We elucidate the case in which the Ablowitz-Ladik (AL)-type discrete nonlinear Schrödinger equation (NLSE) on simple networks (e.g., star graphs and tree graphs) becomes completely integrable just as in the case of a simple one-dimensional (1D) discrete chain. The strength of cubic nonlinearity is different from bond to bond, and networks are assumed to have at least two semi-infinite bonds with one of them working as an incoming bond. The present work is a nontrivial extension of our preceding one [Sobirov et al., Phys. Rev. E 81, 066602 (2010)] on the continuum NLSE to the discrete case. We find (1) the solution on each bond is a part of the universal (bond-independent) AL soliton solution on the 1D discrete chain, but it is multiplied by the inverse of the square root of bond-dependent nonlinearity; (2) nonlinearities at individual bonds around each vertex must satisfy a sum rule; and (3) under findings 1 and 2, there exist an infinite number of constants of motion. As a practical issue, with the use of an AL soliton injected through the incoming bond, we obtain transmission probabilities inversely proportional to the strength of nonlinearity on the outgoing bonds.

The profile of the self-preserving far wake with zero net momentum depends on the effective turbulent viscosity coefficient. The current model is based on the assumption of uniform viscosity in the wake cross section. It predicts the self-similar shape of the wake where the width $W$ depends on the distance $z$ from the body as $W\propto z^{1/5}$ for the axisymmetric case (or $z^{1/4}$ for the plane case). The observed wake width, however, demonstrates the dependence $W\propto z^{\alpha }$ (where $\alpha ⩽1/5$). We generalize the model of a self-preserving far wake for the case of the turbulent viscosity coefficient depending on the radius. Additional integrals of motion allow a new family of self-similar profiles with $\alpha ⩽1/5$.

We apply the Hamilton-Jacobi equation (HJE) formalism to solve the dynamics of the chemical master equation (CME). We found exact analytical expressions (in large system-size limit) for the probability distribution, including explicit expression for the dynamics of variance of distribution. We also give the solution for some simple cases of the model with time-dependent rates. We derived the results of the Van Kampen method from the HJE approach using a special ansatz. Using the Van Kampen method, we give a system of ordinary differential equations (ODEs) to define the variance in a two-dimensional case. We performed numerics for the CME with stationary noise. We give analytical criteria for the disappearance of bistability in the case of stationary noise in one-dimensional CMEs.

From particle simulations of a sheared frictional granular gas, we show that the Coulomb friction can have dramatic effects on orientational correlation as well as on both the translational and angular velocity distribution functions even in the Boltzmann (dilute) limit. The dependence of orientational correlation on friction coefficient ($\mu$) is found to be nonmonotonic, and the Coulomb friction plays a dual role of enhancing or diminishing the orientational correlation, depending on the value of the tangential restitution coefficient (which characterizes the roughness of particles). From the sticking limit (i.e., with no sliding contact) of rough particles, decreasing the Coulomb friction is found to reduce the density and spatial velocity correlations which, together with diminished orientational correlation for small enough $\mu$, are responsible for the transition from non-Gaussian to Gaussian distribution functions in the double limit of small friction ($\mu \to 0$) and nearly elastic particles ($e\to 1$). This double limit in fact corresponds to perfectly smooth particles, and hence the Maxwellian (Gaussian) is indeed a solution of the Boltzmann equation for a frictional granular gas in the limit of elastic collisions and zero Coulomb friction at any roughness. The high-velocity tails of both distribution functions seem to follow stretched exponentials even in the presence of Coulomb friction, and the related velocity exponents deviate strongly from a Gaussian with increasing friction.

Evidence for the parity-breaking nature of the magnetic buoyancy instability in a stably stratified gas is reported. In the absence of rotation, no helicity is produced, but the nonhelical state is found to be unstable to small helical perturbations during the development of the instability. The parity-breaking nature of this magnetohydrodynamic instability appears to be the first of its kind and has properties similar to those in chiral symmetry breaking in biochemistry. Applications to the production of mean fields in galaxy clusters are discussed.

