We study spatial patterns excited by resonant, multifrequency forcing of systems near a Hopf bifurcation to spatially homogeneous oscillations. Our third-order, weakly nonlinear analysis shows that for small amplitudes only stripe patterns or hexagons (up and down) are linearly stable; for larger amplitudes rectangles and super-hexagons may become stable. Numerical simulations show, however, that in the latter regime the third-order analysis is insufficient: superhexagons are unstable. Instead large-amplitude hexagons can arise and be bistable with the weakly nonlinear hexagons.

Nonlinear longitudinal relaxation of a spin in a uniform external dc magnetic field is treated using a master equation for the quasiprobability distribution function of spin orientations in the configuration space of polar and azimuthal angles (analogous to the Wigner phase space distribution for translational motion). The solution of the corresponding classical problem of the rotational Brownian motion of a magnetic moment in an external magnetic field essentially carries over to the quantum regime yielding in closed form the dependence of the longitudinal spin relaxation on the spin size $S$ as well as an expression for the integral relaxation time, which in linear response reduces to that previously given by D. A. Garanin [Phys. Rev. E 55, 2569 (1997)] using the density matrix approach. The nonlinear relaxation is dominated by a single exponential having as time constant the integral relaxation time. Thus a simple description in terms of a Bloch equation holds even for the nonlinear response of a giant spin.

We consider a class of rotationally invariant unitary random matrix ensembles where the eigenvalue density falls off as an inverse power law. Under a scaling appropriate for such power-law densities (different from the scaling required in Gaussian random matrix ensembles), we calculate exactly the two-level kernel that determines all eigenvalue correlations. We show that such ensembles belong to the class of critical ensembles.

The percolation threshold of hard prolate ellipsoids of revolution dispersed in a continuum is obtained as a function of the aspect ratio. First random close packing of ellipsoids is produced by a dropping-and-shaking protocol. Two ellipsoids are regarded as connected when they come sufficiently close. Then a given fraction of ellipsoids selected randomly is removed and percolation of remaining ellipsoids is investigated as the fraction of remaining ellipsoids is varied. It is shown that the critical volume fraction of the colored ellipsoids is a decreasing function of the aspect ratio and that the aspect ratio dependence is well fitted by the inverse of the interaction range determined by the surface area and the radius of gyration of the ellipsoid surface.

We consider the static and dynamical behavior of a flexible polymer chain under steady-state sedimentation using analytic arguments and computer simulations. The model system comprises a single coarse-grained polymer chain of $N$ segments, which resides in a Newtonian fluid as described by the Navier-Stokes equations. The chain is driven into nonequilibrium steady state by gravity acting on each segment. The equations of motion for the segments and the Navier-Stokes equations are solved simultaneously using an immersed boundary method, where thermal fluctuations are neglected. To characterize the chain conformation, we consider its radius of gyration $R_G\left(N\right)$. We find that the presence of gravity explicitly breaks the spatial symmetry leading to anisotropic scaling of the components of $R_G$ with $N$ along the direction of gravity $R_{G,\parallel }$ and perpendicular to it $R_{G,\perp }$, respectively. We numerically estimate the corresponding anisotropic scaling exponents ${\nu }_{\parallel }\approx 0.79$ and ${\nu }_{\perp }\approx 0.45$, which differ significantly from the equilibrium scaling exponent ${\nu }_e=0.588$ in three dimensions. This indicates that on the average, the chain becomes elongated along the sedimentation direction for large enough $N$. We present a generalization of the Flory scaling argument, which is in good agreement with the numerical results. It also reveals an explicit dependence of the scaling exponents on the Reynolds number. To study the dynamics of the chain, we compute its effective diffusion coefficient $D\left(N\right)$, which does not contain Brownian motion. For the range of values of $N$ used here, we find that both the parallel and perpendicular components of $D$ increase with the chain length $N$, in contrast to the case of thermal diffusion in equilibrium. This is caused by the fluid-driven fluctuations in the internal configuration of the polymer that are magnified as polymer size becomes larger.

It is known that an identical delay in all transmission lines can destabilize the macroscopic stationarity of a neural network, causing oscillation. We analyze the collective dynamics of a network whose transmission delays are distributed in time. Here, a neuron is modeled as a discrete-time threshold element that responds in an all-or-nothing manner to a linear sum of signals that arrive after delays assigned to individual transmission lines. Even though transmission delays are distributed in time, a whole network exhibits a single collective oscillation with a period close to the average transmission delay. The collective oscillation cannot only be a simple alternation of the consecutive firing and resting, but also arbitrarily sequenced series of firing and resting, reverberating in a certain period of time. Moreover, the system dynamics can be made quasiperiodic or chaotic by changing the distribution of delays.

We consider the semiclassical ballistic $\sigma$ model as an effective theory describing the quantum mechanics of classically chaotic systems. Specifically, we elaborate on close analogies to the recently developed semiclassical theory of quantum interference in chaotic systems and show how semiclassical “diagrams” involving near action degenerate sets of periodic orbits emerge in the field theoretical description. We further discuss how the universality phenomenon (i.e., the fact that individual chaotic systems behave according to the prescriptions of random matrix theory) can be understood from the perspective of the field theory.

