We suggest a method for generation of random binary sequences of elements 0 and 1, with prescribed correlation properties. It is based on a modification of the widely used convolution method of constructing continuous random processes. Using this method, a binary sequence with a power-law decaying pair correlator can be easily generated.

We study finite-size effects in the variance of the displacement of a tagged particle in the stationary state of the asymmetric simple exclusion process (ASEP) on a ring of size $L$. The process involves hard core particles undergoing stochastic driven dynamics on a lattice. The variance of the displacement of the tagged particle, averaged with respect to an initial stationary ensemble and stochastic evolution, grows linearly with time at both small and very large times. We find that at intermediate times, it shows oscillations with a well defined size-dependent period. These oscillations arise from sliding density fluctuations (SDFs) in the stationary state with respect to the drift of the tagged particle, the density fluctuations being transported through the system by kinematic waves. In the general context of driven diffusive systems, both the Edwards-Wilkinson (EW) and the Kardar-Parisi-Zhang (KPZ) fixed points are unstable with respect to the SDF fixed point, a flow towards which is generated on adding a gradient term to the EW and the KPZ time-evolution equation. We also study tagged particle correlations for a fixed initial configuration, drawn from the stationary ensemble, following earlier work by van Beijeren. We find that the time dependence of this correlation is determined by the dissipation of the density fluctuations. We show that an exactly solvable linearized model captures the essential qualitative features seen in the finite-size effects of the tagged particle correlations in the ASEP. Moreover, this linearized model also provides an exact coarse-grained description of two other microscopic models.

An all-optically driven strategy to govern a liquid-crystalline collective molecular nonlinear oscillator is discussed. It does not require external feedback of any kind while the oscillator and a time-dependent perturbation both are sustained by incident light. Various dynamical regimes such as frequency-locked, quasiperiodic, forced, and chaotic are observed, in agreement with a theoretical approach developed in the limit of the plane-wave approximation.

Integrate-and-fire models are mainstays of the study of single-neuron response properties and emergent states of recurrent networks of spiking neurons. They also provide an analytical base for perturbative approaches that treat important biological details, such as synaptic filtering, synaptic conductance increase, and voltage-activated currents. Steady-state firing rates of both linear and nonlinear integrate-and-fire models, receiving fluctuating synaptic drive, can be calculated from the time-independent Fokker-Planck equation. The dynamic firing-rate response is less easy to extract, even at the first-order level of a weak modulation of the model parameters, but is an important determinant of neuronal response and network stability. For the linear integrate-and-fire model the response to modulations of current-based synaptic drive can be written in terms of hypergeometric functions. For the nonlinear exponential and quadratic models no such analytical forms for the response are available. Here it is demonstrated that a rather simple numerical method can be used to obtain the steady-state and dynamic response for both linear and nonlinear models to parameter modulation in the presence of current-based or conductance-based synaptic fluctuations. To complement the full numerical solution, generalized analytical forms for the high-frequency response are provided. A special case is also identified—time-constant modulation—for which the response to an arbitrarily strong modulation can be calculated exactly.

We use a simple dynamical model of two interacting networks of integrate-and-fire neurons to explain a seemingly paradoxical result observed in epileptic patients indicating that the level of phase synchrony declines below normal levels during the state preceding seizures (preictal state). We model the transition from the seizure free interval (interictal state) to the seizure (ictal state) as a slow increase in the mean depolarization of neurons in a network corresponding to the epileptic focus. We show that the transition from the interictal to preictal and then to the ictal state may be divided into separate dynamical regimes: the formation of slow oscillatory activity due to resonance between the two interacting networks observed during the interictal period, structureless activity during the preictal period when the two networks have different properties, and bursting dynamics driven by the network corresponding to the epileptic focus. Based on this result, we hypothesize that the beginning of the preictal period marks the beginning of the transition of the epileptic network from normal activity toward seizing.

