We review the properties of time intervals between the crossings at a level $M$ of a smooth stationary Gaussian temporal signal. The distribution of these intervals and the persistence are derived within the independent interval approximation (IIA). These results grant access to the distribution of extrema of a general Gaussian process. Exact results are obtained for the persistence exponents and the crossing interval distributions, in the limit of large $\mid M\mid$. In addition, the small-time behavior of the interval distributions and the persistence is calculated analytically, for any $M$. The IIA is found to reproduce most of these exact results, and its accuracy is also illustrated by extensive numerical simulations applied to non-Markovian Gaussian processes appearing in various physical contexts.

We consider the asymmetric simple exclusion process (ASEP) on a semi-infinite chain which is coupled at the end to a reservoir with a particle density that changes periodically in time. It is shown that the density profile assumes a time-periodic sawtoothlike shape. This shape does not depend on initial conditions and is found analytically in the hydrodynamic limit. In a finite system, the stationary state is shown to be governed by effective boundary densities and the extremal flux principle. Effective boundary densities are determined numerically via Monte Carlo simulations and compared with those given by mean-field approach and numerical integration of the hydrodynamic limit equation which is the Burgers equation. Our results extend straightforwardly beyond the ASEP to a wide class of driven diffusive systems with one conserved particle species.

Reaction-diffusion processes can be adopted to model a large number of dynamics on complex networks, such as transport processes or epidemic outbreaks. In most cases, however, they have been studied from a fermionic perspective, in which each vertex can be occupied by at most one particle. While still useful, this approach suffers from some drawbacks, the most important probably being the difficulty to implement reactions involving more than two particles simultaneously. Here we develop a general framework for the study of bosonic reaction-diffusion processes on complex networks, in which there is no restriction on the number of interacting particles that a vertex can host. We describe these processes theoretically by means of continuous-time heterogeneous mean-field theory and divide them into two main classes: steady-state and monotonously decaying processes. We analyze specific examples of both behaviors within the class of one-species processes, comparing the results (whenever possible) with the corresponding fermionic counterparts. We find that the time evolution and critical properties of the particle density are independent of the fermionic or bosonic nature of the process, while differences exist in the functional form of the density of occupied vertices in a given degree class $k$. We implement a continuous-time Monte Carlo algorithm, well suited for general bosonic simulations, which allows us to confirm the analytical predictions formulated within mean-field theory. Our results, at both the theoretical and numerical levels, can be easily generalized to tackle more complex, multispecies, reaction-diffusion processes and open a promising path for a general study and classification of this kind of dynamical systems on complex networks.

The scaling properties of a post-mortem mortar crack surface are investigated. The root mean square of the height fluctuations is found to obey anomalous scaling properties, but with three exponents, two of them characterizing the local roughness ($\zeta \simeq 0.79$ and ${\zeta }_e\simeq 0.41$) and the third one driving the global roughness $\left({\zeta }_g\simeq 1.60\right)$. The critical exponent $\zeta \simeq 0.79$ is conjectured to reflect damage screening occurring for length scales smaller than the process zone size, while the exponent ${\zeta }_e\simeq 0.41$ characterizes roughness at larger length scales, i.e., at length scales where the material can be considered as linear elastic. Finally, we argue that the global roughness exponent could be material dependent contrary to both local roughness exponents ($\zeta \simeq 0.8$ and ${\zeta }_e\simeq 0.4$) which can be considered as universal.

We benchmark the dynamical simplex evolution (DSE) method with several of the currently available algorithms to detect communities in complex networks by comparing correctly identified nodes for different levels of “fuzziness” of random networks composed of well-defined communities. The potential benefits of the DSE method to detect hierarchical substructures in complex networks are discussed.

A simple method, the multiscale minimal Lagrangian map (MMLM) approach, to generate synthetic turbulent vector fields was previously introduced [C. Rosales and C. Meneveau, Phys. Fluids 18, 075104 (2006)]. It was shown that the synthesized fields reproduce many statistical and geometric properties observed in real, isotropic, turbulence. In this paper we investigate if this procedure, which applies a minimal Lagrangian map to deform an initial Gaussian field, can produce also anomalous scaling in the inertial range. It is found that the advection Lagrangian map time scale is crucial in determining anomalous scaling properties. With the sweeping time scale used in the MMLM approach, non-Gaussian statistics and realistic geometric features are reproduced at each scale, but anomalous exponents are not observed; i.e., we observe nearly 1941 Kolmogorov scaling. Conversely, if the appropriate Kolmogorov inertial-range turnover time scale is used in a modified approach [the multiscale turnover Lagrangian map (MTLM) method], fields with realistic anomalous scaling exponents are reproduced. Remarkably, the intermittency and multifractal nature of the energy dissipation is also found to be quite realistic. Finally, the properties of the pressure field derived from the MTLM velocity field are studied and found to be quite realistic also. The results shed new light on what are minimal dynamical requirements for the generation of anomalous scaling and intermittency in turbulent flow: at least one turnover time for small eddies to be sufficiently deformed, as well as the accumulation of spatially correlated deformations across scales.

