We consider the Ising model on the Bethe lattice with aperiodic modulation of the couplings, which has been studied numerically in Phys. Rev. E 77, 041113 (2008). Here we present a relevance-irrelevance criterion and solve the critical behavior exactly for marginal aperiodic sequences. We present analytical formulas for the continuously varying critical exponents and discuss a relationship with the (surface) critical behavior of the aperiodic quantum Ising chain.

A reactive flow, the gelation of sodium acrylate (SA), was carried out in a cuboidal cavity with the top surface undergoing sinusoidal periodic motion. The instantaneous two-dimensional planar velocity fields during gelation were obtained using particle image velocimetry. The experiments were carried out with different plate velocities and different amounts of accelerator (TEMED). The temporal and spatial variations of the velocity components were analyzed. The magnitude of the velocity components was found to decrease with the progress of reaction due to gel formation. The role of mixing on the reaction is understood from the amount of gel formed at different plate velocities. Gel formation patterns are explained in terms of the mixing characteristics of the periodic flow. The periodic variation of point velocities showed the presence of higher harmonics in the flow.

The densities of states are evaluated for very short chain molecules made up of overlapping monomers, using a model which has previously been shown to produce helical structure. The results of numerical calculations are presented for tetramers and pentamers. We show that these models demonstrate behaviors relevant to the behaviors seen in longer, helix-forming chains, particularly “magic numbers” of the overlap parameter, where the derivatives of the densities of states change discontinuously, and a region of bimodal energy probability distributions, reminiscent of a first-order phase transition in a bulk system.

We study the conserved-mass aggregation model with mass-dependent fragmentation on random networks (RNs) and scale-free networks (SFNs) with degree distribution $P\left(k\right)\sim k^{-\gamma }$. In the model, masses isotropically diffuse with unit rate. With rate $\omega$, a mass $m^{\lambda }$ is fragmented from a node with mass $m$, and moves to one of the linked nodes with the equal probability. For $\lambda =0$, the model is known to undergo the condensation phase transitions at a certain criticality ${\rho }_c$. From the mean-field balance equation for an aggregate, we analytically show that the present model exhibits different behavior depending on network structures. The condensation always occurs for $\lambda <0$. For $0<\lambda <1$, finite-sized systems on RNs and SFNs with $\gamma >3$ undergoes the condensation transitions (sharp crossovers) at ${\rho }_c$ which diverges with network size $N$ as $N^{\lambda }$. Hence, in the limit $N\to \infty$, masses uniformly distribute without the condensation (fluid phase). On the other hand, for $\gamma ⩽3$, a crossover ${\lambda }_c=1∕(\gamma -1)$ exists. The condensation always occurs for $\lambda <{\lambda }_c$, while the fluid phase is always stable at any nonzero density for $\lambda ⩾{\lambda }_c$. The phase separation results from the competition between the heterogeneity of network structure and the enhanced chipping by $\lambda$. We numerically confirm all the predictions.

We discuss the consequences of time reversal of the excitation parameters on the stability of a periodically driven dynamic system. The example chosen, electrohydrodynamic convection of nematics, is a well-investigated dissipative pattern-forming system, usually driven with time-periodic external electric fields $E\left(t\right)$. The dynamic model that describes linear stability at onset can be represented by a set of two differential equations for the amplitudes of the dynamic variables. Floquet analysis reveals an astonishing dynamic property: Under time reversal of any periodic excitation function $E\left(t\right)$, the dynamic equations reproduce the same stability thresholds, pattern types, and critical pattern wavelengths, even when the dynamics of all involved system variables are essentially different for the excitations $E\left(t\right)$ and $E(-t)$. Experimental confirmation of these mathematical predictions is provided by the measurement of threshold curves and the analysis of the time-resolved optical characteristics of the convection roll patterns.

In a quantum ratchet accelerator system, a linearly increasing directed current can be dynamically generated without using a biased field. Generic quantum ratchet acceleration with full classical chaos [J. B. Gong and P. Brumer, Phys. Rev. Lett. 97, 240602 (2006)] constitutes a new element of quantum chaos and an interesting violation of a sum rule of classical ratchet transport. Here we propose a simple quantum ratchet accelerator model that can also generate linearly increasing quantum current with full classical chaos. This model does not require a bichromatic lattice potential. It is based on a variant of an on-resonance kicked-rotor system, periodically kicked by two optical lattice potentials of the same lattice constant, but with unequal amplitudes and a fixed phase shift between them. The dependence of the ratchet current acceleration rate on the system parameters is studied in detail. The cold-atom version of our quantum ratchet accelerator model should be realizable by introducing slight modifications to current cold-atom experiments.

