The objective of this work is to evaluate the potential of reduced order models to reproduce the extreme event and predictability characteristics of higher dimensional dynamical systems. A nonlinear toy model is used which contains key features of comprehensive climate models. First, we demonstrate that the systematic stochastic mode reduction strategy leads to a reduced order model with the same extreme value characteristics as the full dynamical models for a wide range of time-scale separations. Second, we find that extreme events in this model follow a generalized Pareto distribution with a negative shape parameter; thus extreme events are bounded in this model. Third, we show that a precursor approach has good forecast skill for extreme events. We then find that the reduced stochastic models capture the predictive skill of extreme events of the full dynamical models well. Consistent with previous studies we also find that the larger the extreme events, the better predictable they are. Our results suggest that systematically derived reduced order models have the potential to be used for the modeling and statistical prediction of weather- and climate-related extreme events and, possibly, in other areas of science and engineering too.

The probability distribution function for an out of equilibrium system may sometimes be approximated by a physically motivated “trial” distribution. A particularly interesting case is when a driven system (e.g., active matter) is approximated by a thermodynamic one. We show here that every set of trial distributions yields an inequality playing the role of a generalization of the second law. The better the approximation is, the more constraining the inequality becomes: this suggests a criterion for its accuracy, as well as an optimization procedure that may be implemented numerically and even experimentally. The fluctuation relation behind this inequality, a natural and practical extension of the Hatano-Sasa theorem, does not rely on the a priori knowledge of the stationary probability distribution.

Modern developments in microscopy and image processing are revolutionizing areas of physics, chemistry, and biology as nanoscale objects can be tracked with unprecedented accuracy. The goal of single-particle tracking is to determine the interaction between the particle and its environment. The price paid for having a direct visualization of a single particle is a consequent lack of statistics. Here we address the optimal way to extract diffusion constants from single trajectories for pure Brownian motion. It is shown that the maximum likelihood estimator is much more efficient than the commonly used least-squares estimate. Furthermore, we investigate the effect of disorder on the distribution of estimated diffusion constants and show that it increases the probability of observing estimates much smaller than the true (average) value.

The behavior of the magnetic currents in one-dimensional Heisenberg $XXZ$ spin chains kept out of equilibrium by boundary driving fields is investigated. In particular, the dependence of the spin currents on the anisotropy parameter $\Delta$ and on the boundary fields is studied both analytically and numerically in the framework of the Lindblad master equation formalism. We show that the spin current can be maximized with appropriate choices of the boundary fields, and for odd system sizes, $N$, we demonstrate the existence of additional symmetries that cause the current to be an odd function of $\Delta$. From direct numerical integrations of the quantum master equation, we find that for an arbitrary $N$ the current $J_z\left(N\right)$ vanishes for $\Delta =0$, while for $\Delta$ negative it alternates its sign with the system size. In the gapless critical region $\left|\Delta \right|<1$, the scaling of the current is shown to be $J_z\left(N\right)\sim 1/N$ while in the gapped region $\left|\Delta \right|>1$ we find that $J_z\left(N\right)\sim \exp (-\alpha N)$. A simple mean-field approach, which predicts rather well the values of $J_z\left(N\right)$ for the gapped region and the values of the absolute current maxima in the critical region, is developed. The existence of two different stationary solutions for the mean-field density matrix in the whole parameter range is also demonstrated.

The propagation of elastic waves in a box under direct shear, filled with glass beads and being sheared at constant rates, is studied experimentally and theoretically. The respective velocities are shown to be essentially unchanged from that in a static granular system under the same pressure and shear stress but without a shear band. Influence of shear band on sound behaviors are also briefly discussed.

