Boolean networks are used to study the large-scale properties of nonlinear systems and are mainly applied to model genetic regulatory networks. A statistical method called the annealed approximation is commonly used to examine the dynamical properties of randomly generated Boolean networks that are created with selected statistical features. However, in the literature there are several variations of the annealed approximation. These approximations cannot be interchangeably used in all cases due to different background assumptions. In this paper, we present the so-called four-state model, derive the different approximations from this model, and make the differences and connections between these approximations explicit. As an application of the presented results, we study the properties of the Boolean networks that are constructed with random functions, canalizing functions, and regulatory functions found in the biological literature.

We investigate the dynamics of a kicked particle in an infinite square well undergoing frequent measurements of energy. For a large class of periodic kicking forces, constant diffusion is found in such a non-Kolmogorov-Arnol’d-Moser system. The influence of a phase shift of the kicking potential on the short-time dynamical behavior is discussed. The general asymptotical measurement-assisted diffusion rate is obtained. The entanglement between the particle and the measuring apparatus is investigated. There exist two distinct dynamical behaviors of entanglement. The bipartite entanglement between the system of interest and the whole spin of the measuring apparatus grows with the kicking steps and it gains a larger value for a more chaotic system. However, the partial entanglement between the system of interest and the partial spin of the measuring apparatus decreases with the kicking steps. The relation between the entanglement and quantum diffusion is also analyzed.

We present a broad family of quantum baker maps that generalize the proposal of Schack and Caves to any even Hilbert space with arbitrary boundary conditions. We identify a structure, common to all maps consisting of a simple kernel perturbed by diffraction effects. This “essential” baker’s map has a different semiclassical limit and can be diagonalized analytically for Hilbert spaces spanned by qubits. In all cases this kernel provides an accurate approximation to the spectral properties—eigenvalues and eigenfunctions—of all the different quantizations.

We analyze the resonant linear and nonlinear transmission through a photonic crystal waveguide side-coupled to a Kerr-nonlinear photonic crystal resonator. First, we extend the standard coupled-mode theory analysis to photonic crystal structures and obtain explicit analytical expressions for the bistability thresholds and transmission coefficients which provide the basis for a detailed understanding of the possibilities associated with these structures. Next, we discuss limitations of standard coupled-mode theory and present an alternative analytical approach based on the effective discrete equations derived using a Green’s function method. We find that the discrete nature of the photonic crystal waveguides allows a geometry-driven enhancement of nonlinear effects by shifting the resonator location relative to the waveguide, thus providing an additional control of resonant waveguide transmission and Fano resonances. We further demonstrate that this enhancement may result in the lowering of the bistability threshold and switching power of nonlinear devices by several orders of magnitude. Finally, we show that employing such enhancements is of paramount importance for the design of all-optical devices based on slow-light photonic crystal waveguides.

We study the modulational instability and spatial pattern formation in extended media, taking the one-dimensional complex Ginzburg-Landau equation with higher-order terms as a perturbation of the nonlinear Schrödinger equation as a model. By stability analysis for the original partial differential equation, we derive its stability condition as well as the threshold for amplitude perturbations and we show how nonlinear higher-order terms qualitatively change the behavior of the system. The analytical results are found to be in agreement with numerical findings. Modulational instability mediates pattern formation through the lattice. The main feature of the traveling plane waves is its disintegration in pulse train during the propagation through the system.

We propose a recursive procedure to estimate the microcanonical density of states in multicanonical Monte Carlo simulations which relies only on measurements of moments of the energy distribution, avoiding entirely the need for energy histograms. This method yields directly a piecewise analytical approximation to the microcanonical inverse temperature $\beta \left(E\right)$ and allows improved control over the statistics and efficiency of the simulations. We demonstrate its utility in connection with recently proposed schemes for improving the efficiency of multicanonical sampling, either with adjustment of the asymptotic energy distribution or with the replacement of single spin flip dynamics with collective updates.

