In string percolation model, the study of colliding systems at high energies is based on a continuum percolation theory in two dimensions, where the number of strings distributed in the surface of interest is strongly determined by the size and energy of the colliding particles. It is also expected that the surface where the disks are lying be finite, defining a system without periodic boundary conditions. In this work, we report modifications to the fraction of the area covered by disks in continuum percolating systems due to a finite number of disks and bounded by different geometries: circle, ellipse, triangle, square, and pentagon, which correspond to the first Fourier modes of the shape fluctuation of the initial state after the particle collision. We find that the deviation of the fraction of area covered by disks from its corresponding value in the thermodynamic limit satisfies a universal behavior, where the free parameters depend on the density profile, number of disks, and shape of the boundary. Consequently, it is also found that the color suppression factor of the string percolation model is modified by a damping function related to the small-bounded effects. Corrections to the temperature and the speed of sound defined in string systems are also shown for small and elliptically bounded systems.

We study two generalizations of the basin of attraction of a stable state, to the case of stochastic dynamics, arbitrary regions, and finite-time horizons. This is done by introducing generalized committor functions and studying soujourn times. We show that the volume of the generalized basin, the basin stability, can be efficiently estimated using Monte Carlo–like techniques, making this concept amenable to the study of high-dimension stochastic systems. Finally, we illustrate in a set of examples that stochastic basins efficiently capture the realm of attraction of metastable sets, which parts of phase space go into long transients in deterministic systems, that they allow us to deal with numerical noise, and can detect the collapse of metastability in high-dimensional systems. We discuss two far-reaching generalizations of the basin of attraction of an attractor. The basin of attraction of an attractor are those states that eventually will get to the attractor. In a generic stochastic system, all regions will be left again; no attraction is permanent. To obtain the equivalent of the basin of attraction of a region we need to generalize the notion to cover finite-time horizons and finite regions. We do so by considering soujourn times, the fraction of time that a trajectory spends in a set, and by generalizing committor functions which arise in the study of hitting probabilities. In a simplified setting we show that these two notions reduce to the normal notions of the basin of attraction in the appropriate limits. We also show that the volume of these stochastic basins can be efficiently estimated for high-dimensional systems at computational cost comparable to that for deterministic systems. To fully illustrate the properties captured by the stochastic basins, we show a set of examples ranging from simple conceptual models to high-dimensional inhomogeneous oscillator chains. These show that stochastic basins efficiently capture metastable attraction, the presence of long transients, that they allow us to deal with numerical and approximation noise, and can detect the collapse of metastability with increasing noise in high-dimensional systems.

In this work, the qualitative structures of traveling waves are investigated in a bidimensional inductor-capacitor network with quadratic nonlinear dispersion. Applying the continuum limit approximation, we show that the dynamics of small-amplitude signals in the network can be governed by a $(2+1)$-dimensional partial differential equation. Using a simple transformation, we reduce the given equation to a nonlinear ordinary differential equation. By means of the phase plane analysis and depending on the wave velocity of the signals that are to propagate in the lattice, we present all phase portraits of the dynamical system. Parametric representations for solitary-wave solutions corresponding to the various phase portrait trajectories under different parameter conditions are derived. The results of our study demonstrate that the nonlinear dispersion in the network leads to a number of interesting solitary-wave profiles, e.g., bright-dark solitons and gray-gray solitons, which have not been observed for the same model when the dispersion is assumed linear. The two-dimensional graphics of all the solutions obtained in this paper are given.

For waves described by the focusing nonlinear Schrödinger equation (FNLS), we present an effective dispersion relation (EDR) that arises dynamically from the interplay between the linear dispersion and the nonlinearity. The form of this EDR is parabolic for a robust family of “generic” FNLS waves and equals the linear dispersion relation less twice the total wave action of the wave in question multiplied by the square of the nonlinearity parameter. We derive an approximate form of this EDR explicitly in the limit of small nonlinearity and confirm it using the wave-number-frequency spectral (WFS) analysis, a Fourier-transform based method used for determining dispersion relations of observed waves. We also show that it extends to the FNLS the universal EDR formula for the defocusing Majda-McLaughlin-Tabak (MMT) model of weak turbulence. In addition, unexpectedly, even for some spatially periodic versions of multisolitonlike waves, the EDR is still a downward shifted linear-dispersion parabola, but the shift does not have a clear relation to the total wave action. Using WFS analysis and heuristic derivations, we present examples of parabolic and nonparabolic EDRs for FNLS waves and also waves for which no EDR exists.

