The discharging of dust particles in the afterglow of plasma with large dust density is studied. We used measured electron and metastable dependencies to calculate the rate describing collection of electrons by dust particles by solving the electron balance equation. This rate is compared with the rate calculated using the orbital motion limited (OML) theory. It is found that the OML theory may not be applied for description of dust charging at large afterglow times, and the energetic electrons generated in metastable-metastable collisions significantly affect charging of dust particles. The time dependence for dust charge is calculated by two different approaches: first, the “standard” approach is used, which assumes that ion and electron fluxes to the dust particles are different in the afterglow. Second, the dust charge is calculated by assuming that desorption of electrons from dust particles is very fast. Both approaches gave similar results for dust charging. In addition, the effects of secondary emission due to ion-dust and metastable-dust collisions on dust discharging are investigated. The main source of dust charging in the late afterglow of plasma with large dust density are the energetic electrons generated in Ar${}^m$ metastable-metastable collisions.

Ensembles of isotropic random matrices are defined by the invariance of the probability measure under the left (and right) multiplication by an arbitrary unitary matrix. We show that the multiplication of large isotropic random matrices is spectrally commutative and self-averaging in the limit of infinite matrix size $N\to \infty$. The notion of spectral commutativity means that the eigenvalue density of a product $\mathit{\text{ABC}}...$ of such matrices is independent of the order of matrix multiplication, for example, the matrix $\mathit{\text{ABCD}}$ has the same eigenvalue density as $\mathit{\text{ADCB}}$. In turn, the notion of self-averaging means that the product of $n$ independent but identically distributed random matrices, which we symbolically denote by $\mathit{\text{AAA}}...$, has the same eigenvalue density as the corresponding power $A^n$ of a single matrix drawn from the underlying matrix ensemble. For example, the eigenvalue density of $\mathit{\text{ABCCABC}}$ is the same as that of $A^2{B}^2{C}^3$. We also discuss the singular behavior of the eigenvalue and singular value densities of isotropic matrices and their products for small eigenvalues $\lambda \to 0$. We show that the singularities at the origin of the eigenvalue density and of the singular value density are in one-to-one correspondence in the limit $N\to \infty$: The eigenvalue density of an isotropic random matrix has a power-law singularity at the origin $\sim {\left|\lambda \right|}^{-s}$ with a power $s\in (0,2)$ when and only when the density of its singular values has a power-law singularity $\sim {\lambda }^{-\sigma }$ with a power $\sigma =s/(4-s)$. These results are obtained analytically in the limit $N\to \infty$. We supplement these results with numerical simulations for large but finite $N$ and discuss finite-size effects for the most common ensembles of isotropic random matrices.

We present a master equation formulation based on a Markovian random walk model that exhibits subdiffusion, classical diffusion, and superdiffusion as a function of a single parameter. The nonclassical diffusive behavior is generated by allowing for interactions between a population of walkers. At the macroscopic level, this gives rise to a nonlinear Fokker-Planck equation. The diffusive behavior is reflected not only in the mean squared displacement [$\langle r^2\left(t\right)\rangle \sim t^{\gamma }$ with $0<\gamma \le 1.5$] but also in the existence of self-similar scaling solutions of the Fokker-Planck equation. We give a physical interpretation of sub- and superdiffusion in terms of the attractive and repulsive interactions between the diffusing particles and we discuss analytically the limiting values of the exponent $\gamma$. Simulations based on the master equation are shown to be in agreement with the analytical solutions of the nonlinear Fokker-Planck equation in all three diffusion regimes.

