A basic question on innovation diffusion is why the growth curve of the adopter population in a large society is often S shaped. From macroscopic, microscopic, and mesoscopic viewpoints, the growth of the adopter population is observed as the growth curve, individual adoptions, and differences among individual adoptions, respectively. The S shape can be explained if an empirical model of the growth curve can be deduced from models of microscopic and mesoscopic structures. However, even the structure of growth curve has not been revealed yet because long-term extrapolations by proposed models of S-shaped curves are unstable and it has been very difficult to predict the long-term growth and final adopter population. This paper studies the S-shaped growth from the viewpoint of social regularities. Simple methods to analyze power laws enable us to extract the structure of the growth curve directly from the growth data of recent basic telecommunication services. This empirical model of growth curve is singular at the inflection point and a logarithmic function of time after this point, which explains the unstable extrapolations obtained using previously proposed models and the difficulty in predicting the final adopter population. Because the empirical S curve can be expressed in terms of two power laws of the regularity found in social performances of individuals, we propose the hypothesis that the S shape represents the heterogeneity of the adopter population, and the heterogeneity parameter is distributed under the regularity in social performances of individuals. This hypothesis is so powerful as to yield models of microscopic and mesoscopic structures. In the microscopic model, each potential adopter adopts the innovation when the information accumulated by the learning about the innovation exceeds a threshold. The accumulation rate of information is heterogeneous among the adopter population, whereas the threshold is a constant, which is the opposite of previously proposed models. In the mesoscopic model, flows of innovation information incoming to individuals are organized as dimorphic and partially clustered. These microscopic and mesoscopic models yield the empirical model of the S curve and explain the S shape as representing the regularities of information flows generated through a social self-organization. To demonstrate the validity and importance of the hypothesis, the models of three level structures are applied to reveal the mechanism determining and differentiating diffusion speeds. The empirical model of S curves implies that the coefficient of variation of the flow rates determines the diffusion speed for later adopters. Based on this property, a model describing the inside of information flow clusters can be given, which provides a formula interconnecting the diffusion speed, cluster populations, and a network topological parameter of the flow clusters. For two recent basic telecommunication services in Japan, the formula represents the variety of speeds in different areas and enables us to explain speed gaps between urban and rural areas and between the two services. Furthermore, the formula provides a method to estimate the final adopter population.

We obtain the location of the Hopf bifurcation threshold for a modified van der Pol oscillator, parametrically driven by a stochastic source and including delayed feedback in both position and velocity. We introduce a multiple scale expansion near threshold, and we solve the resulting Fokker-Planck equation associated with the evolution at the slowest time scale. The analytical results are compared with a direct numerical integration of the model equation. Delay modifies the Hopf frequency at threshold and leads to a stochastic bifurcation that is shifted relative to the deterministic limit by an amount that depends on the delay time, the amplitude of the feedback terms, and the intensity of the noise. Interestingly, stochasticity generally increases the region of stability of the limit cycle.

Detecting nonlinear correlations between time series presents a hard problem for data analysis. We present a generative statistical modeling method for detecting nonlinear generalized synchronization. Truncated Volterra series are used to approximate functional interactions. The Volterra kernels are modeled as linear combinations of basis splines, whose coefficients are estimated via $l_1$ and $l_2$ regularized maximum likelihood regression. The regularization manages the high number of kernel coefficients and allows feature selection strategies yielding sparse models. The method's performance is evaluated on different coupled chaotic systems in various synchronization regimes and analytical results for detecting $m:n$ phase synchrony are presented. Experimental applicability is demonstrated by detecting nonlinear interactions between neuronal local field potentials recorded in different parts of macaque visual cortex.

We study the synchronization region of two unidirectionally coupled FitzHugh-Nagumo systems, in a master-slave configuration, under the influence of external forcing terms. In the particular region where the slave anticipates the dynamics of the master system, we observe that the synchronization is robust to the different types of forcings. We then use the predict-prevent control method to suppress unwanted pulses in the master system by using the information of the slave output. We find that this method is more efficient than the direct control method based on the master dynamics. Finally, we observe that a perfect matching between the parameters of the master and the slave is not necessary for the control to be efficient. Moreover, this parameter mismatch can, in some cases, improve the control.

