The structure of probability currents is studied for the dynamical network after consecutive contraction on two-state, nonequilibrium lattice systems. This procedure allows us to investigate the transition rates between configurations on small clusters and highlights some relevant effects of lattice symmetries on the elementary transitions that are responsible for entropy production. A method is suggested to estimate the entropy production for different levels of approximations (cluster sizes) as demonstrated in the two-dimensional contact process with mutation.

In the first part of this series of two papers [Y. Jiao, F. H. Stillinger, and S. Torquato, Phys. Rev. E 81, 011105 (2010)], we considered the geometrical ambiguity of pair statistics associated with point configurations. Here we focus on the analogous problem for heterogeneous media (materials). Heterogeneous media are ubiquitous in a host of contexts, including composites and granular media, biological tissues, ecological patterns, and astrophysical structures. The complex structures of heterogeneous media are usually characterized via statistical descriptors, such as the $n$-point correlation function $S_n$. An intricate inverse problem of practical importance is to what extent a medium can be reconstructed from the two-point correlation function $S_2$ of a target medium. Recently, general claims of the uniqueness of reconstructions using $S_2$ have been made based on numerical studies, which implies that $S_2$ suffices to uniquely determine the structure of a medium within certain numerical accuracy. In this paper, we provide a systematic approach to characterize the geometrical ambiguity of $S_2$ for both continuous two-phase heterogeneous media and their digitized representations in a mathematically precise way. In particular, we derive the exact conditions for the case where two distinct media possess identical $S_2$, i.e., they form a degenerate pair. The degeneracy conditions are given in terms of integral and algebraic equations for continuous media and their digitized representations, respectively. By examining these equations and constructing their rigorous solutions for specific examples, we conclusively show that in general $S_2$ is indeed not sufficient information to uniquely determine the structure of the medium, which is consistent with the results of our recent study on heterogeneous-media reconstruction [Y. Jiao, F. H. Stillinger, and S. Torquato, Proc. Natl. Acad. Sci. U.S.A. 106, 17634 (2009)]. The analytical examples include complex patterns composed of building blocks bearing the letter “T” and the word “WATER” as well as degenerate stacking variants of the densest sphere packing in three dimensions (Barlow films). Several numerical examples of degeneracy (e.g., reconstructions of polycrystal microstructures, laser-speckle patterns and sphere packings) are also given, which are virtually exact solutions of the degeneracy equations. The uniqueness issue of multiphase media reconstructions and additional structural information required to characterize heterogeneous media are discussed, including two-point quantities that contain topological connectedness information about the phases.

The pressure and energy behavior of the Gaussian core model (GCM) fluid as a function of strain-rate are obtained from nonequilibrium molecular dynamics simulations for a wide range of thermodynamic state points. An analytical dependence of pressure on strain-rate is observed which is in agreement with a Taylor series expansion of pressure in terms of the strain-rate tensor. In contrast, the energy as a function of strain rate is found to be dependent on temperature and density. The different behavior of pressure and energy contradicts mode-coupling theory, which requires the same variation of pressure and energy with respect to the strain-rate. The results for the GCM fluid do not support the hypothesis that the strain-rate exponent for both pressure and energy can be universally represented by a simple linear function of temperature and density.

Dynamic light scattering on a nematic liquid-crystal elastomer was performed as a function of deformation perpendicular to the director and along the director. We show that the relaxation rate of the nematic director fluctuations increases with strain along the director, as expected from the theory of semisoft elasticity. Deformation applied perpendicular to the director, on the other hand, decreases the relaxation rate to a very small value at the onset of the soft elastic response, revealing the existence of a dynamic soft mode. The results are in complete agreement with the theory of semisoft elasticity and allow us to determine all the constants of the model.

Nematic elastomers exhibit a rich elastic response to external stresses. Of particular interest is the semisoft response of elastomers with an anisotropy direction $\left(z\right)$ frozen in by a double cross-linking process. This response is characterized by a stress-strain curve for stresses along $x$ perpendicular to $z$ that rises initially, exhibits a nearly flat plateau between two critical values of strain, and then rises again. This paper explores elastic response in semisoft elastomers as a function of externally applied strain. It derives general Ward identities for elastic moduli and shows that the elastic modulus measuring response to $xz$ shears vanishes at the boundaries of the semisoft plateau whereas moduli measuring response to shears perpendicular to the $xz$ plane do not. It then calculates all relevant moduli in a simple model of elastomers and verifies the general Ward-identity predictions.

