We study effects of spatiotemporal noise and spatial variations on long-lasting transient oscillations in ring networks of unidirectionally coupled bistable elements (neurons), the duration of which increases exponentially with the number of neurons. On the one hand, spatiotemporal noise tends to sustain the transient oscillations. The duration of the oscillations occurring from fixed initial conditions changes nonmonotonically with noise strength and takes the maximum value at intermediate noise strength. Further, the duration of the oscillations is distributed in the form of the second power law and the mean duration increases with the number of neurons in the presence of an optimal noise. On the other hand, spatial variations degrade the exponential increases in the duration of the oscillations with the number of neurons. In the presence of fixed biases in the steady states of the neurons, there is a flat region in the distribution of the duration of the oscillations occurring under random initial conditions and an increase in the mean duration is almost linear with the number of neurons. Further, the duration of the oscillations in an ensemble of the networks with random biases drawing from an identical distribution is distributed in the form of the second power law and the ensemble mean increases in proportion to the five-halves power of the number of neurons.

Application of the fractional calculus to quantum processes is presented. In particular, the quantum dynamics is considered in the framework of the fractional time Schrödinger equation (SE), which differs from the standard SE by the fractional time derivative: $\partial /\partial t\to {\partial }^{\alpha }/\partial t^{\alpha }$. It is shown that for $\alpha =1/2$ the fractional SE is isospectral to a comb model. An analytical expression for the Green’s functions of the systems are obtained. The semiclassical limit is discussed.

We investigate the recently proposed label-propagation algorithm (LPA) for identifying network communities. We reformulate the LPA as an equivalent optimization problem, giving an objective function whose maxima correspond to community solutions. By considering properties of the objective function, we identify conceptual and practical drawbacks of the label-propagation approach, most importantly the disparity between increasing the value of the objective function and improving the quality of communities found. To address the drawbacks, we modify the objective function in the optimization problem, producing a variety of algorithms that propagate labels subject to constraints; of particular interest is a variant that maximizes the modularity measure of community quality. Performance properties and implementation details of the proposed algorithms are discussed. Bipartite as well as unipartite networks are considered.

This paper presents an iterated tabu search (denoted by ITS) algorithm for optimizing the modularity of community structure in complex networks. The proposed algorithm follows a general framework composed of two phases: basic optimization and postrefinement. When the basic optimization cannot improve the modularity any more, a postrefinement procedure is employed to further optimize the objective function with a global view. For both these two phases, iterated tabu search algorithm is employed to optimize the objective function. Computational results show the high effectiveness of the proposed ITS algorithm compared with six state-of-the-art algorithms in the literature. In particular, our ITS algorithm improves the previous best known modularity for several small and medium size networks.

We report theoretically on the interplay between polarization switching and bifurcations to nonlinear dynamics in a vertical-cavity surface-emitting laser (VCSEL) subject to orthogonal optical injection. Qualitatively different bifurcation scenarios leading to polarization switching are found and mapped out in the plane of the injection parameters, i.e., the frequency detuning vs injection strength plane. A Hopf bifurcation mechanism on the two-polarization-mode solution determines the injection-locking boundaries and influences polarization switching induced by optical injection. We furthermore report on a torus bifurcation emerging from a two-linearly polarized (LP) mode time-periodic dynamics before polarization switching and injection locking appear. It corresponds to an interesting combination of relaxation oscillation dynamics in the $x$-LP mode together with wave mixing dynamics in the injected $y$-LP mode. In agreement with recent experiments, we unveil a period-doubling route to chaos that involves both VCSEL orthogonal LP modes. The corresponding region of chaotic dynamics coincides with abrupt changes in the polarization switching boundaries in the plane of the injection parameters.

Lagrangian properties obtained from a particle tracking velocimetry experiment in a turbulent flow at intermediate Reynolds number are presented. Accurate sampling of particle trajectories is essential in order to obtain the Lagrangian structure functions and to measure intermittency at small temporal scales. The finiteness of the measurement volume can bias the results significantly. We present a robust way to overcome this obstacle. Despite no fully developed inertial range, we observe strong intermittency at the scale of dissipation. The multifractal model is only partially able to reproduce the results.

This paper treats the nonequilibrium processes in mixtures of fluids under the assumption that each constituent is characterized by its own velocity and temperature field. First we discuss the concept of the average temperature of mixture based upon considerations that the internal energy of the mixture is the same as in the case of a single-temperature mixture. As a consequence, it is shown that the entropy of the mixture reaches a local maximum in equilibrium. An illustrative example of homogeneous mixtures is given to support the theoretical considerations. Through the procedure of Maxwellian iteration a new constitutive equation for nonequilibrium temperatures of constituents is obtained in a classical limit, together with the Fick’s law for the diffusion flux. These results obtained for $n$-species are in perfect agreement with a recent classical approach of thermodynamics of irreversible processes in multitemperature case due to Gouin and Ruggeri and generalize our previous papers concerning the case of a binary mixture.

