We propose a dynamical gradient network approach to consider the synchronization in the Kuramoto model. Our scheme to adaptively adjust couplings is based on the dynamical gradient networks, where the number of links in each time interval is the same as the number of oscillators, but the links in different time intervals are also different. The gradient network in the $(n+1)$th time interval is decided by the oscillator dynamics in the $n$th time interval. According to the gradient network in the $(n+1)$th time interval, only one inlink’s coupling for each oscillator is adjusted by a small incremental coupling. During the transition to synchronization, the intensities for all oscillators are identical. Direct numerical simulations fully verify that the synchronization in the Kuramoto model is realized effectively, even if there exist delayed couplings and external noise.

We study many interacting Brownian particles under a tilted periodic potential. We numerically measure the linear response coefficient of the density field by applying a slowly varying potential transversal to the tilted direction. In equilibrium cases, the linear response coefficient is related to the intensity of density fluctuations in a universal manner, which is called a fluctuation-response relation. We then report numerical evidence that this relation holds even in nonequilibrium cases. This result suggests that Einstein’s formula on density fluctuations can be extended to driven diffusive systems when the slowly varying potential is applied in a direction transversal to the driving force.

We propose to reconsider the diffusion of atoms in the Knudsen regime in terms of a complex dynamical reflection process. By means of molecular dynamics simulations, we emphasize the asymptotic nature of the cosine law of reflection at the atomic scale, and carefully analyze the resulting strong correlations in the reflection events. A dynamical interpretation of the accommodation coefficient associated with the slip at the wall interface is also proposed. Finally, we show that the first two moments of the stochastic process of reflection depend nonuniformly on the incident angle.

The elastic response of a smectic-$C$ elastomer to three deformations, namely imposed ${\lambda }_{xx}$, ${\lambda }_{xz}$, and ${\lambda }_{zz}$, has been modeled using a nonlinear theory of a nematic elastomer with embedded smectic layers, and with the director tilt (in the $\mathbf{x}$ direction) at a fixed angle with respect to the smectic layer normal ($\mathbf{z}$ direction). The main focus is the elastic response after any soft mode of the sample. It is found that the elastomer contracts in the $\mathbf{x}$ direction under ${\lambda }_{xz}$ shear. On stretching parallel to the layer normal it is found that there is a soft mode that acts to rotate the director toward the $\mathbf{z}$ direction. The deformation of the system after this soft mode can be reduced to shear and elongation in the plane of the layers. We make predictions of the mechanical response of the elastomer, in particular the length of the soft plateau and the asymptotic modulus for the elastomer when stretched parallel to the layer normal. Finally, since monodomain Sm-$C$ elastomers are made by the deformation-induced alignment of polydomains, we describe these important systems. Qualitative behavior of the model is then compared to existing experimental literature on the mechanical alignment of polydomains

We report a generalized scenario for the formation of modulated smectic phases of bent-core molecules based on locally ferroelectric layering and spontaneous splay of the polarization. Twelve phases are proposed, distinguished by neighboring splay stripes with either syn- or antiorder of the polarization and undulation slope, in addition to layer continuity versus layer discontinuity at the intervening defects. We outline the experimental techniques necessary to differentiate among the phases and interpret previous results in the present context, using high resolution x-ray scattering diffraction and block and undulation models of the layer organization to distinguish among the three 2D lattice types which emerge.

Over the last 12 years, the possible existence of a tetrahedratic mesophase, involving a third-rank orientational order parameter and no positional order, has been addressed theoretically and predicted in some cases; no experimental realizations of a purely tetrahedratic phase are known at the time being, but various pieces of evidence suggest that interactions of tetrahedral symmetry do play a significant role in the macroscopic properties of mesophases resulting from banana-shaped (bent-core) mesogens. We address a very simple tetrahedratic mesogenic lattice model, involving continuous interactions; we consider particles possessing $T_d$ symmetry, whose centers of mass are associated with a three-dimensional simple-cubic lattice; the pair potential is taken to be isotropic in orientation space and restricted to nearest-neighboring sites; we let the two orthonormal triads $\left\{{\mathbf{u}}_{\alpha },\alpha =1,2,3\right\}$ and $\left\{{\mathbf{v}}_{\gamma },\gamma =1,2,3\right\}$ define the orientations of a pair of interacting particles; we let the unit vectors ${\mathbf{u}}_{\alpha }$ be combined to yield four unit vectors $\left\{{\mathbf{e}}_j,j=1,2,3,4\right\}$, arranged in a tetrahedral fashion; we let the unit vectors ${\mathbf{v}}_{\gamma }$ be similarly combined to yield the four unit vectors $\left\{{\mathbf{f}}_k,k=1,2,3,4\right\}$; and finally we let $h_{jk}=\left({\mathbf{e}}_j\cdot {\mathbf{f}}_k\right)$. The interaction model studied here is defined by the simplest nontrivial (cubic) polynomial in the scalar products $h_{jk}$, consistent with the assumed symmetry and favoring orientational order; it is, so to speak, the tetrahedratic counterpart of the Lebwohl-Lasher model for uniaxial nematics. The model was investigated by molecular field (MF) theory and Monte Carlo simulations; MF theory predicts a low-temperature, tetrahedrically ordered phase, undergoing a second-order transition to the isotropic phase at higher temperature; on the other hand, available theoretical treatments point to the transition being driven first order by thermal fluctuations. Simulations showed evidence of a first-order transition.