Numerical computations and order-of-magnitude estimates are used to describe the time evolution of a drop of a very viscous liquid of finite electrical conductivity attached to a metallic plate which is suddenly subject to a uniform electric field. Under the action of the electric stresses induced at its surface, the drop elongates in the direction of the field, and charged droplets are emitted when the strength of the field is higher than a certain critical value. A stationary emission mode exists in which the attached drop develops a conical tip and a thin jet, with small droplets emitted from the end of the jet in a process that involves the formation of a long ligament. The flow rate and the electric current carried by the stream of droplets emitted in this mode are determined by the flow and the transfer of charge in the attached drop, in particular in a small region around its tip and in a leading stretch of the jet, where the solution is nearly stationary despite the transient character of the jet further downstream. A simplified analysis of the stationary regions is carried out to elucidate the effects of the physical properties of the liquid (electrical conductivity, permittivity, viscosity, and surface tension), the volume of the drop, and the strength of the applied field. For high electrical conductivities and applied fields well above its critical value, the electrical and viscous stresses are large compared to surface tension stresses, and their balance gives a flow rate proportional to the square of the applied field. The electric current is then that of a stationary electrified jet fed with this flow rate.

Sivashinsky’s [Acta Astron. 4, 1177 (1977)] nonlinear integrodifferential equation for the shape of corrugated one-dimensional flames is ultimately reducible to a $2N$-body problem, involving the $2N$ complex poles of the flame slope. Thual, Frisch, and Hénon [J. Phys. (France) 46, 1485 (1985)] derived singular linear integral equations for the pole density in the limit of large steady wrinkles $(N⪢1)$, which they solved exactly for monocoalesced periodic fronts of highest amplitude of wrinkling and approximately otherwise. Here we solve those analytically for isolated crests, next for monocoalesced, then bicoalesced periodic flame patterns, whatever the (large) amplitudes involved. We compare the analytically predicted pole densities and flame shapes to numerical results deduced from the pole-decomposition approach. Good agreement is obtained, even for moderately large $N\mathrm{s}$. The results are extended to give hints as to the dynamics of supplementary poles. Open problems are evoked.

Interfacial hydrodynamic instabilities arise in a range of chemical systems. One mechanism for instability is the occurrence of unstable density gradients due to the accumulation of reaction products. In this paper we conduct two-dimensional nonlinear numerical simulations for a member of this class of system: the methylene-blue–glucose reaction. The result of these reactions is the oxidation of glucose to a relatively, but marginally, dense product, gluconic acid, that accumulates at oxygen permeable interfaces, such as the surface open to the atmosphere. The reaction is catalyzed by methylene-blue. We show that simulations help to disassemble the mechanisms responsible for the onset of instability and evolution of patterns, and we demonstrate that some of the results are remarkably consistent with experiments. We probe the impact of the upper oxygen boundary condition, for fixed flux, fixed concentration, or mixed boundary conditions, and find significant qualitative differences in solution behavior; structures either attract or repel one another depending on the boundary condition imposed. We suggest that measurement of the form of the boundary condition is possible via observation of oxygen penetration, and improved product yields may be obtained via proper control of boundary conditions in an engineering setting. We also investigate the dependence on parameters such as the Rayleigh number and depth. Finally, we find that pseudo-steady linear and weakly nonlinear techniques described elsewhere are useful tools for predicting the behavior of instabilities beyond their formal range of validity, as good agreement is obtained with the simulations.

Unphysical heating and macroparticle trapping that arise in the numerical modeling of laser wakefield accelerators using particle-in-cell codes are investigated. A dark current free laser wakefield accelerator stage, in which no trapping of background plasma electrons into the plasma wave should occur, and a highly nonlinear cavitated wake with self-trapping, are modeled. Numerical errors can lead to errors in the macroparticle orbits in both phase and momentum. These errors grow as a function of distance behind the drive laser and can be large enough to result in unphysical trapping in the plasma wake. The resulting numerical heating in intense short-pulse laser-plasma interactions grows much faster and to a higher level than the known numerical grid heating of an initially warm plasma in an undriven system. The amount of heating, at least in the region immediately behind the laser pulse, can, in general, be decreased by decreasing the grid size, increasing the number of particles per cell, or using smoother interpolation methods. The effect of numerical heating on macroparticle trapping is less severe in a highly nonlinear cavitated wake, since trapping occurs in the first plasma wave period immediately behind the laser pulse.

Following our recent interest in metamaterial-based devices supporting resonant tunneling, energy squeezing, and supercoupling through narrow waveguide channels and bends, here we analyze the fundamental physical mechanisms behind this phenomenon using a transmission-line model. These theoretical findings extend our theory, allowing us to take fully into account frequency dispersion and losses and revealing the substantial differences between this unique tunneling phenomenon and higher-frequency Fabry-Perot resonances. Moreover, they represent the foundations for other possibilities to realize tunneling through arbitrary waveguide bends, both in $E$ and $H$ planes of polarization, waveguide connections, and sharp abruptions and to obtain analogous effects with geometries arguably simpler to realize.

