A real-space renormalization transformation is constructed for lattices of nonidentical oscillators with dynamics of the general form $d{\varphi }_k/dt={\omega }_k+g{\sum }_l{f}_{lk}\left({\varphi }_l,{\varphi }_k\right)$. The transformation acts on ensembles of such lattices. Critical properties corresponding to a second-order phase transition toward macroscopic synchronization are deduced. The analysis is potentially exact but relies in part on unproven assumptions. Numerically, second-order phase transitions with the predicted properties are observed as $g$ increases in two structurally different two-dimensional oscillator models. One model has smooth coupling $f_{lk}\left({\varphi }_l,{\varphi }_k\right)=\phi \left({\varphi }_l-{\varphi }_k\right)$, where $\phi \left(x\right)$ is nonodd. The other model is pulse coupled, with $f_{lk}\left({\varphi }_l,{\varphi }_k\right)=\delta \left({\varphi }_l\right)\phi \left({\varphi }_k\right)$. Lower bounds for the critical dimensions for different types of coupling are obtained. For nonodd coupling, macroscopic synchronization cannot be ruled out for any dimension $D\ge 1$, whereas in the case of odd coupling, the well-known result that it can be ruled out for $D<3$ is regained.

Results are presented of an experimental investigation into patterned size segregation of binary granular mixtures in a thin rotating drum that is half full. It is observed that streaks of small particles are formed within regions of large ones where the integer number of streaks is fixed over a range of rotation rate of the drum. Different patterns form in adjacent parameter ranges and the dynamics associated with the exchange between neighboring states is analyzed using angular spatiotemporal diagrams. These help to reveal properties of the merging mechanism for streaks of small particles. We report experimental evidence that the merging of streaks is mediated by the movement of a surplus material in a direction opposite to that of the rotation of the drum. The excess material is distributed throughout the pattern and the extra streak eventually disappears.

We discuss a parameter-free scaling relation that yields a complete data collapse for large classes of nonequilibrium growth processes. We illustrate the power of this scaling relation through various growth models, such as the competitive growth model with random deposition and random deposition with surface diffusion or the restricted solid-on-solid model with different nearest-neighbor height differences, as well as through a deposition model with temperature-dependent diffusion. The scaling relation is compared to the familiar Family-Vicsek relation, and the limitations of the latter are highlighted.

Using nonequilibrium molecular-dynamics simulation, we study the shear-rate dependence of pressure and potential energy in a liquid metal subjected to shear. We show that both thermodynamic properties vary according to a power law ${\dot{\gamma }}^{\beta }$ of the shear rate $\dot{\gamma }$, in which the exponent $\beta$ is a simple linear function of temperature and density. Moreover, we establish that the coefficients for this linear law are the same as those previously obtained for a Lennard-Jones fluid by Ge et al. [Phys. Rev. E 67, 061201 (2003)]. This is a strong indication that these coefficients, as well as the linear law for $\beta$, could be applicable to any atomic fluid. It is also an important step toward the determination of a nonequilibrium equation of state, which would predict the value of pressure and energy of a shearing fluid for any state point and any value of the applied shear rate.

Fick’s law for the diffusion of adsorbed atoms (adatoms) on crystal surfaces below roughening is generalized to account for surface reconstruction. In this case, material parameters vary spatially at the microscale, and the coarse graining for crystal steps via Taylor expansions is not strictly applicable. By invoking elements of the theory of composites in one independent space dimension, we homogenize the microscale description to derive the macroscopic adatom flux from step kinetics. This approach relies on a multiscale expansion for the adatom density. The effective surface diffusivity is determined through appropriate discrete averages of microscale kinetic parameters.

