Motivated by recent empirical studies of business firm growth, we develop a dynamic percolation model which captures some of the features of the economical system—i.e., merging and splitting of business firms—represented as aggregates on a $d$-dimensional lattice. We find the steady-state distribution of the aggregate size and explore how this distribution depends on the model parameters. We find that at the critical threshold, the standard deviation of the aggregate growth rates, $\sigma$, increases with aggregate size $S$ as $\sigma \sim S^{\beta }$, where $\beta$ can be explained in terms of the connectedness length exponent $\nu$ and the fractal dimension $d_f$, with $\beta =1∕\left(2\nu d_f\right)\approx 0.20$ for $d=2$ and 0.125 for $d\to \infty$. The distributions of aggregate growth rates have a sharp peak at the center and pronounced wings extending over many standard deviations, giving the distribution a tent-shape form—the Laplace distribution. The distributions for different aggregate sizes scaled by their standard deviations collapse onto the same curve.

We report frequency-locked resonant patterns induced by additive noise in periodically forced reaction-diffusion Brusselator model. In the regime of 2:1 frequency-locking and homogeneous oscillation, the introduction of additive noise, which is colored in time and white in space, generates and sustains resonant patterns of hexagons, stripes, and labyrinths which oscillate at half of the forcing frequency. Both the noise strength and the correlation time control the pattern formation. The system transits from homogeneous to hexagons, stripes, and to labyrinths successively as the noise strength is adjusted. Good frequency-locked patterns are only sustained by the colored noise and a finite time correlation is necessary. At the limit of white noise with zero temporal correlation, irregular patterns which are only nearly resonant come out as the noise strength is adjusted. The phenomenon induced by colored noise in the forced reaction-diffusion system is demonstrated to correspond to noise-induced Turing instability in the corresponding forced complex Ginzburg-Landau equation.

The lattice Boltzmann (LB) method is applied to solve the time-dependent nonlinear Schrödinger (NLS) equation. Through approximating the reaction term at different orders of accuracy, three diffusion-reaction LB schemes are constructed for the cubic NLS equation. A LB initial condition is proposed to include the first-order nonequilibrium distribution function. These LB schemes are used to solve the one-soliton propagation and the homoclinic orbit problems. Detailed simulation results confirm that the high-order reaction term and the LB initial condition are effective in reducing the truncation errors. Compared with the Crank-Nicolson finite difference scheme, the LB scheme is found to give at least comparable and generally more accurate approximation for the cubic NLS equation.

The lattice Boltzmann method (LBM) and its variants have emerged as promising, computationally efficient and increasingly popular numerical methods for modeling complex fluid flow. However, it is acknowledged that the method can demonstrate numerical instabilities, e.g., in the vicinity of shocks. We propose a simple technique to stabilize the LBM by monitoring the difference between microscopic and macroscopic entropy. Populations are returned to their equilibrium states if a threshold value is exceeded. We coin the name Ehrenfests’ steps for this procedure in homage to the vehicle that we use to introduce the procedure, namely, the Ehrenfests’ coarse-graining idea.

We investigate the site percolation transition in two strongly correlated systems in three dimensions: the massless harmonic crystal and the voter model. In the first case we start with a Gibbs measure for the potential $U=(J∕2){\sum }_{⟨x,y⟩}{[\varphi \left(x\right)-\varphi \left(y\right)]}^2$, $x,y∊{\mathbb{Z}}^3$, $J>0$, and $\varphi \left(x\right)∊\mathbb{R}$, a scalar height variable, and define occupation variables ${\rho }_h\left(x\right)=1$ (0) for $\varphi \left(x\right)>h$ $(<h)$. The probability $p$ of a site being occupied is then a function of $h$. In the voter model we consider a stationary measure in which each site is either occupied or empty, with probability $p$. In both cases the truncated pair correlation of the occupation variables, $G(x-y)$, decays asymptotically as ${\mid x-y\mid }^{-1}$. Using some Monte Carlo simulation methods and finite-size scaling we find accurate values of $p_c$ as well as the critical exponents for these systems. The latter are different from that of independent percolation in $d=3$, as expected from the work of Weinrib and Halperin (WH) for the percolation transition of systems with $G\left(r\right)\sim r^{-a}$ [Phys. Rev. B 27, 413 (1983)]. In particular the correlation length exponent $\nu$ is very close to the predicted value of 2, supporting the conjecture by WH that $\nu =\frac{2}{a}$ is exact.