Based on multiple simulation trajectories, which started from dispersively selected initial conformations, the weighted ensemble dynamics method is designed to robustly and systematically explore the hierarchical structure of complex conformational space through the spectral analysis of the variance-covariance matrix of trajectory-mapped vectors. The nondegenerate ground state of the matrix directly predicts the ergodicity of simulation data. The ground state could be adopted as statistical weights of trajectories to correctly reconstruct the equilibrium properties, even though each trajectory only explores part of the conformational space. Otherwise, the degree of degeneracy simply gives the number of metastable states of the system under the time scale of individual trajectory. Manipulation on the eigenvectors leads to the classification of trajectories into nontransition ones within the states and transition ones between them. The transition states may also be predicted without a priori knowledge of the system. We demonstrate the application of the general method both to the system with a one-dimensional glassy potential and with the one of alanine dipeptide in explicit solvent.

We study the reduced fidelity susceptibility ${\chi }_r$ for an $M$-body subsystem of an $N$-body Lipkin-Meshkov-Glick model with $\tau =M/N$ fixed. The reduced fidelity susceptibility can be viewed as the response of subsystem to a certain parameter. In noncritical region, the inner correlation of the system is weak, and ${\chi }_r$ behaves similar with the global fidelity susceptibility ${\chi }_g$, the ratio $\eta ={\chi }_r/{\chi }_g$ depends on $\tau$ but not on $N$. However, at the critical point, the inner correlation tends to be divergent, and we find ${\chi }_r$ approaches ${\chi }_g$ with increasing the $N$. It is interesting to note that, $\eta =1$ in the thermodynamic limit, which means the susceptibilities of the local and global system are the same. Finally, we make numerical computations, and they are in perfect agreement with the analytical predictions.

We consider the Casimir energy in a geometry of an infinite magnetodielectric wedge closed by a circularly cylindrical, perfectly reflecting arc embedded in another magnetodielectric medium, under the condition that the speed of light be the same in both media. An expression for the Casimir energy corresponding to the arc is obtained and it is found that in the limit where the reflectivity of the wedge boundaries tends to unity the finite part of the Casimir energy of a perfectly conducting wedge-shaped sheet closed by a circular cylinder is regained. The energy of the latter geometry possesses divergences due to the presence of sharp corners. We argue how this is a pathology of the assumption of ideal conductor boundaries and that no analogous term enters in the present geometry.

Deterministic diffusive systems, such as the periodic Lorentz gas, multibaker map, as well as spatially periodic systems of interacting particles, have nonequilibrium stationary states with fractal properties when put in contact with particle reservoirs at their boundaries. We study the macroscopic limits of these systems and establish a correspondence between the thermodynamics of the macroscopic diffusion process and the fractality of the stationary states that characterize the phase-space statistics. In particular the entropy production rate is recovered from first principles using a formalism due to Gaspard [J. Stat. Phys. 88, 1215 (1997)]. This paper is the first of two; the second article considers the influence of a uniform external field on such systems.

Galton boards are models of deterministic diffusion in a uniform external field, akin to driven periodic Lorentz gases, here considered in the absence of dissipation mechanism. Assuming a cylindrical geometry with axis along the direction of the external field, the two-dimensional board becomes a model for one-dimensional mass transport along the direction of the external field. This is a purely diffusive process which admits fractal nonequilibrium stationary states under flux boundary conditions. Analytical results are obtained for the statistics of multibaker maps modeling such a nonuniform diffusion process. A correspondence is established between the local phase-space statistics and their macroscopic counterparts. The fractality of the invariant state is shown to be responsible for the positiveness of the entropy production rate.

We propose a formulation to obtain the line shape of a magnetic response with dissipative effects that directly reflects the nature of the environment. Making use of the fact that the time evolution of a response function is described by the same equation as the reduced density operator, we formulate a full description of the complex susceptibility. We describe the dynamics using the equation of motion for the reduced density operator, including the term for the initial correlation between the system and a thermal bath. In this formalism, we treat the full description of non-Markovian dynamics including the initial correlation. We present an explicit and compact formula up to the second order of cumulants, which can be applied in a straightforward way to multiple-spin systems. We also take into account the frequency shift by the system-bath interaction. We study the dependence of the line shape on the type of interaction between the system and the thermal bath. We demonstrate that the present formalism is a powerful tool for investigating various kinds of systems and we show how it is applied to spin systems, including those with up to three spins. We distinguish the contributions of the initial correlation and the frequency shift and make clear the role of each contribution in the Ohmic coupling spectral function. As examples of applications to multispin systems, we obtain the dependence of the line shape on the spatial orientation in relation to the direction of the static field (Nagata-Tazuke effect), including the effects of the thermal environment, in a two-spin system, along with the dependence on the arrangement of a triangle in a three-spin system.

