Neural field models of firing rate activity typically take the form of integral equations with space-dependent axonal delays. Under natural assumptions on the synaptic connectivity we show how one can derive an equivalent partial differential equation (PDE) model that properly treats the axonal delay terms of the integral formulation. Our analysis avoids the so-called long-wavelength approximation that has previously been used to formulate PDE models for neural activity in two spatial dimensions. Direct numerical simulations of this PDE model show instabilities of the homogeneous steady state that are in full agreement with a Turing instability analysis of the original integral model. We discuss the benefits of such a local model and its usefulness in modeling electrocortical activity. In particular, we are able to treat “patchy” connections, whereby a homogeneous and isotropic system is modulated in a spatially periodic fashion. In this case the emergence of a “lattice-directed” traveling wave predicted by a linear instability analysis is confirmed by the numerical simulation of an appropriate set of coupled PDEs.

An unexpected pattern transition has been found experimentally in the transformation from hexagons to stripes caused by an applied anisotropy effect. The particular system studied is the surface instability of a horizontal layer of magnetic liquid in a tilted magnetic field. Two orthogonal Helmholtz pairs of coils provide a vertical and a tangential magnetic field. Whereas the vertical component destabilizes the flat layer, the tangential one preserves its stability. The ensuing surface patterns comprise regular hexagons, anisotropic hexagons, and stripelike ridges. The phase diagram for the tilted field instability is measured using a radioscopic technique. The investigation reveals an interesting effect: the flip from one hexagonal pattern to another under an increasing tangential field component, which is explained in terms of amplitude equations as a saddle-node bifurcation.

Time-delayed feedback methods can be used to control unstable periodic orbits as well as unstable steady states. We present an application of extended time delay autosynchronization introduced by Socolar et al. [Phys. Rev. E 50, 3245 (1994)] to an unstable focus. This system represents a generic model of an unstable steady state which can be found, for instance, in Hopf bifurcation. In addition to the original controller design, we investigate effects of control loop latency and a bandpass filter on the domain of control. Furthermore, we consider coupling of the control force to the system via a rotational coupling matrix parametrized by a variable phase. We present an analysis of the domain of control and support our results by numerical calculations.

In several recent applications, including those aimed at developing thermal interface materials, nanoparticulate systems have been proposed to improve the effective behavior of the system. While nanoparticles by themselves may have low conductivities relative to larger particles owing to interfacial resistance, their use along with larger particles is believed to enhance the percolation threshold leading to better effective behavior overall. One critical challenge in using nanoparticulate systems is the lack of knowledge regarding their thermal conductivity. In this paper, the thermal conductivity of silica clusters (or nanoparticles) as well as nanowires is determined using molecular dynamics (MD) simulations. The equilibrium MD simulations of nanoparticles using Green-Kubo relations are demonstrated to be computationally very expensive and unsuitable for such nanoscaled systems. A nonequilibrium MD method adapted from the study of Müller-Plathe is shown to be faster and more accurate. The method is first demonstrated on bulk amorphous silica (using both cubic and orthorhombic simulation cells) and silica nanowires. The thermal conductivity values are compared to those reported in the literature. The mean thermal conductivity values for bulk silica and silica nanowire were estimated to be $1.2\mathrm{W}∕\mathrm{mK}$ and $1.435\mathrm{W}∕\mathrm{mK}$, respectively. To model nanoparticles, the Müller-Plathe technique is adapted by dividing the cluster into concentric shells so as to capture the naturally radial mode of heat transfer. The mean thermal conductivity value of a 600-atom silica nanoparticle obtained using this approach was $0.589\mathrm{W}∕\mathrm{mK}$. This value is $\sim 50–60\%$ lower than those of bulk silica or silica nanowire.

We employ a successive Monte Carlo renormalization group procedure in the presence of a linearly varying temperature to study the three-state $\left(q=3\right)$ Potts model on square lattices. By matching correlation functions of different lattice sizes at different renormalization levels, a rate exponent $r$ associated with the temperature sweep rate $R$ and the correlation length exponent $\nu$ are obtained. The dynamic critical exponent $z$ is then obtained by the scaling law $z=r-1∕\nu$ derived in our method. The results of $z=2.171\left(62\right)$ for $q=3$ in this work and the previously obtained $z=2.15\left(13\right)$ for $q=2$ seem to support the extension of the “weak universality” hypothesis to dynamic critical behavior. With these calculated exponents, the dynamic scaling forms of both specific heat and order parameter at various $R$ are presented and all the other static critical exponents are determined, which verify our method. Discussions are made on how to improve the accuracy of the estimation of the dynamic critical exponent.