We study the synchronization in the context of network traffic on a $2-d$ communication network with local clustering and geographic separations. The network consists of nodes and randomly distributed hubs where the top five hubs ranked according to their coefficient of betweenness centrality (CBC) are connected by random assortative and gradient mechanisms. For multiple message traffic, messages can trap at the high CBC hubs, and congestion can build up on the network with long queues at the congested hubs. The queue lengths are seen to synchronize in the congested phase. Both complete and phase synchronization are seen, between pairs of hubs. In the decongested phase, the pairs start clearing and synchronization is lost. A cascading master-slave relation is seen between the hubs, with the slower hubs (which are slow to decongest) driving the faster ones. These are usually the hubs of high CBC. Similar results are seen for traffic of constant density. Total synchronization between the hubs of high CBC is also seen in the congested regime. Similar behavior is seen for traffic on a network constructed using the Waxman random topology generator. We also demonstrate the existence of phase synchronization in real internet traffic data.

Previous work has revealed that the synchronizability of a scale-free network tends to be suppressed when its clustering coefficient is increased. We present a theory to explain this phenomenon. Our proposition is that, as the network becomes more strongly clustered, topological loop structure can emerge, generating a set of eigenvalues that are close to zero. As a result, the dynamics of synchronization tends to be dominated by the loop structure. As the clustering coefficient is increased, the size of the dominant loop increases, leading to continuous degradation of the network synchronizability. We provide analysis and numerical evidence to support the proposition and we speculate that the loop structure can provide a platform for controlling dynamical processes on scale-free networks with high clustering coefficients.

The controllability of spatiotemporal chaos is investigated. Contrary to our common sense that there is only one transition point (i.e., a critical control strength) for successful control, we find that actually a transition zone exists, connecting two transition points for the local and global stabilities of the controlled state, respectively. Within the zone, the controllable probability increases from zero to one for random initial conditions. This behavior is found to be very generic and is expected to have a severe consequence in realistic applications in the control of spatiotemporal chaos.

Cross-jet transport of passive scalars in a kinematic model of the meandering laminar two-dimensional incompressible flow which is known to produce chaotic mixing is studied. We develop a method for detecting barriers to cross-jet transport in the phase space which is a physical space for our model. Using tools from the theory of nontwist maps, we construct a central invariant curve and compute its characteristics that may serve as good indicators of the existence of a central transport barrier, its strength, and topology. Computing fractal dimension, length, and winding number of that curve in the parameter space, we study in detail the change in its geometry and its destruction that is caused by local bifurcations and a global bifurcation known as reconnection of separatrices of resonances. Scenarios of reconnection are different for odd and even resonances. The central invariant curves with rational and irrational (noble) values of winding numbers are arranged into hierarchical series which are described in terms of continued fractions. Destruction of central transport barrier is illustrated for two ways in the parameter space: when moving along resonant bifurcation curves with rational values of the winding number and along curves with noble (irrational) values.

The photosensitive chlorine dioxide–iodine–malonic acid reaction-diffusion system has been an experimental paradigm for the study of Turing pattern over the last several years. When subjected to illumination of varied intensity by visible light the patterns undergo changes from spots to stripes, vice versa, and their mixture. We carry out a nonlinear analysis of the underlying model in terms of a Galerkin scheme with finite number of modes to explore the nature of the stability and existence of various modes responsible for the type and crossover of the light-induced patterns.

We derive upper bounds for the number of degrees of freedom of two-dimensional Navier-Stokes turbulence freely decaying from a smooth initial vorticity field $\omega \left(x,y,0\right)={\omega }_0$. This number denoted by $N$ is defined as the minimum dimension such that for $n\ge N$, arbitrary $n$-dimensional balls in phase space centered on the solution trajectory $\omega \left(x,y,t\right)$; for $t>0$, contract under the dynamics of the system linearized about $\omega \left(x,y,t\right)$. In other words, $N$ is the minimum number of greatest Lyapunov exponents whose sum becomes negative. It is found that $N\le C_1\mathrm{Re}$ when the phase space is endowed with the energy norm and $N\le C_2\mathrm{Re}{\left(1+\text{ln}\mathrm{Re}\right)}^{1/3}$ when the phase space is endowed with the enstrophy norm. Here $C_1$ and $C_2$ are constant and Re is the Reynolds number defined in terms of ${\omega }_0$, the system length scale, and the viscosity $\nu$. The linear (or nearly linear) dependence of $N$ on Re is consistent with the estimate for the number of active modes deduced from a recent mathematical bound for the viscous dissipation wave number. This result is in a sharp contrast to the forced case, for which well-known estimates for the Hausdorff dimension $D_H$ of the global attractor scale highly superlinearly with ${\nu }^{-1}$. We argue that the “extra” dependence of $D_H$ on ${\nu }^{-1}$ is not an intrinsic property of the turbulent dynamics. Rather, it is a “removable artifact” brought about by the use of a time-independent forcing as a model for energy and enstrophy injection that drives the turbulence.

