A computational strategy, aimed at evaluating diffraction catastrophes belonging to the $X_9$ family is presented. The approach proposed is based on the use of power series expansions, suitably derived for giving meaningful representation of the whole ${}^0{X}_9$ subfamily, jointly with a powerful sequence transformation algorithm, the so-called Weniger transformation. The convergence features of the above series expansions are investigated, and several numerical experiments are carried out to assess the effectiveness of the retrieving action of the Weniger transformation, as well as the ease of implementation of the whole approach.

The probability distributions of the order parameter for two models in the directed percolation universality class were evaluated. Monte Carlo simulations have been performed for the one-dimensional generalized contact process and the Domany-Kinzel cellular automaton. In both cases, the density of active sites was chosen as the order parameter. The criticality of those models was obtained by solely using the corresponding probability distribution function. It has been shown that the present method, which has been successfully employed in treating equilibrium systems, is indeed also useful in the study of nonequilibrium phase transitions.

We consider steady-state current activity statistics for the one-dimensional totally asymmetric simple exclusion process. With the help of the known operator algebra (for general open boundary conditions), as well as general probabilistic concepts (for the periodic case), we derive and evaluate closed-form expressions for the lowest three moments of the probability distribution function. These are confirmed, to excellent degree of accuracy, by numerical simulations. Further exact expressions and asymptotic approximations are provided for probability distributions and generating functions.

Statistical measures such as coherence, mutual information, or correlation are usually applied to evaluate the interactions between two or more signals. However, these methods cannot distinguish directions of flow between two signals. The capability to detect causalities is highly desirable for understanding the cooperative nature of complex systems. The main objective of this work is to present a linear and nonlinear time-varying parametric modeling and identification approach that can be used to detect Granger causality, which may change with time and may not be detected by traditional methods. A numerical example, in which the exact causal influences relationships, is presented to illustrate the performance of the method for time-varying Granger causality detection. The approach is applied to EEG signals to track and detect hidden potential causalities. One advantage of the proposed model, compared with traditional Granger causality, is that the results are easier to interpret and yield additional insights into the transient directed dynamical Granger causality interactions.

Social organisms at every level of evolutionary complexity live in groups, such as fish schools, locust swarms, and bird flocks. The complex exchange of multifaceted information across group members may result in a spectrum of salient spatiotemporal patterns characterizing collective behaviors. While instances of collective behavior in animal groups are readily identifiable by trained and untrained observers, a working definition to distinguish these patterns from raw data is not yet established. In this work, we define collective behavior as a manifestation of low-dimensional manifolds in the group motion and we quantify the complexity of such behaviors through the dimensionality of these structures. We demonstrate this definition using the ISOMAP algorithm, a data-driven machine learning algorithm for dimensionality reduction originally formulated in the context of image processing. We apply the ISOMAP algorithm to data from an interacting self-propelled particle model with additive noise, whose parameters are selected to exhibit different behavioral modalities, and from a video of a live fish school. Based on simulations of such model, we find that increasing noise in the system of particles corresponds to increasing the dimensionality of the structures underlying their motion. These low-dimensional structures are absent in simulations where particles do not interact. Applying the ISOMAP algorithm to fish school data, we identify similar low-dimensional structures, which may act as quantitative evidence for order inherent in collective behavior of animal groups. These results offer an unambiguous method for measuring order in data from large-scale biological systems and confirm the emergence of collective behavior in an applicable mathematical model, thus demonstrating that such models are capable of capturing phenomena observed in animal groups.

We investigate the evolution of Boolean networks subject to a selective pressure which favors robustness against noise, as a model of evolved genetic regulatory systems. By mapping the evolutionary process into a statistical ensemble and minimizing its associated free energy, we find the structural properties which emerge as the selective pressure is increased and identify a phase transition from a random topology to a “segregated-core” structure, where a smaller and more densely connected subset of the nodes is responsible for most of the regulation in the network. This segregated structure is very similar qualitatively to what is found in gene regulatory networks, where only a much smaller subset of genes—those responsible for transcription factors—is responsible for global regulation. We obtain the full phase diagram of the evolutionary process as a function of selective pressure and the average number of inputs per node. We compare the theoretical predictions with Monte Carlo simulations of evolved networks and with empirical data for Saccharomyces cerevisiae and Escherichia coli.

By analyzing spatio-temporal characteristics of short optical pulses diffusively transmitted through compacted granular materials, we reveal that powder compaction can give rise to strongly anisotropic light diffusion. Our disclosure represents a revision of the understanding of optics of powder compacts. Routes to material characterization and investigation of compression-induced structural anisotropy are opened, and the falsification of isotropic models have implications for quantitative spectroscopy of powder compacts (e.g., pharmaceutical tablets).

We study the ground state (GS) and the phase transition in a hexagonal-close-packed lattice with both $\mathit{XY}$ and Ising models by using extensive Monte Carlo simulation. We suppose the in-plane interaction $J_1$ and interplane interaction $J_2$, both antiferromagnetic. The system is frustrated with two kinds of GS configuration below and above a critical value of $\eta =J_1/J_2$ (${\eta }_c$). For the Ising case, one has ${\eta }_c=0.5$ which separates in-plane ferromagnetic and antiferromagnetic states, while for the $\mathit{XY}$ case ${\eta }_c=1/3$ separates the collinear and noncollinear spin configurations. The phase transition is shown to be of first (second) order for $\eta >(<){\eta }_c$. The phase diagram in the space ($\eta$,$T$) is shown for both cases.