The network formed by ridges in a straightened sheet of crumpled paper is studied using a laser profilometer. Square sheets of paper were crumpled into balls, unfolded, and their height profile measured. From these profiles the imposed ridges were extracted as networks. Nodes were defined as intersections between ridges, and links as the various ridges connecting the nodes. Many network and spatial properties have been investigated. The tail of the ridge length distribution was found to follow a power law, whereas the shorter ridges followed a log-normal distribution. The degree distribution was found to have an exponentially decaying tail, and the degree correlation was found to be disassortative. The facets created by the ridges and the Voronoi diagram formed by the nodes have also been investigated.

We apply random matrix theory to complex networks. We show that nearest neighbor spacing distribution of the eigenvalues of the adjacency matrices of various model networks, namely scale-free, small-world, and random networks follow universal Gaussian orthogonal ensemble statistics of random matrix theory. Second, we show an analogy between the onset of small-world behavior, quantified by the structural properties of networks, and the transition from Poisson to Gaussian orthogonal ensemble statistics, quantified by Brody parameter characterizing a spectral property. We also present our analysis for a protein-protein interaction network in budding yeast.

We consider a percolationlike phenomenon on a generalization of the Barabási-Albert model, where a modification of the growth dynamics directly allows formation of disconnected clusters. The transition is located with high precision by an original numerical technique based on the comparison of the largest and second largest clusters. A careful investigation focusing on finite size scaling allows us to highlight properties which would hardly be accessible by an analytical solution of cluster growth equations in the stationary limit. Our analysis shows that some critical features of the percolation transition are different from those observed in the case of dilution in fully grown networks. At variance with other models of percolation on growing networks we also find evidence that the order parameter approaches zero as a power of the field $p-p_c$ driving the transition, rather than as a stretched exponential. This behavior does not agree with the Berezinskii-Kosterlitz-Thouless scenario found in other similar models. For describing the phase in which a giant cluster develops, a key role is played by the crossover number of nodes $N_x\sim {(p-p_c)}^{-\zeta }$ with $\zeta \simeq 4$. This power law behavior and that of other quantities are conjectured on the basis of scaling arguments and numerical evidence.

We consider a system which consists of a layer of an incompressible binary liquid with a deformable free surface, and a thick solid substrate subjected to a differential heating across it. We investigate the long-wave thermosolutal Marangoni instability in the case of asymptotically small Lewis and Galileo numbers for finite capillary and Biot numbers with the Soret effect taken into account. We find both long-wave monotonic and oscillatory modes of instability in various parameter domains of Biot and Soret numbers. In the domain of finite wave numbers the monotonic instability is found, but the minimum of the monotonic neutral curve is shown to be located in the long-wave region. A set of nonlinear evolution equations is derived for the description of the spatiotemporal dynamics of the oscillatory instability. The weakly nonlinear analysis is carried out for the monotonic instability.

The axisymmetric linear stability of the Taylor-Couette flow with an azimuthal magnetic field is considered. It is shown that a flow with the combination of a linearly unstable rotation and a linearly unstable azimuthal magnetic field can be linearly stable. The flow stabilization takes place for both ideal and dissipative flows. For dissipative flow the stabilization exists only for a combination of counter-rotating cylinders and a counterdirecting azimuthal magnetic field on cylinders. The effect can be important for the problem of a plasma confinement by the magnetic field.

We propose an approach to the problem of the first-passage time. Our method is applicable not only to the Wiener process but also to the non-Gaussian Lévy flights or to more complicated stochastic processes whose distributions are stable. To show the usefulness of the method, we particularly focus on the first-passage time problems in the truncated Lévy flights (the so-called KoBoL processes from Koponen, Boyarchenko, and Levendorskii), in which the arbitrarily large tail of the Lévy distribution is cut off. We find that the asymptotic scaling law of the first-passage time $t$ distribution changes from $t^{-(\alpha +1)∕\alpha }$-law (non-Gaussian Lévy regime) to $t^{-3∕2}$-law (Gaussian regime) at the crossover point. This result means that an ultraslow convergence from the non-Gaussian Lévy regime to the Gaussian regime is observed not only in the distribution of the real time step for the truncated Lévy flight but also in the first-passage time distribution of the flight. The nature of the crossover in the scaling laws and the scaling relation on the crossover point with respect to the effective cutoff length of the Lévy distribution are discussed.

