We obtain compact, exact, analytical expressions for the first-passage-time distribution for a particle diffusing on a planar wedge for special values of the wedge angle. Specifically, we calculate the first-passage-time distribution for the diffusing particle through a planar wedge of angle $\pi ∕n$, where $n$ is an integer. For the cases $n=2$ and $n$ odd, we provide an exact closed-form expression to the first-passage-time distribution while for the remaining cases, we provide it in integral form and evaluate numerically using quadratures. We then show that our results are in good agreement with Markovian simulations in the continuum limit.

We solve analytically the multiple-scattering equations for two-dimensional photonic crystals in the long-wavelength limit. Different approximations of the electric and magnetic susceptibilities are presented from a unified pseudopotential point of view. The nature of the so-called plasmon-polariton bands is clarified. Its frequency as a function of the wire radius is discussed. The corresponding tunable “magnetic surface plasmon” band is pointed out.

High-intensity laser-matter interaction is an efficient method for high-current relativistic electron beam production. At current densities exceeding a several $\mathrm{kA}\mu {\mathrm{m}}^{-2}$, the beam propagation is maintained by an almost complete current neutralization by the target electrons. In such a geometry of two oppositely directed flows, beam instabilities can develop, depending on the target and the beam parameters. The present paper proposes an analytical description of the filamentation instability of an electron beam propagating through an insulator target. It is shown that the collisionless and resistive instabilities enter into competition with the ionization instability. This latter process is dominant in insulator targets where the field ionization by the fast beam provides free electrons for the neutralization current.

The mapping of the nonlinear Schrödinger equation with a random potential on the Fokker-Planck equation is used to calculate the localization length of its stationary states. The asymptotic growth rates of the moments of the wave function and its derivative for the linear Schrödinger equation in a random potential are computed analytically, and resummation is used to obtain the corresponding growth rate for the nonlinear Schrödinger equation and the localization length of the stationary states.

We consider two models with disorder-dominated critical points and study the distribution of clusters that are confined in strips and touch one or both boundaries. For the classical random bond Potts model in the large-$q$ limit, we study optimal Fortuin-Kasteleyn clusters using a combinatorial optimization algorithm. For the random transverse-field Ising chain, clusters are defined and calculated through the strong-disorder renormalization group method. The numerically calculated density profiles close to the boundaries are shown to follow scaling predictions. For the random bond Potts model, we have obtained accurate numerical estimates for the critical exponents and demonstrated that the density profiles are well described by conformal formulas.

The behavior of a heavy tagged intruder immersed in a bath of particles evolving under ballistic annihilation dynamics is investigated. The Fokker-Planck equation for this system is derived and the peculiarities of the corresponding diffusive behavior are worked out. In the long time limit, the intruder velocity distribution function approaches a Gaussian form, but with a different temperature from its bath counterpart. As a consequence of the continuous decay of particles in the bath, the mean-squared displacement increases exponentially in the collision per particle time scale. Analytical results are finally successfully tested against Monte Carlo numerical simulations.

We study the critical dynamics of the generalized Smoluchowski-Poisson system (for self-gravitating Langevin particles) or generalized Keller-Segel model (for the chemotaxis of bacterial populations). These models [P. H. Chavanis and C. Sire, Phys. Rev. E 69, 016116 (2004)] are based on generalized stochastic processes leading to the Tsallis statistics. The equilibrium states correspond to polytropic configurations with index $n$ similar to polytropic stars in astrophysics. At the critical index $n_3=d∕(d-2)$ (where $d⩾2$ is the dimension of space), there exists a critical temperature ${\Theta }_c$ (for a given mass) or a critical mass $M_c$ (for a given temperature). For $\Theta >{\Theta }_c$ or $M<M_c$ the system tends to an incomplete polytrope confined by the box (in a bounded domain) or evaporates (in an unbounded domain). For $\Theta <{\Theta }_c$ or $M>M_c$ the system collapses and forms, in a finite time, a Dirac peak containing a finite fraction $M_c$ of the total mass surrounded by a halo. We study these regimes numerically and, when possible, analytically by looking for self-similar or pseudo-self-similar solutions. This study extends the critical dynamics of the ordinary Smoluchowski-Poisson system and Keller-Segel model in $d=2$ corresponding to isothermal configurations with $n_3\to +\infty$. We also stress the analogy between the limiting mass of white dwarf stars (Chandrasekhar’s limit) and the critical mass of bacterial populations in the generalized Keller-Segel model of chemotaxis.