The subject of this work is the development and implementation of algorithms which accelerate the simulation of early stage tumor growth models. Among the different computational approaches used for the simulation of tumor progression, discrete stochastic models (e.g., cellular automata) have been widely used to describe processes occurring at the cell and subcell scales (e.g., cell-cell interactions and signaling processes). To describe macroscopic characteristics (e.g., morphology) of growing tumors, large numbers of interacting cells must be simulated. However, the high computational demands of stochastic models make the simulation of large-scale systems impractical. Alternatively, continuum models, which can describe behavior at the tumor scale, often rely on phenomenological assumptions in place of rigorous upscaling of microscopic models. This limits their predictive power. In this work, we circumvent the derivation of closed macroscopic equations for the growing cancer cell populations; instead, we construct, based on the so-called “equation-free” framework, a computational superstructure, which wraps around the individual-based cell-level simulator and accelerates the computations required for the study of the long-time behavior of systems involving many interacting cells. The microscopic model, e.g., a cellular automaton, which simulates the evolution of cancer cell populations, is executed for relatively short time intervals, at the end of which coarse-scale information is obtained. These coarse variables evolve on slower time scales than each individual cell in the population, enabling the application of forward projection schemes, which extrapolate their values at later times. This technique is referred to as coarse projective integration. Increasing the ratio of projection times to microscopic simulator execution times enhances the computational savings. Crucial accuracy issues arising for growing tumors with radial symmetry are addressed by applying the coarse projective integration scheme in a cotraveling (cogrowing) frame. As a proof of principle, we demonstrate that the application of this scheme yields highly accurate solutions, while preserving the computational savings of coarse projective integration.

We study the critical effect of an intermittent social distancing strategy on the propagation of epidemics in adaptive complex networks. We characterize the effect of our strategy in the framework of the susceptible-infected-recovered model. In our model, based on local information, a susceptible individual interrupts the contact with an infected individual with a probability $\sigma$ and restores it after a fixed time $t_b$. We find that, depending on the network topology, in our social distancing strategy there exists a cutoff threshold ${\sigma }_c$ beyond which the epidemic phase disappears. Our results are supported by a theoretical framework and extensive simulations of the model. Furthermore we show that this strategy is very efficient because it leads to a “susceptible herd behavior” that protects a large fraction of susceptible individuals. We explain our results using percolation arguments.

We introduce a generalized rumor spreading model and analytically investigate the spreading of rumors on scale-free (SF) networks. In the standard rumor spreading model, each node has an infectivity equal to its degree, and connectivity is uniform across all links. To generalize this model, we introduce an infectivity function that determines the number of simultaneous contacts that a given node (individual) may establish with its connected neighbors and a connectivity strength function (CSF) for the direct link between two connected nodes. These lead to a degree-biased propagation of rumors. For nonlinear functions, this generalization is reflected in the infectivity's exponent $\alpha$ and the CSF's exponent $\beta$. We show that, by adjusting exponents $\alpha$ and $\beta$, the epidemic threshold can be controlled. This feature is absent in the standard rumor spreading model. In addition, we obtain a critical threshold. We show that the critical threshold for our generalized model is greater than that of the standard model on a finite SF network. Theoretically, we show that $\beta =-1$ leads to a maximum spreading of rumors, and computation results on different networks verify our theoretical prediction. Also, we show that a smaller $\alpha$ leads to a larger spreading of rumors. Our results are interesting since we obtain these results regardless of the network topology and configuration.

We consider the ray limit of propagating ultrasound waves in three-dimensional bodies made from a homogeneous, isotropic, elastic material. Using a Monte Carlo approach, we simulate the propagation and proliferation of elastic rays using realistic angle-dependent reflection coefficients, taking into account mode conversion and ray splitting. For a few simple geometries, we analyze the long-time equilibrium distribution, focusing on the energy ratio between compressional and shear waves. Finally, we study the travel time statistics, i.e., the distribution of the amount of time a given trajectory spends as a compressional wave, as compared to the total travel time. These results are intimately related to recent elastodynamics experiments on Coda-wave interferometry by Lobkis and Weaver [Phys. Rev. E 78, 066212 (2008)].

We study spectral statistics in systems with a mixed phase space, in which regions of regular and chaotic motion coexist. Increasing their density of states, we observe a transition of the level-spacing distribution $P\left(s\right)$ from Berry-Robnik to Wigner statistics, although the underlying classical phase-space structure and the effective Planck constant $h_{\mathrm{eff}}$ remain unchanged. This transition is induced by flooding, i.e., the disappearance of regular states due to increasing regular-to-chaotic couplings. We account for this effect by a flooding-improved Berry-Robnik distribution, in which an effectively reduced size of the regular island enters. To additionally describe power-law level repulsion at small spacings, we extend this prediction by explicitly considering the tunneling couplings between regular and chaotic states. This results in a flooding- and tunneling-improved Berry-Robnik distribution which is in excellent agreement with numerical data.

