislocation motion leads to anisotropic statistical properties when the interaction between defects and their motion is influenced by the presence of an external shear flow transverse to the stripe orientation.

We experimentally analyze collective dynamics of a population of 20 electronic Wien-bridge limit-cycle oscillators with a nonlinear phase-shifting unit in the global feedback loop. With an increase in the coupling strength we first observe formation and then destruction of a synchronous cluster, so that the dependence of the order parameter on the coupling strength is not monotonic. After destruction of the cluster the ensemble remains nevertheless coherent, i.e., it exhibits an oscillatory collective mode (mean field). We show that the system is now in a self-organized quasiperiodic state, predicted in Rosenblum and Pikovsky [Phys. Rev. Lett. 98, 064101 (2007)]. In this state, frequencies of all oscillators are smaller than the frequency of the mean field, so that the oscillators are not locked to the mean field they create and their dynamics is quasiperiodic. Without a nonlinear phase-shifting unit, the system exhibits a standard Kuramoto-like transition to a fully synchronous state. We demonstrate a good correspondence between the experiment and previously developed theory. We also propose a simple measure which characterizes the macroscopic incoherence-coherence transition in a finite-size ensemble.

We study the influence of noise on the spatially localized, temporally regular states (stationary, one frequency, two frequencies) in the regime of anomalous dispersion for the cubic-quintic complex Ginzburg-Landau equation as a function of the bifurcation parameter. We find that noise of a fairly small strength $\eta$ is sufficient to reach a chaotic state with exploding dissipative solitons. That means that noise can induce explosions over a fairly large range of values of the bifurcation parameter $\mu$. Three different routes to chaos with exploding dissipative solitons are found as a function of $\mu$. As diagnostic tools we use the separation to characterize chaotic behavior and the energy to detect spatially localized explosive behavior as a function of time.

We consider long-wave Marangoni convection in a liquid layer atop a substrate of low thermal conductivity, heated from below. We demonstrate that the critical perturbations are materialized at the wave number $K\sim \sqrt{\mathrm{Bi}}$, where $\mathrm{Bi}$ is the Biot number which characterizes the weak heat flux from the free surface. In addition to the conventional monotonic mode, a novel oscillatory mode is found. Applying the $K\sim \sqrt{\mathrm{Bi}}$ scaling, we derive a new set of amplitude equations. Pattern selection on square and hexagonal lattices shows that supercritical branching is possible. A large variety of stable patterns is found for both modes of instability. Finite-amplitude one-dimensional solutions of the set, corresponding to either steady or traveling rolls, are studied numerically; a complicated sequence of bifurcations is found in the former case. The emergence of an oscillatory mode in the case of heating from below and stable patterns with finite-amplitude surface deformation are shown in this system for the first time.

The effect of ambient temperature on the parameters of the single-bubble sonoluminescence in sulfuric acid (SA) diluted in water is studied. Using a hydrochemical model, three dominant instabilities of shape, Bjerknes, and diffusion are considered. The phase diagrams of the bubble in the ($R_0$ − $P_a$) space are presented, and the parametric dependence of the light intensity is discussed. In contrast to water, the calculated thermal-bremsstrahlung mechanism of light emission at the fixed degassing condition of high SA concentrations shows that, with increasing the temperature of aqueous SA solutions, the light intensity increases. However, at diluted SA solutions similar to water, the light intensity decreases with increasing the ambient temperature. For 50 wt $\%$ SA, it was observed that the emitted light was almost temperature independent. Furthermore, it is found that, at the fixed temperatures of 20 °C, 10 °C, and 0 °C, the aqueous solutions of 65 wt $\%$, 50 wt $\%$, and 45 wt $\%$ SA, respectively, have the maximum light emission.

The dynamic interactions between a semipermeable membrane and a long, thin layer of liquid beneath it are investigated in the context of drying processes. The membrane separates two aqueous solutions of sugar, and the transport of water across the membrane is driven by concentration and pressure gradients across it. A model is formulated using a long-wave approximation that includes the effects of volume loss due to water transport across the membrane, the incompressibility and bending stiffness of the membrane, and the dynamical effects that arise owing to the viscous stresses generated by the fluid flow. This model is first applied to study the desiccation of a sessile vesicle that is clamped to a rigid substrate and then also to study the behavior of blisters on laminated substrates. For each problem, equilibrium membrane shapes are obtained and their bifurcation structures are described as the sugar concentration above the membrane is varied. It is demonstrated that a wrinkled membrane coarsens to lessen the frequency of wrinkles and that if the membrane is clamped symmetrically so that it meets the substrate at a nonzero angle, then the membrane favors an asymmetric shape as water is drawn out through it.

