We study the phase-ordering kinetics following a quench to a final temperature $T_f$ of the one-dimensional $p$-state clock model. We show the existence of a critical value $p_c=4$, where the properties of the dynamics change. At $T_f=0$, for $p⩽p_c$ the dynamics is analogous to that of the kinetic Ising model, characterized by Brownian motion and annihilation of interfaces. Dynamical scaling is obeyed with the same dynamical exponents and scaling functions of the Ising model. For $p>p_c$, instead, the dynamics is dominated by a texture mechanism analogous to the one-dimensional $XY$ model and dynamical scaling is violated. During the phase-ordering process at $T_f>0$, before equilibration occurs, a crossover between an early $XY$-like regime and a late Ising-like dynamics is observed for $p>p_c$.

The void percolation and conduction problems for equisized overlapping ellipsoids of revolution are investigated using the discretization method. The method is validated by comparing the estimated percolation threshold of spheres with the precise result found in literature. The technique is then extended to determine the threshold of void percolation as a function of the geometric aspect ratio of ellipsoidal particles. The finite element method is also applied to evaluate the equivalent conductivity of the void phase in the system. The results confirm that there are no universalities for void percolation threshold and conductivity in particulate systems, and these properties are clearly dependent on the geometrical shape of particles. As a consequence, void percolation and conduction associated with ellipsoidal particles of large aspect ratio should be treated differently from spheres.

The local scaling behavior of the radial distribution function of the soft sphere or inverse power, $r^{-n}$ potential, fluid leads to a formula for the equation of state. From this formula different analytic forms for the compressibility factor, $Z$, have been derived. In the first, $Z$ is expressed as a product of three functions, the hard sphere equation of state and two other functions incorporating the effects of the potential softness. In the second formula, the compressibility factor is cast in terms of the position and height of the first peak in the radial distribution function. In the final form, $Z$ can be expressed as an exponential function which depends entirely on a combination of the virial coefficients. In each case $Z$ is an explicit expression which has the correct low density limiting behavior and is accurate up to the freezing density for all packing fractions and circa $n⩾12$. Expressions are derived for the various component functions required for the different forms of $Z$, and relations between them are established. The compressibility factor manifests a maximum value or “ridge” when plotted as contours on the density-softness plane. It starts for the softer fluids at lower densities, increases with particle stiffness, and crosses the freezing line at $n\cong 33$. From the compressibility factor other thermodynamic quantities can be obtained and the density-softness dependence of the infinite frequency limit elastic properties been determined. A self-consistent expression is derived for the effective hard sphere packing fraction (or equivalently, diameter), valid for all packing fractions and circa $n>12$. The effective hard-sphere diameter is compared with the formulas of Barker and Henderson, and Wheatley.

By means of two-dimensional contact dynamics simulations, we analyze the vibrational dynamics of a confined granular layer in response to harmonic forcing. We use irregular polygonal grains allowing for strong variability of solid fraction. The system involves a jammed state separating passive (loading) and active (unloading) states. We show that an approximate expression of the packing resistance force as a function of the displacement of the free retaining wall from the jamming position provides a good description of the dynamics. We study in detail the scaling of displacements and velocities with loading parameters. In particular, we find that, for a wide range of frequencies, the data collapse by scaling the displacements with the inverse square of frequency, the inverse of the force amplitude, and the square of gravity. Interestingly, compaction occurs during the extension of the packing, followed by decompaction in the contraction phase. We show that the mean compaction rate increases linearly with frequency up to a characteristic frequency and then it declines in inverse proportion to frequency. The characteristic frequency is interpreted in terms of the time required for the relaxation of the packing through collective grain rearrangements between two equilibrium states.

