Recently, several powerful tools for the reconstruction of stochastic differential equations from measured data sets have been proposed [e.g., Siegert et al., Phys. Lett. A 243, 275 (1998); Hurn et al., J. Time Series Anal. 24, 45 (2003)]. Efficient application of the methods, however, generally requires Markov properties to be fulfilled. This constraint typically seems to be violated on small scales, which frequently is attributed to physical effects. On the other hand, measurement noise such as uncorrelated measurement and discretization errors has large impacts on the statistics of measurements on small scales. We demonstrate that the presence of measurement noise, likewise, spoils Markov properties of an underlying Markov process. This fact is promising for the further development of techniques for the reconstruction of stochastic processes from measured data, since limitations at small scales might stem from artificial noise sources rather than from intrinsic properties of the dynamics of the underlying process. Measurement noise, however, can be controlled much better than the intrinsic dynamics of the underlying process.

Using molecular dynamics simulations, the kinetics of bundle formation for stiff polyelectrolytes such as actin is studied in the solution of multivalent salt. The dominant kinetic mode of aggregation is found to be the case of one end of one rod meeting others at a right angle due to electrostatic interactions. The kinetic pathway to bundle formation involves a hierarchical structure of small clusters forming initially and then feeding into larger clusters, which is reminiscent of the flocculation dynamics of colloids. For the first few cluster sizes, the Smoluchowski formula for the time evolution of the cluster size gives a reasonable account of the results of our simulation without a single fitting parameter. The description using the Smoluchowski formula provides evidence for the aggregation time scale to be controlled by diffusion, with no appreciable energy barrier to overcome.

The response of three coupled FitzHugh-Nagumo neurons, under Gaussian white noise, to a subthreshold periodic signal is studied in this paper. By combining the canard dynamics, chemical coupling, and stochastic resonance together, the information transfer in this neural system is investigated. We find that chemical synaptic coupling is more efficient than the well-known linear coupling (gap junction) for local signal input, i.e., only one of the three neurons is subject to the periodic signal. This weak and local input is common in biological systems for the sake of low energy consumption.

A great number of biological structures are composed of fibers (elastin, collagen, etc.) dispersed on an aqueous matrix in such a complex way that a detailed mechanical analysis based on microconstituents is, for practical purposes, out of reach. Consequently, the preferred approach to the mechanical behavior of these materials is based on setting up of constitutive equations that homogenize the behavior while capturing their main microstructural features. This work presents a simple macroscopic model for fiber-reinforced materials with deformable matrices, especially suited to many biological structural tissues. The constitutive equation is derived by imposing equivalence between the virtual works of both the fiber-reinforced and the equivalent continuum media, showing that it is independent of the control volume used for such equivalence. The model is particularized to incompressible materials, and an extension to orthotropic biological fibers is shown. Numerical simulations of uniaxial tests on silk fibers demonstrate the model’s ability to capture the progressive alignment of the microconstituents under large deformations.

We study synchronization for two unidirectionally coupled networks. This is a substantial generalization of several recent papers investigating synchronization inside a network. We derive analytically a criterion for the synchronization of two networks which have the same (inside) topological connectivity. Then numerical examples are given which fit the theoretical analysis. In addition, numerical calculations for two networks with different topological connections are presented and interesting synchronization and desynchronization alternately appear with increasing value of the coupling strength.

It is shown that the large-deviation statistical quantities of the discrete-time, finite-state Markov process $P_{n+1}^{\left(j\right)}={\sum }_{k=1}^N{H}_{jk}{P}_n^{\left(k\right)}$, where $P_n^{\left(j\right)}$ is the probability for the $j$ state at the time step $n$ and $H_{jk}$ is the transition probability, completely coincide with those from the Kalman map corresponding to the above Markov process. Furthermore, it is demonstrated that, by using simple examples, time correlation functions in finite-state Markov processes can be well described in terms of unstable periodic orbits embedded in the equivalent Kalman maps.