The fractional behavior is presented for a minimal Hamiltonian system of three degrees of freedom which describes reaction processes. The model has a double-well potential where the Arnold web within the well is nonuniform. The survival probability within the well exhibits power law decay in addition to exponential decay. Moreover, the trajectories of the power law decay exhibit $1∕f$ spectra and subdiffusion in the action space, while the trajectories of the exponential decay show Lorentzian spectra and normal diffusion. Transient features of these statistical properties reveal the dynamical connection, i.e., how trajectories approach to (depart from) the Arnold web from (to) the region around the potential saddle. In particular, a wavelet analysis enables us to extract transient features of the resonances. Based on these results, we suggest that resonance junctions including higher-order resonances are important for understanding the dynamical origins of the fractional behavior in reaction processes.

We investigate the effects of axonal time delay when the neuronal oscillators are coupled by sparse and random connections. Using phase-reduced models with general coupling functions, we show that a small fraction of connections with time delay can destabilize synchronous states and induce near-regular wave states. An order parameter is introduced to characterize those states. We analyze the systems using mean-field-type approximation.

The nonlinear stability and dynamic behavior of falling fluid films is studied for flow over a flexible substrate. We use asymptotic methods to deduce governing equations valid in various limits. Long-wave theory is used to derive Benney-like coupled equations for the film thickness and substrate deflection. Weakly nonlinear equations are then derived from these equations that, in the limit of large wall damping and/or large wall tension, reduce to the Kuramoto-Sivashinsky equation. These models break down when inertia becomes more significant, so we also use a long-wave approximation in conjunction with integral theory to derive three strongly coupled nonlinear evolution equations for the film thickness, substrate deflection, and film volumetric flow rate valid at higher Reynolds numbers. These equations, accounting for inertia, capillary, viscous, wall tension, and damping effects, are solved over a wide range of parameters. Our results suggest that decreasing wall damping and/or wall tension can promote the development of chaos in the weakly nonlinear regime and lead to severe substrate deformations in the strongly nonlinear regime; these can give rise to situations in which the free surface and underlying substrate come into contact in finite time.

The sedimentation of hard spheres in a Newtonian solvent is studied as a function of Péclet number in the low-concentration limit. Two functional forms for the sedimentation velocity as a function of particle concentration are realized in the limit of high and low Péclet numbers. We argue that a more ordered phase occurs for large Péclet numbers. Measurements of settling in sheared suspensions support these contentions. Recent explanations of sedimentation in suspensions are examined in light of these results.

The propagation of acoustic waves in a square-lattice phononic crystal slab consisting of a single layer of spherical steel beads in a solid epoxy matrix is studied experimentally. Waves are excited by an ultrasonic transducer and fully characterized on the slab surface by laser interferometry. A complete band gap is found to extend around $300\mathrm{kHz}$, in good agreement with theoretical predictions. The transmission attenuation caused by absorption and band gap effects is obtained as a function of frequency and propagation distance. Well confined acoustic wave propagation inside a line-defect waveguide is further observed experimentally.

We propose a graph method for the analysis of electronic surface states in a complex basis superlattice terminated by surface with a multilayer clad. Using topology theory, we frame an energy equation for calculating the surface modes based on a graph model. Compared to the traditional methods, the present method does not have the problem of numerical instability for finding out the surface states of the superlattices. The other advantage is that the localized states are directly solved by the present method without spurious solutions, which usually exist in the solutions obtained by traditional methods. Numerical examples show removal of the spurious solutions by traditional methods may cause numerically instable and incorrect results.

A simplified version of a classical problem in thermodynamics—the adiabatic piston—is discussed in the framework of kinetic theory. We consider the limit of gases whose relaxation time is extremely fast so that the gases contained in the left and right chambers of the piston are always in equilibrium (that is, the molecules are uniformly distributed and their velocities obey the Maxwell-Boltzmann distribution) after any collision with the piston. Then by using kinetic theory we derive the collision statistics, from which we obtain a set of ordinary differential equations for the evolution of the macroscopic observables (namely, the piston average velocity and position, the velocity variance, and the temperatures of the two compartments). The dynamics of these equations is compared with simulations of an ideal gas and a microscopic model of a gas devised to verify the assumptions used in the derivation. We show that the equations predict an evolution for the macroscopic variables that catches the basic features of the problem. The results here presented recover those derived, using a different approach, by Gruber, Pache, and Lesne [J. Stat. Phys. 108, 669 (2002); Gruber, Pache, and Lesne,J. Stat. Phys.112, 1177 (2003)].

The properties of shear zones forming in slow three-dimensional granular flow are investigated. We simulate a straight version of the split-bottom shear cell. It is shown that the same type of wide shear zones is obtained in the presence as well as in the absence of gravity. We investigate the relaxation of the material toward stationary flow and analyze the stress and the velocity fields. A functional form of the widening of the shear zone inside the bulk is given. We discuss the growth of the region where the material is in the critical state. The growth of this critical zone turns out to be responsible for the initial transient of the shear zone.

