Stationary dynamical systems have invariant measures (or densities) that are characteristic of the particular dynamical system. We develop a method to characterize this density by partitioning the attractor into the smallest regions in phase space that contain information about the structure of the attractor. To accomplish this, we develop a statistic that tells us if we get more information about our data by dividing a set of data points into partitions rather than just lumping all the points together. We use this method to show that not only can we detect small changes in an attractor from a circuit experiment, but we can also distinguish between a large set of numerically generated chaotic attractors designed by Sprott. These comparisons are not limited to chaotic attractors—they should work for signals from any finite-dimensional dynamical system.

The competitive wetting of oil and aqueous electrolytes on solid surfaces depends strongly on the surface charge of the solid-water and the water-oil interface. This charge density is generally not known a priori but changes as ions adsorb or desorb from or to the interfaces, depending on the composition of the fluid and the thickness of thin films of the aqueous phase that frequently arise on hydrophilic surfaces, such as minerals. We analyze the wettability of such systems by coupling standard Derjaguin-Landau-Verwey-Overbeek theory to a linearized charge regulation model. The latter is found to play an important role. By linearizing electrostatic interactions as well, we obtain a fully analytic description of transitions between different wetting scenarios as a function of the surface potentials at infinite separation and the charge regulation parameters of the two interfaces. Depending on the specific values of the regulation parameters, charge regulation is found to extend the parameter range of partial wetting and complete wetting at the expense of pseudopartial wetting and metastable wetting configurations, respectively. A specific implementation of the model is discussed for mica-water-alkane systems that was investigated in recent experiments.

Liquid disordered $\left(L_d\right)$ to liquid ordered $\left(L_o\right)$ phase transition in myristic acid [MyA, ${\text{CH}}_3{\left({\text{CH}}_2\right)}_{12}\text{COOH}$] Langmuir monolayers was studied macroscopically as well as mesoscopically to locate the critical point. Macroscopically, isotherms of the monolayer were obtained across the $20{}^{\circ }\mathrm{C}\text{–}38{}^{\circ }\mathrm{C}$ temperature $\left(T\right)$ range and the critical point was estimated, primarily from the vanishing of the order parameter, at $\approx 38{}^{\circ }\mathrm{C}$. Mesoscopically, domain morphology in the $L_d\text{−}{L}_o$ coexistence regime was imaged using the technique of Brewster angle microscopy (BAM) as a function of $T$ and the corresponding power spectral density function (PSDF) obtained. Monolayer morphology passed from stable circular domains and a sharp peak in PSDF to stable dendritic domains and a divergence of the correlation length as the critical point was approached from below. The critical point was found to be consistent at $\approx 38{}^{\circ }\mathrm{C}$ from both isotherm and BAM results. In the critical regime the scaling behavior of the transition followed the two-dimensional Ising model. Additionally, we obtained a precritical regime, over a temperature range of $\approx 8{}^{\circ }\mathrm{C}$ below $T_c$, characterized by fluctuations in the order parameter at the macroscopic scale and at the mesoscopic scale characterized by unstable domains of fingering or dendritic morphology as well as proliferation of a large number of small sized domains, multiple peaks in the power spectra, and a corresponding fluctuation in the peak $q$ values with $T$. Further, while comparing temperature studies on an ensemble of MyA monolayers with those on a single monolayer, the system was found to be not strictly ergodic in that the ensemble development did not strictly match with the time development in the system. In particular, the critical temperature was found to be lowered in the latter. These results clearly show that the critical behavior in fatty acid monolayer phase transitions have features of both complex and nonequilibrium systems.

