I study a single slow bird moving with a flock of birds of a different and faster (or slower) species. I find that every species of flocker has a characteristic speed $\gamma \ne v_0$, where $v_0$ is the mean speed of the flock such that if the speed $v_s$ of the slow bird equals $\gamma$, it will randomly wander transverse to the mean direction of flock motion far faster than the other birds will: Its mean-squared transverse displacement will grow in $d=2$ with time $t$ like $t^{5/3}$, in contrast to $t^{4/3}$ for the other birds. In $d=3$, the slow bird's mean-squared transverse displacement grows like $t^{5/4}$, in contrast to $t$ for the other birds. If $v_s\ne \gamma$, the mean-squared displacement of the slow bird crosses over from $t^{5/3}$ to $t^{4/3}$ scaling in $d=2$ and from $t^{5/4}$ to $t$ scaling in $d=3$ at a time $t_c$ that scales according to $t_c\propto {|v_s-\gamma |}^{-2}$.

A concept called balanced estimator of diffusion entropy is proposed to detect quantitatively scalings in short time series. The effectiveness is verified by detecting successfully scaling properties for a large number of artificial fractional Brownian motions. Calculations show that this method can give reliable scalings for short time series with length $\sim {10}^2$. It is also used to detect scalings in the Shanghai Stock Index, five stock catalogs, and a total of 134 stocks collected from the Shanghai Stock Exchange Market. The scaling exponent for each catalog is significantly larger compared with that for the stocks included in the catalog. Selecting a window with size 650, the evolution of scaling for the Shanghai Stock Index is obtained by the window's sliding along the series. Global patterns in the evolutionary process are captured from the smoothed evolutionary curve. By comparing the patterns with the important event list in the history of the considered stock market, the evolution of scaling is matched with the stock index series. We can find that the important events fit very well with global transitions of the scaling behaviors.

We study the effect of coupling delay in a regular network with a ring topology and in a more complex network with an all-to-all (global) topology in the presence of impurities (disorder). We find that the coupling delay is capable of inducing phase-coherent chaotic oscillations in both types of networks, thereby enhancing the spatiotemporal complexity even in the presence of $50\%$ of symmetric disorders of both fixed and random types. Furthermore, the coupling delay increases the robustness of the networks up to $70\%$ of disorders, thereby preventing the network from acquiring periodic oscillations to foster disorder-induced spatiotemporal order. We also confirm the enhancement of coherent chaotic oscillations using snapshots of the phases and values of the associated Kuramoto order parameter. We also explain a possible mechanism for the phenomenon of delay-induced coherent chaotic oscillations despite the presence of large disorders and discuss its applications.

Localized structures appear in a wide variety of systems, arising from a pinning mechanism due to the presence of a small-scale pattern or an imposed grid. When there is a separation of length scales, the width of the pinning region is exponentially small and beyond the reach of standard asymptotic methods. We show how this behavior can be obtained using a variational method, for two systems. In the case of the quadratic-cubic Swift-Hohenberg equation, this gives results that are in agreement with recent work using exponential asymptotics. In addition, the method is applied to a discrete system with cubic-quintic nonlinearity, giving results that agree well with numerical simulations.

In analogy to the viscous fingering problem between parallel flat plates, injection-driven flow in nonflat (conical and spherical) Hele-Shaw cells gives rise to ramified interfacial structures. The multiple occurrences of fingertip-splitting events makes the resulting pattern difficult to predict and to control. We calculate expressions for time-dependent injection rates that suppress the development of bifurcated shapes on curved confined environments. Implementation of these injection procedures leads to interfacial shapes with a robust $n$-fold symmetry. A weakly nonlinear approach is used to extract important analytical information about the influence of the geometric and topological features of the nonflat cells on the efficiency of the proposed fingering control processes.

This paper presents the analysis of axisymmetric helical magnetorotational instability (HMRI) in the inviscid limit, which is relevant for astrophysical conditions. The inductionless approximation defined by zero magnetic Prandtl number is adopted to distinguish the HMRI from the standard MRI in the cylindrical Taylor-Couette flow subject to a helical magnetic field. Using a Chebyshev collocation method convective and absolute instability thresholds are computed in terms of the Elsasser number for a fixed ratio of inner and outer radii $\lambda =2$ and various ratios of rotation rates and helicities of the magnetic field. It is found that the extension of self-sustained HMRI modes beyond the Rayleigh limit does not reach the astrophysically relevant Keplerian rotation profile not only in the narrow- but also in the finite-gap approximation. The Keppler limit can be attained only by the convective HMRI mode provided that the boundaries are perfectly conducting. However, this mode requires not only a permanent external excitation to be observable but also has a long axial wavelength, which is not compatible with limited thickness of astrophysical accretion disks.