The Hamiltonian mean-field (HMF) model describes particles on a ring interacting via a cosine interaction, or equivalently, rotors coupled by infinite-range XY interactions. Conceived as a generic statistical mechanical model for long-range interactions such as gravity (of which the cosine is the first Fourier component), it has recently been used to account for self-organization in experiments on cold atoms with long-range optically mediated interactions. The significance of the HMF model lies in its ability to capture the universal effects of long-range interactions and yet be exactly solvable in the canonical ensemble. In this work we consider the quantum version of the HMF model in one dimension and provide a classification of all possible stationary solutions of its generalized Gross-Pitaevskii equation (GGPE), which is both nonlinear and nonlocal. The exact solutions are Mathieu functions that obey a nonlinear relation between the wave function and the depth of the mean-field potential, and we identify them as bright solitons. Using a Galilean transformation these solutions can be boosted to finite velocity and are increasingly localized as the mean-field potential becomes deeper. In contrast to the usual local GPE, the HMF case features a tower of solitons, each with a different number of nodes. Our results suggest that long-range interactions support solitary waves in a novel manner relative to the short-range case.

An increasing global population forces urban planners to construct buildings and infrastructure that is extremely deep and high. Elevators and escalators serve skyscrapers and tunnels, but in an emergency people still have to walk on stairs. Computer simulations can mitigate risks of escape situations. For these situations, pedestrian locomotion models need to match reality well. Motion on stairs, however, is not nearly as well understood as motion in the plane. Publications are scarce and some are contradictory. As a result, movement on stairs is usually modeled by slowing down pedestrians by a fixed factor. But is this justified? And what happens at intermediate landings? This contribution aims to clarify inconclusive results of previous research and provide new information to directly incorporate empirical results into a parsimonious computer model. The algorithms are freely available through an open-source framework. After outlining the shortcomings of existing approaches, we present three experiments, from which we derive requirements for the computer model. Reenacting computer experiments shows the extent to which our model meets our observations. We conclude with an applied example, simulating an evacuation of Germany's famous Neuschwanstein Castle.

Community identification of network components enables us to understand the mesoscale clustering structure of networks. A number of algorithms have been developed to determine the most likely community structures in networks. Such a probabilistic or stochastic nature of this problem can naturally involve the ambiguity in resultant community structures. More specifically, stochastic algorithms can result in different community structures for each realization in principle. In this study, instead of trying to “solve” this community degeneracy problem, we turn the tables by taking the degeneracy as a chance to quantify how strong companionship each node has with other nodes. For that purpose, we define the concept of companionship inconsistency that indicates how inconsistently a node is identified as a member of a community regarding the other nodes. Analyzing model and real networks, we show that companionship inconsistency discloses unique characteristics of nodes, thus we suggest it as a new type of node centrality. In social networks, for example, companionship inconsistency can classify outsider nodes without firm community membership and promiscuous nodes with multiple connections to several communities. In infrastructure networks such as power grids, it can diagnose how the connection structure is evenly balanced in terms of power transmission. Companionship inconsistency, therefore, abstracts individual nodes' intrinsic property on its relationship to a higher-order organization of the network.

A prescribed, horizontal temperature gradient is imposed upon a horizontal liquid layer bounded from above by a deformable, liquid-gas interface and bounded from below by a partial-slip, rigid surface. A steady shear flow driven by thermocapillary motion emerges. This dynamic liquid layer is susceptible to the onset of oblique three-dimensional hydrothermal waves, purely two-dimensional hydrothermal waves, longitudinal traveling waves, and longitudinal rolls depending on the capillary number. A low capillary number analysis finds that surface deformations are destabilizing for all modes of instability. There is a preference for two-dimensional hydrothermal waves when there are surface deformations. Though longitudinal traveling waves are never selected as the preferred mode of instability, these waves offer a convenient way to understand the behavior of oblique hydrothermal waves, which are near-longitudinal. This is especially the case for low capillary numbers, but oblique hydrothermal waves instead tend to align themselves with the direction of flow as the capillary number increases. Surface deformations affect longitudinal waves most significantly out of all the modes of instability, especially for low Prandtl numbers. The typical length scales shorten and the critical Marangoni numbers increase with the capillary number for all types of modes. Notably, the system selects long waves near a critical Prandtl number when the interface is nondeformable and when the layer is subject to partial slip, but this is no longer the case when the upper surface is deformable.