This work is aimed at developing a simple and efficient predictive model and revealing the volume evolution behavior of small sessile droplets evaporating in stick-slip (SS) mode, covering a wide range of initial $\left({\theta }_0\right)$ and receding $\left({\theta }_{\mathrm{rec}}\right)$ contact angles $\left(0{}^{\circ }<{\theta }_{\mathrm{rec}}\le {\theta }_0<180{}^{\circ }\right)$. For this purpose, simple closed-form expressions for the evolutions of the contact angle and contact radius of droplets evaporating in SS mode are first developed, and are then validated with the experimental data in the literature. With the proposed expressions, the volume evolutions of evaporating droplets with different ${\theta }_0$ and ${\theta }_{\mathrm{rec}}$ are predicted and analyzed. The results show that on a hydrophilic surface $\left({\theta }_0\le 90{}^{\circ }\right)$, the volume evolution of droplets in SS mode always occurs in a way between the constant contact radius (CCR) and constant contact angle (CCA) modes (i.e., two extreme modes); however, on a hydrophobic surface $\left({\theta }_0>90{}^{\circ }\right)$, as ${\theta }_{\mathrm{rec}}$ increases from 0 to ${\theta }_0$, the volume evolution of droplets in SS mode first departs from the CCR mode, then approaches the SS mode with ${\theta }_{\mathrm{rec}}=90{}^{\circ }$, and finally returns to the CCA mode. More interestingly, the initially identical droplets evaporating on hydrophobic surfaces with different receding contact angles $\left({\theta }_{\mathrm{rec}}={90}^{\circ }\pm \mathrm{\Delta }\theta \right)$ can even have approximately the same volume evolution behavior. It is also found that the widely accepted 2/3 law is applicable only for volume evolution of droplets in SS mode with large initial contact angle or small contact angle hysteresis, with the predicted deviation less than 1.5% for ${\theta }_0\ge {150}^{\circ }$ or ${\theta }_0\ge {50}^{\circ }$ when ${\theta }_0-{\theta }_{\mathrm{rec}}\le {15}^{\circ }$. Finally, a physical interpretation for the volume evolution behaviors of droplets in SS mode is proposed.

We derive a three-dimensional theory of self-propelled particle swarming in a viscous fluid environment. Our model predicts emergent collective behavior that depends critically on fluid opacity, mechanism of self-propulsion, and type of particle-particle interaction. In “clear fluids” swimmers have full knowledge of their surroundings and can adjust their velocities with respect to the lab frame, while in “opaque fluids” they control their velocities only in relation to the local fluid flow. We also show that “social” interactions that affect only a particle's propensity to swim towards or away from neighbors induces a flow field that is qualitatively different from the long-ranged flow fields generated by direct “physical” interactions. The latter can be short-ranged but lead to much longer-ranged fluid-mediated hydrodynamic forces, effectively amplifying the range over which particles interact. These different fluid flows conspire to profoundly affect swarm morphology, kinetically stabilizing or destabilizing swarm configurations that would arise in the absence of fluid. Depending upon the overall interaction potential, the mechanism of swimming ( e.g., pushers or pullers), and the degree of fluid opaqueness, we discover a number of new collective three-dimensional patterns including flocks with prolate or oblate shapes, recirculating pelotonlike structures, and jetlike fluid flows that entrain particles mediating their escape from the center of mill-like structures. Our results reveal how the interplay among general physical elements influence fluid-mediated interactions and the self-organization, mobility, and stability of new three-dimensional swarms and suggest how they might be used to kinetically control their collective behavior.

A model is presented for the nonlinear interaction between a large-amplitude laser and semiconductor plasma in the semirelativistic quantum regime. The collective behavior of the electrons in the conduction band of a narrow-gap semiconductor is modeled by a Klein-Gordon equation, which is nonlinearly coupled with the electromagnetic (EM) wave through the Maxwell equations. The parametric instabilities involving the stimulated Raman scattering and modulational instabilities are analyzed theoretically and the resulting dispersion relation relation is solved numerically to assess the quantum effects on the instability. The study of the quasi-steady-state solution of the system and direct numerical simulations demonstrate the possibility of the formation of localized EM solitary structures trapped in electrons density holes.