Recently, Høye, Brevik, Ellingsen, and Aarseth [Phys. Rev. E 75, 051127 (2007)] claimed that the use of the Drude dielectric function leads to zero Casimir entropy at zero temperature in accordance with Nernst’s theorem. We demonstrate that their proof is not applicable to metals with perfect crystal lattices having no impurities. Thus there is no contradiction with previous results in the literature proving that the Drude dielectric function violates the Nernst theorem for the Casimir entropy in the case of perfect crystal lattices. We also indicate that the approximation of temperature independent relaxation frequency used in the paper leads to incorrect values of numerical coefficients in the obtained asymptotic expressions for metals with impurities.

In this reply to the preceding Comment [Klimchitskaya and Mostepanenko, Phys. Rev. E77, 023101 (2008)], we summarize and maintain our position that the Drude dispersion relation when inserted in the Lifshitz formula gives a thermodynamically satisfactory description of the Casimir force, also in the limiting case when the relaxation frequency goes to zero (perfect crystals).

The understanding of dynamic failure in amorphous materials via the propagation of free boundaries like cracks and voids must go beyond elasticity theory, since plasticity intervenes in a crucial and poorly understood manner near the moving free boundary. We focus on failure via a cavitation instability in a radially symmetric stressed material and set up the free boundary dynamics taking both elasticity and viscoplasticity into account using the recently proposed athermal shear transformation zone theory. We demonstrate that this theory predicts the existence (in amorphous systems) of fast cavitation modes accompanied by extensive plastic deformations and discuss the revealed physics.

Given a general physical network and measurements of node dynamics, methods are proposed for reconstructing the network topology. We focus on networks whose connections are sparse and where data are limited. Under these conditions, common in many biological networks, constrained optimization techniques based on the $L1$ vector norm are found to be superior for inference of the network connections.

We present the results of the modeling of CO adsorption and catalytic CO oxidation on inhomogeneous Pt(100) surfaces which contain structurally different areas. These areas are formed during the CO-induced transition from a reconstructed phase with hexagonal geometry of the overlayer to a bulklike $\left(1\times 1\right)$ phase with square atomic arrangement. In the present approach, the surface transition is explained in terms of nonequilibrium bistable behavior. The bistable region is characterized by a coexistence of the hexagonal and $\left(1\times 1\right)$ phases and is terminated in a critical bifurcation point which is located at ($T_c\approx 680\text{K}$, $p_{\text{CO}}^c\approx 10\text{Torr}$). Due to increasing fluctuations, the behavior at high temperatures and pressures in the vicinity of this cusp point should be qualitatively different from the hysteresis-type behavior which is typically observed in the experiments under ultrahigh vacuum conditions. On the inhomogeneous surface, we find a regime of nonuniform oscillations characterized by random standing waves of adsorbate concentrations. The resulting spatial deformations of wave fronts allow us to gain deeper insight into the nature of irregular oscillations on Pt(100) surface.

Classical Faraday experiments were conducted on the oscillatory chemical Belousov-Zhabotinsky (BZ) reaction. The vertical periodic modulation of the acceleration induces flows in the system that change the BZ dynamics, and thus the patterns exhibited. The resulting reaction-diffusion-advection system exhibits four different types of pattern for increasing stirring amplitude: deformed targets and spiral waves, filamentary patterns arranged in large-scale vortices, advection phase waves, and finally front annihilation where the medium becomes homogeneous. A wave period analysis of the forced system has been carried out. Contrary to what is expected, i.e., a continuous increase of the wave period with increasing forcing, the period changes dramatically at the boundaries between pattern domains.

Shannon’s capacity theorem is the main concept behind the theory of communication. It says that if the amount of information contained in a signal is smaller than the channel capacity of a physical media of communication, it can be transmitted with arbitrarily small probability of error. This theorem is usually applicable to ideal channels of communication in which the information to be transmitted does not alter the passive characteristics of the channel that basically tries to reproduce the source of information. For an active channel, a network formed by elements that are dynamical systems (such as neurons, chaotic or periodic oscillators), it is unclear if such theorem is applicable, once an active channel can adapt to the input of a signal, altering its capacity. To shed light into this matter, we show, among other results, how to calculate the information capacity of an active channel of communication. Then, we show that the channel capacity depends on whether the active channel is self-excitable or not and that, contrary to a current belief, desynchronization can provide an environment in which large amounts of information can be transmitted in a channel that is self-excitable. An interesting case of a self-excitable active channel is a network of electrically connected Hindmarsh-Rose chaotic neurons.

