Spatial step edge fluctuations on a multicomponent surface of $\mathrm{Al}∕\mathrm{Si}\left(111\right)\text{−}(\sqrt{3}\times \sqrt{3})$ were measured via scanning tunneling microscopy over a temperature range of $720–1070\mathrm{K}$, for step lengths of $L=65–160\mathrm{nm}$. Even though the time scale of fluctuations of steps on this surface varies by orders of magnitude over the indicated temperature range, measured first-passage spatial persistence and survival probabilities are temperature independent. The power law functional form for spatial persistence probabilities is confirmed and the symmetric spatial persistence exponent is measured to be $\theta =0.498\pm 0.062$ in agreement with the theoretical prediction $\theta =1∕2$. The survival probability is found to scale directly with $y∕L$, where $y$ is the distance along the step edge. The form of the survival probabilities agrees quantitatively with the theoretical prediction, which yields exponential decay in the limit of small $y∕L$. The decay constant is found experimentally to be $y_s∕L=0.076\pm 0.033$ for $y∕L\le 0.2$.

Genetic switch systems with mutual repression of two transcription factors are studied using deterministic methods (rate equations) and stochastic methods (the master equation and Monte Carlo simulations). These systems exhibit bistability, namely two stable states such that spontaneous transitions between them are rare. Induced transitions may take place as a result of an external stimulus. We study several variants of the genetic switch and examine the effects of cooperative binding, exclusive binding, protein-protein interactions, and degradation of bound repressors. We identify the range of parameters in which bistability takes place, enabling the system to function as a switch. Numerous studies have concluded that cooperative binding is a necessary condition for the emergence of bistability in these systems. We show that a suitable combination of network structure and stochastic effects gives rise to bistability even without cooperative binding. The average time between spontaneous transitions is evaluated as a function of the biological parameters.

A statistical description of heavy particles suspended in incompressible rough self-similar flows is developed. It is shown that, differently from smooth flows, particles do not form fractal clusters. They rather distribute inhomogeneously with a statistics that only depends on a local Stokes number, given by the ratio between the particles’ response time and the turnover time associated with the observation scale. Particle clustering is reduced by the fluid roughness. Heuristic arguments supported by numerics explain this effect in terms of the algebraic tails of the probability density function of the velocity difference between two particles.

Flow due to a commercially available vibrating quartz fork is studied in gaseous helium, $\mathrm{He}\mathrm{I}$ and $\mathrm{He}\mathrm{II}$, over a wide range of temperatures and pressures. On increasing the driving force applied to the fork, the drag changes in character from laminar (characterized by a linear drive vs velocity dependence) to turbulent (characterized by a quadratic drive vs velocity dependence). We characterize this transition by a critical Reynolds number ${\mathrm{Re}}_{\mathrm{cr}}^{\delta }=U_{\mathrm{cr}}\delta ∕\nu$, where $U_{\mathrm{cr}}$ is the critical velocity, $\nu$ stands for the kinematic viscosity, $\delta =\sqrt{2\nu ∕\omega }$ is the viscous penetration depth, and $\omega$ is the angular frequency of oscillations. We have experimentally verified that the corresponding scaling $U_{cr}\propto \sqrt{\nu \omega }$ holds in a classical viscous fluid over two decades of $\nu$.

Barabasi has shown that the priority-based scheduling rules in single-stage queuing systems (QS) generate fat tail behavior for the task waiting time distributions (WTD). These fat tails are induced by the waiting times of very low priority tasks that stay unserved almost forever as the task priority indices are “frozen in time” (i.e., a task priority is assigned once for all to each incoming task). Here, we study the new dynamic behavior expected when the priority of each incoming task is time-dependent (i.e., “aging mechanisms” are allowed). For two classes of models, namely a population-type model with an age structure and a QS with deadlines assigned to the incoming tasks, which is operated under the “earliest-deadline-first” policy, we are able to extract analytically some relevant characteristics of the task waiting time distribution. As the aging mechanism ultimately assigns high priority to any long waiting tasks, fat tails in the WTD cannot find their origin in the scheduling rule alone, thus showing a fundamental difference between our approach and Barabasi’s class of models.

Starting with a subexcitable net of FitzHugh-Nagumo elements it is shown that parameter variability (diversity) is able to induce pattern formation. These patterns are most coherent for an intermediate variability strength. This effect is similar to the well-known spatiotemporal stochastic resonance generated by additive noise in subexcitable media. Furthermore, variability is able to induce a transition to an excitable behavior of the net. This transition strongly depends on the coupling strength between the elements and it is found only for strong coupling. For weaker coupling one observes a lifetime lengthening of waves propagating through the medium, but the net stays subexcitable.

