When a low-dimensional chaotic attractor is embedded in a three-dimensional space its topological properties are embedding-dependent. We show that there are just three topological properties that depend on the embedding: Parity, global torsion, and knot type. We discuss how they can change with the embedding. Finally, we show that the mechanism that is responsible for creating chaotic behavior is an invariant of all embeddings. These results apply only to chaotic attractors of genus one, which covers the majority of cases in which experimental data have been subjected to topological analysis. This means that the conclusions drawn from previous analyses, for example that the mechanism generating chaotic behavior is a Smale horseshoe mechanism, a reverse horseshoe, a gateau roulé, an $S$-template branched manifold, etc., are not artifacts of the embedding chosen for the analysis.

The analytic model for the evolution of single and multiple bubbles in Rayleigh-Taylor mixing is presented for the system of arbitrary density ratio. The model is the extension of Zufiria’s potential theory, which is based on the velocity potential with point sources. We present solutions for a single bubble, at various stages, from the model and show that the solutions for the bubble velocity and curvature are in good agreement with numerical results. We demonstrate the evolution of multiple bubbles for finite density contrast and investigate dynamics of bubble competition, whereby leading bubbles grow in size at the expense of neighbors. The model shows that the growth coefficient $\alpha$ for the scaling law of the bubble front depends on the Atwood number and increases logarithmically with the initial perturbation amplitude. It is also found that the aspect ratio of the bubble size to the bubble height exhibits a self-similar behavior in the bubble competition process, and its values are insensitive to the Atwood number. The predictions of the model for the similarity parameters are in accordance with experimental and numerical results.

Using grand canonical Monte Carlo simulations, we study the crystallization of colloidal hard spheres under gravity. More specifically, we investigate the nature of the freezing transition as a function of gravity and chemical potential of the hard spheres. We find a discontinuous freezing transition where several fluid layers close to the bottom of the sample freeze simultaneously, i.e., at the same chemical potential. We also find that the number of layers that freezes at the same chemical potential decreases for higher gravitational field strength. Upon increasing the chemical potential further, the crystalline film thickness increases continuously.

Using both numerical simulations and an approximate analytical theory we describe a flexoelectric-induced instability in a thin nematic liquid crystal layer with asymmetric boundary conditions subjected to an applied electric field. The dependence of the threshold value of the electric field on principal material parameters of the nematic liquid crystal and the director distribution in different regions of the cell have been studied in detail numerically. The results have been compared with a simple analytical theory that enables us to obtain explicit expressions for the threshold electric field and the period of modulation above the threshold. It has been found that in the hybrid aligned nematic cell with homeotropic anchoring on one surface and planar homogeneous anchoring on the other surface, a periodic flexoelectric-induced domain structure appears, above a critical threshold, with a chiral director distribution. The director rotates about the alignment axis when moving along a perpendicular direction in the plane of the cell. The absolute value of the threshold field has been found to depend on the direction of the field due to the initial symmetry of the hybrid aligned cell and the presence of flexoelectricity.

A combination of broadband dielectric $({10}^{-2}\mathrm{Hz}–{10}^9\mathrm{Hz})$ and specific heat $({10}^{-3}\mathrm{Hz}–2\times {10}^3\mathrm{Hz})$ spectroscopy is employed to study the molecular dynamics of the glass-forming nematic liquid crystal $E7$ in a wide temperature range. In the region of the nematic phase the dielectric spectra show two relaxation processes which are expected theoretically: The $\delta$ relaxation which corresponds to rotational fluctuations of the molecules around its short axis and the tumbling mode at higher frequencies than the former one. For both processes the temperature dependence of the relaxation rates follows the Vogel-Fulcher-Tammann formula which is characteristic for glassy dynamics. By applying a detailed data analysis, it is shown that close to the glass transition the tumbling mode has a much steeper temperature dependence than the $\delta$ process. The former has a Vogel temperature which is by $30\mathrm{K}$ higher than that of the $\delta$ relaxation. Specific heat spectroscopy gives one relaxation process in its temperature and frequency dependence which has to be assigned to the $\alpha$ relaxation (dynamic glass transition). The unique and detailed comparison of the temperature dependence of the dielectric and the thermal relaxation rates delivers unambiguously that the dielectric tumbling mode has to be related to the dynamic glass transition.