We consider a random walk model that takes into account the velocity distribution of random walkers. Random motion with alternating velocities is inherent to various physical and biological systems. Moreover, the velocity distribution is often the first characteristic that is experimentally accessible. Here, we derive transport equations describing the dispersal process in the model and solve them analytically. The asymptotic properties of solutions are presented in the form of a phase diagram that shows all possible scaling regimes, including superdiffusive, ballistic, and superballistic motion. The theoretical results of this work are in excellent agreement with accompanying numerical simulations.

A general Monte Carlo simulation method of calculating the elastic constants of a polydisperse hard-sphere colloidal crystal is developed. Elastic constants of a size-polydisperse hard-sphere fcc crystal are calculated. The pressure and three elastic constants ($C_{11}$, $C_{12}$, and $C_{44}$) increase significantly with the polydispersity. It was also found from extrapolation that there is a mechanical terminal polydispersity above which a fcc crystal will be mechanically unstable.

Nucleation in supersaturated vapor is investigated with two series of molecular-dynamics simulations in the canonical ensemble. The applied methods are (a) analysis of critical nuclei at moderate supersaturations by simulating equilibria of single droplets with surrounding vapors in small systems; (b) simulation of homogeneous nucleation during condensation with large systems containing ${10}^5–{10}^6$ particles for calculating the nucleation rate of vapors at high supersaturations. For the Lennard-Jones fluid, truncated and shifted at 2.5 times the size parameter, it is shown that the classical nucleation theory underestimates both the nucleation rate and the size of the critical nucleus. A surface property corrected modification of this theory is proposed to consistently cover data on the surface tension of the curved interface, the critical nucleus size, and the nucleation rate.

Emergent pattern formation in self-propelled particle (SPP) systems is extensively studied because it addresses a range of swarming phenomena that occur without leadership. Here we present a dynamic SPP model in which a sensory blind zone is introduced into each particle’s zone of interaction. Using numerical simulations, we discovered that the degradation of milling patterns with increasing blind zone ranges undergoes two distinct transitions, including a spatially non-homogeneous transition that involves cessation of particles' motion caused by broken symmetries in the interaction fields. Our results also show the necessity of nearly complete panoramic sensory ability for milling behavior to emerge in dynamic SPP models, suggesting a possible relationship between collective behavior and the sensory systems of biological organisms.

A path-integral approach is developed for the analysis of spike-triggered average quantities in neurons with voltage-gated subthreshold currents. Using a linearization procedure to reduce the models to the generalized integrate-and-fire form, analytical expressions are obtained in an experimentally relevant limit of fluctuation-driven firing. The influences of voltage-gated channels as well as excitatory and inhibitory synaptic filtering are shown to affect significantly the neuronal dynamics prior to the spike. Analytical forms are given for all relevant physiological quantities, such as the mean voltage triggered to the spike, mean current flowing through voltage-gated channels, and the mean excitatory and inhibitory conductance waveforms prior to a spike. The mathematical results are shown to be in good agreement with numerical simulations of the underlying nonlinear conductance-based models. The method promises to provide a useful analytical tool for the prediction and interpretation of the temporal structure of spike-triggered averages measured experimentally.

We study a discrete stochastic model of a molecular motor. This discrete model can be viewed as a minimal ratchet model. We extend our previous work on this model, by further investigating the constraints imposed by the fluctuation theorem on the operation of a molecular motor far from equilibrium. In this work, we show the connections between different formulations of the fluctuation theorem. One formulation concerns the generating function of the currents while another one concerns the corresponding large deviation function, which we have calculated exactly for this model. A third formulation concerns the ratio of the probability of observing a velocity $v$ to the same probability of observing a velocity $-v$. Finally, we show that all the formulations of the fluctuation theorem can be understood from the notion of entropy production.

The effects of in-plane electric fields on the director structure of cholesteric liquid crystals has been imaged in three dimensions using fluorescence confocal polarizing microscopy. The results show that a liquid crystal lying outside the electrode gap can be significantly affected by stray fields occurring above the electrode surface, resulting in a 90° rotation of the cholesteric helix. Distinct differences between the behavior of cholesterics with positive and negative dielectric anisotropies are observed.

We study the process of formation of spiral waves in a heterogeneous excitable medium under external stimulation, using numerical and analytical methods. We show that in an excitable medium with several heterogeneities with respect to refractory period, fast rotating spiral waves can be generated. These fast spirals are formed as a result of a phenomenon of period decrease, which is the generation by a heterogeneity of waves with a period shorter than the period of the external stimulation.

We use a power expansion representation of plane-elasticity complex potentials due to Kolossov and Muskhelishvili to compute the elastic fields induced by a localized plastic deformation event. Far from its center, the dominant contributions correspond to first-order singularities of quadrupolar and dipolar symmetry which can be associated, respectively, with pure deviatoric and pure volumetric plastic strain of an equivalent circular inclusion. By construction of holomorphic functions from the displacement field and its derivatives, it is possible to define path-independent Cauchy integrals which capture the amplitudes of these singularities. Analytical expressions and numerical tests on simple finite-element data are presented. The development of such numerical tools is of direct interest for the identification of local structural reorganizations, which are believed to be the key mechanisms for plasticity of amorphous materials.