We investigate surface waves on the interface between a thin vapor film and a layer of liquid in the presence of a high steady heat flux. This problem arises when a metal surface heated to a high temperature is immersed into a cold liquid. The general boundary conditions, which take into account the temperature dependence of saturation pressure on the vapor-liquid interface, are derived. These boundary conditions generalize the traditional conditions on the free surface of liquid in the gravity field. The stability of the planar vapor-liquid interface is investigated analytically with linear approximations. The dispersion equation for surface waves on the vapor-liquid interface in the presence of strong heat flux is derived. A number of different, distinct from the classical surface wave problem, effects arise in the problem under consideration. The thermal processes, which occur on the phase boundary and are possible in the absence of gravity force, lead to the generation of weakly decaying periodic waves of low amplitude, whose velocities may exceed significantly those of gravity waves. The heat flux through the interface may cause periodic surface waves of small length (ripple), which are not capillary. The processes of phase transition on the interface are capable of providing the stability of vapor film under a layer of liquid in the gravity field.

We study the interaction of short laser pulses tightly focused in a tiny volume proportional to the cube of the pulse wavelength $\left({\lambda }^3\right)$ with underdense plasma by means of real-geometry particle-in-cell simulations. Underdense plasma irradiated by relatively low-energy ${\lambda }^3$ (and ${\lambda }^2$) laser pulses is shown to be an efficient source of multi-MeV electrons, $\sim 50\mathrm{n}\mathrm{C}∕\mathrm{J}$, and coherent hard x rays, despite a strong pulse diffraction. Transverse wave breaking in the vicinity of the laser focus is found to give rise to an immense electron charge loading to the acceleration phase of a laser wake field. A strong blowout regime provoked by the injected electrons resulting in the distribution of accelerated electrons is found for ${\lambda }^3$ pulses (further electron acceleration driving by ${\lambda }^2$ pulses runs in the usual way). With an increase of pulse energy, wiggling and electron-hose instabilities in the ${\lambda }^3$ pulse wake are recognized in the blowout regime. For higher-energy ${\lambda }^3$ pulses, the injected beams are well modulated and may serve as a good source of coherent x rays.

The Hamiltonian description of the self-consistent interaction between an electromagnetic plane wave and a copropagating beam of charged particles is considered. We show how the motion can be reduced to a one-dimensional Hamiltonian model (in a canonical setting) from the Vlasov-Maxwell Poisson brackets. The reduction to this paradigmatic Hamiltonian model is performed using a Lie algebraic formalism which allows us to preserve the Hamiltonian character at each step of the derivation.

Recently the occurrence probabilities of ground and metastable states of three-dimensional Yukawa clusters with 27 and 31 particles have been analyzed in dusty plasma experiments [D. Block et al., Phys. Plasmas 15, 040701 (2008)]. There it was found that, in many cases, the ground state appeared substantially less frequently than excited states. Here we analyze this question theoretically by means of molecular dynamics (MD) and Monte Carlo simulations and an analytical method based on the canonical partition function. We confirm that metastable states can occur with a significantly higher probability than the ground state. The results strongly depend on the screening parameter of the Yukawa interaction and the damping coefficient used in the MD simulations. The analytical method allows one to gain insight into the mechanisms being responsible for the occurrence probabilities of metastable states in strongly correlated finite systems.

Based on a multidimensional Riemann theta function, the Hirota bilinear method is extended to explicitly construct multiperiodic (quasiperiodic) wave solutions for the $(2+1)$-dimensional Bogoyavlenskii breaking soliton equation. Among these periodic waves, the one-periodic waves are well-known cnoidal waves, their surface pattern is one-dimensional, and often they are used as one-dimensional models of periodic waves in shallow water. The two-periodic (biperiodic) waves are a direct generalization of one-periodic waves, their surface pattern is two dimensional, that is, they have two independent spatial periods in two independent horizontal directions. The two-periodic waves may be considered to represent periodic waves in shallow water without the assumption of one dimensionality. A limiting procedure is presented to analyze asymptotic behavior of the one- and two-periodic waves in details. The exact relations between the periodic wave solutions and the well-known soliton solutions are established. It is rigorously shown that the periodic wave solutions tend to the soliton solutions under a “small amplitude” limit.

A particle tracking code is developed to solve a general time-fractional diffusion equation (FDE), yielding a Lagrangian framework that can track particle dynamics. Extensive simulations demonstrate the efficiency and flexibility of this simple Langevin approach. Many real problems require a vector FDE with variable parameters and multiscaling spreading rates. For these problems, particle tracking is the only viable solution method.

We study the random diffusion model. This is a continuum model for a conserved scalar density field $\varphi$ driven by diffusive dynamics. The interesting feature of the dynamics is that the bare diffusion coefficient $D$ is density dependent. In the simplest case, $D=\overline{D}+D_1\delta \varphi$, where $\overline{D}$ is the constant average diffusion constant. In the case where the driving effective Hamiltonian is quadratic, the model can be treated using perturbation theory in terms of the single nonlinear coupling $D_1$. We develop perturbation theory to fourth order in $D_1$. The are two ways of analyzing this perturbation theory. In one approach, developed by Kawasaki, at one-loop order one finds mode-coupling theory with an ergodic-nonergodic transition. An alternative more direct interpretation at one-loop order leads to a slowing down as the nonlinear coupling increases. Eventually one hits a critical coupling where the time decay becomes algebraic. Near this critical coupling a weak peak develops at a wave number well above the peak at $q=0$ associated with the conservation law. The width of this peak in Fourier space decreases with time and can be identified with a characteristic kinetic length which grows with a power law in time. For stronger coupling the system becomes metastable and then unstable. At two-loop order it is shown that the ergodic-nonergodic transition is not supported. It is demonstrated that the critical properties of the direct approach survive, going to higher order in perturbation theory.