We investigate in depth the synchronization of coupled oscillators on top of complex networks with different degrees of heterogeneity within the context of the Kuramoto model. In a previous paper [Phys. Rev. Lett. 98, 034101 (2007)], we unveiled how for fixed coupling strengths local patterns of synchronization emerge differently in homogeneous and heterogeneous complex networks. Here, we provide more evidence on this phenomenon, extending the previous work to networks that interpolate between homogeneous and heterogeneous topologies. We also introduce details of the path towards synchronization for the evolution of clustering in the synchronized patterns. Finally, we investigate the synchronization of networks with modular structure and conclude that, in these cases, local synchronization is first attained at the most internal level of organization of modules, progressively evolving to the outer levels as the coupling constant is increased. The present work introduces parameters that are proved to be useful for the characterization of synchronization phenomena in complex networks.

Numerical simulations have shown that certain driven nonlinear systems can be characterized by mean-field statistical properties often associated with ergodic dynamics [C. D. Ferguson, W. Klein, and J. B. Rundle, Phys. Rev. E 60, 1359 (1999); D. Egolf, Science 287, 101 (2000)]. These driven mean-field threshold systems feature long-range interactions and can be treated as equilibriumlike systems with statistically stationary dynamics over long time intervals. Recently the equilibrium property of ergodicity was identified in an earthquake fault system, a natural driven threshold system, by means of the Thirumalai-Mountain (TM) fluctuation metric developed in the study of diffusive systems [K. F. Tiampo, J. B. Rundle, W. Klein, J. S. Sá Martins, and C. D. Ferguson, Phys. Rev. Lett. 91, 238501 (2003)]. We analyze the seismicity of three naturally occurring earthquake fault networks from a variety of tectonic settings in an attempt to investigate the range of applicability of effective ergodicity, using the TM metric and other related statistics. Results suggest that, once variations in the catalog data resulting from technical and network issues are accounted for, all of these natural earthquake systems display stationary periods of metastable equilibrium and effective ergodicity that are disrupted by large events. We conclude that a constant rate of events is an important prerequisite for these periods of punctuated ergodicity and that, while the level of temporal variability in the spatial statistics is the controlling factor in the ergodic behavior of seismic networks, no single statistic is sufficient to ensure quantification of ergodicity. Ergodicity in this application not only requires that the system be stationary for these networks at the applicable spatial and temporal scales, but also implies that they are in a state of metastable equilibrium, one in which the ensemble averages can be substituted for temporal averages in studying their spatiotemporal evolution.

We consider transport properties of a double $\delta$-kicked system, in a regime where all the symmetries (spatial and temporal) that could prevent directed transport are removed. We analytically investigate the (nontrivial) behavior of the classical current and diffusion properties and show that the results are in good agreement with numerical computations. The role of dissipation for a meaningful classical ratchet behavior is also discussed.

We model the transport of particles present in a fluid steadily flowing through a porous medium. The porous medium is described by a representative three-dimensional network. The particles are subjected to advection by the flow and to thermal diffusion. We propose to calculate their trajectories with the continuous time random walk framework. This enables us to efficiently sample disordered networks with realistic topology. The method proposed in this paper is general and can be adapted to model dispersion of tracers. It is applied here to simulate the measurement of the flow propagator $P(\underset{̱}{x},\delta t)$ which is defined as the ensemble density distribution of tracer displacements $\underset{̱}{x}$, in a given time interval $\delta t$. It can be extracted from pulsed magnetic field gradient spin echo NMR experiments carried out on porous media while fluid is flowing. Preliminary numerical results show good qualitative agreement with experiments.

