We study partition of networks into basins of attraction based on a steepest ascent search for the node of highest degree. Each node is associated with, or “attracted” to its neighbor of maximal degree, as long as the degree is increasing. A node that has no neighbors of higher degree is a peak, attracting all the nodes in its basin. Maximally random scale-free networks exhibit different behavior based on their degree distribution exponent $\gamma$: For small $\gamma$ (broad distribution) networks are dominated by a giant basin, whereas for large $\gamma$ (narrow distribution) there are numerous basins, with peaks attracting mainly their nearest neighbors. We derive expressions for the first two moments of the number of basins. We also obtain the complete distribution of basin sizes for a class of hierarchical deterministic scale-free networks that resemble random nets. Finally, we generalize the problem to regular networks and lattices where all degrees are equal, and thus the attractiveness of a node must be determined by an assigned weight, rather than the degree. We derive the complete distribution of basins of attraction resulting from randomly assigned weights in one-dimensional chains.

For singular perturbation problems in dynamical systems, various appropriate singular perturbation methods have been proposed to eliminate secular terms appearing in the naive expansion. For example, the method of multiple time scales, the normal form method, center manifold theory, and the renormalization group method are well known. It is shown that all of the solutions of the reduced equations constructed with those methods are exactly equal to the sum of the most divergent secular terms appearing in the naive expansion. For the proof, a method to construct a perturbation solution which differs from the conventional one is presented, where we make use of the theory of the Lie symmetry group.

We study three-dimensional wave patterns on the surface of a film flowing down a uniformly heated wall. Our starting point is a model of four evolution equations for the film thickness $h$, the interfacial temperature $\theta$, and the streamwise and spanwise flow rates, $q$ and $p$, respectively, obtained by combining a gradient expansion with a weighted residual projection. This model is shown to be robust and accurate in describing the competition between hydrodynamic waves and thermocapillary Marangoni effects for a wide range of parameters. For small Reynolds numbers, i.e., in the “drag-gravity regime,” we observe regularly spaced rivulets aligned with the flow and preventing the development of hydrodynamic waves. The wavelength of the developed rivulet structures is found to closely match the one of the most amplified mode predicted by linear theory. For larger Reynolds numbers, i.e., in the “drag-inertia regime,” the situation is similar to the isothermal case and no rivulets are observed. Between these two regimes we observe a complex behavior for the hydrodynamic and thermocapillary modes with the presence of rivulets channeling quasi-two-dimensional waves of larger amplitude and phase speed than those observed in isothermal conditions, leading possibly to solitarylike waves. Two subregions are identified depending on the topology of the rivulet structures that can be either “ridgelike” or “groovelike.” A regime map is further proposed that highlights the influence of the Reynolds and the Marangoni numbers on the rivulet structures. Interestingly, this map is found to be related to the variations of amplitude and speed of the two-dimensional solitary-wave solutions of the model. Finally, the heat transfer enhancement due to the increase of interfacial area in the presence of rivulet structures is shown to be significant.

A reasonable closure of the Monin-Lundgren hierarchy of equations for many-point probability density functions of velocity for decaying homogeneous isotropic turbulence is achieved to treat the first equation for the one-point velocity distribution. As a result, we are naturally led to a two-parameter family of solutions for the distribution, which are Gaussian.

We study the motion of an overdamped test particle in a static periodic potential lacking spatial symmetry under the influence of periodically modulated $\alpha$-stable (Lévy) type noise. Due to the nonthermal character of the driving noise, the particle exhibits a motion with a preferred direction. The additional periodic modulation of the noise asymmetry changes the behavior of the static “Lévy ratchet.” For the fast rate of the noise asymmetry modulation, the Lévy ratchet behaves like the one driven by the symmetric $\alpha$-stable noise. When the modulation period is larger, the nontrivial effects of the noise asymmetry on the behavior of the Lévy ratchet are visible. In particular, the current inversion is observed in the system at hand. The properties of the Lévy ratchet are studied by use of the robust measures of directionality, which are defined regardless of the type of the stochastic driving.

Here we discuss the annealing behavior of an infinite-range $\pm J$ Ising spin glass in the presence of a transverse field using a zero-temperature quantum Monte Carlo method. Within the simulation scheme, we demonstrate that quantum annealing not only helps finding the ground state of a classical spin glass, but can also help simulating the ground state of a quantum spin glass, in particular, when the transverse field is low, much more efficiently.