We report measurements of the frequency-dependent shear moduli of aging colloidal systems that evolve from a purely low-viscosity liquid to a predominantly elastic glass or gel. Using microrheology, we measure the local complex shear modulus $G^{*}\left(\omega \right)$ over a very wide range of frequencies (from $1\mathrm{Hz}\text{to}100\mathrm{kHz}$). The combined use of one- and two-particle microrheology allows us to differentiate between colloidal glasses and gels—the glass is homogenous, whereas the colloidal gel shows a considerable degree of heterogeneity on length scales larger than $0.5\mu \mathrm{m}$. Despite this characteristic difference, both systems exhibit similar rheological behaviors which evolve in time with aging, showing a crossover from a single-power-law frequency dependence of the viscoelastic modulus to a sum of two power laws. The crossover occurs at a time $t_0$, which defines a mechanical transition point. We found that the data acquired during the aging of different samples can be collapsed onto a single master curve by scaling the aging time with $t_0$. This raises questions about the prior interpretation of two power laws in terms of a superposition of an elastic network embedded in a viscoelastic background.

We study random sequential adsorption of polydisperse mixtures of extended objects both on a triangular and on a square lattice. The depositing objects are formed by self-avoiding random walks on two-dimensional lattices. Numerical simulations were performed to determine the influence of the number of mixture components and length of the shapes making the mixture on the kinetics of the deposition process. We find that the late stage deposition kinetics follows an exponential law $\theta \left(t\right)\sim {\theta }_{\text{jam}}-A\exp (-t∕\sigma )$ not only for the whole mixture, but also for the individual components. We discuss in detail how the quantities such as jamming coverage ${\theta }_{\text{jam}}$ and the relaxation time $\sigma$ depend on the mixture composition. Our results suggest that the order of symmetry axis of the shape may exert a decisive influence on adsorption kinetics of each mixture component.

We study quantitative features of complex repetitive DNA in several genomes by studying sequences that are sufficiently long that they are unlikely to have repeated by chance. For each genome we study, we determine the number of identical copies, the “duplication count,” of each sequence of length 40, that is of each “40-mer.” We say a 40-mer is “repeated” if its duplication count is at least 2. We focus mainly on “complex” 40-mers, those without short internal repetitions. We find that we can classify most of the complex repeated 40-mers into two categories: one category has its copies clustered closely together on one chromosome, the other has its copies distributed widely across multiple chromosomes. For each genome and each of the categories above, we compute $N\left(c\right)$, the number of 40-mers that have duplication count $c$, for each integer $c$. In each case, we observe a power-law-like decay in $N\left(c\right)$ as $c$ increases from 3 to 50 or higher. In particular, we find that $N\left(c\right)$ decays much more slowly than would be predicted by evolutionary models where each 40-mer is equally likely to be duplicated. We also analyze an evolutionary model that does reflect the slow decay of $N\left(c\right)$.

We introduce the concept of entropy rate to characterize a diffusion process on a complex network. The entropy rate represents the minimal amount of information necessary to describe the diffusion on the network, and is a quantity extremely sensitive to the network topology and dynamics. By opportunely tuning the kind of diffusion, the entropy rate allows one to extract different properties of the network structure. Moreover, entropy maximization indicates how to design optimal diffusion processes, providing a new theoretical tool with applications to social, technological, and communication systems.

We propose a set of methods for the estimation of the parameters of time-delay systems with a linear filter and nonlinear delay feedback performing periodic oscillations. The methods are based on an analysis of the system response to regular external perturbations and are valid only for systems whose dynamics can be perturbed. The efficiency of the methods is illustrated using both numerical and experimental data.