Mass transport, such as movement of phosphorus in soils and solutes in rivers, is a natural phenomenon and its study plays an important role in science and engineering. It is found that there are numerous practical diffusion phenomena that do not obey the classical advection-diffusion equation (ADE). Such diffusion is called abnormal or superdiffusion, and it is well described using a fractional advection-diffusion equation (FADE). The FADE finds a wide range of applications in various areas with great potential for studying complex mass transport in real hydrological systems. However, solution to the FADE is difficult, and the existing numerical methods are complicated and inefficient. In this study, a fresh lattice Boltzmann method is developed for solving the fractional advection-diffusion equation (LabFADE). The FADE is transformed into an equation similar to an advection-diffusion equation and solved using the lattice Boltzmann method. The LabFADE has all the advantages of the conventional lattice Boltzmann method and avoids a complex solution procedure, unlike other existing numerical methods. The method has been validated through simulations of several benchmark tests: a point-source diffusion, a boundary-value problem of steady diffusion, and an initial-boundary-value problem of unsteady diffusion with the coexistence of source and sink terms. In addition, by including the effects of the skewness $\beta$, the fractional order $\alpha$, and the single relaxation time $\tau$, the accuracy and convergence of the method have been assessed. The numerical predictions are compared with the analytical solutions, and they indicate that the method is second-order accurate. The method presented will allow the FADE to be more widely applied to complex mass transport problems in science and engineering.

In this paper, we will focus on the multiple-relaxation-time (MRT) lattice Boltzmann model for two-dimensional convection-diffusion equations (CDEs), and analyze the discrete effect on the halfway bounce-back (HBB) boundary condition (or sometimes called bounce-back boundary condition) of the MRT model where three different discrete velocity models are considered. We first present a theoretical analysis on the discrete effect of the HBB boundary condition for the simple problems with a parabolic distribution in the $x$ or $y$ direction, and a numerical slip proportional to the second-order of lattice spacing is observed at the boundary, which means that the MRT model has a second-order convergence rate in space. The theoretical analysis also shows that the numerical slip can be eliminated in the MRT model through tuning the free relaxation parameter corresponding to the second-order moment, while it cannot be removed in the single-relaxation-time model or the Bhatnagar-Gross-Krook model unless the relaxation parameter related to the diffusion coefficient is set to be a special value. We then perform some simulations to confirm our theoretical results, and find that the numerical results are consistent with our theoretical analysis. Finally, we would also like to point out the present analysis can be extended to other boundary conditions of lattice Boltzmann models for CDEs.

We describe a method for disentangling giant component and finite cluster contributions to sparse random matrix spectra, using sparse symmetric random matrices defined on Erdős-Rényi graphs as an example and test bed. Our methods apply to sparse matrices defined in terms of arbitrary graphs in the configuration model class, as long as they have finite mean degree.

We address the study of quasiperiodic interactions on a square lattice by using an Ising model with ferromagnetic and antiferromagnetic exchange interactions following a quasiperiodic Fibonacci sequence in both directions of a square lattice. We applied the Monte Carlo method, together with the Metropolis algorithm, to calculate the thermodynamic quantities of the system. We obtained the Edwards–Anderson order parameter $q_{\text{EA}}$, the magnetic susceptibility $\chi$, and the specific heat $c$ in order to characterize the universality class of the phase transition. We also use the finite size scaling method to obtain the critical temperature of the system and the critical exponents $\beta ,\gamma$, and $\nu$. In the low-temperature limit we obtained a spin-glass phase with critical temperature around $T_c\approx 2.274$, and the critical exponents $\beta ,\gamma$, and $\nu$, indicating that the quasiperiodic order induces a change in the universality class of the system. Also, we discovered a spin-glass ordering in a two-dimensional system which is rare and, as far as we know, the unique example is an under-frustrated Ising model.