We propose a convenient formulation of elemental transport coefficients in chemically reacting and plasma flows locally approaching thermodynamic equilibrium. A set of transport coefficients for elemental diffusion velocities, heat flux, and electric current is introduced. These coefficients relate the transport fluxes with the electric field and with the spatial gradients of elemental fractions, pressure, and temperature. The proposed formalism based on chemical elements and fully symmetric with the classical transport theory based on chemical species, is particularly suitable to model mixing and demixing phenomena due to diffusion of chemical elements. The aim of this work is threefold: to define a simple and rigorous framework suitable for numerical implementation, to allow order of magnitude estimations and qualitative predictions of elemental transport phenomena, and to gain a deeper insight into the physics of chemically reacting flows near local equilibrium.

A two-fluid approach is proposed for direct numerical simulation of particle-laden turbulent flows in two-way coupling where the particle Stokes number is small. An Eulerian velocity field is calculated for the particle phase through a truncated series expansion in terms of the velocity and acceleration of the fluid phase [M. R. Maxey, J. Fluid Mech. 174, 441 (1987)]. This expansion is valid for particles with a sufficiently small Stokes number defined as the ratio of particle time constant to the Kolmogorov time scale. The transport equation of the Eulerian concentration field of particles (particle volume fraction) is solved along with the fluid-phase equations for which the effect of the particles on the fluid phase is taken into account through source terms in the momentum equations. For the assessment purposes, particle-laden decaying isotropic turbulence is studied. The results obtained through the proposed two-fluid approach are compared against those obtained by the trajectory approach in which the particle equations are solved in the Lagrangian framework. It is shown that there is a good agreement between various fluid-phase statistics obtained by these approaches for different small Stokes numbers and mean particle concentrations.

We investigate the assembly of colloidal ellipsoids in ac electric fields. Polystyrene latex ellipsoids with aspect ratios 3.0, 4.3, and 7.6 orient with the applied field and, at sufficient field strengths, interact to form particle chains at an angle with respect to the field. The characteristic chain angle decreases with increasing aspect ratio. The angled chains combine laterally to form an open centered rectangular two-dimensional structures belonging to the $c2mm$ plane group. This chaining and assembly behavior is explained based on calculations of the particle pair interactions explicitly accounting for the electric field and shape of the ellipsoids.

Starting from the Caldeira-Leggett model, we derive the equation describing the quantum Brownian motion, which has been originally proposed by Dekker purely from phenomenological basis containing extra anomalous diffusion terms. This nonperturbative approach yields explicit analytical expressions for the temperature dependence of the diffusion constants. At high temperatures, additional momentum diffusion terms are suppressed and classical Langevin equation can be recovered and at the same time positivity of the density matrix is satisfied. At low temperatures, the diffusion constants have a finite positive value. However, below a certain critical temperature, the master equation does not satisfy the positivity condition as proposed by Dekker.

The self-propelling motion of the flexible ferromagnetic swimmer is described. Necessary symmetry breaking is achieved by the buckling instability at field inversion. The characteristics of self-propulsion are in good agreement with the numerical calculations of the Floquet multipliers for the ferromagnetic filament under the action of ac magnetic field. In the low frequency range the power stroke of self-propelling motion is similar to that used by the unicellular green algae chlamydomonas and in the high frequency region the self-propulsion is due to the undulation waves propagating from the free ends perpendicularly to ac magnetic field.

