We study the fatigue fracture of disordered materials by means of computer simulations of a discrete element model. We extend a two-dimensional fracture model to capture the microscopic mechanisms relevant for fatigue and we simulate the diametric compression of a disc shape specimen under a constant external force. The model allows us to follow the development of the fracture process on the macrolevel and microlevel varying the relative influence of the mechanisms of damage accumulation over the load history and healing of microcracks. As a specific example we consider recent experimental results on the fatigue fracture of asphalt. Our numerical simulations show that for intermediate applied loads the lifetime of the specimen presents a power law behavior. Under the effect of healing, more prominent for small loads compared to the tensile strength of the material, the lifetime of the sample increases and a fatigue limit emerges below which no macroscopic failure occurs. The numerical results are in a good qualitative agreement with the experimental findings.

We present a cellular automaton (CA) model for simulating the complex dynamics of stock markets. Within this model, a stock market is represented by a two-dimensional lattice, of which each vertex stands for a trader. According to typical trading behavior in real stock markets, agents of only two types are adopted: fundamentalists and imitators. Our CA model is based on local interactions, adopting simple rules for representing the behavior of traders and a simple rule for price updating. This model can reproduce, in a simple and robust manner, the main characteristics observed in empirical financial time series. Heavy-tailed return distributions due to large price variations can be generated through the imitating behavior of agents. In contrast to other microscopic simulation (MS) models, our results suggest that it is not necessary to assume a certain network topology in which agents group together, e.g., a random graph or a percolation network. That is, long-range interactions can emerge from local interactions. Volatility clustering, which also leads to heavy tails, seems to be related to the combined effect of a fast and a slow process: the evolution of the influence of news and the evolution of agents’ activity, respectively. In a general sense, these causes of heavy tails and volatility clustering appear to be common among some notable MS models that can confirm the main characteristics of financial markets.

We investigate the relationship between dispersion of inertial particles and the pressure field in free shear flows. A three-dimensional temporally developing particle-laden mixing layer and a three-dimensional spatially developing particle-laden plane jet are studied by means of direct numerical simulation. The incompressible Navier-Stokes equations are accurately solved without any turbulence model. The dispersed inertial particles are traced in the Lagrangian framework. It is found that the instantaneous spatial distribution of inertial particles correlates well with the Laplacian of pressure ${\nabla }^{2}p$, and from the statistical viewpoint, the effect of particle size on the fraction of particle number distributed within different flow zones characterized by ${\nabla }^{2}p$ for different particles is negligible when the flow is well developed. The potential explanations and applications for this feature are explored.

The (Michelson) Sivashinsky equation of premixed flames is studied in a rectangular domain in two dimensions. A huge number of two-dimensional (2D) stationary solutions are trivially obtained by the addition of two 1D solutions. With Neumann boundary conditions, it is shown numerically that adding two stable 1D solutions leads to a 2D stable solution. This type of solution is shown to play an important role in the dynamics of the equation with additive noise.

The recently discovered supersymmetric generalizations of the Langevin dynamics and Kramers equation can be utilized for the exploration of free-energy landscapes of systems whose large time-scale separation hampers the usefulness of standard molecular dynamics techniques. The first realistic application is here presented. The system chosen is a minimalist model for a short alanine peptide exhibiting a helix-coil transition.

Permanently stable ground ice is found beneath a permanently frost free surface on Mars, and similar conditions exist in the Antarctic Dry Valleys. This phenomenon is due to a balance of the vapor pressure of the ice with the atmospheric humidity in the presence of large amplitude temperature oscillations. An exactly solvable model example shows that the fraction of time the atmosphere needs to be saturated to stabilize the ice decreases with temperature amplitude. It is estimated that for conditions that prevail on Mars today, the mean temperature needs to be about $5\mathrm{K}$ lower than the frost point temperature for ground ice to be stable. A decomposition method to evaluate the contribution of short term weather events to ground ice stability is developed; when applied to a study site in the Dry Valleys, it reveals that the coldest periods contribute most to stabilization.

Using molecular dynamics simulation we investigate the dynamics in a two-dimensional binary mixture at a low temperature and at high densities. We increase the size ratio of the diameters of the two components or the fraction of the larger particles. Then changeovers occur from polycrystal to glass with proliferation of defects. The relationship between the degree of disorder and the slow dynamics is studied by simultaneous visualization of a disorder variable $D_j$ introduced in our previous paper T. Hamanaka and A. Onuki[Phys. Rev. E 74, 011506 (2006)] and the particle displacement $\mathrm{\Delta }{\mathit{r}}_j$ in a long time interval. In polycrystal, the particles in the grain boundary regions have large $D_j$ and a relatively large mobility, producing significant dynamic heterogeneity on long time scales. In the crossover from polycrystal to glass, the crystalline regions become narrow, but the particles with relatively large $D_j$ still trigger the formation of chainlike particle motions, leading to the dynamic heterogeneity.