A theoretical framework is developed to analyze molecular bond breaking in dynamic force spectroscopy using atomic force microscopy. An analytic expression of the observed bond breaking probability as a function of force is obtained in terms of the relevant physical parameters. The force-ramp mode is discussed in detail, which gives the best framework to extract the relevant physical parameters such as the potential depth and its width, if a set of widely different force-loading rates are used. We also show that the commonly used Ritchie-Evans model is incomplete and that it is only applicable for forces well below the maximum permitted by the potential. Statistical complications arising from the use of constant velocity experiments are discussed in detail.

We present a stochastic model of NMDA ($N$-methyl $d$-aspartate) receptor transport along a dendrite prior to synapse formation. Receptors undergo periods of intracellular kinesin motor-driven transport along microtubules interspersed with periods of cycling to the cell surface through exocytosis or endocytsosis. The stochastic dynamics is reduced to a spatially discrete hopping model that determines the time-dependent distribution of receptors along the dendrite. We also investigate how possible defects in motor-assisted transport can affect this distribution.

Synchronization of the coupled neuronal oscillators with multiple connections of different coupling nature is analyzed using the phase-model reduction method. Each coupling connection contributes to the dynamic behavior of the system in a complex nonlinear fashion. In the phase-model scheme, the contribution of the individual connections can be separated in terms of the effective coupling functions associated with each connection and a linear superposition of them provides the total effective coupling of the coupled system. The case of multiple connections with various conduction time delays is also examined, which is shown to be capable of promoting synchronization over an ensemble of spatially distributed neuronal oscillators in an efficient way.

A spatial scale-free network is introduced and studied, whose motivation originated in the growing Internet as well as airport networks. We argue that in these real-world networks a new node necessarily selects one of its neighboring local nodes for connection and is not controlled by preferential attachment as in the Barabási-Albert (BA) model. This observation is mimicked in our model where the nodes pop up at randomly located positions in the Euclidean space and are connected to one end of the nearest link. In spite of this crucial difference it is observed that the leading behavior of our network is like that of the BA model. Defining the link weight as an algebraic power of its Euclidean length, the weight distribution and the nonlinear dependence of the nodal strength on the degree are analytically calculated. It is claimed that a power law decay of the link weights with time ensures such nonlinear behavior. Switching off the Euclidean space from the same model yields a much simpler definition of the BA model where numerical effort grows linearly with $N$.

Election results are determined by numerous social factors that affect the formation of opinion of the voters, including the network of interactions between them and the dynamics of opinion influence. In this work we study the result of proportional elections using an opinion dynamics model similar to simple opinion spreading over a complex network. Erdös-Rényi, Barabási-Albert, regular lattices, and randomly augmented lattices are considered as models of the underlying social networks. The model reproduces the power law behavior of a number of candidates with a given number of votes found in real elections with the correct slope, a cutoff for a larger number of votes, and a plateau for a small number of votes. It is found that the small world property of the underlying network is fundamental for the emergence of the power law regime.

We study the dynamics of the entanglement between two qubits coupled to a common chaotic environment, described by the quantum kicked rotator model. We show that the kicked rotator, which is a single-particle deterministic dynamical system, can reproduce the effects of a pure dephasing many-body bath. Indeed, in the semiclassical limit the interaction with the kicked rotator can be described as a random phase kick, so that decoherence is induced in the two-qubit system. We also show that our model can efficiently simulate non-Markovian environments.

We consider a dynamical network model in which a number of agents all move on the plane with the same constant absolute velocity. At each time step, each agent’s direction is updated as the average of its direction plus the directions of other agents who can influence it. The influencing capability of each agent is represented by its influencing radius, which is randomly chosen according to a power-law distribution with a scaling exponent between 2 and ∞. As the value of the scaling exponent decreases, the radius distribution becomes more heterogeneous and the network becomes much easier to achieve direction consensus among agents due to the leading roles played by a few hub agents. Furthermore, almost all agents will finally move in the same desired direction in a strong heterogeneous influence network, if and only if a small fraction of hub agents can be controlled to move in the desired direction. These results also reflect the “robust yet fragile” feature of a heterogeneous influence network.

Three-dimensional steady Rayleigh-Bénard convection in a vertical cylinder is investigated by numerical simulation and bifurcation analysis. The complex pattern formation beyond the onset of the convection is presented by a bifurcation diagram. The coexistence of multiple stable states is observed near the threshold of the first bifurcation and two group symmetries are summarized for the corresponding primary branches. The first stable target pattern originates through a subcritical bifurcation. A multiplicity of flow states for the Rayleigh number of $14200$ is validated numerically in comparison with the experiment, and a four-spoke pattern is observed.