Defects play a crucial role in physics of solids, affecting their mechanical, electromagnetic, and chemical properties. However, influence of thermal defects on wave propagation in exothermic reactions (flame fronts) still remains poorly understood at the molecular level. Here, we show that thermal behavior of the defects exhibits essential features of double-step exothermic reactions with preequilibrium. We use experiments with monolayer complex (dusty) plasma and find that it can show a double-step activation thermal behavior, similar to chemically reactive media. Furthermore, we demonstrate capabilities to control flame fronts using defects and the different dynamic regimes of the thermal defects in complex (dusty) plasmas, from a nonactivated one to being sound and self-activated (like in active soft matter). The results suggest that a range of challenging phenomena at the forefront of modern science (e.g., defect activation, flame front dynamics, reaction waves, etc.) can now be experimentally interrogated on a microscopic scale.

Complex spatiotemporal patterns of action potential duration have been shown to occur in many mammalian hearts due to period-doubling bifurcations that develop with increasing frequency of stimulation. Here, through high-resolution optical mapping experiments and mathematical modeling, we introduce a characteristic spatial length of cardiac activity in canine ventricular wedges via a spatiotemporal correlation analysis, at different stimulation frequencies and during fibrillation. We show that the characteristic length ranges from 40 to 20 cm during one-to-one responses and it decreases to a specific value of about 3 cm at the transition from period-doubling bifurcation to fibrillation. We further show that during fibrillation, the characteristic length is about 1 cm. Another significant outcome of our analysis is the finding of a constitutive phenomenological law obtained from a nonlinear fitting of experimental data which relates the conduction velocity restitution curve with the characteristic length of the system. The fractional exponent of $3/2$ in our phenomenological law is in agreement with the domain size remapping required to reproduce experimental fibrillation dynamics within a realistic cardiac domain via accurate mathematical models.

We study correspondence between a phase oscillator network with distributed natural frequencies and a classical $\mathit{XY}$ model at finite temperatures with the same random and frustrated interactions used in the Sherrington-Kirkpatrick model. We perform numerical calculations of the spin glass order parameter $q$ and the distributions of the local fields $P\left(R\right)$, where $R$ is the amplitude of the local field. As a result, we find that the parameter dependences of $P\left(R\right)$ in both models agree fairly well in all ranges of parameters in the spin glass phase and those of $q$ agree at least for lower values of parameters in the spin glass phase, if parameters are normalized by using the previously obtained correspondence relation between two models with the same other types of interactions. Furthermore, we numerically calculate the time evolution of quantities such as the instantaneous local field in the phase oscillator network in order to study the roles of synchronous and asynchronous oscillators. We also study the self-consistent equation of the local fields in the oscillator network and $\mathit{XY}$ model derived by the mean-field approximation.

Models of discrete space and space-time that exhibit continuum-like behavior at large lengths could have profound implications for physics. They may help tame the infinities arising from quantizing gravity, and remove the need for the machinery of the real numbers; a construct with no direct observational support. However, despite many attempts to build discrete space, researchers have failed to produce even the simplest geometries. Here we investigate graphs as the most elementary discrete models of two-dimensional space. We show that if space is discrete, it must be disordered, by proving that all planar lattice graphs exhibit a taxicab metric similar to square grids. We then give an explicit recipe for growing disordered discrete space by sampling a Boltzmann distribution of graphs at low temperature. Finally, we propose three conditions which any discrete model of Euclid's plane must meet: have a Hausdorff dimension of 2, support unique straight lines, and obey Pythagoras' theorem. Our model satisfies all three, resulting in a discrete model in which continuum-like behavior emerges at large lengths.