We incorporate a source of noise into a continuum neural field model by allowing the firing threshold to fluctuate noisily about a mean value, and examine traveling wave front solutions. Under certain conditions we are able to calculate the first and second moments of the distributions of the resulting time varying front speed and shape. This is then compared with more complete numerical solutions. Fluctuations in the wave front speed and in the shape (i.e., fluctuations in activity at particular coordinate positions across the wave front) were found to increase as the magnitude of the fluctuations in firing threshold increased. The mean speed was found to increase as the magnitude of the fluctuations increases. The role of the correlation time for the threshold variation is also investigated. We also study the role of threshold fluctuations in the failure of front propagation, both in the fast and slow varying noise limits.

We study the statistical properties of the sampled networks by a random walker. We compare topological properties of the sampled networks such as degree distribution, degree-degree correlation, and clustering coefficient with those of the original networks. From the numerical results, we find that most of topological properties of the sampled networks are almost the same as those of the original networks for $\gamma \lesssim 3$. In contrast, we find that the degree distribution exponent of the sampled networks for $\gamma >3$ somewhat deviates from that of the original networks when the ratio of the sampled network size to the original network size becomes smaller. We also apply the sampling method to various real networks such as collaboration of movie actor, Worldwide Web, and peer-to-peer networks. All topological properties of the sampled networks are essentially the same as those of the original real networks.

The formation of spatially localized patterns in a system with subcritical instability under feedback control with delay is investigated within the framework of globally controlled Ginzburg-Landau equation. It is shown that feedback control can stabilize spatially localized solutions. With the increase of delay, these solutions undergo oscillatory instability that, for large enough control strength, results in the formation of localized oscillating pulses. With further increase of the delay the solution blows up.

We study the spreading and fingering of drops of silicone oil on a rotating substrate for a range of rotation speeds and drop volumes. The spreading of the drop prior to the onset of fingering is found to follow the theoretically predicted time dependence, but with a large shift in time scale. For the full range of experimental parameters studied, the contact line becomes unstable and fingers develop when the radius of the drop becomes sufficiently large. We study the growth of perturbations around the perimeter of the drop and find the growth rate of the most unstable mode to agree well with the predictions of lubrication theory. The number of fingers which form around the perimeter of the drop is found to be a function of both rotation speed and drop volume, and is also in excellent agreement with theoretical predictions.

A general factorized form of the dielectric function together with a fractional model-based parameter estimation method is used to provide an accurate analytical formula for the complex refractive index in water for the frequency range ${10}^8–{10}^{16}\mathrm{Hz}$. The analytical formula is derived using a combination of a microscopic frequency-dependent rational function for adjusting zeros and poles of the dielectric dispersion together with the macroscopic statistical Fermi-Dirac distribution to provide a description of both the real and imaginary parts of the complex permittivity for water. The Fermi-Dirac distribution allows us to model the dramatic reduction in the imaginary part of the permittivity in the visible window of the water spectrum.

In this paper we consider the stabilization of nonfundamental unstable stationary solutions of the cubic nonlinear Schrödinger equation. Specifically, we study the stabilization of radially symmetric solutions with nodes and asymmetric complex stationary solutions. For the first ones, we find partial stabilization similar to that recently found for vortex solutions while for the later ones stabilization does not seem possible.

Using the Wang-Landau algorithm combined with the analytic method, the density of states of a two-dimensional $XY$ model on square lattices of sizes $16\times 16$, $24\times 24$, and $32\times 32$ is accurately calculated. Thermodynamic quantities, such as internal energy, free energy, entropy, and specific heat are obtained from the resulted density of states by numerical integration. It is found that both the density of states and the extensive thermodynamic quantities obey the scaling law. From the entropy curve, symptoms of phase transition are observed. A general method of calculation of the density of states of continuous models by simulation combined with analytical method is proposed.

The oscillating cylinder of a low-frequency inverted torsion pendulum is immersed into layers of noncohesive granular materials, including fine sand and glass beads. The relative energy dissipation and relative modulus of the granular system versus the amplitude and immersed depth of the oscillating cylinder are measured. A rheological model based on a mesoscopic picture is presented. The experimental results and rheological model indicate that small slides in the inhomogeneous force chains are responsible for the energy dissipation of the system, and the friction of the grains plays two different roles in the mechanical response of sheared granular material: damping the energy and enhancing the elasticity.

