Dichotomous spin dynamics on a pyramidal hierarchical structure (the Bethe lattice) are studied. The system embodies a number of classes, where a class comprises nodes that are equidistant from the root (head node). Weighted links exist between nodes from the same and different classes. The spin (hereafter state) of the head node is fixed. We solve for the dynamics of the system for different boundary conditions. We find necessary conditions so that the classes eventually repudiate or acquiesce in the state imposed by the head node. The results indicate that to reach unanimity across the hierarchy, it suffices that the bottommost class adopts the same state as the head node. Then the rest of the hierarchy will inevitably comply. This also sheds light on the importance of mass media as a means of synchronization between the topmost and bottommost classes. Surprisingly, in the case of discord between the head node and the bottommost classes, the average state over all nodes inclines towards that of the bottommost class regardless of the link weights and intraclass configurations. Hence the role of the bottommost class is signified.

Minute external perturbations, such as temperature variations, can lead to a creep of the fragile structures that are the granular piles. We report a study, resolved in space and time, of the dynamics associated with the slow compaction of a granular column submitted to thermal cycles. Avoiding the thermal dilations of the container, we observe that the material still creeps, even in the absence of external mechanical perturbations. The latter intrinsic creep of the material exhibits several surprising features: the compaction of the material is not homogeneous along the column height, and sudden collapses involve the entire system even if one could expect arches to screen the mechanical interaction between regions far apart from one another. In addition, the analysis of the dependence of the flow rate on the frequency of the temperature cycles suggests that the creep is mainly induced by the propagation of the temperature profile, slow temperature changes being less efficient than rapid ones in making the material flow.

We study a model for a random walk of two classes of particles ($A$ and $B$). Where both species are present in the same site, the motion of $A$'s takes precedence over that of $B$'s. The model was originally proposed and analyzed in Maragakis et al. [Phys. Rev. E 77, 020103(R) (2008)]; here we provide additional results. We solve analytically the diffusion coefficients of the two species in lattices for a number of protocols. In networks, we find that the probability of a $B$ particle to be free decreases exponentially with the node degree. In scale-free networks, this leads to localization of the $B$'s at the hubs and arrest of their motion. To remedy this, we investigate several strategies to avoid trapping of the $B$'s, including moving an $A$ instead of the hindered $B$, allowing a trapped $B$ to hop with a small probability, biased walk toward non-hub nodes, and limiting the capacity of nodes. We obtain analytic results for lattices and networks, and we discuss the advantages and shortcomings of the possible strategies.

Distinct relationships such as activation, inhibition, cooperation, and competition are not established independently but in a correlated manner in complex systems. Thus the patterns of one type of interaction may reflect the impacts of other classes of interactions, but its quantitative understanding remains to be done. Referring to the plant-pollinator mutualistic networks, here we propose and investigate the structural features of a model bipartite network, in which the mutualistic relationship between two different types of nodes is established under the influence of the compatibility among the nodes of the same type. Interestingly, we find that the degree distributions obtained for extremely broad compatibility distributions are similar to those for a constant compatibility, both of which deviate from those for the Gaussian compatibility distributions. We present the analytic arguments to explain this finding. Also the dependence of the topological similarity of two nodes on their compatibility is illustrated. We discuss the application of our findings to complex systems.

For chaotic scattering on quantum graphs, the semiclassical approximation is exact. We use this fact and employ supersymmetry, the color-flavor transformation, and the saddle-point approximation to calculate the exact expression for the lowest and asymptotic expressions in the Ericson regime for all higher correlation functions of the scattering matrix. Our results agree with those available from the random-matrix approach to chaotic scattering. We conjecture that our results hold universally for quantum-chaotic scattering.

Dissipative solitons show a variety of behaviors not exhibited by their conservative counterparts. For instance, a dissipative soliton can remain localized for a long period of time without major profile changes, then grow and become broader for a short time—explode—and return to the original spatial profile afterward. Here we consider the dynamics of dissipative solitons and the onset of explosions in detail. By using the one-dimensional complex Ginzburg-Landau model and adjusting a single parameter, we show how the appearance of explosions has the general signatures of intermittency: the periods of time between explosions are irregular even in the absence of noise, but their mean value is related to the distance to criticality by a power law. We conjecture that these explosions are a manifestation of attractor-merging crises, as the continuum of localized solitons induced by translation symmetry becomes connected by short-lived trajectories, forming a delocalized attractor. As additive noise is added, the extended system shows the same scaling found by low-dimensional systems exhibiting crises [J. Sommerer, E. Ott, and C. Grebogi, Phys. Rev. A 43, 1754 (1991)], thus supporting the validity of the proposed picture.