The problem of wave-number selections in nonequilibrium pattern-forming systems in the presence of noise is investigated. The minimum-action method is proposed to study the noise-induced transitions between the different spatiotemporal states by generalizing the traditional theory previously applied in low-dimensional dynamical systems. The scheme is shown as an example in the stabilized Kuramoto-Sivashinsky equation. The present method allows us to conveniently find the unique noise selected state, in contrast to previous work using direct simulations of the stochastic partial differential equation, where the constraints of the simulation only allow a narrow band to be determined.

We study the relation between number of contributors and product size in Wikipedia and GitHub. In contrast to traditional production, this is strongly probabilistic, but is characterized by two quantitative nonlinear laws: a power-law bound to product size for increasing number of contributors, and the universal collapse of rescaled distributions. A variant of the random-energy model shows that both laws are due to the heterogeneity of contributors, and displays an intriguing finite-size scaling property with no equivalent in standard systems. The analysis uncovers the right intensive densities, enabling the comparison of projects with different numbers of contributors on equal grounds. We use this property to expose the detrimental effects of conflicting interactions in Wikipedia.

Networks with nodes embedded in a metric space have gained increasing interest in recent years. The effects of spatial embedding on the networks' structural characteristics, however, are rarely taken into account when studying their macroscopic properties. Here, we propose a hierarchy of null models to generate random surrogates from a given spatially embedded network that can preserve certain global and local statistics associated with the nodes' embedding in a metric space. Comparing the original network's and the resulting surrogates' global characteristics allows one to quantify to what extent these characteristics are already predetermined by the spatial embedding of the nodes and links. We apply our framework to various real-world spatial networks and show that the proposed models capture macroscopic properties of the networks under study much better than standard random network models that do not account for the nodes' spatial embedding. Depending on the actual performance of the proposed null models, the networks are categorized into different classes. Since many real-world complex networks are in fact spatial networks, the proposed approach is relevant for disentangling the underlying complex system structure from spatial embedding of nodes in many fields, ranging from social systems over infrastructure and neurophysiology to climatology.

The classical game of rock-paper-scissors has inspired experiments and spatial model systems that address the robustness of biological diversity. In particular, the game nicely illustrates that cyclic interactions allow multiple strategies to coexist for long-time intervals. When formulated in terms of a one-dimensional cellular automata, the spatial distribution of strategies exhibits coarsening with algebraically growing domain size over time, while the two-dimensional version allows domains to break and thereby opens the possibility for long-time coexistence. We consider a quasi-one-dimensional implementation of the cyclic competition, and study the long-term dynamics as a function of rare invasions between parallel linear ecosystems. We find that increasing the complexity from two to three parallel subsystems allows a transition from complete coarsening to an active steady state where the domain size stays finite. We further find that this transition happens irrespective of whether the update is done in parallel for all sites simultaneously or done randomly in sequential order. In both cases, the active state is characterized by localized bursts of dislocations, followed by longer periods of coarsening. In the case of the parallel dynamics, we find that there is another phase transition between the active steady state and the coarsening state within the three-line system when the invasion rate between the subsystems is varied. We identify the critical parameter for this transition and show that the density of active boundaries has critical exponents that are consistent with the directed percolation universality class. On the other hand, numerical simulations with the random sequential dynamics suggest that the system may exhibit an active steady state as long as the invasion rate is finite.

Molten sodium bromide has been investigated by quasielastic neutron scattering focusing on the wave vector range around the first structure factor peak. The linewidth of the scattering function shows a narrowing around the wave number of the structure factor peak, known as deGennes narrowing. In a monatomic system, this narrowing or in the time domain slowing down, has been related to a self-diffusion process of the caged particle. Here we show that this methodology can be applied to the molten alkali halide NaBr. The incoherent scattering from the sodium ions at small wave vectors provides the self-diffusion coefficient of sodium and the dynamics of bromine ions can be studied at wave numbers around the structure factor peak. With input from molecular dynamics simulations on the partial structure factors, diffusion coefficients of the bromine ions can be obtained. These experimentally derived diffusion coefficients are in good agreement with molecular dynamics simulation results. This methodology to extract self-diffusion coefficients from coherent quasielastic neutron scattering is applicable to binary fluids in general when one particle dominates the scattering response at the structure factor maximum.

