The effects of the initial height on the temporal persistence probability of steady-state height fluctuations in up-down symmetric linear models of surface growth are investigated. We study the (1+1)-dimensional Family model and the (1+1)- and (2+1)-dimensional larger curvature (LC) model. Both the Family and LC models have up-down symmetry, so the positive and negative persistence probabilities in the steady state, averaged over all values of the initial height $h_0$, are equal to each other. However, these two probabilities are not equal if one considers a fixed nonzero value of $h_0$. Plots of the positive persistence probability for negative initial height versus time exhibit power-law behavior if the magnitude of the initial height is larger than the interface width at saturation. By symmetry, the negative persistence probability for positive initial height also exhibits the same behavior. The persistence exponent that describes this power-law decay decreases as the magnitude of the initial height is increased. The dependence of the persistence probability on the initial height, the system size, and the discrete sampling time is found to exhibit scaling behavior.

We report a molecular dynamics simulation demonstrating that the smectic-$B$ crystalline phase (Cry-$B$), commonly observed in mesogenic systems of anisotropic molecules, can be formed by a system of identical particles interacting via a spherically symmetric potential. The Cry-$B$ phase forms as a result of a first-order transition from an isotropic liquid phase upon isochoric cooling at appropriate number density. Its structure, determined by the design of the pair potential, corresponds to the Cry-$B$ structure formed by elongated particles with the aspect ratio $1.8$. The diffraction pattern and the real-space structure inspection demonstrate dominance of the ABC-type of axial layer stacking. This result opens a general possibility of producing smectic phases using isotropic interparticle interaction both in simulations and in colloidal systems.

In relation to cardiac arrhythmias, heterogeneity of cardiac tissue is one of the most important factors underlying the onset of spiral waves and determining their type. In this paper, we numerically model heterogeneity of realistic size and value and study formation and dynamics of spiral waves around such heterogeneity. We find that the only sustained pattern obtained is a single spiral wave anchored around the heterogeneity. Dynamics of an anchored spiral wave depend on the extent of heterogeneity, and for certain heterogeneity size, we find abrupt regional increase in the period of excitation occurring as a bifurcation. We study factors determining spatial distribution of excitation periods of anchored spiral waves and discuss consequences of such dynamics for cardiac arrhythmias and possibilities for experimental testings of our predictions.

1 kHz, 2 mJ, 45 fs, 800 nm laser pulses were fired into a laboratory diffusion cloud chamber through a subsaturated zone (relative humidity ∼73$\%$, $T$ ∼ 4.3 °C). After 60 min of laser irradiation, an oval-shaped snow pile was observed right below the filament center and weighed ∼12.0 mg. The air current velocity at the edge of the vortices was estimated to be ∼16.5 cm/s. Scattering scenes recorded from the side show that filament-induced turbulence were formed inside the cloud chamber with two vortices below the filament. Two-dimensional simulations of the air flow motion in two cross sections of the cloud chamber confirm that the turbulent vortices exist below the filament. Based upon this simulation, we deduce that the vortices indeed have a three-dimensional elliptical shape. Hence, we propose that inside vortices where the humidity was supersaturated or saturated the condensation nuclei, namely, HNO${}_3$, N${}_2$${}^{+}$, O${}_2$${}^{+}$ and other aerosols and impurities, were activated and grew in size. Large-sized particles would eventually be spun out along the fast moving direction towards the cold plate and formed an oval-shaped snow pile at the end.

Zipf's law is the major regularity of statistical linguistics that has served as a prototype for rank-frequency relations and scaling laws in natural sciences. Here we show that Zipf's law—together with its applicability for a single text and its generalizations to high and low frequencies including hapax legomena—can be derived from assuming that the words are drawn into the text with random probabilities. Their a priori density relates, via the Bayesian statistics, to the mental lexicon of the author who produced the text.

We study a susceptible-vaccinated–infected–recovered (SVIR) epidemic-spreading model with diversity of infection rate of the individuals. By means of analytical arguments as well as extensive computer simulations, we demonstrate that the heterogeneity in infection rate can either impede or accelerate the epidemic spreading, which depends on the amount of vaccinated individuals introduced in the population as well as the contact pattern among the individuals. Remarkably, as long as the individuals with different capability of acquiring the disease interact with unequal frequency, there always exist a cross point for the fraction of vaccinated, below which the diversity of infection rate hinders the epidemic spreading and above which expedites it. The overall results are robust to the SVIR dynamics defined on different population models; the possible applications of the results are discussed.