We show that inhibitory networks composed of two endogenously bursting neurons can robustly display several coexistent phase-locked states in addition to stable antiphase and in-phase bursting. This work complements and enhances our recent result [Jalil, Belykh, and Shilnikov, Phys. Rev. E 81, 045201(R) (2010)] that fast reciprocal inhibition can synchronize bursting neurons due to spike interactions. We reveal the role of spikes in generating multiple phase-locked states and demonstrate that this multistability is generic by analyzing diverse models of bursting networks with various fast inhibitory synapses; the individual cell models include the reduced leech heart interneuron, the Sherman model for pancreatic beta cells, and the Purkinje neuron model.

The effect of spatially randomizing a system of pointlike sources on the propagation of reaction-diffusion fronts is investigated in multidimensions. The dynamics of the reactive front are modeled by superimposing the solutions for diffusion from a single point source. A nondimensional parameter is introduced to quantify the discreteness of the system, based on the characteristic reaction time of sources compared to the diffusion time between sources. The limits to propagation and the average velocity of propagation are expressed as probabilistic quantities to account for the influence of the randomly distributed sources. In random systems, two- and three-dimensional fronts are able to propagate beyond a limit previously found for systems with regularly distributed sources, while a propagation limit in one dimension that is independent of domain size cannot be defined. The dimensionality of the system is seen to have a strong influence on the front propagation velocity, with higher dimensional systems propagating faster than lower dimensional systems. In a three-dimensional system, both the limit to propagation and average front velocity revert to a solution that assumes a spatially continuous source function as the discreteness parameter is increased to the continuum limit. The results indicate that reactive systems are able to exploit local fluctuations in source concentration to extend propagation limits and increase the velocity in comparison to regularly spaced systems.

The dynamics and acoustic emission of two interacting cavitation bubbles exposed to strong acoustic fields with a frequency of 515 KHz are investigated numerically in this paper. After comparing the dynamics of a single bubble excited by the given pressure waves, bubbles with ambient radii of 2 and 5 μm were chosen to be studied to discuss the influence of the mutual bubble-bubble interaction on the dynamic behaviors and acoustic emission of the bubbles. The results show that, aside from the external driving pressure waves, the interaction between the bubbles imposes an extra nonlinear effect on the oscillations of the bubbles and that the dynamics of the smaller bubble could be suppressed gradually with the enhancement of this mutual interaction by decreasing the distance between the bubbles. Moreover, the improvement in the oscillation nonlinearity of the bubbles due to the change in the ambient circumstance could readily be observed from the frequency spectra of the bubbles’ acoustic emission, which interprets the change by exhibiting an appropriate development of the subharmonics, the ultraharmonics, and the broadband component.

We investigate by particle-in-cell simulations in two and three dimensions the laser-plasma interaction and the proton acceleration in multilayer targets where a low-density (“near-critical”) layer of a few-micron thickness is added on the illuminated side of a thin, high-density layer. This target design can be obtained by depositing a “foam” layer on a thin metallic foil. The presence of the near-critical plasma strongly increases both the conversion efficiency and the energy of electrons and leads to enhanced acceleration of protons from a rear side layer via the target normal sheath acceleration mechanism. The electrons of the foam are strongly accelerated in the forward direction and propagate on the rear side of the target, building up a high electric field with a relatively flat longitudinal profile. In these conditions the maximum proton energy is up to three times higher than in the case of the bare solid target.

We perform three-dimensional under-resolved direct numerical simulations of forced compressible turbulence using the smoothed particle hydrodynamics (SPH) method and investigate the Lagrangian intermittency of the resulting hydrodynamic fields. The analysis presented here is motivated by the presence of typical stretched tails in the probability density function (PDF) of the particle accelerations previously observed in two-dimensional SPH simulations of uniform shear flow [Ellero et al., Phys. Rev. E 82, 046702 (2010)]. In order to produce a stationary isotropic compressible turbulent state, the real-space stochastic forcing method proposed by Kida and Orszag is applied, and the statistics of particle quantities are evaluated. We validate our scheme by checking the behavior of the energy spectrum in the supersonic case where the expected Burgers-like scaling is obtained. By discretizing the continuum equations along fluid particle trajectories, the SPH method allows us to extract Lagrangian statistics in a straightforward fashion without the need for extra tracer particles. In particular, Lagrangian PDF of the density, particle accelerations as well as their Lagrangian structure functions and local scaling exponents are analyzed. The results for low-order statistics of Lagrangian intermittency in compressible turbulence demonstrate the implicit subparticle-scale modeling of the SPH discretization scheme.

