We aim at assessing the validity limits of some simplifying hypotheses that, within a Riemmannian geometric framework, have provided an explanation of the origin of Hamiltonian chaos and have made it possible to develop a method of analytically computing the largest Lyapunov exponent of Hamiltonian systems with many degrees of freedom. Therefore, a numerical hypotheses testing has been performed for the Fermi-Pasta-Ulam $\beta$ model and for a chain of coupled rotators. These models, for which analytic computations of the largest Lyapunov exponents have been carried out in the mentioned Riemannian geometric framework, appear as paradigmatic examples to unveil the reason why the main hypothesis of quasi-isotropy of the mechanical manifolds sometimes breaks down. The breakdown is expected whenever the topology of the mechanical manifolds is nontrivial. This is an important step forward in view of developing a geometric theory of Hamiltonian chaos of general validity.

The two scaling relations in absorbing phase transitions, ${\nu }_{\parallel }=\beta ∕\theta$ and $z={\nu }_{\parallel }∕{\nu }_{\perp }$, are studied for a conserved lattice gas model. The critical indices calculated elaborately from the all-sample average density of active particles appear to satisfy both relations. However, the exponent ${\nu }_{\perp }$ calculated from the surviving samples does not appear to be consistent with the value in the thermodynamic limit. This is in contrast with earlier observations [M. Rossi et al., Phys. Rev. Lett. 85, 1803 (2000); S. Lübeck and P. C. Heger, Phys. Rev. E. 68, 056102 (2003)], in that the former scaling relation was claimed to be violated.

Water-soluble clusters made from $7\text{−}\mathrm{nm}$ inorganic nanoparticles have been investigated by small-angle neutron scattering. The internal structure factor of the clusters was derived and exhibited a universal behavior as evidenced by a correlation hole at intermediate wave vectors. Reverse Monte Carlo calculations were performed to adjust the data and provided an accurate description of the clusters in terms of interparticle distance and volume fraction. Additional parameters influencing the microstructure were also investigated, including the nature and thickness of the nanoparticle adlayer.

The global first passage time density of a network is the probability that a random walker released at a random site arrives at an absorbing trap at time $T$. We find simple expressions for the mean global first passage time $⟨T⟩$ for five fractals: the $d$-dimensional Sierpinski gasket, $T$ fractal, hierarchical percolation model, Mandelbrot-Given curve, and a deterministic tree. We also find an exact expression for the second moment $⟨T^2⟩$ and show that the variance of the first passage time, $\mathrm{Var}\left(T\right)$, scales with the number of nodes within the fractal $N$ such that $\mathrm{Var}\left(T\right)\sim N^{4∕\overline{d}}$, where $\overline{d}$ is the spectral dimension.

The crossover behavior of various models exhibiting phase transition to absorbing phase with parity conserving class has been investigated by numerical simulations and cluster mean-field method. In case of models exhibiting $Z_2$ symmetric absorbing phases (the cellular automaton version of the nonequilibrium kinetic Ising model (NEKIMCA) and a stochastic cellular automaton invented by Grassberger, Krause, and von der Twer [J. Phys. A 17, L105 (1984)]) the introduction of an external symmetry breaking field causes a crossover to kink parity conserving models characterized by dynamical scaling of the directed percolation (DP) and the crossover exponent: $1∕\varphi \simeq 0.53\left(2\right)$. In the case of a branching and annihilating random walk model with an even number of offspring (dual to NEKIMCA) the introduction of spontaneous particle decay destroys the parity conservation and results in a crossover to the DP class characterized by the crossover exponent: $1∕\varphi \simeq 0.205\left(5\right)$. The two different kinds of crossover operators cannot be mapped onto each other and the resulting models show a diversity within the DP universality class in one dimension. These subclasses differ in cluster scaling exponents.

We use simulations to investigate velocity fluctuations in dry granular flow. Our system is comprised of mono- and polydisperse sets of spherical grains falling down a vertical chute under the influence of gravity. We find three different classes of velocity distributions depending on factors such as the local density. The class of the velocity distribution depends on whether the grains are in a free-fall, fluid, or glassy state. The analytic form of the distributions match those that have been found by other authors in fairly diverse systems. Here, we have all three present in a single system in steady state. Power-law tails that match recent experiments are also found but in a transition area suggesting they may be an artifact of crossover from one class of velocity distribution to another. We find evidence that the transition from one class to another may correspond to a second order dynamical phase transition in the limit that the vertical flow speed goes to zero.