A theoretical calculation of bulk viscosity has been carried out by deriving an expression for the relaxation time which appears in the formula for bulk viscosity derived by Okumura and Yonezawa. The expression involved a pair distribution function and interaction potential. Numerical results have been obtained over a wide range of densities and temperatures for Lennard-Jones fluids. It is found that our results provide a good description of bulk viscosity as has been judged by comparing the results with nonequilibrium molecular dynamics results. In addition, our results demonstrate the importance of the multiparticle correlation function.

We apply a model for sand dunes to calculate formation of dunes on Mars under the present Martian atmospheric conditions. We find that different dune shapes as those imaged by Mars Global Surveyor could have been formed by the action of sand-moving winds occuring on today’s Mars. Our calculations show, however, that Martian dunes could be only formed due to the higher efficiency of Martian winds in carrying grains into saltation. The model equations are solved to study saltation transport under different atmospheric conditions valid for Mars. We obtain an estimate for the wind speed and migration velocity of barchan dunes at different places on Mars. From comparison with the shape of bimodal sand dunes, we find an estimate for the time scale of the changes in Martian wind regimes.

Nanoscale electron transport through the purple membrane monolayer, a two-dimensional crystal lattice of the transmembrane protein bacteriorhodopsin, is studied by conductive atomic force microscopy. We demonstrate that the purple membrane exhibits nonresonant tunneling transport, with two characteristic tunneling regimes depending on the applied voltage (direct and Fowler-Nordheim). Our results show that the purple membrane can carry significant current density at the nanometer scale, several orders of magnitude larger than previously estimated by macroscale measurements.

Hopfield’s Lyapunov function is used to view the stability and topology of equilibria in neuronal networks for visual rivalry and pattern formation. For two neural populations with reciprocal inhibition and slow adaptation, the dynamics of neural activity is found to include a pair of limit cycles: one for oscillations between states where one population has high activity and the other has low activity, as in rivalry, and one for oscillations between states where both populations have the same activity. Hopfield’s Lyapunov function is used to find the dynamical mechanism for oscillations and the basin of attraction of each limit cycle. For a spatially continuous population with lateral inhibition, stable equilibria are found for local regions of high activity (solitons) and for bound states of two or more solitons. Bound states become stable when moving two solitons together minimizes the Lyapunov function, a result of decreasing activity in regions between peaks of high activity when the firing rate is described by a sigmoid function. Lowering the barrier to soliton formation leads to a pattern-forming instability, and a nonlinear solution to the dynamical equations is found to be given by a soliton lattice, which is completely characterized by the soliton width and the spacing between neighboring solitons. Fluctuations due to noise create lattice vacancies analogous to point defects in crystals, leading to activity which is spatially inhomogeneous.

A contact of a falling spherical particle with another fixed one in an unbounded viscous fluid is theoretically investigated based on a model of adding the contact interaction to the gravitational and hydrodynamic forces. The hydrodynamic interaction between the two particles is dealt with using an extended successive reflection method, with which the complete solution to the exterior velocity field around the two-particle system is constructed on the basis of the general expression given by Lamb, and then the hydrodynamic forces and torques on the two particles are obtained by integrating the fluid stress over each particle surface. The mechanical contact force is characterized by the standard friction theory with a criterion responsible for the transition from pure rolling to rolling with slip. Resorting to the dynamical equations of motion including the gravitational, hydrodynamic, and contact forces, the settling motion of a spherical particle in the vicinity of another fixed one is depicted using the fourth-order Runge-Kutta-Fehlberg method. Compared with the experimental results available in the literature, the theoretical prediction confirms two moving patterns at contact: pure rolling and rolling with slip, analyzes the dependence of the transition from one to another on the static friction coefficient and the contact separation distance between the particle surfaces, and accounts for a limitation of the quasisteady description of two interacting noncolloidal particles.