Local and nonlocal contributions to the total strain rate tensor $S_{ij}$ at any point $\mathbf{x}$ in a flow are formulated from an expansion of the vorticity field in a local spherical neighborhood of radius $R$ centered on $\mathbf{x}$. The resulting exact expression allows the nonlocal (background) strain rate tensor $S_{ij}^B\left(\mathbf{x}\right)$ to be obtained from $S_{ij}\left(\mathbf{x}\right)$. In turbulent flows, where the vorticity naturally concentrates into relatively compact structures, this allows the local alignment of vorticity with the most extensional principal axis of the background strain rate tensor to be evaluated. In the vicinity of any vortical structure, the required radius $R$ and corresponding order $n$ to which the expansion must be carried are determined by the viscous length scale ${\lambda }_{\nu }$. We demonstrate the convergence to the background strain rate field with increasing $R$ and $n$ for an equilibrium Burgers vortex, and show that this resolves the anomalous alignment of vorticity with the intermediate eigenvector of the total strain rate tensor. We then evaluate the background strain field $S_{ij}^B\left(\mathbf{x}\right)$ in direct numerical simulations of homogeneous isotropic turbulence where, even for the limited $R$ and $n$ corresponding to the truncated series expansion, the results show an increase in the expected equilibrium alignment of vorticity with the most extensional principal axis of the background strain rate tensor.

The simplest and most efficient lattice Boltzmann model that is able to recover the Navier-Stokes equations is based on a single-parameter scattering matrix, where the parameter is the first nonzero eigenvalue of the collision matrix. This simple model, based on a single relaxation time, has many shortcomings. Among these is the lack of freedom to extend the model to complex fluids whose stress tensors are characterized by more complicated constitutive relations. The lattice Boltzmann methodology may be generalized by considering the full collision matrix and tuning the matrix elements to obtain the desired macroscopic properties. The generalized hydrodynamics of a generalized lattice Boltzmann equation (LBE) was studied by Lallemand and Luo [Phys. Rev. E 61, 6546 (2000)] by solving the dispersion equation of the linearized LBE. In this paper, an alternative approach to solving the dispersion equation based on a formal perturbation analysis is described. The methodology outlined is systematic, can be readily applied to other lattices, and does not require the reciprocals of the relaxation times to be small.

We introduce a model for diffusion of two classes of particles ($A$ and $B$) with priority: where both species are present in the same site the motion of $A$’s takes precedence over that of $B$’s. This describes realistic situations in wireless and communication networks. In regular lattices the diffusion of the two species is normal, but the $B$ particles are significantly slower due to the presence of the $A$ particles. From the fraction of sites where the $B$ particles can move freely, which we compute analytically, we derive the diffusion coefficients of the two species. In heterogeneous networks the fraction of sites where $B$’s are free decreases exponentially with the degree of the sites. This, coupled with accumulation of particles in high-degree nodes, leads to trapping of the low priority particles in scale-free networks.

The general form of the electrostatic potential around an arbitrarily charged colloid at a flat interface between a dielectric and a screening phase (such as air and water, respectively) is analyzed in terms of a multipole expansion. The leading term is isotropic in the interfacial plane and varies with $d^{-3}$ where $d$ is the in-plane distance from the colloid. The effective interaction potential between two arbitrarily charged colloids is likewise isotropic and proportional to $d^{-3}$, thus generalizing the dipole-dipole repulsion first found for point charges at water interfaces. Anisotropic attractive interaction terms can arise only for higher powers $d^{-n}$ with $n⩾4$. The relevance of these findings for recent experiments is discussed.

We investigate disordered one- and two-dimensional Heisenberg spin lattices across the transition from integrability to quantum chaos from both statistical many-body and quantum-information perspectives. Special emphasis is devoted to quantitatively exploring the interplay between eigenvector statistics, delocalization, and entanglement in the presence of nontrivial symmetries. The implication of the basis dependence of state delocalization indicators (such as the number of principal components) is addressed, and a measure of relative delocalization is proposed in order to robustly characterize the onset of chaos in the presence of disorder. Both standard multipartite and generalized entanglement are investigated in a wide parameter regime by using a family of spin- and fermion-purity measures, their dependence on delocalization and on energy spectrum statistics being examined. A distinctive correlation between entanglement, delocalization, and integrability is uncovered, which may be generic to systems described by the two-body random ensemble and may point to a new diagnostic tool for quantum chaos. Analytical estimates for typical entanglement of random pure states restricted to a proper subspace of the full Hilbert space are also established and compared with random matrix theory predictions.

The time correlation functions for a Gaussian wave-packet preparation of the dissipative harmonic oscillator evolving from three initial conditions for the heat bath are calculated and compared with each other for Ohmic heat baths. The three initial distributions for the bath are the factorized, partially factorized, and unfactorized distributions. Explicit analytical formulas are derived and then used to study the effect of the three initial distributions on the subsequent dynamics. We find that the transient behavior does not depend sensitively on the initial condition as long as the initial Gaussian wave function of the system is centered at the equilibrium point. Differences become noticeable as the center of the wave packet is significantly shifted from the equilibrium point. These observations justify to some extent the prevalent use of factorized initial conditions for studying real time quantum dynamics in dissipative systems. The total energy in the system is also calculated for the three initial states and its relation to features in the decay is pointed out.