The effects of the molecular symmetry on macroscopic properties of conventional and unconventional nematic phases are investigated theoretically. These effects concern: (1) the form of the molecular orientation fluctuations, (2) the thermodynamic behavior and the list of stable ordered phases, (3) the effective symmetry of the molecules in the ordered phases. The order-parameter consists of tensors forming the main harmonics of the fluctuation distribution. In conventional models (valid for uniaxial molecules) a single tensor is sufficient while unconventional models (valid for less symmetric molecules) need several tensors with the same rank. We analyze the qualitative differences arising when the number of equivalent tensors varies. We show how to work out complete models in the general case, and to calculate the sequences of stable phases and the corresponding effective molecular symmetries. This yields, for each molecular group and each tensor rank, a complete classification and a deep insight into the structure of thermotropic nematics. This work generalizes the approach we have applied to polar nematics recently observed in polyester compounds and to unconventional uniaxial and biaxial phases of bent-core materials.

Membrane ruffle fluctuations of amphibian epithelial cells A6 (CCL102) cultured in normal and upside down oriented plates have been analyzed through video microscopy. Our results reveal that their edge ruffle fluctuations exhibit a stochastic dynamics with $1∕f^{\alpha }$ power spectrum over at least two decades at low frequencies and long range correlated, self-affine lateral border profiles. In a few and small areas of the membrane, probably nearby focal contacts, we found periodic oscillations which could be induced by myosin driven contraction of stress fibers. Furthermore, whereas the different gravitational orientations had none or little effect on the structure (power spectra and surface roughness) of these membrane ruffle fluctuations, their dynamic parameters were differentially affected. Indeed, the decay time of ruffles remained unchanged, but the period of lamellipodia oscillations near the focal adhesion points was significantly altered in A6 cells cultured upside down.

Muller’s ratchet is an evolutionary process that has been implicated in the extinction of asexual species, the evolution of mitochondria, the degeneration of the Y chromosome, the evolution of sex and recombination and the evolution of microbes. Here we study the speed of Muller’s ratchet in a population subdivided into many small subpopulations connected by migration, and distributed on a network. We compare the speed of the ratchet in two distinct types of topologies: scale free networks and random graphs. The difference between the topologies is noticeable when the average connectivity of the network and the migration rate is large. In this situation we observe that the ratchet clicks faster in scale free networks than in random graphs. So contrary to intuition, scale free networks are more prone to loss of genetic information than random graphs. On the other hand, we show that scale free networks are more robust to the random extinction than random graphs. Since these complex networks have been shown to describe well real-life systems, our results open a framework for studying the evolution of microbes and disease epidemics.

We address the role of the nature of material disorder in determining the roughness of cracks, which grow by damage nucleation and coalescence ahead of the crack tip. We highlight the role of quenched and annealed disorders in relation to the length scales $d$ and ${\xi }_c$ associated with the disorder and the damage nucleation, respectively. In two related models, one with quenched disorder in which $d\simeq {\xi }_c$, the other with annealed disorder in which $d⪡{\xi }_c$, we find qualitatively different roughening properties for the resulting cracks in two dimensions. The first model results in random cracks with an asymptotic roughening exponent $\zeta \approx 0.5$. The second model shows correlated roughening with $\zeta \approx 0.66$. The reasons for the qualitative difference are rationalized and explained.

We propose a model of network growth that generalizes the deactivation model previously suggested for complex networks. Several topological features of this generalized model, such as the degree distribution and clustering coefficient, have been investigated analytically and by simulations. A scaling behavior of clustering coefficient $C\sim 1∕M$ is theoretically obtained, where $M$ refers to the number of active nodes in the network. We discuss the relationship between the recently observed numerical behavior of clustering coefficient in the coauthor and paper citation networks and our theoretical result. It shows that both of them are induced by deactivation mechanism. By introducing a perturbation, the generated network undergoes a transition from large to small world, meanwhile the scaling behavior of $C$ is conserved. It indicates that $C\sim 1∕M$ is a universal scaling behavior induced by deactivation mechanism.