The probabilistic properties of extreme values in multivariate deterministic dynamical systems are analyzed. It is shown that owing to the intertwining of unstable and stable modes the effect of dynamical complexity on the extremes tends to be masked, in the sense that the cumulative probability distribution of typical variables is differentiable and its associated probability density is continuous. Still, there exist combinations of variables probing the dominant unstable modes displaying singular behavior in the form of nondifferentiability of the cumulative distributions of extremes on certain sets of phase space points. Analytic evaluations and extensive numerical simulations are carried out for characteristic examples of Kolmogorov-type systems, for low-dimensional chaotic flows, and for spatially extended systems.

We propose a one-dimensional lattice model, inspired by population dynamics interaction. The model combines a variable coupling range with the Allee effect. The system is capable of exhibiting pattern formation that is similar to what occurs in similar continuous models for population dynamics. However, the formation features are quite different; in this case the pattern emerges from a disorder state via intermittence. We analytically estimated the selected wavelength of the formed pattern and numerically studied fluctuations around the mean wavelength. We also comment on the relationship between intermittence and the edge of chaos as well as sensitivity to initial conditions. Next, we present an analytical prediction of the influence of the world size on the intermittent regime which is in good agreement with the numerical observations. Moreover, the last calculation provided us an alternative way to compute the pattern wavelength. Finally, we discuss the continuous limit of our lattice model.

We present a comprehensive study of nanoelectromechanical systems in pressurized fluids. Resonant responses and quality factors are monitored in five different gases and one liquid, in pressures ranging from vacuum to 20 MPa, in order to evaluate theoretical models of device-fluid interactions at the nanoscale. The traditional Newell picture of microresonator damping in different pressure regimes is found to be inadequate in describing nanoresonators in general. Damping at intermediate pressure ranges is better physically characterized by a Weissenberg number (which compares oscillation frequencies with fluid relaxation rates) than a Knudsen number (which compares mean free paths with device widths) and most adequately described by the Yakhot and Colosqui model. At high-pressure ranges, two models are found to give good agreement with data: the phenomenological model of vibrating spheres and the Sader and Bhiladvala model for the viscous regime. The latter is also successful in explicitly predicting pressure-dependent behavior of the viscous mass load and damping. We observe significant increases in damping due to the squeezed film (SF) of gas between the device and substrate as well as due to undercut (an unavoidable artifact of the standard fabrication technique); correcting the shape of the devices with a focused ion beam allows us to differentiate these two factors. Application of the SF model accounts well for additional damping at high pressures while only qualitatively agreeing at lower pressures. The extensive data collected allow additional insight into fundamental processes underlying fluid damping at the nanoscale, particularly in the intermediate- and high-pressure regimes.

The effects of a radiation field (RF) on the unstable modes developed in a relativistic electron beam-plasma interaction are investigated assuming that ${\omega }_0>{\omega }_p$, where ${\omega }_0$ is the frequency of the RF and ${\omega }_p$ is the plasma frequency. These unstable modes are parametrically coupled to each other due to the RF and are a mix between two-stream and parametric instabilities. The dispersion equations are derived by the linearization of the kinetic equations for a beam-plasma system as well as the Maxwell equations. In order to highlight the effect of the radiation field we present a comparison of our analytical and numerical results obtained for nonzero RF with those for vanishing RF. Assuming that the drift velocity ${\mathbf{u}}_b$ of the beam is parallel to the wave vector $\mathbf{k}$ of the excitations two particular transversal and parallel configurations of the polarization vector ${\mathbf{E}}_0$ of the RF with respect to $\mathbf{k}$ are considered in detail. It is shown that in both geometries resonant and nonresonant couplings between different modes are possible. The largest growth rates are expected at the transversal configuration when ${\mathbf{E}}_0$ is perpendicular to $\mathbf{k}$. In this case it is demonstrated that, in general, the spectrum of the unstable modes in the $\omega$-$k$ plane is split into two distinct domains with long and short wavelengths, where the unstable modes are mainly sensitive to the beam or the RF parameters, respectively. In the parallel configuration, ${\mathbf{E}}_0\parallel \mathbf{k}$, and at short wavelengths the growth rates of the unstable modes are sensitive to both beam and RF parameters remaining insensitive to the RF at long wavelengths.

We present data on emission of $K$-shell radiation from Ti foils irradiated with subpicosecond pulses of second harmonic radiation (527 nm) from the TARANIS laser system at intensities of up to ${10}^{18}\text{W}{\text{cm}}^{-2}$. The data are used to demonstrate that a resonance absorption type mechanism is responsible for absorption of the laser light and to estimate fast electron temperatures of 30–60 keV that are in broad agreement with expectation from models of absorption for a steep density gradient. Data taken with resin-backed targets are used to demonstrate clear evidence of electron refluxing even at the modest fast electron temperatures inferred.