We present numerical results on spontaneous symmetry breaking strain localization in axisymmetric triaxial shear tests of granular materials. We simulated shear band formation using the three-dimensional distinct element method with spherical particles. We demonstrate that the local shear intensity, the angular velocity of the grains, the coordination number, and the local void ratio are correlated and any of them can be used to identify shear bands; however, the latter two are less sensitive. The calculated shear band morphologies are in good agreement with those found experimentally. We show that boundary conditions play an important role. We discuss the formation mechanism of shear bands in the light of our observations and compare the results with experiments. At large strains, with enforced symmetry, we found strain hardening.

We report results from a heterodyne-detected transient grating experiment on liquid and supercooled water in a wide temperature range, from $-17.5\text{to}90°\mathrm{C}$. The measured signal covers an extremely large time window with an excellent signal-to-noise ratio that enables the investigation in a single experiment of the sound speed and attenuation, thermal diffusivity, and temperature dependence of the dielectric constant. The experimental data clearly show the effect of the density and the temperature fluctuations on the water dielectric function. In order to describe the experimental results, we introduce a comprehensive hydrodynamic model taking into account the coupled density and temperature variables and their relevance in the definition of the spontaneous and forced dielectric variations. We use this model to describe the measured signal in transient grating experiments, including the heating and the electrostrictive sources produced by the laser excitation. The fitting procedure enables the safe extraction of several dynamic proprieties of liquid and supercooled water: the sound velocity and its damping, the thermal diffusivity, and the ratio between the dielectric thermodynamic derivatives. The measured parameters are compared to the literature data and discussed in the complex scenario of water physics.

We consider the influence of disorder on the nonequilibrium steady state of a minimal model for intracellular transport. In this model particles move unidirectionally according to the totally asymmetric exclusion process (TASEP) and are coupled to a bulk reservoir by Langmuir kinetics. Our discussion focuses on localized point defects acting as a bottleneck for the particle transport. Combining analytic methods and numerical simulations, we identify a rich phase behavior as a function of the defect strength. Our analytical approach relies on an effective mean-field theory obtained by splitting the lattice into two subsystems, which are effectively connected exploiting the local current conservation. Introducing the key concept of a carrying capacity, the maximal current that can flow through the bulk of the system (including the defect), we discriminate between the cases where the defect is irrelevant and those where it acts as a bottleneck and induces various novel phases (called bottleneck phases). Contrary to the simple TASEP in the presence of inhomogeneities, many scenarios emerge and translate into rich underlying phase diagrams, the topological properties of which are discussed.

The topological analysis of chaos based on a knot-theoretic characterization of unstable periodic orbits has proven to be a powerful method, however knot theory can only be applied to three-dimensional systems. Still, the core principles upon which this approach is built—determinism and continuity—apply in any dimension. We propose an alternative framework in which these principles are enforced on triangulated surfaces rather than curves, and we show that in dimension 3 our approach numerically predicts the correct topological entropies for periodic orbits of the horseshoe map.

We study a general set of models of social network evolution and dynamics. The models consist of both a dynamics on the network and evolution of the network. Links are formed preferentially between “similar” nodes, where the similarity is defined by the particular process taking place on the network. The interplay between the two processes produces phase transitions and hysteresis, as seen using numerical simulations for three specific processes. We obtain analytic results using mean-field approximations, and for a particular case we derive an exact solution for the network. In common with real-world social networks, we find coexistence of high and low connectivity phases and history dependence.

The hydraulic permeability of single-layer fibrous media is studied through two-dimensional (2D) and three-dimensional Navier-Stokes based flow simulations. As simple representations of such materials, one-dimensional arrays of parallel cylinders as well as two-dimensional arrays of perpendicularly crossing cylinders were used. The distance between the cylinders was either constant (ordered layers) or variable (disordered layers). For both 1D and 2D ordered layers, we propose a geometrical scaling rule for the hydraulic permeability as a function of cylinder radius and solid volume fraction (porosity), which is a modification of a scaling rule previously reported by Clague and co-workers. The proposed modification is based on theoretical considerations and leads to significantly improved correspondence with simulation results. The hydraulic permeability of unstructured layers is found to be higher than that of structured layers of equal porosity for both 1D and 2D arrays. We propose a single parameter that can be easily determined experimentally to characterize the degree of disorder, as well as a generally valid correction factor in the proposed geometrical scaling rule to account for the influence of disorder on the hydraulic permeability.