We study the chaotic scattering through an Aharonov-Bohm ring containing two cavities. One of the cavities has well-separated resonant levels while the other is chaotic, and is treated by random matrix theory. The conductance through the ring is calculated analytically using the supersymmetry method and the quantum fluctuation effects are numerically investigated in detail. We find that the conductance is determined by the competition between the mean and fluctuation parts. The dephasing effect acts on the fluctuation part only. The Breit-Wigner resonant peak is changed to an antiresonance by increasing the ratio of the level broadening to the mean level spacing of the random cavity, and the asymmetric Fano form turns into a symmetric one. For the orthogonal and symplectic ensembles, the period of the Aharonov-Bohm oscillations is half of that for regular systems. The conductance distribution function becomes independent of the ensembles at the resonant point, which can be understood by the mode-locking mechanism. We also discuss the relation of our results to the random walk problem.

We study interactions between the dark solitons of the parametrically driven nonlinear Schrödinger equation, Eq. (1). When the driving strength, $h$, is below $\sqrt{{\gamma }^2+1∕9}$, two well-separated Néel walls may repel or attract. They repel if their initial separation $2z\left(0\right)$ is larger than the distance $2z_u$ between the constituents in the unstable stationary complex of two walls. They attract and annihilate if $2z\left(0\right)$ is smaller than $2z_u$. Two Néel walls with $h$ lying between $\sqrt{{\gamma }^2+1∕9}$ and a threshold driving strength $h_{sn}$ attract for $2z\left(0\right)<2z_u$ and evolve into a stable stationary bound state for $2z\left(0\right)>2z_u$. Finally, the Néel walls with $h$ greater than $h_{sn}$ attract and annihilate—irrespective of their initial separation. Two Bloch walls of opposite chiralities attract, while Bloch walls of like chiralities repel—except near the critical driving strength, where the difference between the like-handed and oppositely handed walls becomes negligible. In this limit, similarly handed walls at large separations repel while those placed at shorter distances may start moving in the same direction or transmute into an oppositely handed pair and attract. The collision of two Bloch walls or two nondissipative Néel walls typically produces a quiescent or moving breather.

The interaction between a Bloch and a Néel wall in the parametrically driven nonlinear Schrödinger equation is studied by following the dissociation of their unstable bound state. Mathematically, the analysis focuses on the splitting of a fourfold zero eigenvalue associated with a pair of infinitely separated Bloch and Néel walls. It is shown that a Bloch and a Néel wall interact as two classical particles, one with positive and the other one with negative mass.

It is theoretically shown that storage and time reversal of light pulses can be achieved in a coupled-resonator optical waveguide by dynamic tuning of the cavity resonances without maintaining the translational invariance of the system. The control exploits the Bloch oscillation motion of a light pulse in the presence of a refractive index ramp, and it is therefore rather different from the mechanism of adiabatic band compression and reversal proposed by Yanik and Fan in recent works [Phys. Rev. Lett., 92, 083901 (2004); 93, 173903 (2004)].

We present an extensive Monte Carlo simulation study on the nonequilibrium kinetics of triangular antiferromagnetic Ising model within the ground state ensemble which consists of sectors, each of which is characterized by a unique value of the string density $p$ through a dimer covering method. Building upon our recent work [Phys. Rev. E 68, 066127 (2003)] where we considered the nonequilibrium relaxation observed within the dominant sector with $p=2∕3$, we here focus on the nonequilibrium kinetics within the minor sectors with $p<2∕3$. The initial configurations are chosen as those in which the strings are straight and evenly distributed. In the minor sectors, we observe a characteristic spatial anisotropy in both equilibrium and nonequilibrium spatial correlations. We observe emergence of a critical relaxation region (in the spatial and temporal domain) which grows as $p$ deviates from $p=2∕3$. Spatial anisotropy appears in the equilibrium spatial correlation with the characteristic length scale ${\xi }_{e,V}\left(p\right)$ diverging with vanishing string density as ${\xi }_{e,V}\left(p\right)\sim p^{-2}$ along the vertical direction, while along the horizontal direction the spatial length scale diverges as ${\xi }_{e,H}\sim p^{-1}$. Analytic forms for the anisotropic equilibrium correlation functions are given. We also find that the spin autocorrelation function $A\left(t\right)$ shows a simple scaling behavior $A\left(t\right)=\mathcal{A}\mathbf{(}t∕{\tau }_A\left(p\right)\mathbf{)}$, where the time scale ${\tau }_A\left(p\right)$ shows a power-law divergence with vanishing $p$ as ${\tau }_A\left(p\right)\sim p^{-\varphi }$ with $\varphi \simeq 4$. These features can be understood in terms of random walk nature of the fluctuations of the strings within the typical separation between neighboring strings.

The rheology of a two-dimensional granular gas under a plane shear is investigated. From the comparison among the discrete element method, the simulation of a set of hydrodynamic equation, and the analytic solution of the steady hydrodynamic equations, it is confirmed that the fluid equations derived from the kinetic theory give us accurate results even in relatively high density cases.

We present a feedback control scheme to stabilize unstable cellular patterns during the directional solidification of a binary alloy. The scheme is based on local heating of cell tips which protrude ahead of the mean position of all tips in the array. The feasibility of this scheme is demonstrated using phase-field simulations and, experimentally, using a real-time image processing algorithm, to track cell tips, coupled with a movable laser spot array device to heat the tips locally. We demonstrate, both numerically and experimentally, that spacings well below the threshold for a period-doubling instability can be stabilized. As predicted by the numerical calculations, cellular arrays become stable with uniform spacing through the feedback control which is maintained with minimal heating.