We consider nonlinear interactions between two colliding laser beams in an electron plasma, accounting for the relativistic electron mass increase in the laser fields and radiation pressure driven electron-acoustic (EA) perturbations that are supported by hot and cold electrons. By using the hydrodynamic and Maxwell equations, we obtain the relevant equations for nonlinearly coupled laser beams and EA perturbations. The coupled equations are then Fourier analyzed to obtain a nonlinear dispersion relation. The latter is numerically solved to show the existence of new classes of the parametric instabilities in the presence of two colliding laser beams in a two-electron plasma. The dynamics of nonlinearly coupled laser beams in our electron plasma is also investigated. The results should be useful in understanding the nonlinear propagation characteristics of multiple electromagnetic beams in laser-produced plasmas as well as in space plasmas.

A precursor is a kind of instability in a nonequilibrium system observed prior to a bifurcation. It has been observed in important natural phenomena such as earthquakes and epileptic seizures. In most cases just qualitative results were reported, with the exception of some quantitative observations of noisy precursors. We report the observation of precursors excited by a deterministic perturbation. Quantitative characterization of this type of precursor is important in predicting a bifurcation in advance. This predicting ability comes from a dynamical structure present on the perturbed system.

In the framework of the thermodynamic perturbation theory for fluids we study how the phase diagram of an isotropic repulsive soft-core attractive potential, where a liquid-liquid phase transition exists in addition to the standard gas-liquid phase transition, changes by varying the parameters of the potential. We show that there are some regions in a potential parameter space where a high-density liquid-liquid critical point can exist in addition to the conventional gas-liquid critical point. It is also found that there is a correlation between the behavior of the phase diagram as a function of width of the repulsive step and the structure of the correlation function of a reference liquid.

Dynamics of acetylene molecules adsorbed in $\mathrm{Na}\text{−}\mathrm{Y}$ zeolite cages is investigated using molecular dynamics simulation. The translational motion of the acetylene molecules is shown to involve three different time scales. “Free particle” type diffusion is observed in short time and small length scale. At long time and large length scale, center of mass motion of acetylene is determined by the zeolitic pore topology. Rotational motion of the acetylene is found to be very fast. Detailed analysis of the intermediate scattering function corresponding to the rotational motion showed large-angle jumps that could be described by an $m$-diffusion model.

We present experimental evidence for a “granular clock.” We use a setup in which vertical vibrations $(10–20\mathrm{Hz})$ of a bidisperse granular medium cause horizontal oscillations (periods in the order of minutes) between two connected compartments. Moreover, we present the first experimental evidence for full segregation of smaller particles into one compartment, leaving the larger ones in the other compartment. Simulations, using a simple model, describe phenomenologically the observed oscillations.

Intracellular molecular machines synthesize molecules, tear apart others, transport materials, transform energy into different forms, and carry out a host of other coordinated processes. Many molecular processes have been shown to work outside of cells, and the idea of harnessing these molecular machines to build nanostructures is attractive. Two examples are microtubules and motor proteins, which aid cell movement, help determine cell shape and internal structure, and transport vesicles and organelles within the cell. These molecular machines work in a stochastic, noisy fashion: microtubules switch randomly between growing and shrinking in a process known as dynamic instability; motor protein movement along microtubules is randomly interrupted by the motor proteins falling off. A common strategy in attempting to gain control over these highly dynamic, stochastic processes is to eliminate some processes (e.g., work with stabilized microtubules) in order to focus on others (interaction of microtubules with motor proteins). In this paper, we illustrate a different strategy for building nanostructures, which, rather than attempting to control or eliminate some dynamic processes, uses them to advantage in building nanostructures. Specifically, using stochastic agent-based simulations, we show how the natural dynamic instability of microtubules can be harnessed in building nanostructures, and discuss strategies for ensuring that “unreliable” stochastic processes yield a robust outcome.

The bubble-free Belousov-Zhabotinsky reaction has been used to study the effects of centrifugal forces on autowave propagation. The reaction parameters were chosen such that the system oscillates naturally creating target waves. In the present study, the system was forced to rotate with a constant velocity around a central axis. In studying the effects of such a forcing on the system, we focused on target dynamics. The system reacts to this forcing in different ways, the most spectacular being a dramatic increase in the period of the target, the effect growing stronger as we move away from the center of rotation. A numerical study was carried out using the two-variable Oregonator model, modified to include convective effects through the diffusion coefficient. The numerical results showed a good qualitative agreement with those of the experiments.