Homeostasis allows living organisms to perform in optimal conditions despite ever-changing surroundings. Dynamically, it corresponds to a stable steady state, and so its quality can be judged by the volume of the corresponding basin of attraction and/or the length of the relaxation time. Motivated by the fact that the vast majority of intracellular processes involve enzymatic reactions and, as some people have suggested, models similar to those of Brownian motors can be used to study them, here we introduce a simple Brownian motor model and use it to gain insight into the relation between efficiency and stability properties previously observed in macroscopic systems. For this, we analyze the existence, uniqueness, and stability of the motor’s steady state; study its thermodynamic process variables, their relation, and their dependence on the model parameters; and compare the Brownian motor relaxation time and thermodynamic properties. Finally, since the steady state is unique and globally stable, we discuss our results from the standpoint of the energetic costs of maintaining a homeostatic state with short relaxation times.

Experimental measurements of the specific heat in glass-forming systems are obtained from the linear response to either slow cooling (or heating) or to oscillatory perturbations with a given frequency about a constant temperature. The latter method gives rise to a complex specific heat with the constraint that the zero frequency limit of the real part should be identified with thermodynamic measurements. Such measurements reveal anomalies in the temperature dependence of the specific heat, including the so called “specific heat peak” in the vicinity of the glass transition. The aim of this paper is to provide theoretical explanations of these anomalies in general and a quantitative theory in the case of a simple model of glass formation. We first present interesting simulation results for the specific heat in a classical model of a binary mixture glass former. We show that in addition to the formerly observed specific heat peak there is a second peak at lower temperatures which was not observable in earlier simulations. Second, we present a general relation between the specific heat, a caloric quantity, and the bulk modulus of the material, a mechanical quantity, and thus offer a smooth connection between the liquid and amorphous solid states. The central result of this paper is a connection between the micromelting of clusters in the system and the appearance of specific heat peaks; we explain the appearance of two peaks by the micromelting of two types of clusters. We relate the two peaks to changes in the bulk and shear moduli. We propose that the phenomenon of glass formation is accompanied by a fast change in the bulk and the shear moduli, but these fast changes occur in different ranges of the temperature. Last, we demonstrate how to construct a theory of the frequency dependent complex specific heat, expected from heterogeneous clustering in the liquid state of glass formers. A specific example is provided in the context of our model for the dynamics of glycerol. We show that the frequency dependence is determined by the same $\alpha$-relaxation mechanism that operates when measuring the viscosity or the dielectric relaxation spectrum. The theoretical frequency dependent specific heat agrees well with experimental measurements on glycerol. We conclude the paper by stating that there is nothing universal about the temperature dependence of the specific heat in glass formers—unfortunately, one needs to understand each case by itself.

Mesoscopic theory for soft-matter systems that combines density functional and statistical field theory is derived by a systematic coarse-graining procedure. For particles interacting with spherically symmetric potentials of arbitrary form, the grand-thermodynamic potential consists of two terms. The first term is associated with microscopic length-scale fluctuations, and has the form of the standard density functional. The second term is associated with mesoscopic length-scale fluctuations, and has the form known from the statistical field theory. For the correlation function between density fluctuations in mesoscopic regions, a pair of equations similar to the Ornstein-Zernicke equation with a new closure is obtained. In the special case of weak ordering on the mesoscopic length scale, the theory takes a form similar to either the Landau-Ginzburg-Wilson (LGW) or the Landau-Brazovskii (LB) field theory, depending on the form of the interaction potential. Microscopic expressions for the phenomenological parameters that appear in the Landau-type theories are obtained. Within the framework of this theory, we obtain a universal sequence of phases: disordered, bcc, hexagonal, lamellar, inverted hexagonal, inverted bcc, disordered, for increasing density well below the close-packing density. The sequence of phases agrees with experimental observations and with simulations of many self-assembling systems. In addition to the above phases, more complex phases may appear depending on the interaction potentials. For a particular form of the short-range attraction long-range repulsion potential, we find the bicontinuous gyroid phase ($Ia3d$ symmetry) that may be related to a network-forming cluster of colloids in a mixture of colloids and nonadsorbing polymers.