A weakly nonlinear approach is used to investigate interfacial pattern formation in a lifting Hele-Shaw cell containing a yield stress fluid surrounded by a fluid of negligible viscosity. By considering the onset of nonlinear effects and the regime in which viscous effects dominate over yield stress, we study how the system responds to changes in two controlling dimensionless parameters: (i) the geometric aspect ratio (ratio of the initially circular radius of the fluid-fluid interface to the initial Hele-Shaw cell plate spacing), and (ii) a yield stress parameter (relative measure of yield stress to viscous forces). Within this context, we discuss how these key factors influence interface stability and finger competition dynamics during early linear and intermediate stages of pattern evolution.

Exact explicit rogue wave solutions of the Sasa-Satsuma equation are obtained by use of a Darboux transformation. In addition to the double-peak structure and an analog of the Peregrine soliton, the rogue wave can exhibit an intriguing twisted rogue-wave pair that involves four well-defined zero-amplitude points. This exotic structure may enrich our understanding on the nature of rogue waves.

We introduce a thermostat based on fluctuating hydrodynamics for dynamic simulations of implicit-solvent coarse-grained models of lipid bilayer membranes. We show our fluctuating hydrodynamics approach captures interesting correlations in the dynamics of lipid bilayer membranes that are missing in simulations performed using standard Langevin dynamics. Our momentum conserving thermostat accounts for solvent-mediated momentum transfer by coupling coarse-grained degrees of freedom to stochastic continuum fields that account for both the solvent hydrodynamics and thermal fluctuations. We present both a general framework and specific methods to couple the particle and continuum degrees of freedom in a manner consistent with statistical mechanics and amenable to efficient computational simulation. For self-assembled vesicles, we study the diffusivity of lipids and their spatial correlations. We find the hydrodynamic coupling yields within the bilayer interesting correlations between diffusing lipids that manifest as a vortex-like structure similar to those observed in explicit-solvent simulations. We expect the introduced fluctuating hydrodynamics methods to provide a way to extend implicit-solvent models for use in a wide variety of dynamic studies.

We provide a simple no-go theorem for ergodicity and the generalized Einstein relation for anomalous diffusion processes. The theorem states that either ergodicity in the sense of equal time and ensemble averaged mean squared displacements (MSD) is broken, and/or the generalized Einstein relation for time averaged diffusivity and mobility is invalid, which is in complete contrast to normal diffusion processes. We also give a general relation for the time averages of drift and MSD for ergodic (in the MSD sense) anomalous diffusion processes, showing that the ratio of these quantities depends on the measurement time. The Lévy walk model is used to exemplify the no-go theorem.

We extend the Achlioptas model for the delay of criticality in the percolation problem. Instead of having a completely random connectivity pattern, we generalize the idea of the two-site probe in the Achlioptas model for connecting smaller clusters, by introducing two models: the first one by allowing any number $k$ of probe sites to be investigated, $k$ being a parameter, and the second one independent of any specific number of probe sites, but with a probabilistic character which depends on the size of the resulting clusters. We find numerically the complete spectrum of critical points and our results indicate that the value of the critical point behaves linearly with $k$ after the value of $k$ $=$ 3. The range $k$ $=$ 2–3 is not linear but parabolic. The more general model of generating clusters with probability inversely proportional to the size of the resulting cluster produces a critical point which is equivalent to the value of $k$ being in the range $k$ $=$ 5–7.

Existence of stable autoresonance (AR) with continuously growing energy is directly connected with the inherent property of nonlinear systems to remain in resonance when the driving frequency varies in time. However, the physical mechanism underlying the transformation of bounded oscillations into AR remains unclear. As this paper demonstrates, the emergence of AR from stable bounded oscillations is basically analogous to the transition from quasilinear to nonlinear oscillations in the time-invariant oscillator driven by an external harmonic excitation with constant frequency, and AR can occur as a result of the loss of stability of the so-called limiting phase trajectory. We obtain the parametric threshold, which determines the transition from bounded oscillations to AR in the time-dependent system. The accuracy of the obtained approximations is confirmed by numerical simulations.

Random encounters in space are central to describing diffusion-limited reactions, animal foraging, search processes, and many other situations in nature. These encounters, however, are often constrained by the capacity of the searcher to detect and/or recognize its target. This can be due to limited binding and perception abilities of the searcher or hiding and avoiding mechanisms used by the target. Hence detection failure upon passage over the target location turns the process into an $n$-passage problem, with $n$ being random. Here we provide a general description of this detection problem for arbitrary dimensions and arbitrary detection constraints. The mean detection time (MDT) for a random searcher embedded in a sea of homogeneously distributed targets is obtained as a function of the target density $\rho$, the size domain $L$, and the effective detection distance $a$. While the scaling with $\rho$ and $L$ is found to be universal and equivalent to that found for the corresponding first-passage problem, the scaling of the MDT on $a$ depends on the specific detection mechanism considered.