We study crystal growth inside an infinite octant on a cubic lattice. The growth proceeds through the deposition of elementary cubes into inner corners. After rescaling by the characteristic size, the interface becomes progressively more deterministic in the long-time limit. Utilizing known results for the crystal growth inside a two-dimensional corner, we propose a hyperbolic partial differential equation for the evolution of the limiting shape. This equation is interpreted as a Hamilton-Jacobi equation, which helps in finding an analytical solution. Simulations of the growth process are in excellent agreement with analytical predictions. We then study the evolution of the subleading correction to the volume of the crystal, the asymptotic growth of the variance of the volume of the crystal, and the total number of inner and outer corners. We also show how to generalize the results to arbitrary spatial dimension.

Conventionally, pairwise relationships between nodes are considered to be the fundamental building blocks of complex networks. However, over the last decade, the overabundance of certain subnetwork patterns, i.e., the so-called motifs, has attracted much attention. It has been hypothesized that these motifs, instead of links, serve as the building blocks of network structures. Although the relation between a network's topology and the general properties of the system, such as its function, its robustness against perturbations, or its efficiency in spreading information, is the central theme of network science, there is still a lack of sound generative models needed for testing the functional role of subgraph motifs. Our work aims to overcome this limitation. We employ the framework of exponential random graph models (ERGMs) to define models based on triadic substructures. The fact that only a small portion of triads can actually be set independently poses a challenge for the formulation of such models. To overcome this obstacle, we use Steiner triple systems (STSs). These are partitions of sets of nodes into pair-disjoint triads, which thus can be specified independently. Combining the concepts of ERGMs and STSs, we suggest generative models capable of generating ensembles of networks with nontrivial triadic $Z$-score profiles. Further, we discover inevitable correlations between the abundance of triad patterns, which occur solely for statistical reasons and need to be taken into account when discussing the functional implications of motif statistics. Moreover, we calculate the degree distributions of our triadic random graphs analytically.

It is well known that complete energy transfer between two weakly coupled linear oscillators occurs only at resonance. If the oscillators are nonlinear, the amplitude dependence of their frequencies may destroy, in general, any eventual resonance. This means that no substantial energy transfer may occur unless, exceptionally, resonance persists during the transfer. In this paper, the self-sustained resonance is considered for an oscillator array consisting of $n$ coupled linear oscillators (a primary system) initially excited by impulse loading and connected to an essentially nonlinear attachment (NLA). Under the condition of resonance, initial energy is transferred to the NLA and then travels back and forth between the linear and nonlinear oscillators. It is shown that the general mechanism of the energy transport is similar to that in the previously studied system of two coupled oscillators but, in contrast to the two degree-of-freedom case, the multidimensional system requires a proper tuning not only for the NLA but for the entire array. In this work, we develop an order-reduction procedure, which allows the separated dynamical analysis for the pair of nonlinearly coupled oscillators and the remaining ($n-1$) linear oscillators. Using simplifications based on the low-order reduced model, we detect an admissible domain of parameters ensuring resonance interaction and then derive a closed-form approximate solution adequately describing the transient processes in the entire system. We obtain explicit approximate solutions for both conservative (complete energy exchange) and dissipative (irreversible energy transfer) systems and then illustrate the theoretical results by an example of the four degree-of-freedom system. Analytical results are confirmed by numerical simulations.

Wave competition is an important phenomenon for pattern formation and pattern control, which are required in many practical applications. In this work we consider the competition of wave trains around interfaces of two one-dimensional oscillatory media, modeled by complex Ginzburg-Landau equations with different system control parameters. The wave trains are generated by pacings at two boundaries of the media. In the presence of both normal wave trains and antiwave trains, extremely rich phenomena can be observed by varying the control parameters of the systems. These rich behaviors can be fully understood and physically well explained, based on the competitions of three kinds of wave trains: the two wave trains generated by external pacings and the wave train generated internally at the interface.