Using Langevin dynamics simulations, we investigate the influence of polymer-pore interactions on the dynamics of biopolymer translocation through nanopores. We find that an attractive interaction can significantly change the translocation dynamics. This can be understood by examining the three components of the total translocation time $\tau \approx {\tau }_1+{\tau }_2+{\tau }_3$ corresponding to the initial filling of the pore, transfer of polymer from the cis side to the trans side, and emptying of the pore, respectively. We find that the dynamics for the last process of emptying of the pore changes from nonactivated to activated in nature as the strength of the attractive interaction increases, and ${\tau }_3$ becomes the dominant contribution to the total translocation time for strong attraction. This leads to nonuniversal dependence of $\tau$ as a function of driving force and chain length. Our results are in good agreement with recent experimental findings, and provide a plausible explanation for the different scaling behavior observed in solid state nanopores vs that for the natural $\alpha$-hemolysin channel.

A Monte Carlo study on ultrasound backscattering by red blood cells (RBCs) is presented for three-dimensional (3D) distributions of particles. The cells were treated as classical spherical particles and accordingly, the Boltzmann distribution was considered to describe probability distribution of energy states of a system composed of such particles. The well-known Metropolis algorithm can generate configurations according to that probability distribution and therefore, was employed in this study to simulate some realizations of both nonaggregating and aggregating RBCs. The study of nonaggregating particles was motivated to compare simulations with existing experimental results and consequently, to validate the model. In the case of aggregating RBCs, the interaction potential between cells was modeled with the Morse potential and the frequency-dependent backscattering coefficient (BSC) was investigated at different hematocrits ($H$, particle volume fractions). The impact of aggregation potential on the spectral slope (SS) was also evaluated. It is shown that BSC increased as the magnitude of aggregating potential was raised and the effect was more pronounced at higher hematocrits. Moreover, spectral slopes at nonaggregating and low aggregating conditions were found to be around 4, which is consistent with the Rayleigh scattering theory. However, it had diminished significantly, particularly at higher hematocrits as the magnitude of the attractive potential energy was raised. For instance, at $H=40\%$ SS dropped from 4.04 for nonaggregating particles to 3.62 at the highest aggregating potential considered in this study. Our results suggest that this 3D model is capable of reflecting the effects of RBC aggregation on BSC and SS.

An important problem in unsupervised data clustering is how to determine the number of clusters. Here we investigate how this can be achieved in an automated way by using interrelation matrices of multivariate time series. Two nonparametric and purely data driven algorithms are expounded and compared. The first exploits the eigenvalue spectra of surrogate data, while the second employs the eigenvector components of the interrelation matrix. Compared to the first algorithm, the second approach is computationally faster and not limited to linear interrelation measures.

Adequate clay minerals considerably affect the macroscopic mechanical behavior of water even at concentrations of a few percent. Thus when $2\mathrm{wt.}\%$ laponite clay mineral nanoparticles are added to water, the resulting colloidal suspension after some time takes on the semisolid characteristics of a jellylike material at room temperature. Cold neutron time-of-flight spectroscopy data are in agreement with the assumption that notwithstanding this macroscopic change, the mobility of the water molecules on intermolecular and intramolecular length scales remains largely unaffected. This observation is discussed in the context of the properties and the role of water in different more or less dilute ionic environments. The result contributes to the ongoing debate of the properties and role of water in living cells.

We investigate the dynamical states of a two-dimensional network of Hindmarsh-Rose spiking neurons, in the vicinity of the current threshold where the single neuron becomes active. Each neuron is electrically coupled with neurons in its close neighborhood. The existence of multistable synchronization states is established and discussed. We also show that, provided adequate initial conditions, the collective behavior is able to keep the network in activity, even for current values far below the activity threshold of the single neuron. A phase diagram of the different network states is presented for a large interval of the coupling-current parameter space.

We derive an analytical expression showing that the fragility of a supercooled liquid is a result of (i) a thermodynamic term depending on change in configurational entropy and (ii) a kinetic term depending on change in the nonexponentiality or “stretching” of the relaxation function, as quantified by the exponent $\beta$ of the Kohlrausch-Williams-Watts (KWW) relaxation function. Our expression indicates that there is not a direct correlation between the non-Arrhenius scaling of liquid viscosity and the nonexponential nature of glassy relaxation. Rather, the temperature dependence of the stretching exponent $\beta$ provides a lower limit for fragility, which can be increased through changes in the configurational entropy. Our result explains the apparent contradiction between those researchers showing a correlation between $\beta$ and fragility and those who question such a correlation due to the spread of the data.