The long-wave (lubrication) approximation governing the evolution of a thin film over a uniformly heated topographical substrate is solved numerically. We study the initial-value problem for a variety of governing dimensionless parameters and topographical substrates. We demonstrate that the dynamics is characterized by a slow relaxation process with continuous coarsening of drops up to a large time where coarsening is terminated and the interface organizes into a series of drops each of which is located in a trough in topography.

A good representation of mesoscopic fluids is required to combine with molecular simulations at larger length and time scales [De Fabritiis et al., Phys. Rev. Lett. 97, 134501 (2006)]. However, accurate computational models of the hydrodynamics of nanoscale molecular assemblies are lacking, at least in part because of the stochastic character of the underlying fluctuating hydrodynamic equations. Here we derive a finite volume discretization of the compressible isothermal fluctuating hydrodynamic equations over a regular grid in the Eulerian reference system. We apply it to fluids such as argon at arbitrary densities and water under ambient conditions. To that end, molecular dynamics simulations are used to derive the required fluid properties. The equilibrium state of the model is shown to be thermodynamically consistent and correctly reproduces linear hydrodynamics including relaxation of sound and shear modes. We also consider nonequilibrium states involving diffusion and convection in cavities with no-slip boundary conditions.

We construct, in discrete two-component systems with cubic nonlinearity, stable states emulating Skyrmions of the classical field theory. In the two-dimensional case, an analog of the baby Skyrmion is built on the square lattice as a discrete vortex soliton of a complex field [whose vorticity plays the role of the Skyrmion’s winding number (WN)], coupled to a radial “bubble” in a real lattice field. The most compact quasi-Skyrmion on the cubic lattice is composed of a nearly planar complex-field discrete vortex and a three-dimensional real-field bubble; unlike its continuum counterpart which must have $\mathrm{WN}=2$, this stable discrete state exists with $\mathrm{WN}=1$. Analogs of Skyrmions in the one-dimensional lattice are also constructed. Stability regions for all these states are found in an analytical approximation and verified numerically. The dynamics of unstable discrete Skyrmions (which leads to the onset of lattice turbulence) and their partial stabilization by external potentials are explored too.

The physical behavior of a class of mesoscopic models for multiphase flows is analyzed in details near interfaces. In particular, an extended pseudopotential method is developed, which permits to tune the equation of state and surface tension independently of each other. The spurious velocity contributions of this extended model are shown to vanish in the limit of high grid refinement and/or high order isotropy. Higher order schemes to implement self-consistent forcings are rigorously computed for $2d$ and $3d$ models. The extended scenario developed in this work clarifies the theoretical foundations of the Shan-Chen methodology for the lattice Boltzmann method and enhances its applicability and flexibility to the simulation of multiphase flows to density ratios up to $O\left(100\right)$.

We present a modified naming game by introducing weights of words in the evolution process. We assign the weight of a word spoken by an agent according to its connectivity, which is a natural reflection of the agent’s influence in population. A tunable parameter is introduced, governing the word weight based on the connectivity of agents. We consider the scale-free topology and concentrate on the efficiency of reaching the final consensus, which is of high importance in the self-organized system. Interestingly, it is found that there exists an optimal parameter value, leading to the fastest convergence. This indicates appropriate hub’s effects favor the achievement of consensus. The evolution of distinct words helps to give a qualitative explanation of this phenomena. Similar nontrivial phenomena are observed in the total memory of agents with a peak in the middle range of parameter values. Other relevant characters are provided as well, including the time evolution of total memory and success rate for different parameter values as well as the average degree of the network, which are helpful for understanding the dynamics of the modified naming game in detail.

We present a continuum formalism for modeling growing random networks under addition and deletion of nodes based on a differential mass balance equation. As examples of its applicability, we obtain new results on the degree distribution for growing networks with a uniform attachment and deletion of nodes, and complete some recent results on growing networks with preferential attachment and uniform removal.

To study the ground state of an axial next-nearest-neighbor Ising chain under transverse field as a function of frustration parameter $\kappa$ and field strength $\Gamma$, we present here two different perturbative analyses. In one, we consider the (known) ground state at $\kappa =0.5$ and $\Gamma =0$ as the unperturbed state and treat an increase of the field from 0 to $\Gamma$ coupled with an increase of $\kappa$ from 0.5 to $0.5+r\Gamma ∕J$ as perturbation. The first-order perturbation correction to eigenvalue can be calculated exactly and we could conclude that there are only two phase-transition lines emanating from the point $\kappa =0.5$, $\Gamma =0$. In the second perturbation scheme, we consider the number of domains of length 1 as the perturbation and obtain the zeroth-order eigenfunction for the perturbed ground state. From the longitudinal spin-spin correlation, we conclude that floating phase exists for small values of transverse field over the entire region intermediate between the ferromagnetic phase and antiphase.