For systems out of equilibrium and subjected to a static bias force it can often be expected that particle transport will usually follow the direction of this bias. However, counterexamples exist where particles exhibit uphill motion (known as absolute negative mobility, ANM), particularly in the case of coupled particles. Examples in single particle deterministic systems are less common. Recently, in one such example, uphill motion was shown to occur for an inertial driven and damped particle in a spatially symmetric periodic potential. The source of this anomalous transport was a combination of two periodic driving signals which together are asymmetric under time reversal. In this paper we investigate the phenomena of ANM for a deterministic particle evolving in a periodic and symmetric potential subjected to an external unbiased periodic driving and nonuniform space-dependent damping. It will be shown that this system exhibits a complicated response behavior as certain control parameters are varied, most notably being enhanced parameter regimes exhibiting ANM as the static bias force is increased. Moreover, the solutions exhibiting ANM are shown to be, at least over intermediate time periods, superdiffusive, in contrast to the solutions that follow the bias where the diffusion is normal.

We investigate dynamical properties of the system of two interacting hard disks within a circular billiard numerically in the case of zero total angular momentum. Varying the radius of two identical disks, we examine chaotic irregularity and ergodicity of the system. Single-particle configuration and velocity distributions are obtained from dynamical trajectory calculations and compared with those in the microcanonical ensemble. We also analyze properties of trajectories by calculating the finite-time maximum Lyapunov exponent and clarify the existence of sticky motions around Kolmogorov-Arnold-Moser (KAM) tori even for small radii of disks. It is shown that the present system is almost ergodic in spite of the existence of tori for small radii of disks since the ratio of tori to the whole phase space is extremely small. On the other hand, a number of tori increase abruptly as the radius of disks increases beyond some value and tori prevent trajectories to run over the phase space uniformly, which makes the ergodicity of the system broken down.

In a Lagrangian fluid approach, an explicit method has been presented previously to obtain an exact nonstationary magnetosonic-type wave solution in compressible magnetized plasmas of arbitrary resistivity showing competition among hydrodynamic convection, magnetic field diffusion, and dispersion [Chakrabarti et al., Phys. Rev. Lett. 106, 145003 (2011)]. The purpose of the present work is twofold: it serves (i) to describe the physical and mathematical background of the involved magnetosonic wave dynamics in more detail, as proposed by our original Letter, and (ii) to present an alternative approach, which utilizes the Lagrangian mass variable as a new spatial coordinate [Schamel, Phys. Rep. 392, 279 (2004)]. The obtained exact nonlinear wave solutions confirm the correctness of our previous results, indicating a collapse of the magnetic field irrespective of the presence of dispersion and resistivity. The mean plasma density, on the other hand, is less singular, showing collapse only when dispersive effects are negligible. These results may contribute to our understanding of the generation of strongly localized magnetic fields (and currents) in plasmas, and they are expected to be of special importance in the astrophysical context of magnetic star formation.

Recently, a hybrid distribution function [Tribeche et al., Phys. Rev. E 85, 037401 (2012)] was proposed to describe a plasma species with an enhanced superthermal component. This combines a Cairns-type “nonthermal” form with the Tsallis theory for nonextensive thermodynamics. Using this alternative model, the propagation of arbitrary amplitude ion acoustic solitary waves in a two-component plasma is investigated. From a careful study of the distribution function it is found that the model itself is valid only for a very restricted range in the $q$-nonextensive parameter and the nonthermality parameter, $\alpha$. Solitary waves, the amplitude and nature of which depend sensitively on both $q$ and $\alpha$, can exist within a narrow range of allowable Mach numbers. Both positive and negative potential structures are found, and coexistence may occur.