In a nonequilibrium system, a constant flux relation (CFR) expresses the fact that a constant flux of a conserved quantity exactly determines the scaling of the particular correlation function linked to the flux of that conserved quantity. This is true regardless of whether mean-field theory is applicable or not. We focus on cluster-cluster aggregation and discuss the consequences of mass conservation for the steady state of aggregation models with a monomer source in the diffusion-limited regime. We derive the CFR for the flux-carrying correlation function for binary aggregation with a general scale-invariant kernel and show that this exponent is unique. It is independent of both the dimension and of the details of the spatial transport mechanism, a property which is very atypical in the diffusion-limited regime. We then discuss in detail the “locality criterion” which must be satisfied in order for the CFR scaling to be realizable. Locality may be checked explicitly for the mean-field Smoluchowski equation. We show that if it is satisfied at the mean-field level, it remains true over some finite range as one perturbatively decreases the dimension of the system below the critical dimension, $d_c=2$, entering the fluctuation-dominated regime. We turn to numerical simulations to verify locality for a range of systems in one dimension which are, presumably, beyond the perturbative regime. Finally, we illustrate how the CFR scaling may break down as a result of a violation of locality or as a result of finite size effects and discuss the extent to which the results apply to higher order aggregation processes.

Molecular dynamics results are reported concerning cooperatively rearranging regions in simulated ${\mathrm{Ni}}_{0.5}{\mathrm{Zr}}_{0.5}$ melts down to $700\mathrm{K}$. Emphasis is laid on discriminating between clusters of mobile atoms (CMA) from low-frequency dynamics and the all-frequency case, where the former characterize fluctuations and relaxations on the scale of the late $\beta$ regime and $\alpha$ decay, while the latter include, in addition, reversible high-frequency vibrations. Separation of the low-frequency part of the dynamics is carried out by low-pass filtering, exploiting the separation of time scales below the critical temperature $T_c$ of the mode-coupling theory. With increasing temperature, the low-frequency and all-frequency dynamics merge in the range of $T_c$ when the separation of time scales disappears. In the low-frequency CMA, the average size of correlated clusters of connected atoms turns out to be nearly one order of magnitude larger than in the all-frequency CMA. The low-frequency CMA appear as local clusters propagating extremely slowly in space with characteristic time scale of $\mu \mathrm{s}$ at $700\mathrm{K}$, the scale of the onset of $\alpha$ decay.

Extensive Brownian dynamics simulations of conformational changes accompanying the overcharging of a dendrimer by an oppositely charged long linear polyelectrolyte (LPE) have been carried out. The simulated results have been compared with the predictions of the Nguen and Shklovskii correlation theory [Physica A 293, 324 (2001)] for impenetrable charged spherical macroion. Dendrimer overcharging is caused by the spatial correlations between the “excess” of the LPE charges adsorbed onto its surface. The simulated LPE-length dependence of the corresponding “correlation” energy is in agreement with the theoretical predictions. Maximum of the LPE adsorption occurs at some critical LPE length $N_{\mathrm{ch}}^c$, and the first order phase transition from completely coiled conformation to the conformation with released tails takes place. The phase transition is accompanied by the drastic increase in the relative fluctuations of the polyelectrolyte size. Upon increasing the linear-chain length above $N_{\mathrm{ch}}^c$, the one-long-tail conformation becomes energetically preferable; the exchange time between the long-tail conformation and the short-tail conformation is very large.

We show that bifurcations in chaotic scattering manifest themselves through the appearance of an infinitely fine-scale structure of singularities in the cross section. These “rainbow singularities” are created in a cascade, which is closely related to the bifurcation cascade undergone by the set of trapped orbits (the chaotic saddle). This cascade provides a signature in the differential cross section of the complex pattern of bifurcations of orbits underlying the transition to chaotic scattering. We show that there is a power law with a universal coefficient governing the sequence of births of rainbow singularities and we verify this prediction by numerical simulations.

The effect of particle-particle collision in decaying homogeneous and isotropic turbulence in the absence of gravity is studied using direct numerical simulation. The objective is to verify the relative importance of the particle-particle collision effect on particle distribution and other characteristics in gas-solid flows with transition from dilute to dense flow. A pseudospectral method coupling with a discrete element method (a hard-sphere approach) is used to simulate the gas turbulence and particle-particle collision dynamics. The effects of particle-particle collisions with different Stokes numbers and mass loadings are investigated. Suggestions for the consideration of the particle-particle collision effect in transition from dilute to dense gas-particle flow are provided. An empirical criterion for demarcation of dilute and dense flow is established. A selective effect of particle-particle collision on particle distribution with particle Stokes numbers higher and lower than unity is found and qualitatively demonstrated.