We present optical rogue wave solutions for a generalized nonlinear Schrodinger equation by using similarity transformation. We have predicted the propagation of rogue waves through a nonlinear optical fiber for three cases: (i) dispersion increasing (decreasing) fiber, (ii) periodic dispersion parameter, and (iii) hyperbolic dispersion parameter. We found that the rogue waves and their interactions can be tuned by properly choosing the parameters. We expect that our results can be used to realize improved signal transmission through optical rogue waves.

We introduce one- and two-dimensional (1D and 2D) models of parity-time ($\mathcal{PT}$)-symmetric couplers with the mutually balanced linear gain and loss applied to the two cores and cubic-quintic (CQ) nonlinearity acting in each one. The 2D and 1D models may be realized in dual-core optical wave guides in the spatiotemporal and spatial domains, respectively. Stationary solutions for $\mathcal{PT}$-symmetric solitons in these systems reduce to their counterparts in the usual coupler. The most essential problem is the stability of the solitons, which become unstable against symmetry breaking with the increase of the energy (norm) and retrieve the stability at still larger energies. The boundary value of the intercore-coupling constant, above which the solitons are completely stable, is found by means of an analytical approximation, based on the cw (zero-dimensional) counterpart of the system. The approximation demonstrates good agreement with numerical findings for the 1D and 2D solitons. Numerical results for the stability limits of the 2D solitons are obtained by means of the computation of eigenvalues for small perturbations, and verified in direct simulations. Although large parts of the soliton families are unstable, the instability is quite weak. Collisions between 2D solitons in the $\mathcal{PT}$-symmetric coupler are studied by means of simulations. Outcomes of the collisions are inelastic but not destructive, as they do not break the $\mathcal{PT}$ symmetry.

Coherent wave propagation in random media results in a characteristic speckle pattern, with spatial intensity correlations with short-range and long-range behavior. Here, we show how the speckle correlation function can be obtained from a ray picture for two representative geometries, namely a chaotic cavity and a random waveguide. Our calculation allows us to study the crossover between a “ray limit” and a “wave limit,” in which the Ehrenfest time ${\tau }_{\mathrm{E}}$ is larger or smaller than the typical transmission time ${\tau }_{\mathrm{D}}$, respectively. Remarkably, long-range speckle correlations persist in the ray limit ${\tau }_{\mathrm{E}}\gg {\tau }_{\mathrm{D}}$.

We investigated the frequency spectra and field distributions of a dielectric square resonator in a microwave experiment. Since such systems cannot be treated analytically, the experimental studies of their properties are indispensable. The momentum representation of the measured field distributions shows that all resonant modes are localized on specific classical tori of the square billiard. Based on these observations a semiclassical model was developed. It shows excellent agreement with all but a single class of measured field distributions that will be treated separately.

A model based on the fractal continuum approach is proposed to describe tracer transport in fractal porous media. The original approach has been extended to treat tracer transport and to include systems with radial and uniform flow, which are cases of interest in geoscience. The models involve advection due to the fluid motion in the fractal continuum and dispersion whose mathematical expression is taken from percolation theory. The resulting advective-dispersive equations are numerically solved for continuous and for pulse tracer injection. The tracer profile and the tracer breakthrough curve are evaluated and analyzed in terms of the fractal parameters. It has been found in this work that anomalous transport frequently appears, and a condition on the fractal parameter values to predict when sub- or superdiffusion might be expected has been obtained. The fingerprints of fractality on the tracer breakthrough curve in the explored parameter window consist of an early tracer breakthrough and long tail curves for the spherical and uniform flow cases, and symmetric short tailed curves for the radial flow case.

