We have simulated the dynamics of a two-dimensional system of hard ellipses by event-oriented molecular dynamics in microcanonical NVE ensemble. Various quantities, namely longitudinal and transverse velocity auto-correlation functions, translational and rotational diffusion mean-squared displacements, pressure, intermediate self-scattering function, radial distribution function, and angular spatial correlation, have been obtained and their dependence on packing fraction is characterized. Despite absence of prominent positional ordering, the orientational degree of freedom behaves nontrivially and exhibits interesting features. Slowing down is observed in the angular part of the motion near isotropic-nematic phase transition. It is shown that above a certain packing fraction the rotational mean-squared displacement exhibits a three-stage temporal regime including a plateau. Comparison to 2D system of hard needles is made and it is shown that from positional viewpoint, the ellipse system is more ordered.

Ventricular fibrillation at the whole heart level is often preceded by the alternation of action potential duration (APD), i.e., alternans, at the cellular level. As proven in many experiments, traditional approaches based on the slope of the restitution curve have not been successful in predicting alternans formation. Recently, a technique has been theoretically developed based on dominant eigenvalue analysis to predict alternans formation in isolated cardiac myocytes. Here, we aimed to demonstrate that this technique can be applied to predict alternans formation at the whole heart level. Optical mapping was performed in Langendorff-perfused hearts from New Zealand white rabbits ($n$ $=$ 4), which were paced at decreasing basic cycle lengths to introduce APD alternans. In each heart, the basic cycle length corresponding to the local onset of alternans, $B^{\mathrm{onset}}$, was determined and two regions of the heart were identified at $B^{\mathrm{onset}}$: one region which exhibited alternans (1:1${}_{\mathrm{alt}}$) and one which did not (1:1). Corresponding two-dimensional eigenvalue (λ) maps were generated using principal component analysis by analyzing action potentials after short perturbations from the steady state, and mean eigenvalues ($\overline{\lambda }$) were calculated separately for the 1:1 and 1:1${}_{\mathrm{alt}}$ regions. We demonstrated that $\overline{\lambda }$ calculated at $B^{\mathrm{onset}}$ was significantly different ($p$<0.05) between the two regions. Our results suggest that this dominant eigenvalue technique can be used to successfully predict the local alternans formation in the heart.

We propose a route to spatiotemporal chaos for one-dimensional stationary patterns, which is a natural extension of the quasiperiodicity route for low-dimensional chaos to extended systems. This route is studied through a universal model of pattern formation. The model exhibits a scenario where stationary patterns become spatiotemporally chaotic through two successive bifurcations. First, the pattern undergoes a subcritical Andronov-Hopf bifurcation leading to an oscillatory pattern. Subsequently, a secondary bifurcation gives rise to an oscillation with an incommensurable frequency with respect to the former one. This last bifurcation is responsible for the spatiotemporally chaotic behavior. The Lyapunov spectrum enables us to identify the complex behavior observed as spatiotemporal chaos, and also from the larger Lyapunov exponents characterize the above instabilities.

We generalize Kirchhoff's point vortex model of two-dimensional fluid motion to a rotor model which exhibits an inverse cascade by the formation of rotor clusters. A rotor is composed of two vortices with like-signed circulations glued together by an overdamped spring. The model is motivated by a treatment of the vorticity equation representing the vorticity field as a superposition of vortices with elliptic Gaussian shapes of variable widths, augmented by a suitable forcing mechanism. The rotor model opens up the way to discuss the energy transport in the inverse cascade on the basis of dynamical systems theory.

This work addresses the stability of a two-dimensional plane layer of a dielectric liquid enclosed in wall bounded cavities of different aspect ratios and subjected to unipolar injection of ions. Numerical simulations have been conducted to investigate the effect of lateral walls, especially in the development of the electroconvective instability. It is found that an unexpected change of the bifurcation nature occurs for certain cavity aspect ratios. We show that above the linear stability threshold for the rest state a supercritical bifurcation arises. This bifurcation takes place at a given value $T_c$${}_1$ of the parameter $T$ (the electric Rayleigh number). Then, a second subcritical bifurcation occurs at a second threshold $T_c$${}_2$, featuring a typical hysteresis loop with an associated nonlinear criterion $T_f$, which is very characteristic of the Coulomb-driven convection. This behavior has been confirmed by different numerical codes based on different numerical methods. The physical mechanism which leads to this situation is analyzed and discussed. The evolution of the bifurcation diagrams with the aspect ratio of the cavity is also provided and analyzed.