Conformational fluctuations of a protein molecule are important to its function, and it is known that environmental molecules, such as water molecules, ions, and ligand molecules, significantly affect the function by changing the conformational fluctuations. However, it is difficult to systematically understand the role of environmental molecules because intermolecular interactions related to the conformational fluctuations are complicated. To identify important intermolecular interactions with regard to the conformational fluctuations, we develop herein (i) distance-independent and (ii) distance-dependent perturbation analyses of the intermolecular interactions. We show that these perturbation analyses can be realized by performing (i) a principal component analysis using conditional expectations of truncated and shifted intermolecular potential energy terms and (ii) a functional principal component analysis using products of intermolecular forces and conditional cumulative densities. We refer to these analyses as intermolecular perturbation analysis (IPA) and distance-dependent intermolecular perturbation analysis (DIPA), respectively. For comparison of the IPA and the DIPA, we apply them to the alanine dipeptide isomerization in explicit water. Although the first IPA principal components discriminate two states (the $\alpha$ state and PPII (polyproline II) $+$ $\beta$ states) for larger cutoff length, the separation between the PPII state and the $\beta$ state is unclear in the second IPA principal components. On the other hand, in the large cutoff value, DIPA eigenvalues converge faster than that for IPA and the top two DIPA principal components clearly identify the three states. By using the DIPA biplot, the contributions of the dipeptide-water interactions to each state are analyzed systematically. Since the DIPA improves the state identification and the convergence rate with retaining distance information, we conclude that the DIPA is a more practical method compared with the IPA. To test the feasibility of the DIPA for larger molecules, we apply the DIPA to the ten-residue chignolin folding in explicit water. The top three principal components identify the four states (native state, two misfolded states, and unfolded state) and their corresponding eigenfunctions identify important chignolin-water interactions to each state. Thus, the DIPA provides the practical method to identify conformational states and their corresponding important intermolecular interactions with distance information.

The exact solution for a reaction-diffusion front propagating in a heterogeneous system of discrete, point-like sources is obtained without resorting to a representation of the sources by a spatially continuous function. When the reaction time is smaller than the characteristic diffusion time between neighboring sources, the front speed predicted by this discrete source model differs from the continuum theory based on the spatial averaging of the heterogeneities. Furthermore, when the sources are regularly distributed in space, discreteness introduces a limit and propagation beyond this limit is only possible in a system with randomly distributed sources via local fluctuations of the concentration. The discrete regime of front propagation is observed experimentally in suspensions of iron particles burning in oxygen-xenon mixtures.

The dynamics of two-dimensional (2D) radially spreading growth fronts of Vero cell colonies was investigated utilizing two types of colonies, namely type I starting from clusters with a small number of cells, which initially exhibited arbitrary-shaped rough growth fronts and progressively approached quasicircular ones as the cell population increased; and type II colonies, starting from a relatively large circular three-dimensional (3D) cell cluster. For large cell population colonies, the fractal dimension of the fronts was $D_F=1.20\pm 0.05$. For low cell populations, the mean colony radius increased exponentially with time, but for large ones the constant radial front velocity $0.20\pm 0.02$ $\mu$m min${}^{-1}$ was reached. Colony spreading was accompanied by changes in both cell morphology and average size, and by the formation of very large cells, some of them multinuclear. Therefore the heterogeneity of colonies increased and local driving forces that set in began to influence the 2D growth front kinetics. The retardation effect related to the exponential to constant radial front velocity transition was assigned to a number of possible interferences including the cell duplication and 3D growth in the bulk of the colony. The dynamic scaling analysis of overhang-corrected rough colony fronts, after arc-radius coordinate system transformation, resulted in roughness exponent $\alpha$ $=$ $0.50\pm 0.05$ and growth exponent $\beta$ $=$ $0.32\pm 0.04$, for arc lengths greater than 100 $\mu$m. This set of scaling exponents agreed with that predicted by the Kardar, Parisi, and Zhang continuous equation. For arc lengths shorter than 2–3 cell diameters, the value $\alpha$ $=$ $0.85\pm 0.05$ would be related to a cell front roughening caused by temporarily membrane deformations occasionally interfered by cell proliferation.