We present an analytical description of free-electron-laser (FEL) oscillations in a perfectly synchronized optical cavity by solving the one-dimensional FEL equations. It is shown that the radiation stored in the cavity eventually evolves into an intense few-cycle optical pulse in the high-gain and low-loss regime despite the lethargy effect. The evolution of the leading slope of the optical pulse, which is defined from the front edge toward the primary peak, is found to play an important role in generating the intense few-cycle optical pulse. The phase space evolution of electrons which interact with the leading slope is solved analytically in a perturbation method, leading to an analytical solution for the optical pulse evolution. The peak amplitude and the pulse length at saturation are found to scale with the electron beam density and optical cavity loss. Those scalings agree well with the intense few-cycle pulses recently observed in a high-power FEL.

An anisotropic lattice model in two spatial dimensions, with on-site and intersite cubic nonlinearities (the Salerno model), is introduced, with emphasis on the case in which the intersite nonlinearity is self-defocusing, competing with on-site self-focusing. The model applies, for example, to a dipolar Bose-Einstein condensate trapped in a deep two-dimensional (2D) optical lattice. Soliton families of two kinds are found in the model: ordinary ones and cuspons, with peakons at the border between them. Stability borders for the ordinary solitons are found, while all cuspons (and peakons) are stable. The Vakhitov-Kolokolov criterion does not apply to cuspons, but for the ordinary solitons it correctly identifies the stability limits. In direct simulations, unstable solitons evolve into localized pulsons. Varying the anisotropy parameter, we trace a transition between the solitons in 1D and 2D versions of the model. In the isotropic model, we also construct discrete vortices of two types, on-site and intersite centered (vortex crosses and squares, respectively), and identify their stability regions. In simulations, unstable vortices in the noncompeting model transform into regular solitons, while in the model with the competing nonlinearities they evolve into localized vortical pulsons, which maintain their topological character. Bound states of regular solitons and vortices are constructed too, and their stability is identified.

The majority of dynamical objects, which demonstrate energy localization in nonlinear lattices, represent quasibreathers rather than strictly time-periodic discrete breathers since, as a rule, there exist certain deviations in vibrational frequencies of the individual particles exceeding the possible numerical errors. We illustrate this idea with the James breathers in the $K_2$-$K_3$-$K_4$ chain and with quasibreathers in the $K_4$ chain. For the latter case, a rigorous investigation of existence and stability of the breathers and quasibreathers is presented. In particular, it is proved that they are stable up to a certain strength of the intersite part of the potential with respect to its on-site part. We conjecture that quasibreathers play a fundamental role in the problem of energy localization in more realistic nonlinear lattices, as well. The difference between breathers and quasibreathers can be characterized by the mean square deviation of the frequencies of individual particles.

The theory of spatial Kerr solitons is extended to colliding beams that are neither almost-exactly copropagating nor almost-exactly counterpropagating. Our new Helmholtz formalism yields results that are consistent with the inherent symmetry of the collision process and that are not predicted by existing paraxial descriptions. Full numerical and approximate analytical results are presented. These show excellent agreement. In particular, Kerr solitons are found to be remarkably robust under nonparaxial collisions.

It has been suggested that Glauber (inflow) and Sznajd (outflow) zero-temperature dynamics for the one-dimensional Ising ferromagnet with nearest-neighbor interactions are equivalent. Here we compare the two dynamics from the analytical and computational points of view. We use the method of mapping an Ising spin system onto the dimer RSA model and show that already this simple mapping allows us to see the differences between inflow and outflow zero-temperature dynamics. Then we investigate both dynamics with synchronous, partially synchronous, and random sequential updating using the Monte Carlo technique and compare both dynamics in the number of persistent spins, clusters, mean relaxation time, and relaxation time distribution.