Over the last few years, renewed interest has been raised by the simplified general interaction models proposed by Straley for mesogenic molecules possessing the $D_{2h}$ symmetry and capable of producing biaxial nematic order. It has already been shown that, in the presence of certain special symmetries, just two out of the four order parameters that are in general necessary, suffice for the description of a biaxial phase. For some other range of parameters, these reducing symmetries do not hold, and, moreover, a mean-field treatment has to be suitably changed into a minimax strategy, still producing a transition to a low-temperature biaxial phase. Upon studying the general parameter range, we identify as a common feature the behavior of a uniaxial order parameter, attaining a local minimum at the biaxial-to-uniaxial transition temperature, and recognizably increasing away from it. This finding is confirmed by a Monte Carlo simulation.

Elastic incoherent neutron scattering (EINS) data can be approximated with a Gaussian function of $q$ in a low $q$ region. However, in a higher $q$ region the deviation from a Gaussian function becomes non-negligible. Protein dynamic properties can be derived from the analyses of the non-Gaussian behavior, which has been experimentally investigated. To evaluate the origins of the non-Gaussian behavior of protein dynamics, we conducted a molecular dynamics (MD) simulation of staphylococcal nuclease. Instead of the ordinary cumulant expansion, we decomposed the non-Gaussian terms into three components: (i) the component originating from the heterogeneity of the mean-square fluctuation, (ii) that from the anisotropy, and (iii) that from higher-order terms such as anharmonicity. The MD simulation revealed various dynamics for each atom. The atomic motions are classified into three types: (i) “harmonic,” (ii) “anisotropic,” and (iii) “anharmonic.” However, each atom has a different degree of anisotropy. The contribution of the anisotropy to the total scattering function averages out due to these differences. Anharmonic motion is described as the jump among multiple minima. The jump distance and the probability of the residence at one site vary from atom to atom. Each anharmonic component oscillates between positive and negative values. Thus, the contribution of the anharmonicity to the total scattering is canceled due to the variations in the anharmonicity. Consequently, the non-Gaussian behavior of the total EINS from a protein can be analyzed by the dynamical heterogeneity.

We have measured the multiscale wrinkling that occurs along the edge of torn plastic sheets. The plastic deformations produced by tearing define a new metric on the sheet, which then relaxes elastically. The resultant patterns of wrinkles correspond to a superposition of waves of different wavelengths. Measurements of the variation of the pattern as a function of the distance from the edge reveal a set of transitions, each of which adds a new mode to the cascade. The wavelengths $\lambda$ in the cascade depend on both a geometrical length scale $L_{geo}$ given by the metric near the sheet’s edge, and the sheet thickness $t$: $\lambda \propto t^{0.3}{L}_{geo}^{0.7}$. This scaling implies vanishingly short wavelengths in the limit $t\to 0$. A possible geometrical origin of this behavior is discussed. Finally, we show that our measurement and analysis techniques are applicable to the study of some wavy patterns of leaves. These measurements reveal that the intrinsic geometry of a wavy leaf resembles that of the torn plastic sheets. This supports the possibility that some leaves form waves through a spontaneous wrinkling, rather than through an explicit three-dimensional construction.

We investigate three-dimensional Turing patterns in two-component reaction diffusion systems. The FitzHugh-Nagumo equation, the Brusselator, and the Gray-Scott model are solved numerically in three dimensions. Several interconnected structures of domains as well as lamellar, hexagonal, and spherical domains are obtained as stable motionless equilibrium patterns. The relative stability of these structures is studied analytically based on the reduction approximation. The relation with the microphase-separated structures in block copolymers is also discussed.

A self-consistent model of the ambipolar diffusion of electrons and ions in complex (dusty) plasmas accounting for the local electric fields, the dust grain charging process, and the interaction of the plasma particles with the dust grains and neutrals is presented. The dependence of the diffusion coefficient on the interaction of the electrons and ions with the dust grains as well as with the neutrals are investigated. It is shown that increase of the dust density leads to a reduction of the diffusion scale length, and this effect is enhanced at higher electron densities. The dependence of the diffusion scale length on the neutral gas pressure is found to be given by a power law, where the absolute value of the power exponent decreases with increase of the dust density. The electric field gradient and its effects are shown to be significant and should thus be taken into account in studies of complex plasmas with not very small dust densities. The possibility of observing localized coherent dissipative nonlinear dust ion-acoustic structures in an asymmetrically discharged double plasma is discussed.