A simple analytical model for the planar radio-frequency (rf) sheath in capacitive discharges is developed that is based on the assumptions of a step profile for the electron front, charge exchange collisions with constant cross sections, negligible ionization within the sheath, and negligible ion dynamics. The continuity, momentum conservation, and Poisson equations are combined in a single integro-differential equation for the square of the ion drift velocity, the so called sheath equation. Starting from the kinetic Boltzmann equation, special attention is paid to the derivation and the validity of the approximate fluid equation for momentum balance. The integrals in the sheath equation appear in the screening function which considers the relative contribution of the temporal mean of the electron density to the space charge in the sheath. It is shown that the screening function is quite insensitive to variations of the effective sheath parameters. The two parameters defining the solution are the ratios of the maximum sheath extension to the ion mean free path and the Debye length, respectively. A simple general analytic expression for the screening function is introduced. By means of this expression approximate analytical solutions are obtained for the collisionless as well as the highly collisional case that compare well with the exact numerical solution. A simple transition formula allows application to all degrees of collisionality. In addition, the solutions are used to calculate all static and dynamic quantities of the sheath, e.g., the ion density, fields, and currents. Further, the rf Child-Langmuir laws for the collisionless as well as the collisional case are derived. An essential part of the model is the a priori knowledge of the wave form of the sheath voltage. This wave form is derived on the basis of a cubic charge-voltage relation for individual sheaths, considering both sheaths and the self-consistent self-bias in a discharge with arbitrary symmetry. The externally applied rf voltage is assumed to be sinusoidal, although the model can be extended to arbitrary wave forms, e.g., for dual-frequency discharges. The model calculates explicitly the cubic correction parameter in the charge-voltage relation for the case of highly asymmetric discharges. It is shown that the cubic correction is generally moderate but more pronounced in the collisionless case. The analytical results are compared to experimental data from the literature obtained by laser electric field measurements of the mean and dynamic fields in the capacitive sheath for various gases and pressures. Very good agreement is found throughout.

An adaptive mesh in phase space (AMPS) methodology has been developed for solving multidimensional kinetic equations by the discrete velocity method. A Cartesian mesh for both configuration ($r$) and velocity ($v$) spaces is produced using a “tree of trees” (ToT) data structure. The $r$ mesh is automatically generated around embedded boundaries, and is dynamically adapted to local solution properties. The $v$ mesh is created on-the-fly in each $r$ cell. Mappings between neighboring $v$-space trees is implemented for the advection operator in $r$ space. We have developed algorithms for solving the full Boltzmann and linear Boltzmann equations with AMPS. Several recent innovations were used to calculate the discrete Boltzmann collision integral with dynamically adaptive $v$ mesh: the importance sampling, multipoint projection, and variance reduction methods. We have developed an efficient algorithm for calculating the linear Boltzmann collision integral for elastic and inelastic collisions of hot light particles in a Lorentz gas. Our AMPS technique has been demonstrated for simulations of hypersonic rarefied gas flows, ion and electron kinetics in weakly ionized plasma, radiation and light-particle transport through thin films, and electron streaming in semiconductors. We have shown that AMPS allows minimizing the number of cells in phase space to reduce the computational cost and memory usage for solving challenging kinetic problems.

We adapt the compression algorithm of Weinstein, Auerbach, and Chandra from eigenvectors of spin lattice Hamiltonians to eigenvectors of light-front field-theoretic Hamiltonians. The latter are approximated by the standard discrete light-cone quantization technique, which provides a matrix representation of the Hamiltonian eigenvalue problem. The eigenvectors are represented as singular value decompositions of two-dimensional arrays, indexed by transverse and longitudinal momenta, and compressed by truncation of the decomposition. The Hamiltonian is represented by a rank-four tensor that is decomposed as a sum of contributions factorized into direct products of separate matrices for transverse and longitudinal interactions. The algorithm is applied to a model theory to illustrate its use.

We present a form of the discrete dipole approximation for electromagnetic scattering computations in time domain. We show that the introduction of complex frequencies, through the Laplace transform, significantly improves the computation time. We also show that the Laplace transform and its inverse can be combined to extract the field inside a scatterer at a real resonance frequency.

We develop a classical density functional theory (DFT) for two-site associating fluids in spatially homogeneous external fields which exhibit orientational inhomogeneities. The Helmholtz free-energy functional is obtained using Wertheim's thermodynamic perturbation theory and the orientational distribution function is obtained using DFT in the canonical ensemble. It is shown that an orientating field significantly enhances association by ordering the molecules, thereby reducing the entropic penalty of association. It is also shown that association enhances the orientational order for fixed field strength.