The global renormalization-group theory (GRGT) for fluids is derived starting with the square-gradient approximation for the Helmholtz free energy functional such that any mean-field free energy density and direct correlation function can be employed. The new derivation uses Wilson's functions for representing density fluctuations, thereby relaxing the assumption of cosine variation of density fluctuations used in earlier approaches. The generality of the present approach is shown by deriving the relationships to the earlier developments. A qualitative way to infer the free parameters in the present form of GRGT is also suggested. The new theory is applied to square-well fluids of ranges $1.5$ and $3.0$ (in units of hard-sphere diameter) and Lennard-Jones fluids. It is shown that the present theory produces a flat isotherm in the two-phase region. Thus the theory accounts for fluctuations at all length scales and avoids the use of Maxwell's construction. An analysis of the liquid-vapor phase diagrams and the critical constants obtained for different potentials shows that, with a mean-field free energy density that is accurate away from the critical region and an appropriate coarse graining length for the mean-field theory, GRGT can provide results in good agreement with the simulation and experimental results.

Schloegl's second model on a ($d$ ≥ 2)-dimensional hypercubic lattice involves: (i) spontaneous annihilation of particles with rate $p$ and (ii) autocatalytic creation of particles at vacant sites at a rate proportional to the number of suitable pairs of neighboring particles. This model provides a prototype for nonequilibrium discontinuous phase transitions. However, it also exhibits nontrivial generic two-phase coexistence: Stable populated and vacuum states coexist for a finite range, $p_f\left(d\right)<p<p_e\left(d\right),$ spanned by the orientation-dependent stationary points for planar interfaces separating these states. Analysis of interface dynamics from kinetic Monte Carlo simulation and from discrete reaction-diffusion equations (dRDEs) obtained from truncation of the exact master equation, reveals that $p_{e\left(f\right)}\sim 0.2113765+c_{e\left(f\right)}/d$ as $d\to \infty ,$ where $\Delta c=c_e-c_f\approx 0.014$. A metastable populated state persists above $p_e$($d$) up to a spinodal $p=p_s\left(d\right)$, which has a well-defined limit $p_s(d\to \infty )=\frac{1}{4}$. The dRDEs display artificial propagation failure, absent in the stochastic model due to fluctuations. This feature is amplified for increasing $d$, thus complicating our analysis.

Based on previous works on polymers by Bauer et al. [Phys. Rev. E 61, 1755 (2000)], this paper describes a capacitative method for measuring the dynamical expansion coefficient of a viscous liquid. Data are presented for the glass-forming liquid tetramethyl tetraphenyl trisiloxane (DC704) in the ultraviscous regime. Compared to the method of Bauer et al., the dynamical range has been extended by making time-domain experiments and by making very small and fast temperature steps. The modeling of the experiment presented in this paper includes the situation in which the capacitor is not full because the liquid contracts when cooling from room temperature down to around the glass-transition temperature, which is relevant when measuring on a molecular liquid rather than a polymer.

Dye-doped cholesteric liquid crystals with a helical pitch of the order of a wavelength have a strong effect on the fluorescence properties of dye molecules. This is a promising system for realizing tunable lasers at low cost. We apply a plane wave model to simulate the spontaneous emission from a layer of cholesteric liquid crystal. We simulate the spectral and angle dependence and the polarization of the emitted light as a function of the order parameter of the dye in the liquid crystal. Measurements of the angle dependent emission spectra and polarization are in good agreement with the simulations.

In this paper, we experimentally study the dynamics of slow oscillations in Purkinje neurons in vitro, and derive a strong association with a forced parametric oscillator model. We observed the precise rhythmicity of these oscillations in Purkinje neurons, as well as a dynamic tunability of this oscillation using a photoswitchable compound. We found that this slow oscillation can be induced in every Purkinje neuron measured, having periods ranging between 10 and 25 s. Starting from a Hodgkin-Huxley model, we demonstrate that this oscillation can be externally modulated, and that the neurons will return to their intrinsic firing frequency after the forced oscillation is concluded. These findings signify an additional timing functional role of tunable oscillations within the cerebellum, as well as a dynamic control of a time scale in the brain in the range of seconds.

We present a method for the testing of significance when evaluating the coherence of two oscillatory time series that may have variable amplitude and frequency. It is based on evaluating the self-correlations of the time series. We demonstrate our approach by the application of wavelet-based coherence measures to artificial and physiological examples. Because coherence measures of this kind are strongly biased by the spectral characteristics of the time series, we evaluate significance by estimation of the characteristics of the distribution of values that may occur due to chance associations in the data. The expectation value and standard deviation of this distribution are shown to depend on the autocorrelations and higher order statistics of the data. Where the coherence value falls outside this distribution, we may conclude that there is a causal relationship between the signals regardless of their spectral similarities or differences.