In this paper we report on a theoretical investigation of the nonlinear dynamics of the polarization of multitransverse mode vertical-cavity surface-emitting lasers (VCSELs) under current modulation. Special attention is given to the comparison with a previously studied case of single-transverse mode VCSEL emitting in two orthogonal polarizations. The consideration of spatial effects in VCSEL modifies the polarization dynamics that accompanies the period doubling route to chaos for large modulation amplitudes. Depending on the modulation parameters, the excitation of a higher order transverse mode may either induce chaotic pulsing in an otherwise regularly pulsating VCSEL, or induce a time-periodic pulsing dynamics in an otherwise chaotic VCSEL. Bifurcation diagrams obtained for different modulation frequencies, several values of the dichroism, and different transverse mode characteristics allow us to identify the different scenarios of polarization dynamics in a directly modulated VCSEL. Temporal analysis of carrier number radial profile reveals considerable changes for the multitransverse mode case only constituting the physical origin of the reported changes in the temporal and polarization dynamics.

Using a standard pore-level model, which includes both viscous and capillary forces, we have studied the injection of a viscous, nonwetting fluid into a two-dimensional porous medium saturated with a less viscous, wetting fluid, i.e., drainage with favorable viscosity ratios, $M⩾1$. We have observed a crossover from fractal capillary fingering to standard compact flow at a characteristic time, which decreases with increased capillary number and/or viscosity ratio. We have tested an earlier prediction for the dependence of this crossover upon viscosity ratio and capillary number using our data for a wide-but-physical range of capillary numbers and viscosity ratios. We find good agreement between the predicted behavior and our results from pore-level modeling. Furthermore, we show that this agreement is not affected by changes in the random distribution of pore throat radii or by changes in the coordination number, suggesting that the prediction is universal, i.e., valid for any porous medium structure, as expected from the general nature of the derivation of the prediction. Furthermore, this agreement indicates that the prediction correctly accounts for dependence of the flow upon capillary number and viscosity ratios, thereby enabling predictions for interfacial advance and width as well as saturation and fractional flow profiles. Also this agreement supports the validity of the general theoretical development lending credence to the three-dimensional predictions.

A nonlinear model inspired by the tribological problem of a thin solid lubricant layer between two sliding periodic surfaces is used to analyze the novel phenomenon of hysteresis at pinning or depinning around a moving state rather than around a statically pinned state. The cycling of an external driving force $F_{\text{ext}}$ is used as a simple means to destroy and then to recover the dynamically pinned state previously discovered for the lubricant center-of-mass velocity. Depinning to a freely sliding state occurs either directly, with a single jump, or through a sequence of discontinuous transitions. The intermediate sliding steps are reminiscent of phase-locked states and stick-slip motion in static friction, and can be interpreted in terms of the appearance of traveling density defects in an otherwise regular arrangement of kinks. Repinning occurs more smoothly, through the successive disappearance of different traveling defects. The resulting bistability and multistability regions may also be accessed by varying mechanical parameters other than $F_{\text{ext}}$. The hysteretic phenomena are confined to the underdamped dynamics, and the overdamped dynamics of the same model is generally not hysteretic, much like in static friction.

We analyze the properties of the contact process with long-range interactions by the use of a kinetic ensemble in which the total number of particles is strictly conserved. In this ensemble, both annihilation and creation processes are replaced by a unique process in which a particle of the system chosen at random leaves its place and jumps to an active site. The present approach is particularly useful for determining the transition point and the nature of the transition, whether continuous or discontinuous, by evaluating the fractal dimension of the cluster at the emergence of the phase transition. We also present another criterion appropriate to identify the phase transition that consists of studying the system in the supercritical regime, where the presence of a “loop” characterizes the first-order transition. All results obtained by the present approach are in full agreement with those obtained by using the constant rate ensemble, supporting that, in the thermodynamic limit the results from distinct ensembles are equivalent.