The equation governing quadratic and cubic transparent dispersion within the framework of the slowly varying envelope approximation is shown to admit an infinite-energy uniformly moving Airy wave packet solution, as well as a square-integrable accelerating Airy solution. Some insight is provided regarding the local acceleration dynamics in the latter case and comparisons are made with the “accelerating” beam solution introduced by Siviloglou and Christodoulides and experimentally demonstrated by Siviloglou, Broky, Dogariu, and Christodoulides recently. It is shown, in particular, that under certain parametrizations, the presence of cubic dispersion can increase the “depth of penetration” of a wave packet. In other words, a pulse can propagate for a larger range without sustaining significant dispersive distortion than in the presence of quadratic dispersion alone. Finally, imaging properties of accelerating airy wave packets are discussed.

The restrictions of analyticity, relativistic (Born) rigidity, and negligible $O\left(a\right)$ terms involved in the evaluation of the self-electromagnetic force on an extended charged sphere of radius $a$ are explicitly revealed and taken into account in order to obtain a classical equation of motion of the extended charge that is both causal and conserves momentum-energy. Because the power-series expansion used in the evaluation of the self-force becomes invalid during transition time intervals immediately following the application and termination of an otherwise analytic externally applied force, a transition force must be included during each of these two transition time intervals to remove the noncausal pre-acceleration and pre-deceleration from the solution to the equation of motion without the transition forces. Although the exact time dependence of each transition force is not known, the effect of each transition force on the solution to the equation of motion can be determined to within a single unknown constant, namely the change in velocity of the charge across the transition interval. For the extended charged sphere, the changes in velocity across the transition intervals can be chosen to maintain conservation of momentum-energy in the causal solutions to the equation of motion within the restrictions of relativistic rigidity and negligible $O\left(a\right)$ terms under which the equation of motion is derived. However, regardless of the values chosen for the changes in the velocity across the transition intervals, renormalization of the electrostatic mass to a finite value as the radius of the charge approaches zero introduces a violation of momentum-energy conservation into the causal solutions to the equation of motion of the point charge if the magnitude of the external force becomes too large. That is, the causal classical equation of motion of a point charge with renormalized mass experiences a high acceleration catastrophe.

We present an amplitude equation for sand ripples under oscillatory flow in a situation where the sand is moving in a narrow channel and the height profile is practically one dimensional. The equation has the form $h_t=-ϵ(h-\overline{h})+\mathbf{(}{\left(h_x\right)}^2-1\mathbf{)}{h}_{xx}-h_{xxxx}+\delta {\mathbf{\left(}{\left(h_x\right)}^2\mathbf{\right)}}_{xx}$ which, due to the first term, is neither completely local (it has long-range coupling through the average height $\overline{h}$) nor has local sand conservation. We argue that this is reasonable and show that the equation compares well with experimental observations in narrow channels. We focus in particular on the so-called doubling transition, a secondary instability caused by the sudden decrease in the amplitude of the water motion, leading to the appearance of a new ripple in each trough. This transition is well reproduced for sufficiently large $\delta$ (asymmetry between trough and crest). We finally present surprising experimental results showing that long-range coupling is indeed seen in the initial details of the doubling transition, where in fact two small ripples are initially formed, followed by global symmetry breaking removing one of them.

We study the dynamics of the isotropic-nematic phase transition caused by an applied electric field at the time scales of dielectric relaxation. In the classic Landau-Khalatnikov theory of the phase transition dynamics, the nematic (nonpolar) order parameter is an instantaneous function of the applied field. We demonstrate that, when the field is changing faster than the time of dielectric relaxation, the induced polar order dynamics influences the dynamics of the nonpolar order parameter. We develop a model based on the Langevin equation to describe the simultaneous dynamics of both polar and nonpolar order parameters; the model is supported by experiment.

Hydrogen-terminated intrinsic diamond is a most unusual insulator since in contact with air or electrolyte it develops a hole accumulation layer just below the surface. When immersed in electrolyte the hole concentration responds to ionic charges and potentials and this response can be monitored by measuring the conductance of the hole accumulation layer without interference form the bulk conductivity. This feature has been widely used for chemical sensors in the form of solution-gate field effect transistors (SGFET). Here we analyze the charge and the potential profiles in the diamond and the electrolyte, as well as the static differential capacitance of the diamond electrode under controlled potential conditions. From this analysis we derive expressions for the transfer characteristics of diamond-based SGFET’s that faithfully describe experimental data presented here as well. This holds in particular for the threshold region of the transfer characteristics that can only be modeled if the unusual semiconducting properties of the diamond electrode are taken into account properly. From fits to our data we derive (among other things) a value of $\chi =-0.50\pm 0.02\mathrm{eV}$ for the electron affinity of the hydrogen-terminated diamond surface in contact with aqueous electrolyte.