We investigate the collision of adhesive viscoelastic spheres in quasistatic approximation where the adhesive interaction is described by the Johnson, Kendall, and Roberts (JKR) theory. The collision dynamics, based on the dynamic contact force, describes both restitutive collisions quantified by the coefficient of restitution $\epsilon$ as well as aggregative collisions, characterized by the critical aggregative impact velocity $g_{\mathrm{cr}}$. Both quantities $\epsilon$ and $g_{\mathrm{cr}}$ depend sensitively on the impact velocity and particle size. Our results agree well with laboratory experiments.

The effect of the nonmesogenic solutes cyclohexane (CH), biphenyl (BP), and water $\left(W\right)$ on the nematic-isotropic $\left(N\text{−}I\right)$ and the nematic–smectic-$A$ $\left(N\text{-Sm}A\right)$ phase transitions in the liquid crystal octylcyanobiphenyl (8CB) has been studied by means of adiabatic scanning calorimetry. For BP and CH a linear decrease of both transition temperatures was observed with increasing solute mole fraction. For water the transition temperature stayed nearly constant (after a slight decrease for mole fractions of water up to 0.1) over the entire investigated region. For all concentrations of biphenyl and water studied the $N\text{-Sm}A$ transition remains second order. For $8\text{CB}+\text{CH}$, however, crossover from second order to first order is observed at a tricritical point of the mole fraction $x$ of CH around 0.046. For water it was found that for mole fractions of water above 0.1 phase separation between an 8CB rich and a water rich phase occurred. The changes in the transition temperatures and differences between the three systems, and in particular the crossover from second order to first order in the $8\text{CB}+\text{CH}$ system, is explained in terms of a mean-field free energy density expression including coupling terms of the solutes mole fraction with the nematic and smectic-$A$ order parameters.

In this study we have fabricated eight different liquid-crystal lasers using the same gain medium but different homologues from the bimesogenic series $\alpha$-($2^{\prime }$, 4-difluorobiphenyl-$4^{\prime }$-yloxy)-$\omega$-(4-cyanobiphenyl-$4^{\prime }$-yloxy)alkanes, whereby the number of methylene units in the spacer chain varied from $n=5$ to $n=12$. To quantify the performance of these lasers, the threshold energy and the slope efficiency were extracted from the input-output characteristics of each laser. A clear odd-even effect was observed when both the excitation threshold and the slope efficiency were plotted as a function of the number of methylene units in the spacer chain. In all cases, the bimesogen lasers for which $n$ is even exhibit lower threshold energies and higher slope efficiencies than those for which $n$ is odd. These results are then interpreted in terms of the macroscopic physical properties of the liquid-crystalline compounds. In accordance with a previous study [S. M. Morris, A. D. Ford, M. N. Pivnenko, O. Hadeler, and H. J. Coles, Phys. Rev. E. 74, 061709 (2006)], a combination of a large birefringence and high order parameters are found, in the most part, to correlate with low-threshold energy and high slope efficiency. This indicates that the threshold and slope efficiency are dominated by the host macroscopic properties as opposed to intermolecular interactions between the dye and the liquid crystal. However, certain differences in the slope efficiency could not be explained by the birefringence and order parameter values alone. Instead, we find that the slope efficiency is further increased by increasing the elastic constants of the liquid-crystal host so as to decrease the scattering losses incurred by local distortions in the director field under high-energy optical excitation.

We describe a model of cytoskeletal mechanics based on the force-induced conformational change of protein cross-links in a stressed polymer network. Slow deformation of simulated networks containing cross-links that undergo repeated, serial domain unfolding leads to an unusual state—with many cross-links accumulating near the critical force for further unfolding. This state is robust to thermalization and does not occur in similar protein unbinding based simulations. Moreover, we note that the unusual configuration of near-critical protein cross-links in the fragile state provides a physical mechanism for the chemical transduction of cell-level mechanical strain and extra-cellular matrix stiffness.

A phase-field model for vesicles including hydrodynamics was presented in two and three dimensions [T. Biben and C. Misbah, Phys. Rev. E 67, 031908 (2003); T. Biben, K. Kassner, and C. Misbah, Phys. Rev. E 72, 041921 (2005)]. A particularly important feature for vesicles is that their membrane is locally incompressible. In these works a tension field defined everywhere in the bulk was introduced in order to fulfill local membrane inextensibility. Here we reconsider the original model by treating the phase field as a thermodynamic variable and develop a picture which is consistent with the second law of thermodynamics. This enables us to write the phase-field evolution equations in terms of a thermodynamical potential. This potential acquires, at global equilibrium, a Lyapunov functional character. The goal of this paper is twofold: (i) The first and primary goal is purely conceptual, in that we can write down a first and second principle for membranes, from which the evolution equations follow, thanks to the evaluation of the entropy production and the use of concepts of irreversible thermodynamics. (ii) Due to the monotonous character of the evolution of the functional (at global equilibrium), we expect this formulation to be more appropriate for numerical studies. The formalism developed to account for the local incompressibility of the membrane is believed to offer a systematic framework in order to include naturally other physical ingredients, as briefly discussed here and demonstrated in future works.