The statistics of the multidimensional Gaussian point process are discussed in connection with the spacing statistics of eigenvalues of $2\times 2$ random matrices. We consider the three-dimensional Gaussian point process when two of the coordinates of a point are randomly chosen from a Gaussian distribution having a mean of zero and a variance of ${\sigma }^2=1$ but the third coordinate is chosen from a Gaussian distribution having a variance in the range of $0⩽{\sigma }_3^2⩽1$. The probability density function associated with a random point being at a distance $r$ from the origin is shown to be closely related to the nearest-neighbor spacing distribution of eigenvalues coming from an ensemble of $2\times 2$ matrices defined by the French-Kota–Pandey-Mehta two-matrix model of random matrix theory. An elementary explanation of this result is given.

Tunneling processes in frustrated total internal reflection are reexamined in order to compare the results already obtained from adopting a procedure based on a transition-element analysis with those recently reported on the basis of a statistical method applied to a Brownian-like motion. A close agreement between the two approaches can be established, at optical and microwave scales, when suitable values of the involved parameters are adopted.

In this, the fourth of a series of papers [the first three papers were Phys. Rev. E 68, 016103 (2003), 68, 036133 (2003), and Phys. Lett. A 334, 12 (2005)] on the response of overdamped noisy bistable systems subject to an asymmetrizing constant signal superimposed on a time-sinusoidal driving signal, we obtain analytic expressions for the power spectral density of the response, including a detailed theoretical analysis of the power spectrum. The results are valid for any two-state system, however the specific case of the Duffing (or standard quartic) potential is considered in detail. The stochastic dynamics are confined to the weak noise limit (periodic signal amplitude much greater than noise intensity), i.e., when the response of the system to the external periodic field is strongly nonlinear.

We study the residence and local times for a Brownian particle confined by reflecting boundaries, and propose a general solution to the problem of finding the related probability distribution. Its Fourier transform (characteristic function) and Laplace transform (survival probability) are obtained in a compact matrix form involving the Laplace operator eigenbasis. Explicit combinatorial relations are derived for the moments, and the probability distribution is shown to be nearly Gaussian when the exploration time is long enough. When the eigenbasis (or a part of it) is known, the numerical computation of the residence time distributions is straightforward and accurate. The present approach can also be applied to investigate other functionals of reflected Brownian motion describing, in particular, restricted diffusion in an external field or potential (e.g., nuclei diffusing in an inhomogeneous magnetic field). Theoretical results for the local times are confronted with Monte Carlo simulations on the unit interval, disk, and sphere.

We study the classical dimer model on a square lattice with a single vacancy by developing a graph-theoretic classification of the set of all configurations which extends the spanning tree formulation of close-packed dimers. With this formalism, we can address the question of the possible motion of the vacancy induced by dimer slidings. We find a probability $57∕4-10\sqrt{2}$ for the vacancy to be strictly jammed in an infinite system. More generally, the size distribution of the domain accessible to the vacancy is characterized by a power law decay with exponent $9∕8$. On a finite system, the probability that a vacancy in the bulk can reach the boundary falls off as a power law of the system size with exponent $1∕4$. The resultant weak localization of vacancies still allows for unbounded diffusion, characterized by a diffusion exponent that we relate to that of diffusion on spanning trees. We also implement numerical simulations of the model with both free and periodic boundary conditions.

We present the results of experiments studying the shape of craters formed by the normal impact of a solid spherical projectile into a deep noncohesive granular bed at low energies. The resultant impact crater surfaces are accurately digitized using laser profilometry, allowing for the detailed investigation of the crater shape. We find that these impact craters are very nearly hyperbolic in profile. Crater radii and depths are dependent on impact energy, as well as the projectile density and size. The precise crater shape is a function of the crater aspect ratio. While the dimensions of the crater are highly dependent on the impact energy, we show that the energy required to excavate the crater is only a tiny fraction (0.1%–0.5%) of the kinetic energy of the projectile.

Like-charged colloidal spheres dispersed in de-ionized water are supposed to repel each other. Instead, artifact-corrected video microscopy measurements reveal an anomalous long-ranged like-charge attraction in the interparticle pair potential when the spheres are confined to a layer by even a single-charged glass surface. These attractions can be masked by electrostatic repulsions at low ionic strengths. Coating the bounding surfaces with a conducting gold layer suppresses the attraction. These observations suggest a possible mechanism for the anomalous confinement-induced attractions.