A magnetron source of silver clusters captured by an argon flow with the quadrupole mass filter is used for the analysis of charged clusters after an orifice of the magnetron chamber, and the size distribution function follows from the analysis of clusters deposited on a silicon substrate by an atomic force microscope. Cluster charge near an orifice results from attachment of ions of a secondary plasma that is a tail of a magnetron plasma, and the cluster charge is mostly positive. The character of passage of a buffer gas flow with metal clusters through an orifice is studied both theoretically and experimentally. Assuming the cone shape of the drift chamber near the orifice, we analyze drift of charged clusters in a buffer gas flow towards the orifice if the electric field inside the drift chamber is created by charged rings on the cone surface. Under experimental conditions, when an equilibrium between the buffer gas flow and cluster flux is violated, a typical voltage of rings and parameters of corona discharge for cluster charging are estimated if the electric field does not allow for clusters to reach walls of the drift chamber. The number density of clusters near the orifice is estimated that increases both due to violation of an equilibrium for the cluster flux inside the buffer gas flow and owing to focusing of the cluster by the electric field that is created by electrodes located near walls and due to diffusion motion of clusters. Processes of cluster charging in the magnetron chamber are analyzed.

This paper shows that optimal control techniques can be used to avoid some pattern formation in a Rayleigh-Bénard problem with horizontal temperature gradient. Appropriate thermal boundary conditions determined by these techniques lead to new strong controlled basic states with reduced pattern and for which the thermoconvective instability is avoided.

The Green-Kubo-Nakano formula should be modified in relativistic hydrodynamics because of the problem of acausality and the breaking of sum rules. In this Rapid Communication, we propose a formula to calculate the transport coefficients of causal hydrodynamics based on the projection operator method. As concrete examples, we derive the expressions for the diffusion coefficient, the shear viscosity coefficient, and corresponding relaxation times.

We discuss recently introduced numerical linked-cluster (NLC) algorithms that allow one to obtain temperature-dependent properties of quantum lattice models, in the thermodynamic limit, from exact diagonalization of finite clusters. We present studies of thermodynamic observables for spin models on square, triangular, and kagomé lattices. Results for several choices of clusters and extrapolations methods, that accelerate the convergence of NLCs, are presented. We also include a comparison of NLC results with those obtained from exact analytical expressions (where available), high-temperature expansions (HTE), exact diagonalization (ED) of finite periodic systems, and quantum Monte Carlo simulations. For many models and properties NLC results are substantially more accurate than HTE and ED.

We discuss the application of a recently introduced numerical linked-cluster (NLC) algorithm to strongly correlated itinerant models. In particular, we present a study of thermodynamic observables: chemical potential, entropy, specific heat, and uniform susceptibility for the $t\text{−}J$ model on the square lattice, with $J∕t=0.5$ and 0.3. Our NLC results are compared with those obtained from high-temperature expansions (HTE) and the finite-temperature Lanczos method (FTLM). We show that there is a sizeable window in temperature where NLC results converge without extrapolations whereas HTE diverges. Upon extrapolations, the overall agreement between NLC, HTE, and FTLM is excellent in some cases down to $0.25t$. At intermediate temperatures NLC results are better controlled than other methods, making it easier to judge the convergence and numerical accuracy of the method.

We present exact and explicit results for the thermodynamic properties (isochores, isotherms, isobars, response functions, velocity of sound) of a quantum gas in dimensions $\mathcal{D}⩾1$ and with fractional exclusion statistics $0⩽g⩽1$ connecting bosons $(g=0)$ and fermions $(g=1)$. In $\mathcal{D}=1$ the results are equivalent to those of the Calogero-Sutherland model. Emphasis is given to the crossover between bosonlike and fermionlike features, caused by aspects of the statistical interaction that mimic long-range attraction and short-range repulsion. A phase transition along the isobar occurs at a nonzero temperature in all dimensions. The $T$ dependence of the velocity of sound is in simple relation to isochores and isobars. The effects of soft container walls are accounted for rigorously for the case of a pure power-law potential.