Time-dependent simulations of two-dimensional isothermal $\mathrm{Ni}\text{−}\mathrm{Cu}$ dendrites are simulated using a phase-field model solved with a finite-difference adaptive mesh refinement technique. Dendrite tip velocity selection is examined and found to exhibit a transition between two markedly different regimes as undercooling is increased. At low undercooling, the dendrite tip growth rate is consistent with the kinetics of the classical Stefan problem, where the interface is assume to be in local equilibrium. At high undercooling, the growth velocity selected approaches a linear dependence on melt undercooling, consistent with the continuous growth kinetics of Aziz et al. and with a one-dimensional steady-state phase-field asymptotic analysis of Ahmad et al. [Phys. Rev. E 58, 3436 (1998)]. Our simulations are also consistent with other previously observed behaviors of dendritic growth as undercooling is increased. These include the transition of dendritic morphology to absolute stability and nonequilibrium solute partitioning. Our results show that phase-field models of solidification, which inherently contain a nonzero interface width, can be used to study the dynamics of complex solidification phenomena involving both equilibrium and nonequilibrium interface growth kinetics.

The inverse light-scattering problem for polymer dispersed liquid crystal film is considered. An optical method to retrieve the tensor order parameter of liquid crystal droplets arranged in a monolayer has been developed. The method is based on the measurement of coherent transmission coefficients of film at normal illumination and comparison with the results of the direct light-scattering problem solution. To find the coherent transmission coefficient, the anomalous diffraction approach, and the approximation of effective refractive indices are used. The dependence of the tensor order parameter on the applied voltage for elongated liquid crystal droplets with the rigidly fixed poles is retrieved by the available experimental data. The obtained results are in correlation with the results of known visual observations of liquid crystal droplet structure and calculations of the director field in droplets with rigidly-fixed poles.

We investigate the dynamics of monodisperse colloidal polystyrene particles suspended in solutions of the semiflexible polymer filamentous actin, over a range of filament lengths that either exceed or are substantially less than the particle radius. The filament length is controlled by the capping protein gelsolin, and particle surface chemistries that minimize the adsorption of filaments are used. The particle dynamics are measured on short time scales using diffusing wave spectroscopy. A sharp transition in the initial particle diffusivity marks the expected shift from a dilute to a tightly entangled polymer network as the filament average length increases. In both the dilute and entangled regimes, the measured particle dynamics are compared with the theories of rodlike and semiflexible polymer solution rheology using the generalized Stokes-Einstein relationship. In the dilute limit, the particle dynamics are in good agreement with theory. However, in the tightly entangled regime, the particle response is consistent with polymer depleted near the surfaces of the particles. The magnitude of the depletion layer thickness depends strongly on particle size and weakly on filament length. This behavior is in agreement with nonlocal entropic repulsions and the loss of conformational entropy associated with rodlike molecules near impenetrable particles. These results illustrate the use of microrheology as a method to investigate local structure and dynamics in colloid-polymer solutions.

The naming game is a model of nonequilibrium dynamics for the self-organized emergence of a linguistic convention or a communication system in a population of agents with pairwise local interactions. We present an extensive study of its dynamics on complex networks, that can be considered as the most natural topological embedding for agents involved in language games and opinion dynamics. Except for some community structured networks on which metastable phases can be observed, agents playing the naming game always manage to reach a global consensus. This convergence is obtained after a time generically scaling with the population’s size $N$ as $t_{\mathit{conv}}\sim N^{1.4\pm 0.1}$, i.e., much faster than for agents embedded on regular lattices. Moreover, the memory capacity required by the system scales only linearly with its size. Particular attention is given to heterogenous networks, in which the dynamical activity pattern of a node depends on its degree. High-degree nodes have a fundamental role, but require larger memory capacity. They govern the dynamics acting as spreaders of (linguistic) conventions. The effects of other properties, such as the average degree and the clustering, are also discussed.