Optimization is essential in many scientific and economical areas, but it is often too complex to be tackled by simple straightforward calculations or by trial and error. Two well-known methods to find low-lying minima in such complex systems are simulated annealing and the genetic algorithm. In these methods artificial fluctuations control the probability of the system to overcome a local minimum having a certain depth. Here we present a complementary scheme that is based on the nudged-elastic-band method ordinarily used to find saddle points and we apply the scheme to find the most stable isomers of the phosphorus $P_4$, $P_8$ molecules and the corresponding molecules of $A{s}_n$, $S{b}_n$, and $B{i}_n$ ($n=4,8$) in the framework of the density functional theory. In the case of $n=8$ we have found stable and metastable configurations, some of which are new and have similar energies. As a by-product we obtained an upper bound for the energy barriers between these configurations.

We show that the diffusion anisotropy due to the asymmetry of the particle can be extracted from the trajectory data without the information of the particle orientation. The subject of analysis is typical in single-particle tracking (SPT) experiments, and the analysis is based on the large-deviation principle in mathematics. We consider the model system of Langevin equations in two dimensions where a particle diffusion shows anisotropy depending on a single parameter defined by the two diffusion coefficients in the perpendicular directions of the frame fixed to the particle. We show how the large-deviation quantities depend on this parameter so that it can be used for the detection of the diffusion anisotropy. We also illustrate how the discreteness of the sampling interval in the SPT and the finiteness of the number of samples influence the results of the analysis.

In this work we study a two-dimensional XXZ-Ising spin-1/2 model with quartic interactions. The model is composed of a two-dimensional lattice of edge-sharing unitary cells, where each cell consists of two triangular prisms, converging in a basal plane with four Ising spin-1/2 (open circles); the apical positions are also occupied by four Heisenberg spin-1/2 (solid circles). Interaction of the base plane containing the multispin Ising interaction has the parameter $J_4$, and the other pairwise interactions have parameter $J$. For the proposed model we construct the phase diagram at zero temperature and give all possible spin configurations. In addition, we investigate two regions where the model can be solved exactly, the free fermion condition (FFC) and the symmetrical eight-vertex condition (SEVC). For this purpose we perform a straightforward mapping for a zero-field eight-vertex model. The necessary conditions for the equivalence are analyzed for all ranges of the interaction parameters. Unfortunately, the present model does not satisfy the FFC unless the trivial case; however, it was possible to give a region where the model can be solved approximately. We study the SEVC and verify that this condition is always satisfied. We also explore and discuss the critical conditions giving the region where these critical points are relevant.

We study the threshold distribution function of the coherent-noise model for the case of infinite number of agents. This function is piecewise constant with a finite number of steps $n$. The latter exhibits a $1/f$ behavior as a function of the order of occurrence of an avalanche and hence versus natural time. An analytic expression of the expectation value $\mathcal{E}\left(S\right)$ for the size $S$ of the next avalanche is obtained and used for the prediction of the next avalanche. Apart from $\mathcal{E}\left(S\right)$, the number of steps $n$ can also serve for this purpose. This enables the construction of a similar prediction scheme which can be applied to real earthquake aftershock data.

We quantitatively assess the effect of density fluctuation modes on the dynamics of a tagged polymer in an unentangled melt. To this end, we develop a density-based mode-coupling theory (dMCT) using the Mori-Zwanzig approach and projecting the fluctuating force onto pair-density fluctuation modes. The effect of dynamical density fluctuations on the center-of-mass (c.m.) dynamics is also analyzed based on a perturbative approach and we show that dMCT and perturbation techniques yield identical results. The c.m. velocity autocorrelation function (c.m. VAF) exhibits a slow power law relaxation in the time range between the monomer time $t_1$ and the Rouse relaxation time $t_N$. We obtain an analytical expression for the c.m. VAF in terms of molecular parameters. In particular, the c.m. VAF scales as $-N^{-1}{t}^{-5/4}$ (where $N$ is the number of monomer units per chain) in the relevant time regime. The results are qualitatively accounted for by the dynamical correlation hole effect. The predicted $-t^{-5/4}$ dependence of the c.m. VAF is supported by data of non-momentum-conserving computer simulations. However, the comparison shows that the theory significantly underestimates the amplitude of the effect. This issue is discussed and an alternative approach is addressed in the second part of this series [Farago et al., Phys. Rev. E 85, 051807 (2012), the following paper].