One of the most attractive features of diamond is its robust $p$-type surface conductivity that develops spontaneously under atmospheric conditions on hydrogen-terminated samples. An electrochemical charge transfer between diamond and an air-borne redox couple has been suggested to be responsible for the spontaneous appearance of surface-near holes. We present direct proof for the redox activity of the diamond surface by measuring $p\mathrm{H}$-dependent open circuit potentials and quasistatic polarization curves for hydrogen-terminated and partially oxidized diamond electrodes. Under open circuit conditions we find in fact a mixed (or corrosion) potential that is consistent with the simultaneous equilibriation of the electrode versus both the hydrogen-hydronium and the oxygen-hydroxyl redox couple. Our data show extremely long-time constants for establishing the redox equilibrium and very low exchange current densities making the identification and characterization of the redox process a demanding experimental task.

Monte Carlo simulations of the Asakura-Oosawa model for colloid-polymer mixtures confined between two parallel repulsive structureless walls are presented and analyzed in the light of current theories on capillary condensation and interface localization transitions. Choosing a polymer-to-colloid size ratio of $q=0.8$ and studying ultrathin films in the range of $D=3$ to $D=10$ colloid diameters thickness, grand canonical Monte Carlo methods are used; phase transitions are analyzed via finite size scaling, as in previous work on bulk systems and under confinement between identical types of walls. Unlike the latter work, inequivalent walls are used here: While the left wall has a hard-core repulsion for both polymers and colloids, at the right-hand wall an additional square-well repulsion of variable strength acting only on the colloids is present. We study how the phase separation into colloid-rich and colloid-poor phases occurring already in the bulk is modified by such a confinement. When the asymmetry of the wall-colloid interaction increases, the character of the transition smoothly changes from capillary condensation type to interface localization type. For very thin films (i.e., for $D=3$) and a suitable choice of the wall-colloid interactions, evidence is found that the critical behavior falls in the universality class of the two-dimensional Ising model. Otherwise, we observe crossover scaling between different universality classes (namely, the crossover from the three-dimensional to the two-dimensional Ising model universality class). The colloid and polymer density profiles across the film in the various phases are discussed, as well as the correlation of interfacial fluctuations in the direction parallel to the confining walls. The broadening of the interface between the coexisting colloid-rich and polymer-rich phases (located parallel to the confining walls) is understood in terms of capillary wave fluctuations. The experimental observability of all these phenomena is briefly discussed.

In this paper different arguments are put forward to explain why two-particle diffusion is depleted in the direction of stratification of a stably stratified turbulence. Kinematic simulations (KSs) which reproduce that depletion are used to shed light on the responsible mechanisms. The local horizontal divergence is studied and comparisons are made with two-dimensional kinematic simulation. The probability density function of the horizontal divergence of the velocity field is not a Dirac distribution in the presence of stratification but a Gaussian and this Gaussian does not depend on the Froude number. The number of stagnation points in the KS of three-dimensional strongly stratified turbulence is found virtually identical to what it is in KS of three-dimensional isotropic turbulence. However, the root mean square horizontal and vertical stagnation point velocities of the stratified turbulence are both larger than their counterparts in isotropic turbulence that latter getting progressively smaller as the Reynolds number increases. Therefore, the strong stratification destroys the persistence of the stagnation points. The main reason for the depletion, however, seems to have to be sought in the effect of stratification on the strain rate tensor. The stratification does lead to a depletion of the average square strain rate tensor, as well as of all average square strain rate eigenvalues. We conclude that it is these effects of stratification on the strain rate tensor that explain the depletion of the horizontal turbulent pair diffusion.

We revisit the problem of the three-soliton collisions in the weakly perturbed sine-Gordon equation and develop an effective three-particle model allowing us to explain many interesting features observed in numerical simulations of the soliton collisions. In particular, we explain why collisions between two kinks and one antikink are observed to be practically elastic or strongly inelastic depending on relative initial positions of the kinks. The fact that the three-soliton collisions become more elastic with an increase in the collision velocity also becomes clear in the framework of the three-particle model. The three-particle model does not involve internal modes of the kinks, but it gives a qualitative description to all the effects observed in the three-soliton collisions, including the fractal scattering and the existence of short-lived three-soliton bound states. The radiationless energy exchange between the colliding solitons in weakly perturbed integrable systems takes place in the vicinity of the separatrix multi-soliton solutions of the corresponding integrable equations, where even small perturbations can result in a considerable change in the collision outcome. This conclusion is illustrated through the use of the reduced three-particle model.