We study the cascading dynamics immediately before and immediately after 219 market shocks. We define the time of a market shock $T_c$ to be the time for which the market volatility $V\left(T_c\right)$ has a peak that exceeds a predetermined threshold. The cascade of high volatility “aftershocks” triggered by the “main shock” is quantitatively similar to earthquakes and solar flares, which have been described by three empirical laws—the Omori law, the productivity law, and the Bath law. We analyze the most traded 531 stocks in U.S. markets during the 2 yr period of 2001–2002 at the 1 min time resolution. We find quantitative relations between the main shock magnitude $M\equiv {\text{log}}_{10}V\left(T_c\right)$ and the parameters quantifying the decay of volatility aftershocks as well as the volatility preshocks. We also find that stocks with larger trading activity react more strongly and more quickly to market shocks than stocks with smaller trading activity. Our findings characterize the typical volatility response conditional on $M$, both at the market and the individual stock scale. We argue that there is potential utility in these three statistical quantitative relations with applications in option pricing and volatility trading.

Analytical description of propagation phenomena on random networks has flourished in recent years, yet more complex systems have mainly been studied through numerical means. In this paper, a mean-field description is used to coherently couple the dynamics of the network elements (such as nodes, vertices, individuals, etc.) on the one hand and their recurrent topological patterns (such as subgraphs, groups, etc.) on the other hand. In a susceptible-infectious-susceptible (SIS) model of epidemic spread on social networks with community structure, this approach yields a set of ordinary differential equations for the time evolution of the system, as well as analytical solutions for the epidemic threshold and equilibria. The results obtained are in good agreement with numerical simulations and reproduce the behavior of random networks in the appropriate limits which highlights the influence of topology on the processes. Finally, it is demonstrated that our model predicts higher epidemic thresholds for clustered structures than for equivalent random topologies in the case of networks with zero degree correlation.

Adaptive networks have been recently introduced in the context of disease propagation on complex networks. They account for the mutual interaction between the network topology and the states of the nodes. Until now, existing models have been analyzed using low complexity analytical formalisms, revealing nevertheless some novel dynamical features. However, current methods have failed to reproduce with accuracy the simultaneous time evolution of the disease and the underlying network topology. In the framework of the adaptive susceptible-infectious-susceptible (SIS) model of Gross et al. [Phys. Rev. Lett. 96, 208701 (2006)], we introduce an improved compartmental formalism able to handle this coevolutionary task successfully. With this approach, we analyze the interplay and outcomes of both dynamical elements, process and structure, on adaptive networks featuring different degree distributions at the initial stage.

Magnetic field generation by dynamo action is often studied within the theoretical framework of magnetohydrodynamics (MHD). However, for sufficiently diffuse media, the Hall effect may become non-negligible. We present results from three-dimensional simulations of the Hall-MHD equations subjected to random nonhelical forcing. We study the role of the Hall effect in the dynamo efficiency for different values of the Hall parameter. For small values of the Hall parameter, the small-scale dynamo is more efficient, displaying faster growth and saturating at larger amplitudes of the magnetic field. For larger values of the Hall parameter, saturation of the magnetic field is reached at smaller amplitudes than in the MHD case. We also study energy transfer rates among spatial scales and show that the Hall effect produces a reduction of the direct energy cascade at scales larger than the Hall scale, therefore leading to smaller energy dissipation rates. Finally, we present results stemming from simulations at large magnetic Prandtl numbers, which is the relevant regime in the hot and diffuse interstellar medium. In the range of magnetic Prandtl numbers considered, the Hall effect moves the peak of the magnetic energy spectrum as well as other relevant magnetic length scales toward the Hall scale.

Discrete nonlinear Schrödinger model is a nonlinear lattice model used to investigate different nonlinear phenomena arising in many physical contexts. In this work we used this model to observe linear traveling phonon scattering by time-periodic discrete breathers in the piecewise smooth (PWS) version of nonlinearity. In one-dimensional system the single-channel scattering is found to be elastic (with the incoming and outgoing fluxes of energy being equal to each other). Considering one-site symmetric breather solution we are able to calculate the exact expression for the transmission coefficient $\left(T\right)$ or the ratio of the transmitted to incident flux of energy in such a system using transfer-matrix formalism. From the condition of perfect transmission ($T=1$, and hence reflection coefficient $R=0$) or the ratio of reflected to incident flux of energy for elastic scattering is zero, our observation shows that perfect transmission happens to appear at the threshold of a localized mode, which occurs at the band edge of the extended eigenmodes (plane waves). We have also presented the results obtained from the condition of perfect reflection. The advantage of using the PWS version of nonlinearity in the model is that all the results derived are exact. Numerical simulations complement our results.