The relaxation of a nonequilibrium solid to a fluid is determined by observing the positional order parameter in Monte Carlo simulations, and discussed based on diffusion processes in the hard-particle systems. From the cumulant expansion up to the second order, the relation between the positional order parameter $\Psi$ and the mean square displacement $⟨u_i^2⟩$ is obtained to be $\Psi \sim \exp (-{\mathbf{K}}^2⟨u_i^2⟩∕2d)$ with a reciprocal vector $\mathbf{K}$ and the dimension of the system $d$. On the basis of this relation, the positional order should decay exponentially as $\Psi \sim \exp (-{\mathbf{K}}^{2}Dt)$ when the system involves normal diffusion with a diffusion constant $D$. A diffusion process with swapping positions of particles is also discussed. The swapping of particles contributes to the higher orders of the cumulants, and swapping positions allows particles to diffuse without destroying the positional order while the normal diffusion destroys it.

We studied the structure and the dynamics of a nanocolloidal silica gel dispersed in an organic solvent [octylcyanobiphenyl (8CB)] as a function of the silica density by x-ray intensity fluctuation spectroscopy (XIFS). The silica density of the dispersed aerosil gel samples ranged from $0.03\text{to}0.20\mathrm{g}{\mathrm{cm}}^{-3}$ and the autocorrelation of the silica scattering was probed over the $q$ range from $0.03\text{to}0.15{\mathrm{nm}}^{-1}$ (corresponding to length scales from $42\text{to}209\mathrm{nm}$) at a constant room temperature at which 8CB is in the smectic-$A$ phase. The gel structure has a fractal dimension in this density range of $d_f\simeq 2.15$. The time autocorrelation functions of the gels show clear density-dependent and complex dynamics. The gel relaxation times are very long and become bimodal with nonergodic character for densities from $0.10\text{to}0.16\mathrm{g}{\mathrm{cm}}^{-3}$. In this same density range, the fluctuation contrast (strength) is a minimum while the relaxation time becomes independent of wave vector. Together, these results indicate that there is a narrow silica density range for these gels in which the dynamics changes dramatically. This suggests a complex phase diagram for the dynamics of aerosil gels as a function of densification.

Nematic liquid crystals possess three different phases: isotropic, uniaxial, and biaxial. The ground state of most nematics is either isotropic or uniaxial, depending on the external temperature. Nevertheless, biaxial domains have been frequently identified, especially close to defects or external surfaces. In this paper we show that any spatially varying director pattern may be a source of biaxiality. We prove that biaxiality arises naturally whenever the symmetric tensor $\mathbf{S}=(\mathbf{\nabla }\mathbf{n}){(\mathbf{\nabla }\mathbf{n})}^T$ possesses two distinct nonzero eigenvalues. The eigenvalue difference may be used as a measure of the expected biaxiality. Furthermore, the corresponding eigenvectors indicate the directions in which the order tensor $\mathbf{Q}$ is induced to break the uniaxial symmetry about the director $\mathbf{n}$. We apply our general considerations to some examples. In particular we show that, when we enforce homeotropic anchoring on a curved surface, the order tensor becomes biaxial along the principal directions of the surface. The effect is triggered by the difference in surface principal curvatures.

We report on neutron spin echo experiments on hydrogen-bonded polymers and compare the experimentally found dynamical structure factor with theoretical predictions. Surprisingly, we find that in the melt phase the expected scaling of the Rouse dynamics is not satisfied. We propose an explanation based upon the large spatial volume occupied by the connecting groups. When the effects of these bulky groups on the local friction are taken into account, the usual scaling behavior is restored.

Higher order corrections to the Born approximation in perturbation theory are derived in order to improve its performance with the experiments in slablike geometry. A general expression of the $n\text{th}$ order correction to absorption is developed. The cross talking between absorption and scattering is given. The convergence for higher orders of perturbation analysis for absorbing inclusions was studied. Second order absorption and scattering contributions to the transmitted flux are discussed by analyzing the data from forward simulations. The validity of the results is proven in the experiments with phantoms simulating breast tumors. The significant improvement for the fitted values of the absorption is observed. The alternative application of developed formalism as the first order theory to treat the multiple inclusions is suggested.

Complex systems can be characterized by classes of equivalency of their elements defined according to system specific rules. We propose a generalized preferential attachment model to describe the class size distribution. The model postulates preferential growth of the existing classes and the steady influx of new classes. According to the model, the distribution changes from a pure exponential form for zero influx of new classes to a power law with an exponential cut-off form when the influx of new classes is substantial. Predictions of the model are tested through the analysis of a unique industrial database, which covers both elementary units (products) and classes (markets, firms) in a given industry (pharmaceuticals), covering the entire size distribution. The model’s predictions are in good agreement with the data. The paper sheds light on the emergence of the exponent $\tau \approx 2$ observed as a universal feature of many biological, social and economic problems.