Cascading failures in the internet have attracted recent attention due to their unpredictability and destructive consequences. Exploring the failure behavior patterns is necessary because they can provide effective intervention approaches to prevent huge network disasters. To analyze internet flow behaviors during cascading failures (chain reactions in router and link failures), we characterize the internet as two coupled networks, the router network and the flow network. Here the flow network is an abstract representation of data correlations obtained from the router network. We use this coupled network to build a cascading failure model for studying flow transmission and competition, which is reflected in bandwidth competition given by limited link capacity. We first study the dependency between routers and flows to explore the flow transmission efficiency when a failure event occurs. Moreover, we find that rerouting enables flow competition area (the number of flows with which one flow has a competitive relationship) to initially remain stable during a failure episode, but that it then quickly drops due to poor physical network connectivity. Additionally, in the early stage after the failure event, the degree of flow competition sharply increases because of the growing number of the flows and congestion. Subsequently, the flow competition decreases due to the failure of flow transmission.

In this paper we demonstrate that new phase defects of the Gross-Pitaevskii equation (GPE) can be produced as a Aharonov-Bohm effect resulting from pure phase twist injection on existing defects. This is a phenomenon that has physical justification in the hydrodynamic interpretation of GPE. Here we give an analytical proof of its effects by using Fermi-Walker transport and Biot-Savart induction law. An analytical derivation of the dispersion relation is derived from the superposition of phase twist on the fundamental state. Since the extra twist corresponds to a topological change of the total linking number of the system, we show that the production of new defects is just another manifestation of the Aharonov-Bohm effect. We propose a laboratory experiment for Bose-Einstein condensates to test this phenomenon and to show that it can have useful applications in science and technology.

Determining design principles that boost the robustness of interdependent networks is a fundamental question of engineering, economics, and biology. It is known that maximizing the degree correlation between replicas of the same node leads to optimal robustness. Here we show that increased robustness might also come at the expense of introducing multiple phase transitions. These results reveal yet another possible source of fragility of multiplex networks that has to be taken into the account during network optimization and design.

We propose a model for a chain vibrating in three dimensions, with first neighbors anharmonic interatomic potential, which depends on their distance, and subjected to an external tension. In the framework of the nonlinear fluctuating hydrodynamic theory, which was successfully applied to one-dimensional chains, we obtain a heat mode, two longitudinal, and four transverse sound modes. We compute their spatiotemporal correlations comparing the theoretical results with molecular dynamics simulations, finding a good agreement for high temperatures. We find that the transverse sound modes behave diffusively, meanwhile the heat and longitudinal sound modes behave superdiffusively, exploring their possible scaling functions and characteristic exponents.

How often should a random walker change its direction of motion in order to maximize correlation in velocity fluctuations over a finite time interval? We address this optimal diffusion problem in the context of the one-dimensional persistent random walk, where we evaluate the correlation and mutual information in velocity trajectories as a function of the persistence level and the observation time. We find the optimal persistence level corresponds to the average number of direction reversals asymptotically scaling as the square root of the observation time. This square-root scaling law makes the relative growth between the average number of direction reversals and the persistence length invariant with respect to changes in the overall time duration of the random walk.

We model the diffusion of a tracer that interacts with the internal surface of a porous medium formed by a packing of solid spheres. The tracer executes a lattice random walk in which hops from surface to bulk sites and hops on the surface have small probabilities compared to hops from bulk sites; those probabilities are related to bulk and surface diffusion coefficients and to a desorption rate. A scaling approach distinguishes three regimes of steady state diffusion, which are confirmed by numerical simulations. If the product of desorption rate and sphere diameter is large, dominant bulk residence is observed and the diffusion coefficient is close to the bulk value. If that product is small and the surface mobility is low, the tracers are adsorbed most of the time but most hops are executed in the bulk. However, for high surface mobility, there is a nontrivial regime of dominant surface displacement, since the connectivity of solid walls allows the tracers to migrate to long distances while they are adsorbed. In this regime, we observe rounded tracer paths on the sphere walls, which are qualitatively similar to those of a recent experiment on polystyrene particle diffusion. The calculated average residence times are proportional to the bulk and surface densities of an equilibrium ensemble of noninteracting tracers, and the relation between those densities sets the adsorption isotherm. Simulations performed with initially uniform (nonequilibrium) distribution of tracers in the pores show other nontrivial results in cases of dominant surface residence: slow increase of the mean-square displacement at short times, since the tracer has not explored a homogeneous medium, and a remarkable slowdown between the first encounter with a solid wall and the first hop from that point. Relations between our results and other models of diffusion and adsorption in porous media are discussed.