We demonstrate with a thought experiment that fitness-based population dynamical approaches to evolution are not able to make quantitative, falsifiable predictions about the long-term behavior of some evolutionary systems. A key characteristic of evolutionary systems is the ongoing endogenous production of new species. These novel entities change the conditions for already existing species. Even Darwin’s Demon, a hypothetical entity with exact knowledge of the abundance of all species and their fitness functions at a given time, could not prestate the impact of these novelties on established populations. We argue that fitness is always a posteriori knowledge—it measures but does not explain why a species has reproductive success or not. To overcome these conceptual limitations, a variational principle is proposed in a spin-model-like setup of evolutionary systems. We derive a functional which is minimized under the most general evolutionary formulation of a dynamical system, i.e., evolutionary trajectories causally emerge as a minimization of a functional. This functional allows the derivation of analytic solutions of the asymptotic diversity for stochastic evolutionary systems within a mean-field approximation. We test these approximations by numerical simulations of the corresponding model and find good agreement in the position of phase transitions in diversity curves. The model is further able to reproduce stylized facts of timeseries from several man-made and natural evolutionary systems. Light will be thrown on how species and their fitness landscapes dynamically coevolve.

As a social media, online social networks play a vital role in the social information diffusion. However, due to its unique complexity, the mechanism of the diffusion in online social networks is different from the ones in other types of networks and remains unclear to us. Meanwhile, few works have been done to reveal the coupled dynamics of both the structure and the diffusion of online social networks. To this end, in this paper, we propose a model to investigate how the structure is coupled with the diffusion in online social networks from the view of weak ties. Through numerical experiments on large-scale online social networks, we find that in contrast to some previous research results, selecting weak ties preferentially to republish cannot make the information diffuse quickly, while random selection can achieve this goal. However, when we remove the weak ties gradually, the coverage of the information will drop sharply even in the case of random selection. We also give a reasonable explanation for this by extra analysis and experiments. Finally, we conclude that weak ties play a subtle role in the information diffusion in online social networks. On one hand, they act as bridges to connect isolated local communities together and break through the local trapping of the information. On the other hand, selecting them as preferential paths to republish cannot help the information spread further in the network. As a result, weak ties might be of use in the control of the virus spread and the private information diffusion in real-world applications.

In order to establish a cost-effective strategy to simulate complex flows in continuum to slip and transitional regimes, the present study assesses the performance of a lattice Boltzmann method (LBM) formerly discussed by the present authors’ group [Niu et al., Phys. Rev. E 76, 036711 (2007)]. This LBM is based on a diffuse scattering wall boundary condition, a regularization procedure, and an effective relaxation time associated with the Knudsen number. The present assessment is on its regularization procedure and third-order truncated system based on the two-dimensional twenty-one discrete velocity (D2Q21) model for the Cartesian lattices. The test flow cases are force-driven Poiseuille flows, the Couette flows and a flow around a square cylinder situated in a nanochannel. For producing the reference data of the square cylinder flow, the molecular dynamics simulation using Lennard-Jones potential is also performed. Although the flow profiles and the slip velocities of the Poiseuille flows and the Couette flows are more accurately predicted by the third-order truncated system, the general velocity profiles around the square cylinder are also well predicted by the second-order truncated system based on the two-dimensional nine discrete velocity (D2Q9) model. It is also confirmed that without the regularization process, the entire flow field prediction suffers unphysical momentum oscillations around the square cylinder.

We study the response of a model microelectrochemical cell to a large ac voltage of frequency comparable to the inverse cell relaxation time. To bring out the basic physics, we consider the simplest possible model of a symmetric binary electrolyte confined between parallel-plate blocking electrodes, ignoring any transverse instability or fluid flow. We analyze the resulting one-dimensional problem by matched asymptotic expansions in the limit of thin double layers and extend previous work into the strongly nonlinear regime, which is characterized by two features—significant salt depletion in the electrolyte near the electrodes and, at very large voltage, the breakdown of the quasiequilibrium structure of the double layers. The former leads to the prediction of “ac capacitive desalination” since there is a time-averaged transfer of salt from the bulk to the double layers, via oscillating diffusion layers. The latter is associated with transient diffusion limitation, which drives the formation and collapse of space-charge layers, even in the absence of any net Faradaic current through the cell. We also predict that steric effects of finite ion sizes (going beyond dilute-solution theory) act to suppress the strongly nonlinear regime in the limit of concentrated electrolytes, ionic liquids, and molten salts. Beyond the model problem, our reduced equations for thin double layers, based on uniformly valid matched asymptotic expansions, provide a useful mathematical framework to describe additional nonlinear responses to large ac voltages, such as Faradaic reactions, electro-osmotic instabilities, and induced-charge electrokinetic phenomena.