We investigated how the finiteness of the length of a sequence affects the phase transition that takes place at the DNA melting temperature. For this purpose, we modified the transfer integral method to adapt it to the calculation of both extensive (partition function, entropy, specific heat, etc.) and nonextensive (order parameter and average separation between paired bases) thermodynamic quantities of finite sequences with open boundary conditions, and applied the modified procedure to two different dynamical models. We characterized in some detail the three effects that take place when the length of the sequence is decreased, namely, (i) the decrease of the critical temperature, (ii) the decrease of the peak values of all quantities that diverge at the thermodynamic limit but remain finite for finite sequences, like the specific heat and the correlation length, and (iii) the broadening of the temperature range over which the transition affects the dynamics of the system. We also performed a finite size scaling analysis of the two models and showed that the singular part of the free energy can indeed be expressed in terms of a homogeneous function. However, Josephson’s identity is satisfied for none of the investigated models, so that the derivation of the characteristic exponents which appear, for example, in the expression of the specific heat requires some care.

Interpretive and predictive tools are needed to assist in the understanding of cell invasion processes. Cell invasion involves cell motility and proliferation, and is central to many biological processes including developmental morphogenesis and tumor invasion. Experimental data can be collected across a wide range of scales, from the population scale to the individual cell scale. Standard continuum or discrete models used in isolation are insufficient to capture this wide range of data. We develop a discrete cellular automata model of invasion with experimentally motivated rules. The cellular automata algorithm is applied to a narrow two-dimensional lattice and simulations reveal the formation of invasion waves moving with constant speed. The simulation results are averaged in one dimension—these data are used to identify the time history of the leading edge to characterize the population-scale wave speed. This allows the relationship between the population-scale wave speed and the cell-scale parameters to be determined. This relationship is analogous to well-known continuum results for Fisher’s equation. The cellular automata algorithm also produces individual cell trajectories within the invasion wave that are analogous to cell trajectories obtained with new experimental techniques. Our approach allows both the cell-scale and population-scale properties of invasion to be predicted in a way that is consistent with multiscale experimental data. Furthermore we suggest that the cellular automata algorithm can be used in conjunction with individual data to overcome limitations associated with identifying cell motility mechanisms using continuum models alone.

A two step method is proposed to reduce colored noise for chaotic data in the local phase space. With the observation that the energy of colored noise is mainly distributed in a low dimensional subspace, a noise dominated subspace is first estimated by the energy distribution of colored noise. At step 1, for the reference phase point, the components projected into the noise dominated subspace are deleted and the enhanced data are reconstructed with the remaining components. The residual error of the output of step 1 tends to distribute on each direction uniformly. So at step 2, the local projection method is further applied to the output of step 1, treating the residual error as white noise. Experiments show that our method performs well in eliminating colored noise for chaotic data.

We investigate experimentally and theoretically the dynamics of a semiconductor laser subject to filtered optical feedback. Depending on the feedback strength we find dynamical regimes with different dependence on the feedback phase. In particular, the influence of the feedback phase on cw emission and on frequency oscillations is characterized experimentally. We also measure the dependence of the filter mirror distance on the frequency oscillations. In general, good agreement between experiment and theory is found.