Birth-death processes often exhibit an oscillatory behavior. We investigate a particular case where the oscillation cycles are marginally stable on the mean-field level. An iconic example of such a system is the Lotka-Volterra model of predator-prey interaction. Fluctuation effects due to discreteness of the populations destroy the mean-field stability and eventually drive the system toward extinction of one or both species. We show that the corresponding extinction time scales as a certain power-law of the population sizes. This behavior should be contrasted with the extinction of models stable in the mean-field approximation. In the latter case the extinction time scales exponentially with size.

A Fortuin-Kasteleyn cluster on a torus is said to be of type $\left\{a,b\right\},a,b∊\mathbb{Z}$, if it is possible to draw a curve belonging to the cluster that winds $a$ times around the first cycle of the torus as it winds $-b$ times around the second. Even though the $Q$-Potts models make sense only for $Q$ integers, they can be included into a family of models parametrized by $\beta =\sqrt{Q}$ for which the Fortuin-Kasteleyn clusters can be defined for any real $\beta ∊\left(0,2\right]$. For this family, we study the probability $\pi \left(\left\{a,b\right\}\right)$ of a given type of clusters as a function of the torus modular parameter $\tau ={\tau }_r+i{\tau }_i$. We compute the asymptotic behavior of some of these probabilities as the torus becomes infinitely thin. For example, the behavior of $\pi \left(\left\{1,0\right\}\right)$ is studied for ${\tau }_i\to \infty$. Exponents describing these behaviors are defined and related to weights $h_{r,s}$ of the extended Kac table for $r$ and $s$ integers, but also half-integers. Numerical simulations are also presented. Possible relationship with recent works and conformal loop ensembles is discussed.

We studied free-surface gravity-driven recirculating flows of cohesionless granular materials down a rough inclined plane and overflowing a wall normal to the incoming flow and to the bottom. We performed two-dimensional spherical particle discrete element simulations using a linear damped spring law between particles with a Coulomb failure criterion. High-frequency force fluctuations were observed. This paper focuses on the mean steady force exerted by the flow on the obstacle versus the macroscopic inertial number of the incoming flow, where the inertial number measures the ratio between a macroscopic deformation time scale and an inertial time scale. A triangular stagnant zone is formed upstream of the obstacle and sharply increases the mean force at low incoming inertial numbers. A simple hydrodynamic model based on depth-averaged momentum conservation is proposed. This analytical model predicts the numerical data fairly well and allows us to quantify the different contributions to the mean force on the wall. Beyond this model, our study provides an example of the ability of simple hydrodynamic approaches to describe the macroscopic behavior of an assembly of discrete particles not only in terms of kinematics but also in terms of forces.

We report on the phase behavior of a binary charged sphere mixture of size ratio $\Gamma =0.68$ and charge ratio $\Lambda \approx 1$ as a function of composition $p$ and number density $n$. For $p=0.1–0.3$ we observe freezing at densities well below the freezing densities of the pure components. At all compositions our data indicate the formation of substitutional alloy crystals of body centered cubic structure. No indications for compound formation were observed. Rather, our findings point at the first observation of an upper azeotrope. Measurements of the crystallization kinetics reveal a combined density and composition dependence of growth velocities and nucleation rate densities, with small but significant anomalies at $p^{\ast }=0.2$. These correlated deviations can be rationalized within classical theories of solidification and suggest an increased similarity between melt and solid for this particular composition.

We derive analytical formulas for the firing rate of integrate-and-fire neurons endowed with realistic synaptic dynamics. In particular, we include the possibility of multiple synaptic inputs as well as the effect of an absolute refractory period into the description. The latter affects the firing rate through its interaction with the synaptic dynamics.

We study the complexity of stable waves in unidirectional bistable coupled map lattices as a test tube to spatial chaos of traveling patterns in open flows. Numerical calculations reveal that, grouping patterns into sets according to their velocity, at most one set of waves has positive topological entropy for fixed parameters. By using symbolic dynamics and shadowing, we analytically determine velocity-dependent parameter domains of existence of pattern families with positive entropy. These arguments provide a method to exhibit chaotic sets of stable waves with arbitrary velocity in extended systems.

To detect directional couplings from time series various measures based on distances in reconstructed state spaces were introduced. These measures can, however, be biased by asymmetries in the dynamics’ structure, noise color, or noise level, which are ubiquitous in experimental signals. Using theoretical reasoning and results from model systems we identify the various sources of bias and show that most of them can be eliminated by an appropriate normalization. We furthermore diminish the remaining biases by introducing a measure based on ranks of distances. This rank-based measure outperforms existing distance-based measures concerning both sensitivity and specificity for directional couplings. Therefore, our findings are relevant for a reliable detection of directional couplings from experimental signals.