The main characteristics of biased greedy random walks (BGRWs) on two-dimensional lattices with real-valued quenched disorder on the lattice edges are studied. Here the disorder allows for negative edge weights. In previous studies, considering the negative-weight percolation (NWP) problem, this was shown to change the universality class of the existing, static percolation transition. In the presented study, four different types of BGRWs and an algorithm based on the ant colony optimization heuristic were considered. Regarding the BGRWs, the precise configurations of the lattice walks constructed during the numerical simulations were influenced by two parameters: a disorder parameter $\rho$ that controls the amount of negative edge weights on the lattice and a bias strength $B$ that governs the drift of the walkers along a certain lattice direction. The random walks are “greedy” in the sense that the local optimal choice of the walker is to preferentially traverse edges with a negative weight (associated with a net gain of “energy” for the walker). Here, the pivotal observable is the probability that, after termination, a lattice walk exhibits a total negative weight, which is here considered as percolating. The behavior of this observable as function of $\rho$ for different bias strengths $B$ is put under scrutiny. Upon tuning $\rho$, the probability to find such a feasible lattice walk increases from zero to 1. This is the key feature of the percolation transition in the NWP model. Here, we address the question how well the transition point ${\rho }_{\mathrm{c}}$, resulting from numerically exact and “static” simulations in terms of the NWP model, can be resolved using simple dynamic algorithms that have only local information available, one of the basic questions in the physics of glassy systems.

We investigate the stochastic motion of a Brownian particle in the harmonic potential with a time-dependent force constant. It may describe the motion of a colloidal particle in an optical trap where the potential well is formed by a time-dependent field. We use the path integral formalism to solve the Langevin equation and the associated Fokker-Planck (Kramers) equation. Rigorous relations are derived to generate the probability density function for the time-dependent nonequilibrium work production beyond the overdamped limit. We find that the work distribution exhibits an exponential tail with a power-law prefactor, accompanied by an interesting oscillatory feature (multiple pseudo-locking-unlocking transitions) due to the inertial effect. Some exactly solvable cases are discussed in the overdamped limit.

Achlioptas processes, a class of percolation models which can lead to rich critical phenomena, including the well-known explosive percolation, have attracted much attention in recent years. In this paper, we show that, in a three-vertex Achlioptas process, two giant clusters emerge after the percolation transition with size fluctuations in different realizations, and the choice of the connecting vertex in the smaller cluster depends on a probability parameter $p$, the increase of which can make the transition sharper. Using finite-size scaling analysis, we can determine the critical point $r_c$ and critical exponents $\eta$, $1/\nu$, and $\beta$ through Monte Carlo simulations. Comparison of such exponents for different giant clusters indicates that their critical nature is always the same. However, when link choice is strongly biased, it is surprising that the scaling relation $\eta =\beta /\nu$ is violated, and the data collapse for scaling function diverges. Furthermore, by inspecting the variance of exponents with $p$, three distinct scaling phases are classified for different parameter intervals according to the divergence scaling function, which suggests an inconsistent scaling form in the critical window with the supercritical region. The study on the criticality and scaling behavior of multiple giant clusters in an Achlioptas process, in particular, the discovery of three scaling phases that depend on the parameter $p$, may help us in finding a complete scaling theory for the Achlioptas-process percolation and give insight into understanding the accelerating nature of the phase transition for Achlioptas processes once reaching criticality.

We study the patterns generated in finite-time sweeps across symmetry-breaking bifurcations in individual-based models. Similar to the well-known Kibble-Zurek scenario of defect formation, large-scale patterns are generated when model parameters are varied slowly, whereas fast sweeps produce a large number of small domains. The symmetry breaking is triggered by intrinsic noise, originating from the discrete dynamics at the microlevel. Based on a linear-noise approximation, we calculate the characteristic length scale of these patterns. We demonstrate the applicability of this approach in a simple model of opinion dynamics, a model in evolutionary game theory with a time-dependent fitness structure, and a model of cell differentiation. Our theoretical estimates are confirmed in simulations. In further numerical work, we observe a similar phenomenon when the symmetry-breaking bifurcation is triggered by population growth.