Using fully minimized fundamental measure functionals, we investigate free energies, vacancy concentrations, and density distributions for bcc, fcc, and hcp hard-sphere crystals. Results are complemented by an approach due to Stillinger, which is based on expanding the crystal partition function in terms of the number $n$ of free particles while the remaining particles are frozen at their ideal lattice positions. The free energies of fcc and hcp and one branch of bcc agree well with Stillinger's approach truncated at $n=2$. A second branch of bcc solutions features rather spread-out density distributions around lattice sites and large equilibrium vacancy concentrations and is presumably linked to the shear instability of the bcc phase. Within fundamental measure theory and the Stillinger approach ($n=2$), hcp is more stable than fcc by a free energy per particle of about 0.001$k_{\mathrm{B}}T$. In previous simulation work, the reverse situation has been found, which can be rationalized in terms of effects due to a correlated motion of at least five particles in the Stillinger picture.

The suprachiasmatic nucleus (SCN) is a principal circadian clock in mammals, which controls physiological and behavioral daily rhythms. The SCN has two main features: Maintaining a rhythmic cycle of approximately 24 h in the absence of a light-dark cycle (free-running period) and the ability to entrain to external light-dark cycles. Both free-running period and range of entrainment vary from one species to another. To understand this phenomenon, we investigated the diversity of a free-running period by the distribution of coupling strengths in our previous work [Phys. Rev. E 80, 030904(R) (2009)]. In this paper we numerically found that the dispersion of intrinsic periods among SCN neurons influence the entrainment range of the SCN, but has little influence on the free-running periods under constant darkness. This indicates that the dispersion of coupling strengths determines the diversity in free-running periods, while the dispersion of intrinsic periods determines the diversity in the entrainment range. A theoretical analysis based on two coupled neurons is presented to explain the results of numerical simulations.

Most real-world networks are not isolated. In order to function fully, they are interconnected with other networks, and this interconnection influences their dynamic processes. For example, when the spread of a disease involves two species, the dynamics of the spread within each species (the contact network) differs from that of the spread between the two species (the interconnected network). We model two generic interconnected networks using two adjacency matrices, A and B, in which A is a $2N\times 2N$ matrix that depicts the connectivity within each of two networks of size $N$, and B a $2N\times 2N$ matrix that depicts the interconnections between the two. Using an $N$-intertwined mean-field approximation, we determine that a critical susceptible-infected-susceptible (SIS) epidemic threshold in two interconnected networks is $1/{\lambda }_1(A+\alpha B)$, where the infection rate is $\beta$ within each of the two individual networks and $\alpha \beta$ in the interconnected links between the two networks and ${\lambda }_1(A+\alpha B)$ is the largest eigenvalue of the matrix $A+\alpha B$. In order to determine how the epidemic threshold is dependent upon the structure of interconnected networks, we analytically derive ${\lambda }_1(A+\alpha B)$ using a perturbation approximation for small and large $\alpha$, the lower and upper bound for any $\alpha$ as a function of the adjacency matrix of the two individual networks, and the interconnections between the two and their largest eigenvalues and eigenvectors. We verify these approximation and boundary values for ${\lambda }_1(A+\alpha B)$ using numerical simulations, and determine how component network features affect ${\lambda }_1(A+\alpha B)$. We note that, given two isolated networks $G_1$ and $G_2$ with principal eigenvectors $x$ and $y$, respectively, ${\lambda }_1(A+\alpha B)$ tends to be higher when nodes $i$ and $j$ with a higher eigenvector component product $x_i{y}_j$ are interconnected. This finding suggests essential insights into ways of designing interconnected networks to be robust against epidemics.

We report measurements of the transmitted coherent (ensemble-averaged) wave resulting from the interaction of an ultrasonic shock wave with a two-dimensional random medium. Despite multiple scattering, the coherent waveform clearly shows the steepening that is typical of nonlinear harmonic generation. This is taken advantage of to measure the elastic mean free path and group velocity over a broad frequency range (2–15 MHz) in only one experiment. Experimental results are found to be in good agreement with a linear theoretical model taking into account spatial correlations between scatterers. These results show that nonlinearity and multiple scattering are both present, yet uncoupled.

We have recently shown that nonspecifically bound lac repressors are spatially inhomogeneous in E. coli cells and depends upon the location of its encoding gene and the DNA compaction state [Kuhlman and Cox, Mol. Syst. Biol. 8, 610 (2012)]. Here we model this inhomogeneity as a consequence of diffusion within and exchange between two distinct intracellular phases: the condensed chromosomal DNA and an extrachromosomal compartment, the cytoplasm. We discuss the consequences of this model for the target search process.