A statistical segment scale theory for the physical aging of polymer glasses is proposed and applied. The approach is based on a nonlinear stochastic Langevin equation of motion and the concept of an effective free energy which quantifies temporary localization, collective barriers, and the activated segment hopping process. The key collective structural variable that plays the role of the dynamic order parameter for aging is the experimentally measurable nanometer and longer wavelength amplitude of density fluctuations, $S_0$. The degree of local cooperativity, and the bare activation energy of the high-temperature Arrhenius process, are determined in the molten state by utilizing experimental measurements of the glass temperature and dynamic crossover time, respectively. A first-order kinetic equation with a time varying rate is proposed for the temporal evolution of $S_0$ which is self-consistently and nonlinearly coupled with the mean segmental relaxation time. The theory has been applied to study physical aging of the $\alpha$ relaxation time, shear relaxation modulus, amplitude of density fluctuations, cohesive energy, absolute yield stress, and fictive temperature of polymethylmethacrylate and other glasses over a range of temperatures. Temperature-dependent logarithmic and effective power-law aging is predicted at intermediate times. Time-aging time superposition is found for the mechanical relaxation function. A strongly asymmetric aging response is predicted for up and down temperature jump experiments. Comparison of the approach with the classic phenomenological model is presented.

Propulsion by actin polymerization is widely used in cell motility. Here, we investigate a model of the brush range of an actin gel close to a propelled object, describing the force generation and the dynamics of the propagation velocity. We find transitions between stable steady states and relaxation oscillations when the attachment rate of actin filaments to the obstacle is varied. The oscillations set in at small values of the attachment rate via a homoclinic bifurcation. A second transition from a stable steady state to relaxation oscillations, found for higher values of the attachment rate, occurs via a supercritical Hopf bifurcation. The behavior of the model near the second transition is similar that of a system undergoing a canard explosion. Consequently, we observe excitable dynamics also. The model further exhibits bistability between stationary states or stationary states and limit cycles. Therefore, the brush of actin filament ends appears to have a much richer dynamics than was assumed until now.

We study the critical properties of the Lipkin-Meshkov-Glick model in terms of the fidelity susceptibility. By using the Holstein-Primakoff transformation, we obtain explicitly the critical exponent of the fidelity susceptibility around the second-order quantum phase transition point. Our results provide a rare analytic case for the fidelity susceptibility in describing the universality class in quantum critical behavior. The different critical exponents in two phases are nontrivial results, indicating that the fidelity susceptibility is not always extensive.

Langmuir films at the air-water interface exhibit a variety of surface phases which arise primarily due to the molecular interaction governed by intermolecular separation. We have studied the thermodynamical aspects of Langmuir monolayers of amphiphilic functionalized gold nanoparticles (AGNs) at the air-water $\left(A\text{−}W\right)$ interface. Interestingly, the AGN monolayer exhibits phases like gas, a low-ordered liquid $\left(L_1\right)$, a high-ordered liquid $\left(L_2\right)$, and a collapsed state. We find that the first-order phase transition between $L_1$ and $L_2$ vanishes above a critical temperature of $28.4°\mathrm{C}$. Surprisingly, for a range of higher temperatures ($⩾29.4°\mathrm{C}$ and $⩽36.3°\mathrm{C}$), the $L_1$ phase undergoes a transition to a bilayer of the $L_2$ phase before entering into the collapsed state.

When immiscible wetting and nonwetting fluids move in parallel in a porous medium, an instability may occur at sufficiently high capillary numbers so that interfaces between the fluids initially held in place by the porous medium are mobilized. A boundary zone containing bubbles of both fluids evolves, which has a well-defined thickness. This zone moves at constant average speed toward the nonwetting fluid. A diffusive current of bubbles of nonwetting fluid into the wetting fluid is set up.

We use a clustering signature, based on a recently introduced generalization of the clustering coefficient to directed networks, to analyze 16 directed real-world networks of five different types: social networks, genetic transcription networks, word adjacency networks, food webs, and electric circuits. We show that these five classes of networks are cleanly separated in the space of clustering signatures due to the statistical properties of their local neighborhoods, demonstrating the usefulness of clustering signatures as a classifier of directed networks.

We take a dynamical-systems approach to study the qualitative dynamical aspects of the tidal locking of the rotation of secondary celestial bodies with their orbital motion around the primary. We introduce a minimal model including the essential features of gravitationally induced elastic deformation and tidal dissipation that demonstrates the details of the energy transfer between the orbital and rotovibrational degrees of freedom. Despite its simplicity, our model can account for both synchronization into the 1:1 spin-orbit resonance and the circularization of the orbit as the only true asymptotic attractors, together with the existence of relatively long-lived metastable orbits with the secondary in $p:q$ synchronous rotation.