Motivated by the observed coordination of nearby beating cilia, we use a scale model experiment to show that hydrodynamic interactions can cause synchronization between rotating paddles driven at constant torque in a very viscous fluid. Synchronization is only observed when the shafts supporting the paddles have some flexibility. The phase difference in the synchronized state depends on the symmetry of the paddles. We use the method of regularized Stokeslets to model the paddles and find excellent agreement with the experimental observations. We also use a simple analytic theory based on far-field approximations to derive scaling laws for the synchronization time as a function of paddle separation.

The rotationally covariant renormalization group equations of motion for the density wave amplitudes in the phase field crystal model are shown to follow from a dynamical equation driven by an effective free energy density that we derive. We show that this free energy can be written purely as a function of the strain tensor and thence derive the corresponding equations governing the nonlinear elastic response.

Nonlinear water waves interacting with quasi-one-dimensional nonuniformly periodic bed profiles are studied numerically in the deep-water regime with the help of approximate equations for envelopes of the forward and backward waves. Spontaneous formation of localized two-dimensional wave structures is observed in the numerical experiments, which looks essentially as a wave collapse.

We investigate the role of the learning rate in a Kuramoto Model of coupled phase oscillators in which the coupling coefficients dynamically vary according to a Hebbian learning rule. According to the Hebbian theory, a synapse between two neurons is strengthened if they are simultaneously coactive. Two stable synchronized clusters in antiphase emerge when the learning rate is larger than a critical value. In such a fast learning scenario, the network eventually constructs itself into an all-to-all coupled structure, regardless of initial conditions in connectivity. In contrast, when learning is slower than this critical value, only a single synchronized cluster can develop. Extending our analysis, we explore whether self-development of neuronal networks can be achieved through an interaction between spontaneous neural synchronization and Hebbian learning. We find that self-development of such neural systems is impossible if learning is too slow. Finally, we demonstrate that similar to the acquisition and consolidation of long-term memory, this network is capable of generating and remembering stable patterns.

Finite-dimensional wave turbulence refers to the chaotic dynamics of interacting wave “clusters” consisting of finite number of connected wave triads with exact three-wave resonances. We examine this phenomenon using the example of atmospheric planetary (Rossby) waves. It is shown that the dynamics of the clusters is determined by the types of connections between neighboring triads within a cluster; these correspond to substantially different scenarios of energy flux between different triads. All the possible cases of the energy cascade termination are classified. Free and forced chaotic dynamics in the clusters are investigated: due to the huge fluctuations of the energy exchange between resonant triads these two types of evolution have a lot in common. It is confirmed that finite-dimensional wave turbulence in finite wave systems is fundamentally different from kinetic wave turbulence in infinite systems; the latter is described by wave-kinetic equations that account for interactions with overlapping quasiresonances of finite amplitude waves. The present results are directly applicable to finite-dimensional wave turbulence in any wave system in finite domains with three-mode interactions as encountered in hydrodynamics, astronomy, plasma physics, chemistry, medicine, etc.

A graphic processing unit (GPU) implementation of the multicomponent lattice Boltzmann equation with multirange interactions for soft-glassy materials [“glassy” lattice Boltzmann (LB)] is presented. Performance measurements for flows under shear indicate a GPU/CPU speed up in excess of 10 for ${1024}^2$ grids. Such significant speed up permits to carry out multimillion time-steps simulations of ${1024}^2$ grids within tens of hours of GPU time, thereby considerably expanding the scope of the glassy LB toward the investigation of long-time relaxation properties of soft-flowing glassy materials.

The graph theoretic concept of maximal independent set arises in several practical problems in computer science as well as in game theory. A maximal independent set is defined by the set of occupied nodes that satisfy some packing and covering constraints. It is known that finding minimum and maximum-density maximal independent sets are hard optimization problems. In this paper, we use cavity method of statistical physics and Monte Carlo simulations to study the corresponding constraint satisfaction problem on random graphs. We obtain the entropy of maximal independent sets within the replica symmetric and one-step replica symmetry breaking frameworks, shedding light on the metric structure of the landscape of solutions and suggesting a class of possible algorithms. This is of particular relevance for the application to the study of strategic interactions in social and economic networks, where maximal independent sets correspond to pure Nash equilibria of a graphical game of public goods allocation.

We study the nonequilibrium entropy of the heat transport problem by performing molecular dynamics simulations of a quasi-one-dimensional gas of hard disks in the steady state. The entropy density, flux, and production rate, associated with the entropy balance of the system, are obtained from the numerically measured velocity distributions, based on the kinetic theory analysis of the Boltzmann entropy. We obtain an equilibriumlike Clausius relation from the computer experiments which, in turn, fulfills the generalized Gibbs relation for spatially inhomogeneous states.