Position-jump processes are used for the mathematical modeling of spatially extended chemical and biological systems with increasing frequency. A large subset of the literature concerning such processes is concerned with modeling the effect of stochasticity on reaction-diffusion systems. Traditionally, computational domains have been divided into regular voxels. Molecules are assumed well mixed within each of these voxels and are allowed to react with other molecules within the same voxel or to jump to neighboring voxels with predefined transition rates. For a variety of reasons implementing position-jump processes on irregular grids is becoming increasingly important. However, it is not immediately clear what form an appropriate irregular partition of the domain should take if it is to allow the derivation of mean molecular concentrations that agree with a given partial differential equation for molecular concentrations. It has been demonstrated, in one dimension, that the Voronoi domain partition is the appropriate method with which to divide the computational domain. In this Brief Report, we investigate theoretically the propriety of the Voronoi domain partition as an appropriate method to partition domains for position-jump models in higher dimensions. We also provide simulations of diffusion processes in two dimensions in order to corroborate our results.

Periodically sheared colloids at low densities demonstrate a dynamical phase transition from an inactive to active phase as the strain amplitude is increased. The inactive phase consists of no collisions (contacts) between particles in the steady state limit, while in the active phase collisions persist. To investigate this system at higher densities, we construct and study a conserved-particle-number contact process with three-body interactions, which are potentially more likely than two-body interactions at higher densities. For example, consider one active (diffusing) particle colliding with two inactive (nondiffusing) particles such that they become active and consider spontaneous inactivation. In mean field, this system exhibits a continuous dynamical phase transition. Simulations on square lattices also indicate a continuous transition with exponents similar to those measured for the conserved lattice gas (CLG) model. In contrast, the three-body interaction requiring two active particles to activate one inactive particle exhibits a discontinuous transition. Finally, inspired by kinetically constrained models of the glass transition, we investigate the “caging effect” at even higher particle densities to look for a second dynamical phase transition back to an inactive phase. Square lattice simulations suggest a continuous transition with a new set of exponents differing from both the CLG model and what is known as directed percolation, indicating a potentially new universality class for a contact process with a conserved particle number.

We investigate ordering of granular rods in a container subject to vibrations in a gravitational field as a function of number density of the rods. We study rods with three different length to diameter aspect ratios $A_r=$ 5, 10, and 15. The measurements are performed in three dimensions using x-ray computer tomography to visualize the rods in the entire container. We first discuss a method to extract the position and orientation of the rods from the scans which enables us to obtain statistical measures of the degree of order in the packing. We find that the rods with $A_r=5$ phase separate into domains with vertical and horizontal orientation as the number density of the rods is increased, whereas, for $A_r=10$ and 15 the rods are predominately oriented vertically in layers. By calculating two-point spatial correlation functions, we further show that long range hexagonal order occurs within a layer when the rods are oriented along the vertical axis. Thus, our experiments find that long range order increases rapidly in granular rods with growing anisotropy.

Vibrating walls, frequently employed to maintain the temperature (i.e., average velocity) in a granular gas, modify the system strongly, rendering it dissimilar to a molecular one in various aspects. As evidenced by microgravity experiments employing a quasi-two-dimensional (quasi-2D) rectangular box and by 2D simulations, the one-peak velocity distribution is split into two, rendering the stress both nonuniform and anisotropic—without a shear flow and in the absence of gravitation. To account for this, granular hydrodynamics (as first proposed by Haff and later derived employing the kinetic theory) is generalized by introducing two additional variables, with one accounting for the distance between the two peaks and a second for the difference between the average velocities along different directions. The hydrodynamic theory thus generalized relates the velocity distribution to the stress, yielding results that agree with experiments and simulations.

