The average avalanche size in the Watts model of threshold dynamics on random networks of arbitrary degree distribution is determined analytically. Existence criteria for global cascades are shown to depend sensitively on the size of the initial seed disturbance. The dependence of cascade size upon the mean degree $z$ of the network is known to exhibit several transitions—these are typically continuous at low $z$ and discontinuous at high $z$; here it is demonstrated that the low-$z$ transition may in fact be discontinuous in certain parameter regimes. Connections between these results and the zero-temperature random-field Ising model on random graphs are discussed.

In recent experiments the transverse normalized rms emittance of laser-accelerated MeV ion beams was found to be $<0.002\mathrm{mm}\mathrm{mrad}$, which is at least 100 times smaller than the emittance of thermal ion sources used in accelerators [T. E. Cowan et al., Phys. Rev. Lett. 92, 204801 (2004)]. We investigate the origin for the low emittance of laser-accelerated proton beams by studying several candidates for emittance-growth mechanisms. As our main tools, we use analytical models and one- and two-dimensional particle-in-cell simulations that have been modified to include binary collisions between particles. We find that the dominant source of emittance is filamentation of the laser-generated hot electron jets that drive the ion acceleration. Cold electron-ion collisions that occur before ions are accelerated contribute less than ten percent of the final emittance. Our results are in qualitative agreement with the experiment, for which we present a refined analysis relating emittance to temperature, a better representative of the fundamental beam physics.

Solitary waves bifurcated from edges of Bloch bands in two-dimensional periodic media are determined both analytically and numerically in the context of a two-dimensional nonlinear Schrödinger equation with a periodic potential. Using multiscale perturbation methods, the envelope equations of solitary waves near Bloch bands are analytically derived. These envelope equations reveal that solitary waves can bifurcate from edges of Bloch bands under either focusing or defocusing nonlinearity, depending on the signs of the second-order dispersion coefficients at the edge points. Interestingly, at edge points with two linearly independent Bloch modes, the envelope equations lead to a host of solitary wave structures, including reduced-symmetry solitons, dipole-array solitons, vortex-cell solitons, and so on—many of which have not been reported before to our knowledge. It is also shown analytically that the centers of envelope solutions can be positioned at only four possible locations at or between potential peaks. Numerically, families of these solitary waves are directly computed both near and far away from the band edges. Near the band edges, the numerical solutions spread over many lattice sites, and they fully agree with the analytical solutions obtained from the envelope equations. Far away from the band edges, solitary waves are strongly localized, with intensity and phase profiles characteristic of individual families.

Elegant integration schemes of second and fourth order for simulations of rigid-body systems are presented which treat translational and rotational motion on the same footing. This is made possible by a recent implementation of the exact solution of free rigid-body motion. The two schemes are time reversible and symplectic, and exactly respect conservation principles for both the total linear and the angular momentum vectors. Simulations of simple test systems show that the second-order scheme is stable and conserves all constants of the motion to high precision. Furthermore, the schemes are demonstrated to be more accurate and efficient than existing methods, except for high densities, in which case the second-order scheme performs at least as well, showing their general applicability. Finally, it is demonstrated that the fourth-order scheme is more efficient than the second-order scheme provided the time step is smaller than a system-dependent threshold value.

A trapping mechanism is observed and proposed as the origin of the anomalous behavior recently discovered in transport properties of overdamped ratchets subject to an external oscillatory drive in the presence of quenched noise. In particular, this mechanism is shown to appear whenever the quenched disorder strength is greater than a threshold value. The minimum disorder strength required for the existence of traps is determined by studying the trap structure in a disorder configuration space. An approximation to the trapping probability density function in a disordered region of finite length included in an otherwise perfect ratchet lattice is obtained. The mean velocity of the particles and the diffusion coefficient are found to have a nonmonotonic dependence on the quenched noise strength due to the presence of the traps.

We investigate tracer advection in fractal media with finite correlation length. The process is superdiffusive at early times and transforms to a classical diffusion regime later. At large time the spatial tail of the concentration has a two-stage structure with a long-distant part corresponding to superdiffusive regime.