In this paper, we clearly demonstrate that the electric potential and the magnetic field can contain different information about current sources in three-dimensional conducting media. Expressions for the magnetic fields of electric dipole and quadrupole current sources immersed in an infinite conducting medium are derived, and it is shown that two different point dipole distributions that are electrically equivalent have different magnetic fields. Although measurements of the electric potential are not sufficient to determine uniquely the characteristics of a quadrupolar source, the radial component of the magnetic field can supply the additional information needed to resolve these ambiguities and to determine uniquely the configuration of dipoles required to specify the electric quadrupoles. We demonstrate how the process can be extended to even higher-order terms in an electrically silent series of magnetic multipoles. In the context of a spherical brain source model, it has been mathematically demonstrated that the part of the neuronal current generating the electric potential lives in the orthogonal complement of the part of the current generating the magnetic potential. This implies a mathematical relationship of complementarity between electroencephalography and magnetoencephalography, although the theoretical result in question does not apply to the nonspherical case [G. Dassios, Math. Med. Biol. 25, 133 (2008)]. Our results have important practical applications in cases where electrically silent sources that generate measurable magnetic fields are of interest. Moreover, electrically silent, magnetically active moments of higher order can be useful when cancellation due to superposition of fields can occur, since this situation leads to a substantial reduction in the measurable amplitude of the signal. In this context, information derived from magnetic recordings of electrically silent, magnetically active multipoles can supplement electrical recordings for the purpose of studying the physiology of the brain. Magnetic fields of the electric multipole sources in a conducting medium surrounded by an insulating spherical shell are also presented and the relevance of this calculation to cardiographic and encephalographic experimentation is discussed.

Data assimilation in dynamical networks is intrinsically challenging. A method is introduced for the tracking of heterogeneous networks of oscillators or excitable cells in a nonstationary environment, using a homogeneous model network to expedite the accurate reconstruction of parameters and unobserved variables. An implementation using ensemble Kalman filtering to track the states of the heterogeneous network is demonstrated on simulated data and applied to a mammalian brain network experiment. The approach has broad applicability for the prediction and control of biological and physical networks.

Colloidal particles embedded in the cytoplasm of living mammalian cells have been found to display remarkable heterogeneity in their amplitude of motion. However, consensus on the significance and origin of this phenomenon is still lacking. We conducted experiments on Hmec-1 cells loaded with about 100 particles to reveal the intracellular particle dynamics as a function of both location and time. Central quantity in our analysis is the amplitude $\left(A\right)$ of the individual mean-squared displacement (iMSD), averaged over a short time. Histograms of $A$ were measured, (1) over all particles present at the same time and (2) for individual particles as a function of time. Both distributions showed significant broadening compared to particles in Newtonian liquid, indicating that the particle dynamics varies with both location and time. However, no systematic dependence of $A$ on intracellular location was found. Both the (strong) spatial and (weak) temporal variations were further analyzed by correlation functions of $A$. Spatial cross correlations were rather weak down to interparticle distances of $1\mu \text{m}$, suggesting that the precise intracellular probe distribution is not crucial for observing a dynamic behavior that is representative for the whole cell. Temporal correlations of $A$ decayed at $\sim 10\text{s}$, possibly suggesting an intracellular reorganization at this time scale. These findings imply (1) that both individual particle dynamics and the ensemble averaged behavior in a given cell can be measured if there are enough particles per cell and (2) that the amplitude and power-law exponent of iMSDs can be used to reveal local dynamics. We illustrate this by showing how superdiffusive and subdiffusive behaviors may be hidden under an apparently diffusive global dynamics.

The continuum percolation of circular cylinders has been studied for various values of the aspect ratio $b^{\prime }$. The percolation threshold is shown to have a maximum for $b^{\prime }\approx 2$ when the cylinder length is equal to its diameter. Other quantities such as the average intersection volume and the porosity also possess a maximum for this value.