It has been difficult to establish a clear connection between the hydrophobic interaction among small molecules typically studied in molecular simulations (a weak, oscillatory force) and that found between large, macroscopic surfaces in experiments (a strong, monotonic force). Here, we show that both types of interaction can emerge with a simple, core-softened water model that captures water's unique pairwise structure. As in hydrophobic hydration, we find that the hydrophobic interaction manifests a length-scale dependence, exhibiting distinct driving forces in the molecular and macroscopic regimes. Moreover, the ability of this simple model to capture both regimes suggests that several features of the hydrophobic force can be understood merely through water's pair correlations.

We derive the “conservation law” for nematic polymers in tensorial form valid for quadrupolar orientational order, in contradistinction to the conservation law in the case of polar orientational order. Due to microscopic differences in the coupling between the orientational field deformations and the density variations for polar and quadrupolar order, we find that the respective order parameters satisfy fundamentally distinct constraints. Being necessarily scalar in its form, the tensorial conservation law is obtained straightforwardly from the gradients of the polymer nematic tensor field and connects the spatial variation of this tensor field with density variations. We analyze the differences between the polar and the tensorial forms of the conservation law, present some explicit orientational fields that satisfy the tensorial constraint, and discuss the role of singular “hairpins,” which do not affect the local quadrupolar order of polymer nematics, but nevertheless influence its gradients.

A nonlinear data assimilation technique is applied to determine and track effective connections between ensembles of cultured spinal cord neurons measured with multielectrode arrays. The method is statistical, depending only on confidence intervals, and requiring no form of arbitrary thresholding. In addition, the method updates connection strengths sequentially, enabling real-time tracking of nonstationary networks. The ensemble Kalman filter is used with a generic spiking neuron model to estimate connection strengths as well as other system parameters to deal with model mismatch. The method is validated on noisy synthetic data from Hodgkin-Huxley model neurons before being used to find network connections in the neural culture recordings.

This paper reintroduces the network reliability polynomial, introduced by Moore and Shannon [Moore and Shannon, J. Franklin Inst. 262, 191 (1956)], for studying the effect of network structure on the spread of diseases. We exhibit a representation of the polynomial that is well suited for estimation by distributed simulation. We describe a collection of graphs derived from Erdős-Rényi and scale-free-like random graphs in which we have manipulated assortativity-by-degree and the number of triangles. We evaluate the network reliability for all of these graphs under a reliability rule that is related to the expected size of a connected component. Through these extensive simulations, we show that for positively or neutrally assortative graphs, swapping edges to increase the number of triangles does not increase the network reliability. Also, positively assortative graphs are more reliable than neutral or disassortative graphs with the same number of edges. Moreover, we show the combined effect of both assortativity-by-degree and the presence of triangles on the critical point and the size of the smallest subgraph that is reliable.

In this paper, a study on the directed particle current on flexible surfaces excited by a harmonic wave is reported. The proposed theory considers three different models for the kinematics of the surface, namely the Euler-Bernoulli, Timoshenko, and Rayleigh surface wave models. The particle-surface interaction terms in the theory incorporate Coulomb friction and inelastic collision between the particle and the surface. Three possible phases of motion, namely sticking, sliding, and jumping, are considered, and the phase transition boundaries are estimated analytically for a general surface model. The effect of various parameters on the particle current and certain statistical features of the particle motion are then studied numerically. Remarkably, the particle current spectra exhibit, in addition to resonance modes, antiresonance and secondary resonance modes and transversal zero crossings. These features have interesting implications for the particle dynamics in terms of dynamic jamming states and particle eddies, which are pointed out. Under certain restricted conditions, averaging calculations are also performed and compared with the corresponding numerical simulations.