We address the problem of recollisions in cooling granular gases. To this aim, we dynamically construct the interaction network in a granular gas, using the sequence of collisions collected in an event driven simulation of inelastic hard disks from time $0$ until time $t$. The network is decomposed into its $k$-core structure: particles in a core of index $k$ have collided at least $k$ times with other particles in the same core. The difference between cores $k+1$ and $k$ is the so-called $k$-shell, and the set of all shells is a complete and nonoverlapping decomposition of the system. Because of energy dissipation, the gas cools down: its initial spatially homogeneous dynamics, characterized by the Haff law, i.e., a $t^{-2}$ energy decay, is unstable toward a strongly inhomogeneous phase with clusters and vortices, where energy decays as $t^{-1}$. The clear transition between those two phases appears in the evolution of the $k$-shells structure in the collision network. In the homogeneous state the $k$-shell structure evolves as in a growing network with a fixed number of vertices and randomly added links: the shell distribution is strongly peaked around the most populated shell, which has an index $k_{\max }\sim 0.9⟨d⟩$ with $⟨d⟩$ the average number of collisions experienced by a particle. During the final nonhomogeneous state a growing fraction of collisions is concentrated in small, almost closed, communities of particles: $k_{\max }$ is no more linear in $⟨d⟩$ and the distribution of shells becomes extremely large developing a power-law tail $\sim k^{-3}$ for high shell indexes. We conclude proposing a simple algorithm to build a correlated random network that reproduces, with few essential ingredients, the whole observed phenomenology, including the $t^{-1}$ energy decay. It consists of two kinds of collisions (links): single random collisions with any other particle and long chains of recollisions with only previously encountered particles. The algorithm disregards the exact spatial arrangement of clusters, suggesting that the observed stringlike structures are not essential to determine the statistics of recollisions and the energy decay.

We study the spreading characteristics of a reactive-wetting system of mercury (Hg) droplets on silver (Ag) films in room temperature. This is done using our recently developed method for reconstructing the dynamical three-dimensional shape of spreading droplets from two-dimensional microscope images [A. Be’er and Y. Lereah, J. Microsc. 208, 148 (2002)]. We study the time evolution of the droplet radius and its contact angle, and find that the spreading process consists of two stages: (i) the “bulk propagation” regime, controlled by chemical reaction on the surface, and (ii) the “fast-flow” regime, which occurs within the metal film as well as on the surface and consists of both reactive and diffusive propagation. We show that the transition time between the two main time regimes depends solely on the thickness of the Ag film. We also discuss the chemical structure of the intermetallic compound formed in this process.

Aperiodic spike sequences, characterized by an almost constant spike amplitude and randomly distributed interspike intervals, are studied experimentally in an electrochemical cell. In the first set of experiments, these aperiodic spike trains are converted to regular spike sequences using periodic forcing. Subsequently, synchronization of two such irregular spike time series is achieved for an appropriate bidirectional coupling. Finally, reproducibility of these irregular spike profiles is evoked by virtue of an externally superimposed stochastic stimulus.

We consider the rewiring of a bipartite graph using a mixture of random and preferential attachment. The full mean-field equations for the degree distribution and its generating function are given. The exact solution of these equations for all finite parameter values at any time is found in terms of standard functions. It is demonstrated that these solutions are an excellent fit to numerical simulations of the model. We discuss the relationship between our model and several others in the literature, including examples of urn, backgammon, and balls-in-boxes models, the Watts and Strogatz rewiring problem, and some models of zero range processes. Our model is also equivalent to those used in various applications including cultural transmission, family name and gene frequencies, glasses, and wealth distributions. Finally some Voter models and an example of a minority game also show features described by our model.

The generalized synchronization of fractional order systems is investigated, including the synchronization between two different fractional order systems with the same order, the synchronization between two different fractional order systems with no orders the same, and the synchronization between a classical system and its corresponding fractional order system with mismatched parameters. The mechanism for the occurrence of generalized synchronization of fractional order systems is clarified, the necessary and sufficient conditions are given, and several methods to detect generalized synchronization are discussed. The relationship between generalized synchronization and the equivalence of fractional order systems is also considered.

We describe a convective instability of a stratified ferrofluid subject to a magnetic field. The ferrofluid is described as binary mixture featuring two important dimensionless numbers the barometric one $B$ and the magnetic barometric one $B_m$. At the field strength, where the latter exceeds the former, instability sets in. We discuss the growth rate of linear perturbations and the amplitude of nonlinear stationary convection rolls. The latter is governed by the balance of concentration advection and sedimentation.