Hyperbolicity is an important concept in dynamical system theory; however, we know little about the hyperbolicity of concrete physical systems including fluid motions governed by the Navier-Stokes equations. Here, we study numerically the hyperbolicity of the Navier-Stokes equation on a two-dimensional torus (Kolmogorov flows) using the method of covariant Lyapunov vectors developed by Ginelli et al. [Phys. Rev. Lett. 99, 130601 (2007)]. We calculate the angle between the local stable and unstable manifolds along an orbit of chaotic solution to evaluate the hyperbolicity. We find that the attractor of chaotic Kolmogorov flows is hyperbolic at small Reynolds numbers, but that smaller angles between the local stable and unstable manifolds are observed at larger Reynolds numbers, and the attractor appears to be nonhyperbolic at a certain Reynolds numbers. Also, we observed some relations between these hyperbolic properties and physical properties such as time correlation of the vorticity and the energy dissipation rate.

A general description of cylindrical electromagnetic waves propagating in nonlinear and inhomogeneous media is given by deducing cylindrical coupled-wave equations. Based on the cylindrical coupled-wave equations, we analyze second-harmonic generation (SHG) of some special cases of inhomogeneity, and find that the inhomogeneity of the first- and second-order polarization can influence the amplitude of the SHG. From a different point of view, exact solutions of cylindrical electromagnetic waves propagating in a nonlinear medium with a special case of inhomogeneity have been obtained previously. We show that cylindrical SHG in an inhomogeneous and nonlinear medium can also be deduced from exact solutions. As verification, we compare the results obtained from the two different methods and find that descriptions of SHG by the coupled-wave equations are in good agreement with the exact solutions.

We investigate numerically the effect of the competition of disorder, nonlinearity, and boundaries on the Anderson localization of light waves in finite-size, one-dimensional waveguide arrays. Using the discrete Anderson–nonlinear Schrödinger equation, the propagation of the mode amplitudes up to some finite distance is monitored. The analysis is based on the calculated localization length and the participation number, two standard measures for the statistical description of Anderson localization. For relatively weak disorder and nonlinearity, a higher disorder strength is required to achieve the same degree of localization at the edge than in the interior of the array, in agreement with recent experimental observations in the linear regime. However, for relatively strong disorder and/or nonlinearity, this behavior is reversed and it is now easier to localize an excitation at the edge than in the interior.

An anomalous transport phenomenon termed absolute negative mobility (ANM) was observed in a vibrational motor, where an additional time-periodic signal filled the role usually played by noise in a Brownian motor. Within a tailored parameter regime, the ANM behavior is maximized at two regimes upon variation of the bias. The observed ANM still survives at a wide range of the driving strength and angular frequency of the additional signal.

Using molecular dynamics simulations, we calculate fluctuations and responses for steadily sheared hard spheres over a wide range of packing fractions $\phi$ and shear strain rates $\dot{\gamma }$, using two different methods to dissipate energy. To a good approximation, shear stress and density fluctuations are related to their associated response functions by a single effective temperature $T_{\mathrm{eff}}$ that is equal to or larger than the kinetic temperature $T_{\mathrm{kin}}$. We find a crossover in the relationship between the relaxation time $\tau$ and the the nondimensionalized effective temperature $T_{\mathrm{eff}}/p{\sigma }^3$, where $p$ is the pressure and $\sigma$ is the sphere diameter. In the solid response regime, the behavior at a fixed packing fraction satisfies $\tau \dot{\gamma }\propto \exp (-cp{\sigma }^3/T_{\mathrm{eff}})$, where $c$ depends weakly on $\phi$, suggesting that the average local yield strain is controlled by the effective temperature in a way that is consistent with shear transformation zone theory. In the fluid response regime, the relaxation time depends on $T_{\mathrm{eff}}/p{\sigma }^3$ as it depends on $T_{\mathrm{kin}}/p{\sigma }^3$ in equilibrium. This regime includes both near-equilibrium conditions where $T_{\mathrm{eff}}\simeq T_{\mathrm{kin}}$ and far-from-equilibrium conditions where $T_{\mathrm{eff}}\ne T_{\mathrm{kin}}$. We discuss the implications of our results for systems with soft repulsive interactions.

We analyze bulk diffusion and transport through hydrophobic nanochannels of nanoparticles (NPs) with different hydrophobic-hydrophilic patterns achieved by coating a fraction of the NP sites with positive or negative charges via explicit solvent molecular dynamics simulations. Ten different charge pattern types including Janus charged-hydrophobic NPs are studied. The cationic NPs are more affected by the patterns and have higher diffusion constants and fluxes than their anionic NPs counterparts. The NP-water interaction dependence on surface pattern and field strength explains these observations. The NP-water Coulomb interaction of anionic NPs in the bulk, which are much stronger than the hydrophobic NP-water interactions, are stronger for NPs with higher localized charge, and stronger than in the cationic NPs counterparts. The diffusion and transport of anionic NPs such as proteins and protein charge ladders with the same total charge but different surface charge patterns are slowest for the highest localized charge pattern, which also adsorb strongest onto surfaces. Our model demonstrates the separation (by reverse osmosis, capillary electrophoresis, or chromatography) of cationic NPs, including proteins with equal net charge but different surface charge distributions.