In multistable regimes, noise can create “multistate hopping intermittency,” i.e., intermittent transitions among coexisting stable attractors. We demonstrate that a small periodic perturbation can significantly control such hopping intermittency. By “control” we imply a qualitative change in the probability distribution of occupation in the phase space around the stable attractors. In other words, if the uncontrolled system exhibits a preference to stay around a given attractor (say “$A$”) in comparison to another attractor (say “$B$”), the control perturbation creates a contrasting scenario so that attractor $B$ is most frequently visited and consequently, the occupation probability becomes maximum around $B$ instead of $A$. The control perturbation works in the following way: It destroys attractor $A$ by boundary crisis while attractor $B$ remains stable. As a result, even if the system is pushed by noise into the erstwhile basin of attractor $A$, the system does not remain there for long and therefore stays longer around attractor $B$. Significantly, such a change in the intermittent scenario can be obtained by a small-amplitude and slow-periodic perturbation. The control is theoretically demonstrated with two standard models, namely, Lorenz equations (for autonomous systems), and the periodically driven, damped Toda oscillator (for nonautonomous systems). Recent experiments with a cavity-loss modulated $\mathrm{C}{\mathrm{O}}_2$ laser and an analog circuit of Lorenz equations have validated our theoretical demonstrations excellently.

When a sample of fluid of finite size is displaced in a porous medium by another miscible fluid, viscous fingering may occur when the two fluids have different viscosities. Depending whether the sample is more or less viscous than the carrier fluid, the log-mobility ratio $R$ [defined as $R=\ln ({\mu }_2∕{\mu }_1)$ where ${\mu }_2$ and ${\mu }_1$ are the viscosities of the sample and of the carrier] is respectively positive or negative. In the case of a linear displacement of a finite slice of fluid, $R>0$ leads to fingering of the rear interface of the sample where the less viscous carrier invades the more viscous sample. If $R<0$, it is on the contrary the frontal interface of the sample that develops fingers where the less viscous sample displaces the more viscous bulk solution. We investigate here numerically the differences in fingering dynamics between the positive and negative log-mobility ratio cases leading to the growth of fingers against or along the direction of the flow, respectively. To do so, we integrate Darcy’s law coupled to a convection-diffusion equation for the concentration of a solute ruling the viscosity of the finite-size sample. The statistical moments of the solute’s concentration distribution are analyzed as a function of dimensionless parameters of the problem such as the length of the slice $l$, the log-mobility ratio $R$, and the ratio between transverse and axial dispersion coefficients $ϵ$. We find that, on average, the mixing zones and the width of the sample broadening due to fingering are larger for negative $R$ than for positive $R$. This is due to the fact that fingers travel quicker in the flow direction than against the flow. Relevance of our results are discussed for interpretation of experimental results obtained in chromatographic separation and for understanding conditions of enhanced spreading of contaminants in aquifers.

It is shown that for super intense laser pulses propagating in a hot plasma, the action of the radiation reaction force (appropriately incorporated into the equations of motion) causes strong bulk plasma motion with the kinetic energy raised even to relativistic values; the increase in bulk energy is accompanied by a corresponding cooling (intense cooling) of the plasma. The effects are demonstrated through explicit analytical calculations.

The notion of duality between the hydrodynamic and kinetic (ghost) variables of lattice kinetic formulations of the Boltzmann equation is introduced. It is suggested that this notion can serve as a guideline in the design of matrix versions of the lattice Boltzmann equation in a physically transparent and computationally efficient way.

Recently Reis and Phillips [Phys. Rev. E 77, 026702 (2008)] proposed a perturbative method to solve the dispersion equation derived from the linearized lattice Boltzmann equation. We will demonstrate that the method proposed by Reis and Phillips is a reinvention of an existing method. We would also like to refute a number of claims made by Reis and Phillips.

In this reply to the comment by Lallemand and Luo, we defend our assertion that the alternative approach for the solution of the dispersion relation for a generalized lattice Boltzmann dispersion equation [T. Reis and T. N. Phillips, Phys. Rev. E 77, 026702 (2008)] is mathematically transparent, elegant, and easily justified. Furthermore, the rigorous perturbation analysis used by Reis and Phillips does not require the reciprocals of the relaxation parameters to be small.

We discuss elastic effects due to lattice strain which are a new key ingredient in the theory of dendritic growth for solid-solid transformations. Both thermal and elastic fields are eliminated by Green’s function techniques, and a closed nonlinear integro-differential equation for the evolution of the interface is derived. We find dendritic patterns even without the anisotropy of the surface energy required by classical dendritic growth theory. In this sense, elastic effects serve as a new selection mechanism.