High-resolution three-dimensional numerical simulations are carried out for hydrothermal waves in a thermocapillary liquid bridge with Prandtl number $\text{Pr}=4$ and length-to-radius aspect ratio $\Gamma =0.66$. The flow topology is analyzed using Poincaré sections in a frame of reference co-rotating with the phase velocity of the wave. We find regions of regular and chaotic motion. The regular regions are shown to be of key importance for dissipative structures of transported particles. Suspended particles which are passively advected in the bulk, but experience dissipation in a thin layer below the free surface, can rapidly form dissipative structures, also called particle accumulation structures. The shape and the formation time of the particulate structures are determined by the location of the invariant tori of the flow field with respect to the sub-surface layer in which the dissipation of the particle motion acts. The results from a hard-wall particle–free-surface interaction model are in good agreement with experimental observations.

We present a set of uniform polynomial equations that provides multidimensional on-lattice higher-order models of the lattice Boltzmann theory while keeping compact the number of discrete velocities. As examples, we explicitly derive one-, two-, and three-dimensional on-lattice models applicable to describing thermal compressible flows of the accuracy levels of the Navier-Stokes equations with smaller numbers of discrete velocities in comparison to the existing models. We demonstrate the accuracy and stability of the one-, two-, and three-dimensional models by using the Riemann problem.

We propose improved measurement methods of interparticle force between nematic colloids. Although various methods have been utilized for the force measurement, the comparison between the forces obtained by different methods has not been reported. In the frequently used method called the “free-release” method, the hydrodynamic interaction between moving particles has a serious influence on the measurement. In this study we modified those measurement methods by taking the long-ranged hydrodynamic interaction into account. The evaluated forces have been compared with that obtained by the dual beam “optical trap” method, which is free from the hydrodynamic effect. The agreement between them is quantitatively fairly good.

In two recent papers [Phys. Rev. E 85, 025302(R) (2012) and Phys. Rev. E 85, 056314 (2012)], the authors proposed fractal continuum hydrodynamics and its application to model fluid flows in fractally permeable reservoirs. While in general providing a certain advancement of continuum mechanics modeling of fractal media to fluid flows, some results and statements to previous works need clarification. We first show that the nonlocal character those authors alleged in our paper [Proc. R. Soc. A 465, 2521 (2009)] actually does not exist; instead, all those works are in the same general representation of derivative operators differing by specific forms of the line coefficient $c_1$. Next, the claimed generalization of the volumetric coefficient $c_3$ is, in fact, equivalent to previously proposed product measures when considering together the separate decomposition of $c_3$ on each coordinate. Furthermore, the modified Jacobian proposed in the two commented papers does not relate the volume element between the current and initial configurations, which henceforth leads to a correction of the Reynolds’ transport theorem. Finally, we point out that the asymmetry of the Cauchy stress tensor resulting from the conservation of the angular momentum must not be ignored; this aspect motivates a more complete formulation of fractal continuum models within a micropolar framework.

The aim of this Reply is to elucidate the difference between the fractal continuum models used in the preceding Comment and the models of fractal continuum flow which were put forward in our previous articles [Phys. Rev. E 85, 025302(R) (2012); 85, 056314 (2012)]. In this way, some drawbacks of the former models are highlighted. Specifically, inconsistencies in the definitions of the fractal derivative, the Jacobian of transformation, the displacement vector, and angular momentum are revealed. The proper forms of the Reynolds’ transport theorem and angular momentum principle for the fractal continuum are reaffirmed in a more illustrative manner. Consequently, we emphasize that in the absence of any internal angular momentum, body couples, and couple stresses, the Cauchy stress tensor in the fractal continuum should be symmetric. Furthermore, we stress that the approach based on the Cartesian product measured and used in the preceding Comment cannot be employed to study the path-connected fractals, such as a flow in a fractally permeable medium. Thus, all statements of our previous works remain unchallenged.

By means of Monte Carlo simulations and a finite-size scaling analysis, we find a critical line of an $n$-component Eulerian face-cubic model on the square lattice and the simple cubic lattice in the region $v>1$, where $v$ is the bond weight. The phase transition belongs to the Ising universality class independent of $n$. The critical properties of the phase transition can also be captured by the percolation of the complement of the Eulerian graph.