We consider a family of models describing the evolution under selection of a population whose dynamics can be related to the propagation of noisy traveling waves. For one particular model that we shall call the exponential model, the properties of the traveling wave front can be calculated exactly, as well as the statistics of the genealogy of the population. One striking result is that, for this particular model, the genealogical trees have the same statistics as the trees of replicas in the Parisi mean-field theory of spin glasses. We also find that in the exponential model, the coalescence times along these trees grow like the logarithm of the population size. A phenomenological picture of the propagation of wave fronts that we introduced in a previous work, as well as our numerical data, suggest that these statistics remain valid for a larger class of models, while the coalescence times grow like the cube of the logarithm of the population size.

The continuous time random walk (CTRW) is a natural generalization of the Brownian random walk that allows the incorporation of waiting time distributions $\psi \left(t\right)$ and general jump distribution functions $\eta \left(x\right)$. There are two well-known fluid limits of this model in the uncoupled case. For exponential decaying waiting times and Gaussian jump distribution functions the fluid limit leads to the diffusion equation. On the other hand, for algebraic decaying waiting times $\psi \sim t^{-\left(1+\beta \right)}$ and algebraic decaying jump distributions $\eta \sim x^{-\left(1+\alpha \right)}$ corresponding to Lévy stable processes, the fluid limit leads to the fractional diffusion equation of order $\alpha$ in space and order $\beta$ in time. However, these are two special cases of a wider class of models. Here we consider the CTRW for the most general Lévy stochastic processes in the Lévy-Khintchine representation for the jump distribution function and obtain an integrodifferential equation describing the dynamics in the fluid limit. The resulting equation contains as special cases the regular and the fractional diffusion equations. As an application we consider the case of CTRWs with exponentially truncated Lévy jump distribution functions. In this case the fluid limit leads to a transport equation with exponentially truncated fractional derivatives which describes the interplay between memory, long jumps, and truncation effects in the intermediate asymptotic regime. The dynamics exhibits a transition from superdiffusion to subdiffusion with the crossover time scaling as ${\tau }_c\sim {\lambda }^{-\alpha ∕\beta }$, where $1∕\lambda$ is the truncation length scale. The asymptotic behavior of the propagator (Green’s function) of the truncated fractional equation exhibits a transition from algebraic decay for $t\ll {\tau }_c$ to stretched Gaussian decay for $t\gg {\tau }_c$.

By use of conformal field theory, we discover several exact factorizations of higher-order density correlation functions in critical two-dimensional percolation. Our formulas are valid in the upper half-plane, or any conformally equivalent region. We find excellent agreement of our results with high-precision computer simulations. There are indications that our formulas hold more generally.

We consider the ferromagnetic large-$q$ state Potts model in complex evolving networks, which is equivalent to an optimal cooperation problem, in which the agents try to optimize the total sum of pair cooperation benefits and the supports of independent projects. The agents are found to be typically of two kinds: A fraction of $m$ (being the magnetization of the Potts model) belongs to a large cooperating cluster, whereas the others are isolated one man’s projects. It is shown rigorously that the homogeneous model has a strongly first-order phase transition, which turns to second-order for random interactions (benefits), the properties of which are studied numerically on the Barabási-Albert network. The distribution of finite-size transition points is characterized by a shift exponent, $1∕{\tilde{\nu }}^{\prime }=0.26\left(1\right)$, and by a different width exponent, $1∕{\nu }^{\prime }=0.18\left(1\right)$, whereas the magnetization at the transition point scales with the size of the network, $N$, as $m\sim N^{-x}$, with $x=0.66\left(1\right)$.

Structures formed by a colloidal suspension of silica particles in 4-methyl cyclohexanol have been analyzed in the presence of an electric field. The formation of chains of particles was detected using an elliptical mirror to collect scattered light and a nearly matched refractive index between particles and solvent. A numerical method has been developed to obtain the size distribution of chains and their kinetics of formation from the record of a two-dimensional map of scattered light. We have compared the experimental size distribution to the prediction of a statistical theory based on a minimization of the free energy of a gas of chains. This theory quite well reproduced the experimental results for small chains but overestimates the tail of the distribution at high field. A saturation of the average size of chains versus the electric field was observed experimentally instead of a continuous growth as would be expected from aggregation under dipolar forces. A kinetic model, taking into account both capture and escape rates of a particle at the extremity of a chain, was shown to reproduce well the experimental growth of the average size of chains with time.