Slowly driven elastic interfaces, such as domain walls in dirty magnets, contact lines wetting a nonhomogeneous substrate, or cracks in brittle disordered material proceed via intermittent motion, called avalanches. Here we develop a field-theoretic treatment to calculate, from first principles, the space-time statistics of instantaneous velocities within an avalanche. For elastic interfaces at (or above) their (internal) upper critical dimension $d\ge d_{\mathrm{uc}}$ ($d_{\mathrm{uc}}=2,4$ respectively for long-ranged and short-ranged elasticity) we show that the field theory for the center of mass reduces to the motion of a point particle in a random-force landscape, which is itself a random walk [Alessandro, Beatrice, Bertotti, and Montorsi (ABBM) model]. Furthermore, the full spatial dependence of the velocity correlations is described by the Brownian-force model (BFM) where each point of the interface sees an independent Brownian-force landscape. Both ABBM and BFM can be solved exactly in any dimension $d$ (for monotonous driving) by summing tree graphs, equivalent to solving a (nonlinear) instanton equation. We focus on the limit of slow uniform driving. This tree approximation is the mean-field theory (MFT) for realistic interfaces in short-ranged disorder, up to the renormalization of two parameters at $d=d_{\mathrm{uc}}$. We calculate a number of observables of direct experimental interest: Both for the center of mass, and for a given Fourier mode $q$, we obtain various correlations and probability distribution functions (PDF's) of the velocity inside an avalanche, as well as the avalanche shape and its fluctuations (second shape). Within MFT we find that velocity correlations at nonzero $q$ are asymmetric under time reversal. Next we calculate, beyond MFT, i.e., including loop corrections, the one-time PDF of the center-of-mass velocity $\dot{u}$ for dimension $d<d_{\mathrm{uc}}$. The singularity at small velocity $\mathcal{P}\left(\dot{u}\right)\sim 1/{\dot{u}}^{\mathsf{a}}$ is substantially reduced from $\mathsf{a}=1$ (MFT) to $\mathsf{a}=1-\frac{2}{9}(4-d)+...$ (short-ranged elasticity) and $a=1-\frac{4}{9}(2-d)+...$ (long-ranged elasticity). We show how the dynamical theory recovers the avalanche-size distribution, and how the instanton relates to the response to an infinitesimal step in the force.

We obtain estimates for critical nematic liquid crystal (LC) temperatures under the action of a slowly varying temperature-dependent control variable. We show that biaxiality has a negligible effect within our model and that these delay estimates are well described by a purely uniaxial model. The static theory predicts two critical temperatures: the supercooling temperature below which the isotropic phase loses stability and the superheating temperature above which the ordered nematic states do not exist. In contrast to the static problem, the isotropic phase exhibits a memory effect below the supercooling temperature in the dynamic framework. This delayed loss of stability is independent of the rate of change of temperature and depends purely on the initial value of the temperature. We also show how our results can be used to improve estimates for LC material constants.

The mechanisms and structural elements of the Rayleigh-Taylor instability whose evolution results in the occurrence of the collapse have been studied in the scope of the rotating shallow water model with horizontal density gradient. Analysis of the instability mechanism shows that two collapse scenarios are possible. One scenario implies anisotropic collapse during which the contact area of a collapsing fragment with the bottom contracts into a spinning segment. The other implies isotropic contracting of the area into a point. The rigorous integral criteria and power laws of collapses are found.

We perform an asymptotic analysis of electrohydrodynamic (EHD) flow of nonpolar liquid subjected to an external, nonuniform electric field. The domain of interest covers the bulk as well as the thin dissociation layers (DSLs) near the electrodes. Outer (i.e., bulk) equations for the ion transport in hierarchical order of perturbation parameters can be expressed in linear form, whereas the inner (i.e., DSL) equations take a nonlinear form. We derive a simple formula in terms of various parameters which can be used to estimate the relative importance of the DSL-driven flow compared with the bulk-driven flow. EHD flow over a pair of cylindrical electrodes is then solved asymptotically and numerically. It is found that in large geometric scale and high ion concentration the EHD flow is dominated by the bulk-charge-induced flow. As the scale and concentration are decreased, the DSL-driven slip velocity increases and the resultant flow tends to dominate the domain and finally leads to flow reversal. We also conduct a flow-visualization experiment to verify the analysis and attain good agreement between the two results with parameter tuning. We finally show, based on the comparison of experimental and numerical solutions, that the rate of free-ion generation (dissociation) should be less than the one predicted from the existing formula.