We propose a surface reaction model for $\mathrm{NO}$ reduction with ${\mathrm{NH}}_3$ on a $\mathrm{Pt}\left(100\right)$ single crystal catalyst surface and we explore it by carrying out Monte Carlo simulations. Our model includes experimentally observed realistic features such as adsorbate-induced surface phase transition, structure-dependent sticking coefficients and reactivity, desorption probabilities, and surface diffusion of adsorbed species. We discuss similarities found while comparing the available experimental data and our model as reactant ratio and temperature vary. Simulations qualitatively reproduce the kinetic oscillations observed in reaction rates and surface coverages. Also, the essential role of the adsorbate-induced phase transition regarding the appearance of kinetic oscillations is discussed.

The gas-isotropic liquid-nematic liquid phase behavior of the Stockmayer fluid is studied using molecular dynamics simulation together with a mean field lattice model. We obtain coexistence curves of the Stockmayer fluid over a wide range of dipole strengths, temperatures, and densities, including the transition from the isotropic liquid to the ferroelectric liquid. In our simulations we do not observe the disappearance of the isotropic gas-isotropic liquid coexistence at high dipole strength contrary to earlier findings based on Monte Carlo techniques. Even though the formation of reversible dipole chains strongly affects the location of the critical point, it does not lead to its disappearance. These results are supported by a mean field lattice model which yields good qualitative, and in parts quantitative, agreement with our simulations. In addition, we also investigate the gas-isotropic liquid phase behavior for different polarizabilities.

Using Monte Carlo simulation techniques, we study the ferromagnetic order-disorder phase transition in Ising spin fluids with hard-core Yukawa interaction truncated at various cutoff radii $r_c$. We focus our interest on the dependence of critical quantities such as the Binder cumulant and various exponent ratios on the value of $r_c$, and on the question whether the Fisher-renormalized exponents expected for such systems can be observed in the simulations. It turns out that the corrections to scaling decaying with a rather small exponent prevent reaching the asymptotic region with the computational power available. Thus, we observe only effective exponents, with different (nonuniversal) values depending on the cutoff radius. The same behavior is also found for the critical Binder cumulant. Nevertheless, an exact investigation of the effective susceptibility exponent ${\gamma }_{\mathrm{eff}}$ as a function of temperature seems to point towards a Fisher-renormalized value. For two selected cutoff radii, the critical temperature is determined more accurately using, in addition to the cumulant crossing technique, the scanning technique and the shifting technique, taking into account corrections to scaling. Simulations of Ising fluids with constant cutoff radius and varying Yukawa-tail screening lengths $\lambda$ also show a nonuniversal dependence of $U_c$ on $\lambda$. Finally, we have performed simulations of the Ising lattice model with increasing number of couplings which show the expected asymptotic behavior, independent of the range of interactions.

Surface transfer coefficients are determined by nonequilibrium molecular dynamics simulations for a Lennard-Jones fluid with a long-range spline potential. In earlier work [A. Røsjorde et al., J. Colloid Interface Sci. 240, 355 (2001); J. Xu et al., ibid. 299, 452 (2006)], using a short-range Lennard-Jones spline potential, it was found that the resistivity coefficients to heat and mass transfer agreed rather well with the values predicted by kinetic theory. For the long-range Lennard-Jones spline potential considered in this paper we find significant discrepancies from the values predicted by kinetic theory. In particular the coupling coefficient, and as a consequence the heat of transfer on the vapor side of the surface are much larger. Thermodynamic data for the liquid-vapor equilibrium confirmed the law of corresponding states for the surface, when it is described as an autonomous system. The importance of these findings for modelling phase transitions is discussed.