Liesegang patterns emerge from precipitation processes and may be used to build bulk structures at submicrometer length scales. Thus they have significant potential for technological applications provided adequate methods of control can be devised. Here we describe a simple, physically realizable pattern control based on the notion of driven precipitation, meaning that the phase separation is governed by a guiding field such as, for example, a temperature or pH field. The phase separation is modeled through a nonautonomous Cahn-Hilliard equation whose spinodal is determined by the evolving guiding field. Control over the dynamics of the spinodal gives control over the velocity of the instability front that separates the stable and unstable regions of the system. Since the wavelength of the pattern is largely determined by this velocity, the distance between successive precipitation bands becomes controllable. We demonstrate the above ideas by numerical studies of a one-dimensional system with a diffusive guiding field. We find that the results can be accurately described by employing a linear stability analysis (pulled-front theory) for determining the velocity–local-wavelength relationship. From the perspective of the Liesegang theory, our results indicate that the so-called revert patterns may be naturally generated by diffusive guiding fields.

The local and nonlocal characteristics of triad enstrophy transfer in the enstrophy inertial range of generalized two-dimensional turbulence, so-called $\alpha$ turbulence, are investigated using direct numerical simulations, with a special emphasis on $\alpha =1$, 2, and 3. The enstrophy transfer via nonlocal triad interactions dominates the transfer dynamics in the enstrophy inertial range, irrespective of $\alpha$. However, the contributions from more local interactions to the total enstrophy transfer increase as $\alpha$ decreases. The results are discussed in connection with the local and nonlocal transition of the enstrophy transfer at $\alpha =2$ expected from the phenomenological scaling theory. The specific nature of the enstrophy transfer in surface quasigeostrophic turbulence $(\alpha =1)$ is also discussed.

Electric fields in a plasma that conducts a high-current pulse are measured as a function of time and space. The experiment is performed using a coaxial configuration, in which a current rising to 160 kA in 100 ns is conducted through a plasma that prefills the region between two coaxial electrodes. The electric field is determined using laser spectroscopy and line-shape analysis. Plasma doping allows for three-dimensional spatially resolved measurements. The measured peak magnitude and propagation velocity of the electric field is found to match those of the Hall electric field, inferred from the magnetic-field front propagation measured previously.

We study electron transport in a one-dimensional molecular lattice chain. The molecules are linked by Morse interaction potentials. The electronic degree of freedom, expressed in terms of a tight binding system, is coupled to the longitudinal displacements of the molecules from their equilibrium positions along the axis of the lattice. More specifically, the distance between two sites influences in an exponential fashion the corresponding electronic transfer matrix element. We demonstrate that when an electron is injected in the undistorted lattice it causes a local deformation such that a compression results leading to a lowering of the electron’s energy below the lower edge of the band of linear states. This corresponds to self-localization of the electron due to a polaronlike effect. Then, if a traveling soliton lattice deformation is launched a distance apart from the electron’s position, upon encountering the polaronlike state it captures the latter dragging it afterwards along its path. Strikingly, even when the electron is initially uniformly distributed over the lattice sites a traveling soliton lattice deformation gathers the electronic amplitudes during its traversing of the lattice. Eventually, the electron state is strongly localized and moves coherently in unison with the soliton lattice deformation. This shows that for the achievement of coherent electron transport we need not start with the polaronic effect.

Genetic competence is a phenotypic state of a bacterial cell in which it is capable of importing DNA, presumably to improve survival under stress. Motivated by several colony-level known responses, we present a model for the influence of quorum sensing on the transition to competence of B. Subtilis. Coupling to the external signal creates an effective inhibitory mechanism, which results in anticorrelation between the cycles of adjacent cells. We show that this is consistent with recent experimental measurements and propose measurement methods to verify the role of quorum-sensing signals.