The two-dimensional (2D) growth dynamics of HeLa (cervix cancer) cell colonies was studied following both their growth front and the pattern morphology evolutions utilizing large population colonies exhibiting linearly and radially spreading fronts. In both cases, the colony profile fractal dimension was $d_f=1.20\pm 0.05$ and the growth fronts displaced at the constant velocity $0.90\pm 0.05$ $\mu$m min${}^{-1}$. Colonies showed changes in both cell morphology and average size. As time increased, the formation of large cells at the colony front was observed. Accordingly, the heterogeneity of the colony increased and local driving forces that set in began to influence the dynamics of the colony front. The dynamic scaling analysis of rough colony fronts resulted in a roughness exponent $\alpha$ $=$ $0.50\pm 0.05$, a growth exponent $\beta$ $=$ $0.32\pm 0.04$, and a dynamic exponent $z=1.5\pm 0.2$. The validity of this set of scaling exponents extended from a lower cutoff $l_c\approx 60$ $\mu$m upward, and the exponents agreed with those predicted by the standard Kardar-Parisi-Zhang continuous equation. HeLa data were compared with those previously reported for Vero cell colonies. The value of $d_f$ and the Kardar-Parisi-Zhang-type 2D front growth dynamics were similar for colonies of both cell lines. This indicates that the cell colony growth dynamics is independent of the genetic background and the tumorigenic nature of the cells. However, one can distinguish some differences between both cell lines during the growth of colonies that may result from specific cooperative effects and the nature of each biosystem.

An experimental demonstration of reaction-driven viscous fingering developing when a more viscous solution of a reactant $A$ displaces a less viscous miscible solution of another reactant $B$ is presented. In the absence of reaction, such a displacement of one fluid by another less mobile one is classically stable. However, a simple $A+B\to C$ reaction can destabilize this interface if the product $C$ is either more or less viscous than both reactant solutions. Using the pH dependence of the viscosity of some polymer solutions, we provide experimental evidence of both scenarios. We demonstrate quantitatively that reactive viscous fingering results from the buildup in time of nonmonotonic viscosity profiles with patterns behind or ahead of the reaction zone, depending on whether the product is more or less viscous than the reactants. The experimental findings are backed up by numerical simulations.

This study is an overview of network topology metrics and a computational approach to analyzing graph topology via multiple-metric analysis on graph ensembles. The paper cautions against studying single metrics or combining disparate graph ensembles from different domains to extract global patterns. This is because there often exists considerable diversity among graphs that share any given topology metric, patterns vary depending on the underlying graph construction model, and many real data sets are not actual statistical ensembles. As real data examples, we present five airline ensembles, comprising temporal snapshots of networks of similar topology. Wikipedia language networks are shown as an example of a nontemporal ensemble. General patterns in metric correlations, as well as exceptions, are discussed by representing the data sets via hierarchically clustered correlation heat maps. Most topology metrics are not independent and their correlation patterns vary across ensembles. In general, density-related metrics and graph distance-based metrics cluster and the two groups are orthogonal to each other. Metrics based on degree-degree correlations have the highest variance across ensembles and cluster the different data sets on par with principal component analysis. Namely, the degree correlation, the $s$ metric, their elasticities, and the rich club moments appear to be most useful in distinguishing topologies.

Random walks have been successfully used to measure user or object similarities in collaborative filtering (CF) recommender systems, which is of high accuracy but low diversity. A key challenge of a CF system is that the reliably accurate results are obtained with the help of peers' recommendation, but the most useful individual recommendations are hard to be found among diverse niche objects. In this paper we investigate the direction effect of the random walk on user similarity measurements and find that the user similarity, calculated by directed random walks, is reverse to the initial node's degree. Since the ratio of small-degree users to large-degree users is very large in real data sets, the large-degree users' selections are recommended extensively by traditional CF algorithms. By tuning the user similarity direction from neighbors to the target user, we introduce a new algorithm specifically to address the challenge of diversity of CF and show how it can be used to solve the accuracy-diversity dilemma. Without relying on any context-specific information, we are able to obtain accurate and diverse recommendations, which outperforms the state-of-the-art CF methods. This work suggests that the random-walk direction is an important factor to improve the personalized recommendation performance.

We investigate two coupled oscillators, each of which shows an attracting heteroclinic cycle in the absence of coupling. The two heteroclinic cycles are nonidentical. Weak coupling can lead to the elimination of the slowing-down state that asymptotically approaches the heteroclinic cycle for a single cycle, giving rise to either quasiperiodic motion with separate frequencies from the two cycles or periodic motion in which the two cycles are synchronized. The synchronization transition, which occurs via a Hopf bifurcation, is not induced by the commensurability of the two cycle frequencies but rather by the disappearance of the weaker frequency oscillation. For even larger coupling the motion changes via a resonant heteroclinic bifurcation to a slowing-down state corresponding to a single attracting heteroclinic orbit. Coexistence of multiple attractors can be found for some parameter regions. These results are of interest in ecological, sociological, neuronal, and other dynamical systems, which have the structure of coupled heteroclinic cycles.