An efficient algorithm is presented to simulate the $\mathrm{O}\left(N\right)$ loop model on the square lattice for arbitrary values of $N>0$. The scheme combines the worm algorithm with a new data structure to resolve both the problem of loop crossings and the necessity of counting the number of loops at each Monte Carlo update. With the use of this scheme, the line of critical points (and other properties) of the O($N$) model on the square lattice for $0<N\le 2$ have been determined.

The effect of the Coulomb interactions on the crossover between mean-field and Ising critical behavior in ionic fluids is studied using the Ginzburg criterion. We consider the charge-asymmetric primitive model supplemented by short-range attractive interactions in the vicinity of the gas-liquid critical point. The model without Coulomb interactions exhibiting typical Ising critical behavior is used to calibrate the Ginzburg temperature of the systems comprising electrostatic interactions. Using the collective variables method, we derive a microscopic-based effective Hamiltonian for the full model. We obtain explicit expressions for all the relevant Hamiltonian coefficients within the framework of the same approximation, i.e., the one-loop approximation. Then we consistently calculate the reduced Ginzburg temperature $t_{\mathrm{G}}$ for both the purely Coulombic model (a restricted primitive model) and the purely nonionic model (a hard-sphere square-well model) as well as for the model parameters ranging between these two limiting cases. Contrary to the previous theoretical estimates, we obtain the reduced Ginzburg temperature for the purely Coulombic model to be about 20 times smaller than for the nonionic model. For the full model including both short-range and long-range interactions, we show that $t_{\mathrm{G}}$ approaches the value found for the purely Coulombic model when the strength of the Coulomb interactions becomes sufficiently large. Our results suggest a key role of Coulomb interactions in the crossover behavior observed experimentally in ionic fluids as well as confirm the Ising-like criticality in the Coulomb-dominated ionic systems.

Local time is the measure of how much time a random walk has visited a given position. In multiple scattering media, where waves are diffuse, local time measures the sensitivity of the waves to the local medium's properties. Local variations of absorption, velocity, and scattering between two measurements yield variations in the wave field. These variations are proportional to the local time of the volume where the change happened and the amplitude of variation. The wave field variations are measured using correlations and can be used as input in a inversion algorithm to produce variation maps. The present article gives the expression of the local time in dimensions one, two, and three and an expression of its fluctuations, in order to perform such inversions and estimate their accuracy.

We study melting of a face-centered crystalline solid consisting of polydisperse Lennard-Jones spheres with Gaussian polydispersity in size. The phase diagram reproduces the existence of a nearly temperature invariant terminal polydispersity (${\delta }_t$ ≃ 0.11), with no signature of reentrant melting. The absence of reentrant melting can be attributed to the influence of the attractive part of the potential upon melting. We find that at terminal polydispersity the fractional density change approaches zero, which seems to arise from vanishingly small compressibility of the disordered phase. At constant temperature and volume fraction the system undergoes a sharp transition from crystalline solid to the disordered amorphous or fluid state with increasing polydispersity. This has been quantified by second- and third-order rotational invariant bond orientational order, as well as by the average inherent structure energy. The translational order parameter also indicates a similar sharp structural change at δ ≃ 0.09 in case of $T^{*}$ $=$ 1.0, ϕ  $=$ 0.58. The free energy calculation further supports the sharp nature of the transition. The third-order rotationally invariant bond order shows that with increasing polydispersity, the local cluster favors a more icosahedral arrangement and the system loses its local crystalline symmetry. Interestingly, the value of structure factor $S$($k$) of the amorphous phase at δ ≃ 0.10 (just beyond the solid-liquid transition density at $T^{*}$ $=$ 1) becomes 2.75, which is below the value of 2.85 required for freezing given by the empirical Hansen-Verlet rule of crystallization, well known in the theory of freezing.