For heterogeneous media with piecewise constant complex permittivity on regular domains, we show that the dyadic Green’s function has the same singular part as the corresponding free space dyadic Green’s function on every domain of constant permittivity. We give two important applications of this property, namely the distorted-wave Born approximation for composite media and the calculation of the single photon decay rate in spontaneous emission.

Large-scale collective motion emerging in a monolayer of vertically vibrated elongated particles is studied. The motion is characterized by recurring swirls, with the characteristic scale exceeding several times the size of an individual particle. Our experiments identified a small horizontal component of the oscillatory acceleration of the vibrating plate in combination with orientation-dependent bottom friction (with respect to horizontal acceleration) as a source for the swirl formation. We developed a continuum model operating with the velocity field and local alignment tensor, which is in qualitative agreement with the experiment.

We study the generalized diffusion-limited aggregates, grown from two proximal nucleation seeds placed at distance $d$ lattice units and investigate the probability $p\left(d\right)$ that these aggregates get connected. We vary the sticking probability to get a range of aggregate geometry from fractal to compact one. For fractal aggregates, $p\left(d\right)$ decays rapidly with $d$, while for compact ones, the decay is so slow that $p\left(d\right)\approx 1$ for all practical distances. We experimentally demonstrate similar behavior for viscous fingering patterns with two injection points and electrochemical deposits grown on two cathodes. Our observations along with previous results on competitive growth suggest a common underlying principle.

Curved textures of ferroelectric smectic-$C^{*}$ liquid crystals produce space charge when they involve divergence of the spontaneous polarization field. Impurity ions can partially screen this space charge, reducing long-range interactions to local ones. Through studies of the textures of islands on very thin free-standing smectic films, we see evidence of this effect, in which materials with a large spontaneous polarization have static structures described by a large effective bend elastic constant. To address this issue, we calculated the electrostatic free energy of a free-standing film of ferroelectric liquid crystal, showing how the screened Coulomb interaction contributes a term to the effective bend elastic constant, in the static long-wavelength limit. We report experiments which support the main features of this model.

Many viruses self-assemble from a large number of identical capsid proteins with long flexible N-terminal tails and single-stranded (ss) RNA. We study the role of the strong Coulomb interaction of positive N-terminal tails with ssRNA in the kinetics of in vitro virus self-assembly. Capsid proteins stick to the unassembled chain of ssRNA (which we call an “antenna”) and slide on it toward the assembly site. We show that at excess of capsid proteins such one-dimensional diffusion accelerates self-assembly more than ten times. On the other hand at excess of ssRNA, the antenna slows self-assembly down. Several experiments are proposed to verify the role of the ssRNA antenna.

The linear stage of stability is studied for a two-component fractional reaction-diffusion system. It is shown that, with a certain value of the fractional derivative index, a different type of instability occurs. The linear stability analysis shows that the system becomes unstable toward perturbations of finite wave number. As a result, inhomogeneous oscillations with this wave number become unstable and lead to nonlinear oscillations which result in spatial oscillatory structure formation. A computer simulation of a Bonhoeffer–van der Pol type of reaction-diffusion system with fractional time derivatives is performed.

The effective permeability of composites embedded with self-similar fractal-like tree networks is studied. The effective permeability tensor of the composites is derived and is found to be related to microstructures of the networks. The present results show that the larger the ratio of successive branching channel diameters, the higher the effective permeability; the higher the relative surface porosity of the tree networks and matrix, the higher the effective permeability; the denser the tree networks, the lower the effective permeability; the longer the branches, the lower the effective permeability. It is found that the dimensionless effective permeability of composites scales as the diameter exponent by $K_{e,y}^{+}\sim {\beta }^{\alpha m}$, with $\alpha \approx 4$. It is also found that when $m>1$ and $n{\beta }^4∕\gamma <1$, the effective permeability $K_{e,y}^{+}$ scales as the iteration $m$, $\ln K_{e,y}^{+}∕m\sim \ln (n{\beta }^4∕\gamma )$. The fractal-like tree networks can significantly increase the effective permeability of the composites compared to the traditional parallel nets under properly chosen structural parameters.