We compared three formulations of the discrete dipole approximation (DDA) for simulation of light scattering by particles with refractive indices $m=10+10i$, $0.1+i$, and $1.6+0.01i$. These formulations include the filtered coupled dipoles (FCD), the lattice dispersion relation (LDR) and the radiative reaction correction. We compared the number of iterations required for the convergence of the iterative solver (proportional to simulation time) and the accuracy of final results. We showed that the LDR performance for $m=10+10i$ is especially bad, while the FCD is a good option for all cases studied. Moreover, we analyzed the detailed structure of DDA errors and the spectrum of the DDA interaction matrix to understand the performance of the FCD. In particular, this spectrum, obtained with the FCD for particles smaller than the wavelength, falls into the bounds, physically implied for the spectrum of the infinite-dimensional integral scattering operator, contrary to two other DDA formulations. Finally, such extreme refractive indices can now be routinely simulated using modern desktop computers using the publicly available ADDA code, which includes an efficient implementation of the FCD.

In contrast with most Monte Carlo integration algorithms, which are used to estimate ratios, the replica gas identities recently introduced by Adib enable the estimation of absolute integrals and partition functions using multiple copies of a system and normalized transition functions. Here, an optimized form is presented. After generalizing a replica gas identity with an arbitrary weighting function, we obtain a functional form that has the minimal asymptotic variance for samples from two replicas and is provably good for a larger number. This equation is demonstrated to improve the convergence of partition function estimates in a two-dimensional Ising model.

We study an inertial Brownian particle moving in a symmetric periodic substrate, driven by a zero-mean biharmonic force and correlated thermal noise. The Brownian motion is described in terms of a generalized Langevin equation with an exponentially correlated Gaussian noise term, obeying the fluctuation-dissipation theorem. We analyze impact of nonzero correlation time of thermal noise on transport properties of the Brownian particle. We identify regimes where the increase of the correlation time intensifies long-time transport of the Brownian particle. The opposite effect is also found: longer correlation time reduces the stationary velocity of the particle. The correlation time induced multiple current reversal is detected. We reveal that thermal noise of nonzero correlation time can radically enhance long-time velocity of the Brownian particle in regimes where in the white noise limit the velocity is extremely small. All transport properties can be tested in the setup consisting of a resistively and capacitively shunted Josephson junction device.

We show that an atomic system in a periodically modulated optical trap displays an ideal mean-field symmetry-breaking transition. The symmetry is broken with respect to time translation by the modulation period. We describe experimental observations and develop a full microscopic theory of the observed critical phenomena. The transition is explained as resulting from the interplay of the long-range interatomic interaction and nonequilibrium fluctuations in the strongly modulated system. The observations, including anomalous fluctuations in the symmetry broken phase, are fully described by the theory.

We study the behavior of the inverse participation ratio and the localization transition in infinitely large random matrices through the cavity method. Results are shown for two ensembles of random matrices: Laplacian matrices on sparse random graphs and fully connected Lévy matrices. We derive a critical line separating localized from extended states in the case of Lévy matrices. Comparison between theoretical results and diagonalization of finite random matrices is shown.

Experimental and theoretical studies are made of Brownian particles trapped in a periodic potential, which is very slightly tilted due to gravity. In the presence of fluctuations, these will trigger a measurable average drift along the direction of the tilt. The magnitude of the drift varies with the ratio between the bias force and the trapping potential. This can be closely compared to a theoretical model system, based on a Fokker-Planck-equation formalism. We show that the level of control and measurement precision we have in our system, which is based on cold atoms trapped in a three-dimensional dissipative optical lattice, makes the experimental setup suitable as a testbed for fundamental statistical physics. We simulate the system with a very simplified and general classical model, as well as with an elaborate semiclassical Monte Carlo simulation. In both cases, we achieve good qualitative agreement with experimental data.

We implement a binary collision approximation to study pulse propagation in a chain of o-rings. In particular, we arrive at analytic results from which the pulse velocity is obtained by simple quadrature. The predicted pulse velocity is compared to the velocity obtained from the far more resource-intensive numerical integration of the equations of motion. We study chains without precompression, chains precompressed by a constant force at the chain ends (constant precompression), and chains precompressed by gravity (variable precompression). The application of the binary collision approximation to precompressed chains provides an important generalization of a successful theory that had up to this point only been implemented to chains without precompression, that is, to chains in a sonic vacuum.

In a static magnetic field paramagnetic and nonmagnetic colloids immersed in a ferrofluid self-assemble into fluctuating colloidal flowers. Adsorption and desorption of nonmagnetic petals to larger paramagnetic cores and changes in the petal conformation around the paramagnetic core induce a fluctuating dynamics. We track the motion of colloidal petals on the paramagnetic core. Adsorption and desorption of petals occur on a larger time scale than the rotational diffusion of the petals. Magnetic dipole interactions split the motion of the petals into different modes of rotational diffusion. Modes of rotational diffusion that change the petal conformation are suppressed compared to the conformation invariant rotational diffusion of all petals. The suppression of higher modes of rotational diffusion results in a subdiffusive dynamics of the individual petals.

We study the nonequilibrium aging dynamics in a system of quasihard spheres at large density by means of computer simulations. We find that, after a sudden quench to large density, the relaxation time initially increases exponentially with the age of the system. After a surprisingly large crossover time, the system enters the asymptotic aging regime characterized by a nearly linear increase in the relaxation time with age. In this aging regime, single-particle motion is strongly non-Fickian, with a mean-squared displacement increasing subdiffusively, associated with broad non-Gaussian tails in the distribution of particle displacements. We find that the system ages through temporally intermittent relaxation events, and a detailed finite-size analysis of these collective dynamic fluctuations reveals that these events are not spanning the entire system, but remain spatially localized.