A mode-coupling theory (MCT) version (called hMCT thereafter) of a recently presented theory [Farago, Meyer, and Semenov, Phys. Rev. Lett. 107, 178301 (2011)] is developed to describe the diffusional properties of a tagged polymer in a melt. The hMCT accounts for the effect of viscoelastic hydrodynamic interactions (VHIs), that is, a physical mechanism distinct from the density-based MCT (dMCT) described in the first paper of this series. The two versions of the MCT yield two different contributions to the asymptotic behavior of the center-of-mass velocity autocorrelation function (c.m. VAF). We show that in most cases the VHI mechanism is dominant; for long chains and prediffusive times it yields a negative tail $\propto$$-N^{-1/2}{t}^{-3/2}$ for the c.m. VAF. The case of non-momentum-conserving dynamics (Langevin or Monte Carlo) is discussed as well. It generally displays a distinctive behavior with two successive relaxation stages: first $-N^{-1}{t}^{-5/4}$ (as in the dMCT approach), then $-N^{-1/2}{t}^{-3/2}$. Both the amplitude and the duration of the first $t^{-5/4}$ stage crucially depend on the Langevin friction parameter $\gamma$. All results are also relevant for the early time regime of entangled melts. These slow relaxations of the c.m. VAF, thus account for the anomalous subdiffusive regime of the c.m. mean square displacement widely observed in numerical and experimental works.

We address the role of topology in the energy transport process that occurs in networks of photosynthetic complexes. We take inspiration from light-harvesting networks present in purple bacteria and simulate an incoherent dissipative energy transport process on more general and abstract networks, considering both regular structures (Cayley trees and hyperbranched fractals) and randomly generated ones. We focus on the the two primary light-harvesting complexes of purple bacteria, i.e., the LH1 and LH2, and we use network-theoretical centrality measures in order to select different LH1 arrangements. We show that different choices cause significant differences in the transport efficiencies, and that for regular networks, centrality measures allow us to identify arrangements that ensure transport efficiencies which are better than those obtained with a random disposition of the complexes. The optimal arrangements strongly depend on the dissipative nature of the dynamics and on the topological properties of the networks considered, and depending on the latter, they are achieved by using global versus local centrality measures. For randomly generated networks, a random arrangement of the complexes already provides efficient transport, and this suggests the process is strong with respect to limited amount of control in the structure design and to the disorder inherent in the construction of randomly assembled structures. Finally, we compare the networks considered with the real biological networks and find that the latter have in general better performances, due to their higher connectivity, but the former with optimal arrangements can mimic the real networks' behavior for a specific range of transport parameters. These results show that the use of network-theoretical concepts can be crucial for the characterization and design of efficient artificial energy transport networks.

In capillary electrophoresis, sample ions migrate along a microcapillary filled with a background electrolyte under the influence of an applied electric field. If the sample concentration is sufficiently high, the electrical conductivity in the sample zone could differ significantly from the background. Under such conditions, the local migration velocity of sample ions becomes concentration-dependent, resulting in a nonlinear wave that exhibits shocklike features. If the nonlinearity is weak, the sample concentration profile, under certain simplifying assumptions, can be shown to obey Burgers’ equation [Ghosal and Chen, Bull. Math. Biol. 72, 2047 (2010)], which has an exact analytical solution for arbitrary initial condition. In this paper, we use a numerical method to study the problem in the more general case where the sample concentration is not small in comparison to the concentration of background ions. In the case of low concentrations, the numerical results agree with the weakly nonlinear theory presented earlier, but at high concentrations, the wave evolves in a way that is qualitatively different.

We study the scaling properties of discontinuous maps by analyzing the average value of the squared action variable $I^2$. We focus our study on two dynamical regimes separated by the critical value $K_c$ of the control parameter $K$: the slow diffusion ($K<K_c$) and the quasilinear diffusion ($K>K_c$) regimes. We found that the scaling of $I^2$ for discontinuous maps when $K\ll K_c$ and $K\gg K_c$ obeys the same scaling laws, in the appropriate limits, as Chirikov's standard map in the regimes of weak and strong nonlinearity, respectively. However, due to the absence of Kolmogorov-Arnold-Moser tori, we observed in both regimes that $I^2\propto n{K}^{\beta }$ for $n\gg 1$ ($n$ being the $n$th iteration of the map) with $\beta \approx 5/2$ when $K\ll K_c$ and $\beta \approx 2$ for $K\gg K_c$.