We report an experimental and numerical study of the spontaneous emergence of spectral incoherent solitons through supercontinuum generation in a two zero-dispersion wavelengths photonic crystal fiber. By using a simple experimental setup, we show that the highly nonlinear regime of supercontinuum generation is characterized by the emergence of a spectral incoherent soliton in the low-frequency edge of the supercontinuum spectrum. We show that a transition occurs from the discrete spectral incoherent soliton to its continuous counterpart as the power of the laser is increased. Contrary to conventional solitons, spectral incoherent solitons do not exhibit a confinement in the space-time domain, but solely in the frequency domain. These incoherent structures owe their existence to the noninstantaneous nature of the nonlinear Raman effect and, more specifically, to the causality property underlying the Raman response function.

The formation, stability, and dynamics of solitons are considered in an optical system with focusing nonlinearity and active clusters. It is shown that stochastic generation of solitons is possible in such a system provided the cluster properties are properly designed.

Jamming phenomena on a square lattice are investigated for two different models of anisotropic random sequential adsorption (RSA) of linear $k$-mers (particles occupying $k$ adjacent adsorption sites along a line). The length of a $k$-mer varies from 2 to 256. The effect of $k$-mer alignment on the jamming threshold is examined. For completely ordered systems where all the $k$-mers are aligned along one direction (e.g., vertical), the obtained simulation data are very close to the known analytical results for one-dimensional systems. In particular, the jamming threshold tends to the Rényi's parking constant for large $k$. In the other extreme case, when $k$-mers are fully disordered, our results correspond to the published results for short $k$-mers. It was observed that for partially oriented systems the jamming configurations consist of the blocks of vertically and horizontally oriented $k$-mers ($v$ and $h$ blocks, respectively) and large voids between them. The relative areas of different blocks and voids depend on the order parameter $s$, $k$-mer length, and type of the model. For small $k$-mers ($k⩽4$), denser configurations are observed in disordered systems as compared to those of completely ordered systems. However, longer $k$-mers exhibit the opposite behavior.

New entropy measures have been recently introduced for the quantification of the complexity of networks. Most of these entropy measures apply to static networks or to dynamical processes defined on static complex networks. In this paper we define the entropy rate of growing network models. This entropy rate quantifies how many labeled networks are typically generated by the growing network models. We analytically evaluate the difference between the entropy rate of growing tree network models and the entropy of tree networks that have the same asymptotic degree distribution. We find that the growing networks with linear preferential attachment generated by dynamical models are exponentially less than the static networks with the same degree distribution for a large variety of relevant growing network models. We study the entropy rate for growing network models showing structural phase transitions including models with nonlinear preferential attachment. Finally, we bring numerical evidence that the entropy rate above and below the structural phase transitions follows a different scaling with the network size.

Viscous fingering is a well-known hydrodynamic instability that sets in when a less viscous fluid displaces a more viscous fluid. When the two fluids are miscible, viscous fingering introduces disorder in the velocity field and exerts a fundamental control on the rate at which the fluids mix. Here we analyze the characteristic signature of the mixing process in viscously unstable flows, by means of high-resolution numerical simulations using a computational strategy that is stable for arbitrary viscosity ratios. We propose a reduced-order model of mixing, which, in the spirit of turbulence modeling and in contrast with previous approaches, recognizes the fundamental role played by the mechanical dissipation rate. The proposed model captures the nontrivial interplay between channeling and creation of interfacial area as a result of viscous fingering.

An approximation to the Fourier transform (FT) of the Kohlrausch function (stretched exponential) with shape parameter $0<\beta ⩽1$ is presented by using Havriliak-Negami-like functions. Mathematical expressions to fit their parameters $\alpha$, $\gamma$, and $\tau$, as functions of $\beta$ ($0<\beta ⩽1$ and $1<\beta <2$) are given, which allows a quick identification in the frequency domain of the corresponding shape factor $\beta$. Reconstruction via fast Fourier transform of frequency approximants to time domain are shown as good substitutes in short times though biased in long ones (increasing discrepancies as $\beta \to 1$). The method is proposed as a template to commute time and frequency domains when analyzing complex data. Such a strategy facilitates intensive algorithmic search of parameters while adjusting the data of one or several Kohlrausch-Williams-Watts relaxations.