We study universality in the kinetics of spinodal phase separation in unstable thin liquid films, via simulations of the thin film equation. It is shown that, in addition to morphology and free energy, the number density of local maxima in the film profile can also be used to identify the early, late, and intermediate stages of spinodal phase separation. A universal curve between the number density of local maxima and rescaled time describes the kinetics of the early stage in $d=2$ and 3. The Lifshitz-Slyozov exponent of $-1/3$ describes the kinetics of the late stage in $d=2$ even in the absence of coexisting equilibrium phases.

We report electric field dependence of the anchoring transition in a mesogen on cooling in a cell with perfluoropolymer treated surfaces. Below a crossover voltage $V_{co}$ the transition is discontinuous between planar and homeotropic alignments, and as the temperature is lowered, the transition temperature decreases quadratically with the field. Above $V_{co}$ the transition is continuous between planar and tilted alignments, the transition temperature decreasing essentially linearly with the rms field. We develop a simple model to account for these results and argue that the higher field regime corresponds to a temperature driven inverse Freedericksz transition in which the director orientation starts tilting at the weakly anchored surfaces while the tilt angle remains zero at the midplane of the cell.

We study azimuthal gliding of the easy axis that occurs in nematic liquid crystals brought in contact with the photoaligned substrate (initially irradiated azo-dye film) under the action of reorienting UV light combined with in-plane electric field. For irradiation with the linearly polarized light, dynamics of easy axis reorientation is found to be faster as compared to the case of nonpolarized light. Another effect is that it slows down with the initial irradiation dose used to prepare the azo-dye film. This effect is interpreted by using the previously suggested phenomenological model. We present the theoretical results computed by solving the torque balance equations of the model that agree very well with the experimental data.

In most models of the spread of disease over contact networks it is assumed that the probabilities per unit time of disease transmission and recovery from disease are constant, implying exponential distributions of the time intervals for transmission and recovery. Time intervals for real diseases, however, have distributions that in most cases are far from exponential, which leads to disagreements, both qualitative and quantitative, with the models. In this paper, we study a generalized version of the susceptible-infected-recovered model of epidemic disease that allows for arbitrary distributions of transmission and recovery times. Standard differential equation approaches cannot be used for this generalized model, but we show that the problem can be reformulated as a time-dependent message passing calculation on the appropriate contact network. The calculation is exact on trees (i.e., loopless networks) or locally treelike networks (such as random graphs) in the large system size limit. On non-tree-like networks we show that the calculation gives a rigorous bound on the size of disease outbreaks. We demonstrate the method with applications to two specific models and the results compare favorably with numerical simulations.

The distribution of wealth among individuals in real society can be well described by the Pareto principle or “80-20 rule.” How does such heterogeneity in initial wealth distribution affect the emergence of public cooperation, when individuals, the rich and the poor, engage in a collective-risk enterprise, not to gain a profit but to avoid a potential loss? Here we address this issue by studying a simple but effective model based on threshold public goods games. We analyze the evolutionary dynamics for two distinct scenarios, respectively: one with fair sharers versus defectors and the other with altruists versus defectors. For both scenarios, particularly, we in detail study the dynamics of the population with dichotomic initial wealth—the rich versus the poor. Moreover, we demonstrate the possible steady compositions of the population and provide the conditions for stability of these steady states. We prove that in a population with heterogeneous wealth distribution, richer individuals are more likely to cooperate than poorer ones. Participants with lower initial wealth may choose to cooperate only if all players richer than them are cooperators. The emergence of pubic cooperation largely relies on rich individuals. Furthermore, whenever the wealth gap between the rich and the poor is sufficiently large, cooperation of a few rich individuals can substantially elevate the overall level of social cooperation, which is in line with the well-known Pareto principle. Our work may offer an insight into the emergence of cooperative behavior in real social situations where heterogeneous distribution of wealth among individual is omnipresent.