An Ising-like model of proteins is used to investigate the mechanical unfolding of the green fluorescent protein along different directions. When the protein is pulled from its ends, we recover the major and minor unfolding pathways observed in experiments. Upon varying the pulling direction, we find the correct order of magnitude and ranking of the unfolding forces. Exploiting the direction dependence of the unfolding force at equilibrium, we propose a force sensor whose luminescence depends on the applied force.

A unique coarse-grained modeling scheme that combines a systematic, solvent-free multiscale coarse-graining algorithm for a complex macromolecule with an existing coarse-grained solvent model is proposed. We show that this procedure efficiently and reliably describes the interactions for complex macromolecules, using the specific example of dendrimers binding phenanthrenes in water. The experimentally measured binding capacity is predicted by the unique coarse-grained modeling approach; the conditions for this simulation are beyond what could be reasonably simulated with an all-atom molecular dynamics simulation.

Community structure exists in many real-world networks and has been reported being related to several functional properties of the networks. The conventional approach was partitioning nodes into communities, while some recent studies start partitioning links instead of nodes to find overlapping communities of nodes efficiently. We extended the map equation method, which was originally developed for node communities, to find link communities in networks. This method is tested on various kinds of networks and compared with the metadata of the networks, and the results show that our method can identify the overlapping role of nodes effectively. The advantage of this method is that the node community scheme and link community scheme can be compared quantitatively by measuring the unknown information left in the networks besides the community structure. It can be used to decide quantitatively whether or not the link community scheme should be used instead of the node community scheme. Furthermore, this method can be easily extended to the directed and weighted networks since it is based on the random walk.

Tunneling essentially different from instanton-type tunneling, say noninstanton tunneling, is studied from both semiclassical and quantum viewpoints. Taking a periodically perturbed rounded-off step potential for which the instanton-type tunneling is substantially prohibited, we analyze change of the tunneling probability with change of the perturbation frequency based on the stable-unstable manifold-guided tunneling (SUMGT) theory, which we have recently introduced. In the large and small limits of the frequency, the tunneling rate rapidly decays, but it is markedly enhanced in an intermediate range. We will also make a quantum interpretation of the noninstanton tunneling by using an exactly solvable model—a periodically perturbed right-angled step potential. Analysis with this model shows that SUMGT is considered as a sort of photoassisted tunneling through a large energy gap induced with absorbing a huge number quanta, which is completely different from the instanton-type tunneling. Both approaches from the semiclassical and quantum viewpoints complement each other to cause a better understanding of noninstanton tunneling.

Entrainment by a rotating magnetic field of a ferrofluid contained in a cylinder is studied on the basis of spin-diffusion theory. The equations for flow velocity and spin velocity, coupled to Maxwell's equations of magnetostatics, are solved in first-harmonic approximation under the assumption that the magnetic field is small compared to the saturation magnetization. The solution leads to a coupled set of nonlinear integral equations, which can be solved numerically by iteration in a recursive scheme by use of the analytic lowest order perturbation theory solution as the initial state. At a critical applied field, the recursive scheme shows bifurcation. At sufficiently high field, the solution with the lower rate of dissipation shows flow in the direction opposite to the rotating applied field.

At the point of pinch-off of an underwater air bubble, the speed of water rushing in diverges. Previous studies that assumed radial flow throughout showed that the local axial shape is two smoothly connected, slender cones that transition very slowly (logarithmically) to a cylindrical segment. Our simulations show that even with initially radial flow, a transient vertical flow develops with comparable speeds. Bernoulli pressure draws water into the singularity region while incompressibility forces it away from the neck minimum, generating significant vertical flows that rapidly slenderize and symmetrize the collapse region. This transition is due to a different mechanism, occurring much faster than previously expected. Vertical flows dictate the neck shape evolution.