Symmetrically coupled oscillators represent a limiting case for studying the dynamics of natural systems. Therefore, we here investigate the effect of coupling asymmetry on delay-induced oscillation death (OD) in coupled nonlinear oscillators. It is found that the asymmetrical coupling substantially enlarges the domain of the OD island in the parameter space. Specifically, when the intensity of asymmetry is enhanced by turning down the value of the coupling asymmetry parameter $\alpha$, the OD island gradually expands along two directions of both the coupling delay and the coupling strength. The expansion behavior of the OD region is well characterized by a power law scaling, $R={\alpha }^{\gamma }$ with $\gamma \approx -1.19$. The minimum value of the intrinsic frequency, for which OD is possible, monotonically decreases with decreasing $\alpha$ and saturates around a constant value in the limit of $\alpha \to 0$. The generality of the conducive effect of coupling asymmetry is confirmed in a numerical study of two delay-coupled chaotic Rössler oscillators. Our findings shed an improved light on the understanding of dynamics in asymmetrically delay-coupled systems.

In the present study, we focus on alterations in flow physics as a consequence of interactions between patterned-wettability gradients on microfluidic substrates with modulated surface charge distributions, giving rise to an intricate electrohydrodynamic coupling over small scales. We demonstrate that by exploiting such intricate coupling, it may be possible to pattern vortices occurring in the fluidic confinement by exploiting an interplay between the Navier slip and electro-osmotic transport. Our studies do reveal that the resultant flow structure originating out of the spatially periodic variations in the surface charge and surface wettability may depend critically on several independently tunable controlling parameters, such as the amplitudes and frequencies of the respective patterning functions, the phase shift between the two, an asymmetry factor, and the channel height to Debye length ratio. We show that judicious choices with regard to the combinations of these parameters may result in significant augmentations in the corresponding mixing efficiency without any appreciable compromise in the net microfluidic throughput. Furthermore, our studies reveal an optimum patterning frequency, which results in the most efficient microfluidic mixing within the constraints of achieving a desired volumetric flow rate. Our results also demonstrate that the net flow rate is maximized when the surface wettability variation functions and surface charge-density functions are in phase, whereas mixing is best facilitated when they are in opposite phase. In practice, therefore, one may select an intermediate value of the phase angle depending on the extent of compromise necessary between flow rate and mixing characteristics, yielding far-ranging scientific and technological advances toward an improved design of miniaturized fluidic devices of practical relevance.

A simple analytical fluid dynamic model is developed for evaporative electron cooling in a low-pressure decaying plasma and compared to a two-dimensional simulation and experimental data for the particular case of argon. Measured electron temperature and density developments are fully reproduced by the ab initio model and the simulation. Further, it is shown that in the late afterglow thermalization of electrons occurs by coupling to the ion fluid via Coulomb collisions at sufficiently high electron densities and not by coupling to the neutral background.

The heat radiated by objects smaller than or comparable in size to the thermal wavelength can be very different from the classical blackbody radiation as described by the Planck and Stefan-Boltzmann laws. We use methods based on scattering of electromagnetic waves to explore the dependence on size, shape, and material properties. In particular, we explore the radiation from a long cylinder at uniform temperature, discussing in detail the degree of polarization of the emitted radiation. If the radius of the cylinder is much smaller than the thermal wavelength, the radiation is polarized parallel to the cylindrical axis and becomes perpendicular when the radius is comparable to the thermal wavelength. For a cylinder of uniaxial material (a simple model for carbon nanontubes), we find that the influence of uniaxiality on the polarization is most pronounced if the radius is larger than a few micrometers, and quite small for the submicrometer sizes typical for nanotubes.

Droplet breakup of many Newtonian fluids is well described by current experiments, theory, and simulations. Breakup in complex fluids where interactions between mesoscopic structural features can affect the flows remains poorly understood and a burgeoning area of research. Here, we report on our investigations of droplet breakup in thermotropic liquid crystals. We investigate breakup in the smectic, nematic, and isotropic phases of 4-cyano 4-octylbiphenyl (8CB) and the nematic and isotropic phases of 4-cyano 4-pentylbiphenyl (5CB). The experiment consists of varying the ambient temperature to control liquid crystalline phase and imaging breakup using a fast video camera at up to $110000$ frames/s. We expand on previous work [John R. Savage et al., Soft Matter 6, 892 (2010)] that shows breakup in the smectic phase is symmetric, producing no satellite droplets, and is well described by a similarity solution for a shear-thinning power-law fluid. We show that in the nematic phase the breakup occurs in two stages. In the first stage, the breakup is symmetric and the power-law exponent for the minimum radius dependence on the time left to breakup is $1.2<n<1.9$. In the second stage the drop develops two minima and the minimum radii shrink with a power-law exponent $0.6<n<1$. We find that the exponents vary with temperature across the nematic phase. These results are surprising because rheological measurements of 8CB and 5CB in the nematic phases indicate Newtonian behavior that cannot account for the observed breakup dynamics. Finally, in the isotropic phase, the exponents are consistent with theoretical predictions and experiments for Newtonian fluid breakup in the inertial viscous regime.