Contact processes on complex networks are a recent subject of study in nonequilibrium statistical physics and they are also important to applied fields such as epidemiology and computer and communication networks. A basic issue concerns finding an optimal strategy for spreading. We provide a universal strategy that, when a basic quantity in the contact process dynamics, the contact probability determined by a generic function of its degree $W\left(k\right)$, is chosen to be inversely proportional to the node degree, i.e., $W\left(k\right)\sim k^{-1}$, spreading can be maximized. Computation results on both model and real-world networks verify our theoretical prediction. Our result suggests the determining role played by small-degree nodes in optimizing spreading, in contrast to the intuition that hub nodes are important for spreading dynamics on complex networks.

We study dynamic phenomena in a large social network of nearly $3\times {10}^4$ individuals who interact in the large virtual world of a massive multiplayer online role playing game. On the basis of a database received from the online game server, we examine the structure of the friendship network and human dynamics. To investigate the relation between networks of acquaintances in virtual and real worlds, we carried out a survey among the players. We show that, even though the virtual network did not develop as a growing graph of an underlying network of social acquaintances in the real world, it influences it. Furthermore we find very interesting scaling laws concerning human dynamics. Our research shows how long people are interested in a single task and how much time they devote to it. Surprisingly, exponent values in both cases are close to $-1$. We calculate the activity of individuals, i.e., the relative time daily devoted to interactions with others in the artificial society. Our research shows that the distribution of activity is not uniform and is highly correlated with the degree of the node, and that such human activity has a significant influence on dynamic phenomena, e.g., epidemic spreading and rumor propagation, in complex networks. We find that spreading is accelerated (an epidemic) or decelerated (a rumor) as a result of superspreaders’ various behavior.

The agreement of past measurements of elastodynamic scattering fidelity with predictions based on random matrix theory, even in bodies with ostensibly regular ray dynamics, has motivated further measurements with a precision capable of detecting fine deviations. Two aluminum blocks are studied, one rectangular, one irregular. As previously, it is found that fidelity decays more slowly in the irregular object. It is further shown that the time dependence of that decay corresponds well with predictions based on random matrix theory and the Gaussian orthogonal ensemble and that predictions based on Poissonian model statistics do not correspond. In some cases deviations from theory are seen that are consistent with partially broken reflection symmetries and inhomogeneous dissipation.

We performed large eddy simulations (LES) of the turbulent natural convection of an electrically conductive fluid (water with 7% ${\mathrm{Na}}_2\mathrm{S}{\mathrm{O}}_4$ electrolyte solution) in a moderate (4:4:1) aspect ratio enclosure heated from below and cooled from above and subjected to external nonuniformly distributed electromagnetic fields. Different configurations with permanent magnets (located under the lower thermally active wall, $B_0=1\mathrm{T}$) and different strengths of imposed dc electric currents ($I=0–10\mathrm{A}$) were compared to the case of pure thermal convection in the turbulent regime, $\mathrm{Ra}={10}^7$, $\mathrm{Pr}=7$. It is demonstrated that the electromagnetic forcing of the boundary layers caused significant reorganization of flow and turbulence structures producing significant enhancement of the wall-heat transfer (up to 188% for a configuration with 35 magnets and an applied dc current of $10\mathrm{A}$).

Particle sedimentation in the vicinity of a fixed horizontal vortex with time-dependent intensity can be chaotic, provided gravity is sufficient to displace the particle cloud while the vortex is off or weak. This “stretch, sediment, and fold” mechanism is close to the so-called blinking vortex effect, which is responsible for chaotic transport of perfect tracers, except that in the present case the vortex motion is replaced by gravitational settling. In the present work this phenomenon is analyzed for heavy Stokes particles moving under the sole effect of gravity and of a linear drag. The vortex is taken to be a fixed isolated point vortex, the intensity of which varies under the effect of either boundary conditions or volume force. When the unsteadiness of the vortex is weak and the free-fall velocity is of the order of the fluid velocity, and the particle response time is small, the particle motion equation can be written asymptotically as a perturbed Hamiltonian system, the phase portrait of which displays a homoclinic trajectory. A homoclinic bifurcation is therefore likely to occur, and the contribution of particle inertia to the occurrence of this bifurcation is analyzed asymptotically by using Melnikov’s method.