We present numerical simulations and experiments on the horizontal transport and segregation of binary granular mixtures with different sizes and/or different densities in a vertically vibrated container with a sawtooth-shaped base. The larger particles migrate to the positive or negative end of the container, depending on the ratios of the diameters and the densities of two kinds of particles, the vibrating frequencies, and the accelerations. In particular, horizontal segregation occurred even if all the particles have the same size but different densities.

By calculating the linear response of packings of soft frictionless disks to quasistatic external perturbations, we investigate the critical scaling behavior of their elastic properties and nonaffine deformations as a function of the distance to jamming. Averaged over an ensemble of similar packings, these systems are well described by elasticity, while in single packings we determine a diverging length scale ${\ell }^{\ast }$ up to which the response of the system is dominated by the local packing disorder. This length scale, which we observe directly, diverges as $1/\Delta z$, where $\Delta z$ is the difference between contact number and its isostatic value, and appears to scale identically to the length scale which had been introduced earlier in the interpretation of the spectrum of vibrational modes. It governs the crossover from isostatic behavior at the small scale to continuum behavior at the large scale; indeed we identify this length scale with the coarse graining length needed to obtain a smooth stress field. We characterize the nonaffine displacements of the particles using the displacement angle distribution, a local measure for the amount of relative sliding, and analyze the connection between local relative displacements and the elastic moduli.

We have recently developed a kinetic model for the nucleation mechanism of protein folding (NMPF) in terms of ternary nucleation by using the first passage time analysis. A protein was considered as a random heteropolymer consisting of hydrophobic, hydrophilic (some of which are negatively or positively ionizable), and neutral beads. The main idea of the NMPF model consisted of averaging the dihedral potential in which a selected residue is involved over all possible configurations of all neighboring residues along the protein chain. The combination of the average dihedral, effective pairwise (due to Lennard-Jones-type and electrostatic interactions), and confining (due to the polymer connectivity constraint) potentials gives rise to an overall potential around the cluster that, as a function of the distance from the cluster center, has a double-well shape. This allows one to evaluate the protein folding time. In the original NMPF model hydrogen bonding was not taken into account explicitly. To improve the NMPF model and make it more realistic, in this paper we modify our (previously developed) probabilistic hydrogen bond model and combine it with the former. Thus, a contribution due to the disruption of hydrogen bond networks around the interacting particles (cluster of native residues and residue in the protein unfolded part) appears in the overall potential field around a cluster. The modified model is applied to the folding of the same model proteins that were examined in the original model: a short protein consisting of 124 residues (roughly mimicking bovine pancreatic ribonuclease) and a long one consisting of 2500 residues (as a representative of large proteins with superlong polypeptide chains), at $p\text{H}=8.3$, 7.3, and 6.3. The hydrogen bond contribution now plays a dominant role in the total potential field around the cluster (except for very short distances thereto where the repulsive energy tends to infinity). It is by an order of magnitude stronger for hydrophobic residues than for hydrophilic ones. The range of “residue-cluster” distances, at which the hydrogen bond effect exists, is twice as long for hydrophobic residues as for hydrophilic ones.

Symbolic dynamics offers a powerful technique to relate the structure and dynamics of complex networks. We contrast the predictions of two methods of symbolic dynamics for the analysis of monotonic networks suggested by models of genetic control systems.

We present a method to derive implicit solvent models of electrolyte solutions from all-atom descriptions; providing analytical expressions of the thermodynamic and structural properties of the ions consistent with the underlying explicit solvent representation. Effective potentials between ions in solution are calculated to perform perturbation theory calculations, in order to derive the best possible description in terms of charged hard spheres. Applying this method to NaCl solutions yields excellent agreement with the all-atom model, provided ion association is taken into account.

We present a general framework, applicable to a broad class of random walks on complex networks, which provides a rigorous lower bound for the mean first-passage time of a random walker to a target site averaged over its starting position, the so-called global mean first-passage time (GMFPT). This bound is simply expressed in terms of the equilibrium distribution at the target and implies a minimal scaling of the GMFPT with the network size. We show that this minimal scaling, which can be arbitrarily slow, is realized under the simple condition that the random walk is transient at the target site and independently of the small-world, scale-free, or fractal properties of the network. Last, we put forward that the GMFPT to a specific target is not a representative property of the network since the target averaged GMFPT satisfies much more restrictive bounds.