We consider the three-dimensional (3D) Gross-Pitaevskii or nonlinear Schrödinger equation with a quasi-2D square-lattice potential (which corresponds to the optical lattice trapping a self-attractive Bose-Einstein condensate, or, in some approximation, to a photonic-crystal fiber, in terms of nonlinear optics). Stable 3D solitons, with embedded vorticity $S=1$ and 2, are found by means of the variational approximation and in a numerical form. They are built, basically, as sets of four fundamental solitons forming a rhombus, with phase shifts $\pi S∕2$ between adjacent sites, and an empty site in the middle. The results demonstrate two species of stable 3D solitons, which were not studied before, viz., localized vortices (“spinning light bullets,” in terms of optics) with $S>1$, and vortex solitons (with any $S\ne 0$) supported by a lattice in the 3D space. Typical scenarios of instability development (collapse or decay) of unstable localized vortices are identified too.

Reaction networks in the bulk and on surfaces are widespread in physical, chemical, and biological systems. In macroscopic systems, which include large populations of reactive species, stochastic fluctuations are negligible and the reaction rates can be evaluated using rate equations. However, many physical systems are partitioned into microscopic domains, where the number of molecules in each domain is small and fluctuations are strong. Under these conditions, the simulation of reaction networks requires stochastic methods such as direct integration of the master equation. However, direct integration of the master equation is infeasible for complex networks, because the number of equations proliferates as the number of reactive species increases. Recently, the multiplane method, which provides a dramatic reduction in the number of equations, was introduced [Lipshtat and Biham, Phys. Rev. Lett. 93, 170601 (2004)]. The reduction is achieved by breaking the network into a set of maximal fully connected subnetworks (maximal cliques). Lower-dimensional master equations are constructed for the marginal probability distributions associated with the cliques, with suitable couplings between them. In this paper, we test the multiplane method and examine its applicability. We show that the method is accurate in the limit of small domains, where fluctuations are strong. It thus provides an efficient framework for the stochastic simulation of complex reaction networks with strong fluctuations, for which rate equations fail and direct integration of the master equation is infeasible. The method also applies in the case of large domains, where it converges to the rate equation results.

The critical properties of the Frank model of spontaneous chiral synthesis are discussed by applying results from the field theoretic renormalization group (RG). The long time and long wavelength features of this microscopic reaction scheme belong to the same universality class as multicolored directed percolation processes. Thus the following RG fixed points (FPs) govern the critical dynamics of the Frank model for $d<4$: one unstable FP that corresponds to complete decoupling between the two enantiomers, a saddle point that corresponds to symmetric interspecies coupling, and two stable FPs that individually correspond to unidirectional couplings between the two chiral molecules. These latter two FPs are associated with the breakdown of mirror or chiral symmetry. In this simplified model of molecular synthesis, homochirality is a natural consequence of the intrinsic reaction noise in the critical regime, which corresponds to extremely dilute chemical systems.

Using the Langevin equation we develop the model of a stochastic process subject to a given time-dependent regulatory mechanism. The effects of this nonstationarity on the statistical properties of the time series, i.e., on global and conditional probability densities and on the moments of the distribution, are derived. Application of these results on simple model trends allows one to approximate cardiological data and thus to explain effects recently observed in the reconstruction of the deterministic part of the Langevin equation for time series of heart rate.

The two-body interaction in dilute solutions of polymer chains in good solvents can be modeled by means of effective bounded potentials, the simplest of which being that of penetrable spheres (PSs). In this paper we construct two simple analytical theories for the structural properties of PS fluids: a low-temperature (LT) approximation, that can be seen as an extension to PSs of the well-known solution of the Percus-Yevick (PY) equation for hard spheres, and a high-temperature (HT) approximation based on the exact asymptotic behavior in the limit of infinite temperature. Monte Carlo simulations for a wide range of temperatures and densities are performed to assess the validity of both theories. It is found that, despite their simplicity, the HT and LT approximations exhibit a fair agreement with the simulation data within their respective domains of applicability, so that they complement each other. A comparison with numerical solutions of the PY and the hypernetted-chain approximations is also carried out, the latter showing a very good performance, except inside the core at low temperatures.