Hysteresis in disordered systems originates in a plethora of metastable states. Previous works focused on their distribution inside the hysteresis. In contrast, an analysis of their range of metastability is proposed. This model, designed to catch the main features of fluid adsorption in porous materials, shows strong evidence that, in the thermodynamic limit, despite that metastable states of finite range can be found, they are exponentially dominated by those infinitely localized states.

We demonstrate the existence of an orbital Coulomb phase as the exact ground state of a $p$-orbital exchange Hamiltonian on the diamond lattice. The Coulomb phase is an emergent state characterized by algebraic dipolar correlations and a gauge structure resulting from local constraints (ice rules) of the underlying lattice models. For most ice models on the pyrochlore lattice, these local constraints are a direct consequence of minimizing the energy of each individual tetrahedron. On the contrary, the orbital ice rules are emergent phenomena resulting from the quantum orbital dynamics. We show that the orbital ice model exhibits an emergent geometrical frustration by mapping the degenerate quantum orbital ground states to the spin-ice states obeying the 2-in-2-out constraints on the pyrochlore lattice. We also discuss possible realization of the orbital ice model in optical lattices with $p$-band fermionic cold atoms.

We present an analytical method, rooted in the nonperturbative renormalization group, that allows one to calculate the critical exponents and the correlation and response functions of the Kardar-Parisi-Zhang (KPZ) growth equation in all its different regimes, including the strong-coupling one. We analyze the symmetries of the KPZ problem and derive an approximation scheme that satisfies the linearly realized ones. We implement this scheme at the minimal order in the response field, and show that it yields a complete, qualitatively correct phase diagram in all dimensions, with reasonable values for the critical exponents in physical dimensions. We also compute in one dimension the full (momentum and frequency dependent) correlation function, and the associated universal scaling function. We find a very satisfactory quantitative agreement with the exact result from Prähofer and Spohn [J. Stat. Phys. 115, 255 (2004)]. In particular, we obtain for the universal amplitude ratio $g_0\simeq 1.149\left(18\right)$, to be compared with the exact value $g_0=1.1504...$ (the Baik and Rain [J. Stat. Phys. 100, 523 (2000)] constant). We emphasize that all these results, which can be systematically improved, are obtained with sole input the bare action and its symmetries, without further assumptions on the existence of scaling or on the form of the scaling function.

Manifestations of space fractional quantum mechanics (SFQM), as it was formulated by Laskin [Phys. Rev. E 62, 3135 (2000)], are deemed to offer a better physical interpretation of Lévy flight statistics on a quantum mechanical level. We start with the SFQM Schrödinger equation characterized by a Lévy flight index $\alpha \in (1,2)$, perform a Wigner transform, and draw the limit $\frac{\hslash }{E\tau }\to 0$ (i.e., let the observed energy scale $E$ go to infinity in comparison to the quantization given by $\hslash /\tau$). In order to obtain classical transport equations two possible substitutions for the terms ${\left|p\right|}^{\alpha }$ and $|p^{\prime }{|}^{\alpha }$ which appear in von Neumann's equation are presented. It is demonstrated that they conform to the criteria for a successful Wigner transform. Their benefits and caveats are discussed in detail. We find, that, indeed, SFQM manifests itself in an anomalous kinetic term of the free particle's motion and, assuming an external potential diagonal in momentum space for the sake of simplicity, in corresponding anomalous terms in the resulting drift current. All our results reduce to the classical forms in the limit $\alpha =2$.

For a broad class of random walks with anisotropic scattering kernels and absorption, we derive explicit formulas that allow expressing the moments of the collision number $n_V$ performed in a volume $V$ as a function of the particle equilibrium distribution. Our results apply to arbitrary domains $V$ and boundary conditions, and allow assessing the hitting statistics for systems where the typical displacements are comparable to the domain size, so that the diffusion limit is possibly not attained. An example is discussed for one-dimensional random flights with exponential displacements, where analytical calculations can be carried out.

The way diseases spread through schools, epidemics through countries, and viruses through the internet is crucial in determining their risk. Although each of these threats has its own characteristics, its underlying network determines the spreading. To restrain the spreading, a widely used approach is the fragmentation of these networks through immunization, so that epidemics cannot spread. Here we develop an immunization approach based on optimizing the susceptible size, which outperforms the best known strategy based on immunizing the highest-betweenness links or nodes. We find that the network's vulnerability can be significantly reduced, demonstrating this on three different real networks: the global flight network, a school friendship network, and the internet. In all cases, we find that not only is the average infection probability significantly suppressed, but also for the most relevant case of a small and limited number of immunization units the infection probability can be reduced by up to $55\%$.