We present phase diagrams of the single red blood cell and biconcave capsule dynamics in dilute suspension using three-dimensional numerical simulations. The computational geometry replicates an in vitro linear shear flow apparatus. Our model includes all essential properties of the cell membrane, namely, the resistance against shear deformation, area dilatation, and bending, as well as the viscosity difference between the cell interior and suspending fluids. By considering a wide range of shear rate and interior-to-exterior fluid viscosity ratio, it is shown that the cell dynamics is often more complex than the well-known tank-treading, tumbling, and swinging motion and is characterized by an extreme variation of the cell shape. As a result, it is often difficult to clearly establish whether the cell is swinging or tumbling. Identifying such complex shape dynamics, termed here as “breathing” dynamics, is the focus of this article. During the breathing motion at moderate bending rigidity, the cell either completely aligns with the flow direction and the membrane folds inward, forming two cusps, or it undergoes large swinging motion while deep, craterlike dimples periodically emerge and disappear. At lower bending rigidity, the breathing motion occurs over a wider range of shear rates, and is often characterized by the emergence of a quad-concave shape. The effect of the breathing dynamics on the tank-treading-to-tumbling transition is illustrated by detailed phase diagrams which appear to be more complex and richer than those of vesicles. In a remarkable departure from the vesicle dynamics, and from the classical theory of nondeformable cells, we find that there exists a critical viscosity ratio below which the transition is independent of the viscosity ratio, and dependent on shear rate only. Further, unlike the reduced-order models, the present simulations do not predict any intermittent dynamics of the red blood cells.

The properties of solitary and shock structures associated with nonlinear ion modes in a dense plasma with strongly coupled nondegenerate ions and degenerate electron fluids are presented. For this purpose, we have used the viscoelastic fluid model for the ions, the inertialess electron momentum equation with weakly and ultrarelativistic pressure laws for the degenerate electron fluids, and Poisson's equation to derive the Burgers and Kortweg–de Vries equations. Possible stationary solutions of the latter are the shock and solitary structures, respectively. It is found that the speed, amplitude, and width of the shock and solitary waves critically depend on the strong coupling between ions and electron degeneracy effects. The relevance of our investigation to the role of localized excitations in dense astrophysical objects is briefly discussed.

Change points in time series are perceived as isolated singularities where two regular trends of a given signal do not match. The detection of such transitions is of fundamental interest for the understanding of the system’s internal dynamics or external forcings. In practice observational noise makes it difficult to detect such change points in time series. In this work we elaborate on a Bayesian algorithm to estimate the location of the singularities and to quantify their credibility. We validate the performance and sensitivity of our inference method by estimating change points of synthetic data sets. As an application we use our algorithm to analyze the annual flow volume of the Nile River at Aswan from 1871 to 1970, where we confirm a well-established significant transition point within the time series.

The notion of group entropy is proposed. It enables the unification and generaliztion of many different definitions of entropy known in the literature, such as those of Boltzmann-Gibbs, Tsallis, Abe, and Kaniadakis. Other entropic functionals are introduced, related to nontrivial correlation laws characterizing universality classes of systems out of equilibrium when the dynamics is weakly chaotic. The associated thermostatistics are discussed. The mathematical structure underlying our construction is that of formal group theory, which provides the general structure of the correlations among particles and dictates the associated entropic functionals. As an example of application, the role of group entropies in information theory is illustrated and generalizations of the Kullback-Leibler divergence are proposed. A new connection between statistical mechanics and zeta functions is established. In particular, Tsallis entropy is related to the classical Riemann zeta function.

The thermodynamic behavior of a strongly charged polyelectrolyte chain immersed in a salt-free solution is studied using replica-exchange Monte-Carlo simulations. The results reveal that the chain can assume a variety of conformations, and it undergoes two phase transitions upon cooling. The first transition is identified as a continuous counterion condensation transition while the second one as a first-order coil-globule transition. In the globular state, the counterions and the charged chain segments are densely packed forming a three-dimensional Wigner crystal.

We explore how ligand-receptor binding kinetics can be controlled by tethering the receptor to the end of a flexible polymer. The tether confines the diffusive motion of the receptor thus influencing the rate at which it captures ligands that are free in solution. We compute steady-state collision rates between ligand and receptor for this “tethered-capture” mechanism using a combination of analytic and numerical techniques. In doing so, we uncover a dimensionless control parameter, the “opacity,” that determines under what conditions and to what extent a tether regulates the ligand-receptor collision rate. We compute the opacity for a number of different tethering scenarios that appear in biology and use these results to predict the affect of changing the length and flexibility of the tether on the rate at which ligands are captured from solution.