In this work, we have established the evaporation-liquid flow coupling mechanism by which sessile nanofluid droplets on a hydrophobic substrate evaporate and agglomerate to form unique morphological features under controlled external heating. It is well understood that evaporation coupled with internal liquid flow controls particle transport in a spatiotemporal sense. Flow characteristics inside the heated droplet are investigated and found to be driven by the buoyancy effects. Velocity magnitudes are observed to increase by an order at higher temperatures with similar looking flow profiles. The recirculating flow induced particle transport coupled with collision of particles and shear interaction between them leads to the formation of dome shaped viscoelastic shells of different dimensions depending on the surface temperature. These shells undergo sol-gel transition and subsequently undergo buckling instability leading to the formation of daughter cavities. With an increase in the surface temperature, droplets exhibit buckling from multiple sites over a larger sector in the top half of the droplet. Irrespective of the initial nanoparticle concentration and substrate temperature, growth of a daughter cavity (subsequent to buckling) inside the droplet is found to be controlled by the solvent evaporation rate from the droplet periphery and is shown to exhibit a universal trend.

The continuum description of granular flows is still a challenge despite their importance in many geophysical and industrial applications. We extend previous works, which have explored steady flow properties, by focusing on unsteady flows accelerating or decelerating down an inclined plane in the simple shear configuration. We solve the flow kinematics analytically, including predictions of evolving velocity and stress profiles and the duration of the transient stage. The solution shows why and how granular materials reach steady flow on slopes steeper than the angle of repose and how they decelerate on shallower slopes. The model might facilitate development of natural hazard assessment and may be modified in the future to explore unsteady granular flows in different configurations.

This paper analyzes fracturing in inhomogeneous media under dry and fully saturated conditions. We adopt a central force model with continuous damage to study avalanche behavior in a two-dimensional truss lattice undergoing dilation. Multiple fractures can develop at once and a power-law distribution of the avalanche size is observed. The values for the power-law exponent are compared with the ones found in the literature and scale-free behavior is suggested. The fracture evolves intermittently in time because only some avalanches correspond to fracture advancement. A fully saturated model with continuous damage based on the extended Biot's theory is developed and avalanche behavior is studied in the presence of fluid, varying the fluid boundary conditions. We show that power-law behavior is destroyed when the fluid flux governs the problem. Fluid pressure behavior during intermittent crack tip advancement is studied for the continuous-damage fully saturated model. It is found that when mechanical loading prevails, the pressure rises when the crack advances, while when fluid loading prevails, the pressure drops when the crack advances.

A theory of reciprocating contacts for linear viscoelastic materials is presented. Results are discussed for the case of a rigid sphere sinusoidally driven in sliding contact with a viscoelastic half-space. Depending on the size of the contact, the frequency and amplitude of the reciprocating motion, and on the relaxation time of the viscoelastic body, we establish that the contact behavior may range from the steady-state viscoelastic solution, in which traction forces always oppose the direction of the sliding rigid punch, to a more elaborate trend, which is due to the strong interaction between different regions of the path covered during the reciprocating motion. Practical implications span a number of applications, ranging from seismic engineering to biotechnology.

The response of Rayleigh-Bénard convection in a horizontal fluid layer to time-periodic heating of its horizontal boundaries with a mixture of two frequencies is analyzed numerically. The ratio of the two forcing frequencies and the mixing angle of the amplitudes of modulation provide a control on the instability of the system. In addition to the existence of well-known harmonic and subharmonic instability responses under modulation, the time-periodic oscillation of the boundary temperatures of the fluid-layer with two frequencies results in more bicritical states in comparison to the single-frequency excitation. The onset of instability depends strongly on the modulation parameters and the Prandtl number of the fluid.