In this work we investigate knowledge acquisition as performed by multiple agents interacting as they infer, under the presence of observation errors, respective models of a complex system. We focus the specific case in which, at each time step, each agent takes into account its current observation as well as the average of the models of its neighbors. The agents are connected by a network of interaction of Erdős-Rényi or Barabási-Albert type. First, we investigate situations in which one of the agents has a different probability of observation error (higher or lower). It is shown that the influence of this special agent over the quality of the models inferred by the rest of the network can be substantial, varying linearly with the respective degree of the agent with different estimation error. In case the degree of this agent is taken as a respective fitness parameter, the effect of the different estimation error is even more pronounced, becoming superlinear. To complement our analysis, we provide the analytical solution of the overall performance of the system. We also investigate the knowledge acquisition dynamic when the agents are grouped into communities. We verify that the inclusion of edges between agents (within a community) having higher probability of observation error promotes the loss of quality in the estimation of the agents in the other communities.

The “six degrees of separation” between any two individuals on Earth has become emblematic of the “small world” theme, even though the information conveyed via a chain of human encounters decays very rapidly with increasing chain length, and diffusion of information via this process may be very inefficient in large human organizations. The information flow on a communication network in a large organization, the University of Oslo, has been studied by analyzing email records. The records allow for quantification of communication intensity across organizational levels and between organizational units (referred to as “modules”). We find that the number of email messages within modules scales with module size to the power of $1.29\pm .06$, and the frequency of communication between individuals decays exponentially with the number of links required upward in the organizational hierarchy before they are connected. Our data also indicates that the number of messages sent by administrative units is proportional to the number of individuals at lower levels in the administrative hierarchy, and the “divergence of information” within modules is associated with this linear relationship. The observed scaling is consistent with a hierarchical system in which individuals far apart in the organization interact little with each other and receive a disproportionate number of messages from higher levels in the administrative hierarchy.

We consider a dissipative oval-like shaped billiard with a periodically moving boundary. The dissipation considered is proportional to a power of the velocity $V$ of the particle. The three specific types of power laws used are: (i) $F\propto -V$; (ii) $F\propto -V^2$ and (iii) $F\propto -V^{\delta }$ with $1<\delta <2$. In the course of the dynamics of the particle, if a large initial velocity is considered, case (i) shows that the decay of the particle’s velocity is a linear function of the number of collisions with the boundary. For case (ii), an exponential decay is observed, and for $1<\delta <2$, an powerlike decay is observed. Scaling laws were used to characterize a phase transition from limited to unlimited energy gain for cases (ii) and (iii). The critical exponents obtained for the phase transition in the case (ii) are the same as those obtained for the dissipative bouncer model. Therefore near this phase transition, these two rather different models belong to the same class of universality. For all types of dissipation, the results obtained allow us to conclude that suppression of the unlimited energy growth is indeed observed.

We investigate the dynamics of a ball bouncing on a vibrated elastic membrane. Beyond the classical solid-solid case, we study the effect of introducing new degrees of freedom by allowing substrate oscillations. The forcing frequency of the vibration strongly influences the different thresholds between the dynamical states. The simple model proposed gives good agreement between the experiments and the analytical expression for the threshold at which the ball begins to bounce. Numerical simulations permit to qualitatively recover the experimental phase diagram. Finally, we discuss how this simple system can give new insights in the recent experimental studies on bouncing droplets.

Deformation and aggregation of bubbles in magnetic fluid (ferrofluid) can be observed at high resolution by x-ray phase-contrast imaging. Images of gas bubbles in water-based ferrofluid (EMG-607/707) reveal that bubbles with diameters of a few hundreds of microns deform only slightly in applied fields up to 0.2 T, becoming prolate along the field direction. Also, neighboring bubbles readily attract one another along the field direction, forming linear chains of two or more bubbles. Comparison of experimentally measured bubble trajectories with direct numerical simulations and theoretical predictions shows that aggregation of bubbles under an externally applied field is driven by the attractive magnetophoretic force resulting from the induced fields of the bubbles. Direct numerical simulations were performed with a volume-of-fluid code that incorporates a multiple-color function scheme, to suppress numerical bubble merger, as well as Maxwell stresses as an interfacial force.

The kinetic equation for a single-component uniform plasma in a magnetic field is analytically solved by the moment method. The linear system of ordinary differential equations for the moments is decomposed into subsystems of lower dimensions by a geometric method. The eigensystem of each subsystem shows that parallel moments decay monotonically, but perpendicular $l\text{th}$ harmonic moments decay while oscillating with the $l,l-2,\dots$,-th harmonics of gyrofrequency. A generalization to a multicomponent plasma is discussed.