The main goal of the paper is to find the absolute maximum of the width of the separatrix chaotic layer as function of the frequency of the time-periodic perturbation of a one-dimensional Hamiltonian system possessing a separatrix, which is one of the major unsolved problems in the theory of separatrix chaos. For a given small amplitude of the perturbation, the width is shown to possess sharp peaks in the range from logarithmically small to moderate frequencies. These peaks are universal, being the consequence of the involvement of the nonlinear resonance dynamics into the separatrix chaotic motion. Developing further the approach introduced in the recent paper by Soskin et al. [Phys. Rev. E 77, 036221 (2008)], we derive leading-order asymptotic expressions for the shape of the low-frequency peaks. The maxima of the peaks, including in particular the absolute maximum of the width, are proportional to the perturbation amplitude times either a logarithmically large factor or a numerical, still typically large, factor, depending on the type of system. Thus, our theory predicts that the maximal width of the chaotic layer may be much larger than that predicted by former theories. The theory is verified in simulations. An application to the facilitation of global chaos onset is discussed.

We present an investigation of the statistics of velocity gradient related quantities, in particular energy dissipation rate and enstrophy, along the trajectories of fluid tracers and of heavy/light particles advected by a homogeneous and isotropic turbulent flow. The refined similarity hypothesis (RSH) proposed by Kolmogorov and Oboukhov in 1962 is rephrased in the Lagrangian context and then tested along the particle trajectories. The study is performed on state-of-the-art numerical data resulting from numerical simulations up to ${\text{Re}}_{\lambda }\sim 400$ with ${2048}^3$ collocation points. When particles have small inertia, we show that the Lagrangian formulation of the RSH is well verified for time lags larger than the typical response time ${\tau }_p$ of the particle. In contrast, in the large inertia limit when the particle response time approaches the integral time scale of the flow, particles behave nearly ballistic, and the Eulerian formulation of RSH holds in the inertial range.

We study the response of dry granular materials to external stress using experiment, simulation, and theory. We derive a Ginzburg-Landau functional that enforces mechanical stability and positivity of contact forces. In this framework, the elastic moduli depend only on the applied stress. A combination of this feature and the positivity constraint leads to stress correlations whose shape and magnitude are extremely sensitive to the nature of the applied stress. The predictions from the theory describe the stress correlations for both simulations and experiments semiquantitatively.

We study the statistics of the work done in a zero temperature quench of the coupling constant in the Dicke model describing the interaction between an ensemble of two level systems and a single bosonic mode. When either the final or the initial coupling constants approach the critical coupling ${\lambda }_{\text{c}}$ that separates the normal and superradiant phases of the system, the probability distribution of the work done displays singular behavior. The average work tends to diverge as the initial coupling parameter is brought closer to the critical value ${\lambda }_{\text{c}}$. In contrast, for quenches ending close to criticality, the distribution of work has finite moments but displays a sequence of edge singularities. This contrasting behavior is related to the difference between the processes of compression and expansion of a particle subject to a sudden change in its confining potential. We confirm this by studying in detail the time-dependent statistics of other observables, such as the quadratures of the photons and the total occupation of the bosonic modes.

We present an exact and Monte Carlo renormalization group (MCRG) study of semiflexible polymer chains on an infinite family of the plane-filling (PF) fractals. The fractals are compact, that is, their fractal dimension $d_f$ is equal to 2 for all members of the fractal family enumerated by the odd integer $b\left(3\le b<\infty \right)$. For various values of stiffness parameter $s$ of the chain, on the PF fractals (for $3\le b\le 9$), we calculate exactly the critical exponents $\nu$ (associated with the mean squared end-to-end distances of polymer chain) and $\gamma$ (associated with the total number of different polymer chains). In addition, we calculate $\nu$ and $\gamma$ through the MCRG approach for $b$ up to 201. Our results show that for each particular $b$, critical exponents are stiffness dependent functions, in such a way that the stiffer polymer chains (with smaller values of $s$) display enlarged values of $\nu$, and diminished values of $\gamma$. On the other hand, for any specific $s$, the critical exponent $\nu$ monotonically decreases, whereas the critical exponent $\gamma$ monotonically increases, with the scaling parameter $b$. We reflect on a possible relevance of the criticality of semiflexible polymer chains on the PF family of fractals to the same problem on the regular Euclidean lattices.

A power-law distribution is found in the density profile of reacting systems $A+B\to C+D$ and $2A\to 2C$ under a flow in two and three dimensions. Different densities of reactants $A$ and $B$ are fixed at both ends. For the reaction $A+B$, the concentration of reactants asymptotically decay in space as $x^{-1/2}$ and $x^{-3/4}$ in two dimensions and three dimensions, respectively. For $2A$, it decays as $\text{log}\left(x\right)/x$ in two dimensions. The decay of $A+B$ is explained considering the effect of segregation of reactants in the isotropic case. The decay for $2A$ is explained by the marginal behavior of two-dimensional diffusion. A logarithmic divergence of the diffusion constant with system size is found in two dimensions.