Using a generalized order-parameter gradient expansion within density functional theory, we derive a phase-field-crystal model for liquid crystals composed of apolar particles in three spatial dimensions. Both the translational density and the orientational direction and ordering are included as order parameters. Different terms involving gradients in the order parameters in the resulting free-energy functional are compared to the macroscopic Ginzburg-Landau approach as well as to the hydrodynamic description for liquid crystals. Our approach provides microscopic expressions for all prefactors in terms of the particle interactions. Our phase-field-crystal model generalizes the conventional phase-field-crystal model of spherical particles to orientational degrees of freedom and can be used as a starting point to explore phase transitions and interfaces for various liquid-crystalline phases.

Living cells are continually exposed to environmental signals that vary in time. These signals are detected and processed by biochemical networks, which are often highly stochastic. To understand how cells cope with a fluctuating environment, we therefore have to understand how reliably biochemical networks can transmit time-varying signals. To this end, we must understand both the noise characteristics and the amplification properties of networks. In this paper, we use information theory to study how reliably signaling cascades employing autoregulation and feedback can transmit time-varying signals. We calculate the frequency dependence of the gain-to-noise ratio, which reflects how reliably a network transmits signals at different frequencies. We find that the gain-to-noise ratio may differ qualitatively from the power spectrum of the output, showing that the latter does not directly reflect signaling performance. Moreover, we find that autoactivation and autorepression increase and decrease the gain-to-noise ratio for all of frequencies, respectively. Positive feedback specifically enhances information transmission at low frequencies, while negative feedback increases signal fidelity at high frequencies. Our analysis not only elucidates the role of autoregulation and feedback in naturally occurring biological networks, but also reveals design principles that can be used for the reliable transmission of time-varying signals in synthetic gene circuits.

The Eigen quasispecies model in a periodically moving sharp-peak landscape considered in previous seminal works [M. Nilsson and N. Snoad, Phys. Rev. Lett. 84, 191 (2000)] and [C. Ronnewinkel et al., in Theoretical Aspects of Evolutionary Computing, edited by L. Kallel, B. Naudts, and A. Rogers (Springer-Verlag, Heidelberg, 2001)] is analyzed in greater detail. We show here, through a more rigorous analysis, that results in those papers are qualitatively correct. In particular, we obtain a phase diagram for the existence of a quasispecies with the same shape as in the above cited paper by C. Ronnewinkel et al., with upper and lower thresholds for the mutation rate between which a quasispecies may survive. A difference is that the upper value is larger and the lower value is smaller than the previously reported ones, so that the range for quasispecies existence is always larger than thought before. The quantitative information provided might also be important in understanding genetic variability in virus populations and has possible applications in antiviral therapies. The results in the quoted papers were obtained by studying the populations only at some few genomes. As we will show, this amounts to diagonalizing a $3\times 3$ matrix. Our work is based instead in a different division of the population allowing a finer control of the populations at various relevant genetic sequences. The existence of a quasispecies will be related to Perron-Frobenius eigenvalues. Although huge matrices of sizes $2^{\ell }$, where $\ell$ is the genome length, may seem necessary at a first look, we show that such large sizes are not necessary and easily obtain numerical and analytical results for their eigenvalues.

The results of an experimental study of pattern formation on sandy bottom under the action of regular harmonic surface waves are reported. It is found that two modes of pattern formation occur: sand ripples form uniformly on the whole bottom or from localized nucleation sites. In the second regime, the ripples appear in isolated regions (patches) increasing in size, and front propagation speed is measured. A simple dynamical model based on the Ginzburg-Landau equation is proposed to explain the characteristics of patches.

A model for epidemic spreading on rewiring networks is introduced and analyzed for the case of scale free steady state networks. It is found that contrary to what one would have naively expected, the rewiring process typically tends to suppress epidemic spreading. In particular it is found, that as in static networks under a mean-field approximation, rewiring networks with degree distribution exponent $\gamma >3$ exhibit a threshold in the infection rate below which epidemics die out in the steady state. However the threshold is higher in the rewiring case. For $2<\gamma \le 3$ no such threshold exists, but for small infection rate the steady state density of infected nodes (prevalence) is smaller for rewiring networks.

The fractal and the small-world properties of complex networks are systematically studied both in the box-covering (BC) and the cluster-growing (CG) measurements. We elucidate that complex networks possessing the fractal (small-world) nature in the BC measurement are always fractal (small world) even in the CG measurement and vice versa, while the fractal dimensions $d_{\text{B}}$ by the BC measurement and $d_{\text{C}}$ by the CG measurement are generally different. This implies that two structural properties of networks, fractality and small worldness, cannot coexist in the same length scale. These properties can, however, crossover from one to the other by varying the length scale. We show that the crossover behavior in a network near the percolation transition appears both in the BC and CG measurements and is scaled by a unique characteristic length $\xi$.