We show that the application of a time periodic field driving a Brownian particle between the two lobes of a two-dimensional bilobal enclosure results in a hysteresis loop in the variation of integrated probability of residence of the particle as a function of the field. The confinement of the particle is characterized by symmetry breaking of the hysteresis loop, and the area of the loop exhibits a turnover with variation of frequency of the field. This dynamical hysteresis is geometry controlled, entropic in nature, and amenable to theoretical analysis with a two-state model.

The irreversibility of a stationary time series can be quantified using the Kullback-Leibler divergence (KLD) between the probability of observing the series and the probability of observing the time-reversed series. Moreover, this KLD is a tool to estimate entropy production from stationary trajectories since it gives a lower bound to the entropy production of the physical process generating the series. In this paper we introduce analytical and numerical techniques to estimate the KLD between time series generated by several stochastic dynamics with a finite number of states. We examine the accuracy of our estimators for a specific example, a discrete flashing ratchet, and investigate how close the KLD is to the entropy production depending on the number of degrees of freedom of the system that are sampled in the trajectories.

In the classical mean-field description of the electric double layer, known as the Poisson-Boltzmann model, ions interact exclusively through their Coulomb potential. Ion specificity can arise through solvent-mediated, nonelectrostatic interactions between ions. We employ the Yukawa pair potential to model the presence of nonelectrostatic interactions. The combination of Yukawa and Coulomb potential on the mean-field level leads to the Poisson-Helmholtz-Boltzmann model, which employs two auxiliary potentials: one electrostatic and the other nonelectrostatic. In the present work we apply the Poisson-Helmholtz-Boltzmann model to ionic mixtures, consisting of monovalent cations and anions that exhibit different Yukawa interaction strengths. As a specific example we consider a single charged surface in contact with a symmetric monovalent electrolyte. From the minimization of the mean-field free energy we derive the Poisson-Boltzmann and Helmholtz-Boltzmann equations. These nonlinear equations can be solved analytically in the weak perturbation limit. This together with numerical solutions in the nonlinear regime suggests an intricate interplay between electrostatic and nonelectrostatic interactions. The structure and free energy of the electric double layer depends sensitively on the Yukawa interaction strengths between the different ion types and on the nonelectrostatic interactions of the mobile ions with the surface.

We investigate the connection between the theory of complete scaling and a field-theoretic (FT) treatment of asymmetric fluid criticality. To facilitate the comparison, we develop an equation of state from a simplified form of the complete scaling transformations and systematically compare this equation of state with the equation of state generated by a FT treatment of an asymmetric Landau-Ginzburg-Wilson Hamiltonian. We find, with care in interpretation, that these two approaches may be read as equivalent up to terms involving an independent higher-order asymmetric correction-to-scaling exponent.

We derive the fractional master equation with space-dependent anomalous exponent. We analyze the asymptotic behavior of the corresponding lattice model both analytically and by Monte Carlo simulation. We show that the subdiffusive fractional equations with constant anomalous exponent $\mu$ in a bounded domain $[0,L]$ are not structurally stable with respect to the nonhomogeneous variations of parameter $\mu$. In particular, the Gibbs-Boltzmann distribution is no longer the stationary solution of the fractional Fokker-Planck equation whatever the space variation of the exponent might be. We analyze the random distribution of $\mu$ in space and find that in the long-time limit, the probability distribution is highly intermediate in space and the behavior is completely dominated by very unlikely events. We show that subdiffusive fractional equations with the nonuniform random distribution of anomalous exponent is an illustration of a “Black Swan,” the low probability event of the small value of the anomalous exponent that completely dominates the long-time behavior of subdiffusive systems.