A study of the transport coefficients of a system of elastic hard disks based on the use of Helfand-Einstein expressions is reported. The self-diffusion, the viscosity, and the heat conductivity are examined with averaging techniques especially appropriate for event-driven molecular dynamics algorithms with periodic boundary conditions. The density and size dependence of the results are analyzed, and comparison with the predictions from Enskog’s theory is carried out. In particular, the behavior of the transport coefficients in the vicinity of the fluid-solid transition is investigated and a striking power law divergence of the viscosity with density is obtained in this region, while all other examined transport coefficients show a drop in that density range in relation to the Enskog’s prediction. Finally, the deviations are related to shear band instabilities and the concept of dilatancy.

We study the intermittency and noise of dislocation systems undergoing shear deformation. Simulations of a simple two-dimensional discrete dislocation dynamics model indicate that the deformation rate exhibits a power spectrum scaling of the type $1∕f^{\alpha }$. The noise exponent is far away from a Lorentzian, with $\alpha \approx 1.5$. This result is directly related to the way the durations of avalanches of plastic deformation activity scale with their size.

Some recent developments for the validation of nonlinear models built from data are reviewed. Besides giving an overall view of the field, a procedure is proposed and investigated based on the concept of dissipative synchronization between the data and the model, which is very useful in validating models that should reproduce dominant dynamical features, like bifurcations, of the original system. In order to assess the discriminating power of the procedure, four well-known benchmarks have been used: namely, Duffing-Ueda, Duffing-Holmes, and van der Pol oscillators, plus the Hénon map. The procedure, developed for discrete-time systems, is focused on the dynamical properties of the model, rather than on statistical issues. For all the systems investigated, it is shown that the discriminating power of the procedure is similar to that of bifurcation diagrams—which in turn is much greater than, say, that of correlation dimension—but at a much lower computational cost.

Time series from complex systems with interacting nonlinear and stochastic subsystems and hierarchical regulations are often multiscaled. In devising measures characterizing such complex time series, it is most desirable to incorporate explicitly the concept of scale in the measures. While excellent scale-dependent measures such as $ϵ$ entropy and the finite size Lyapunov exponent (FSLE) have been proposed, simple algorithms have not been developed to reliably compute them from short noisy time series. To promote widespread application of these concepts, we propose an efficient algorithm to compute a variant of the FSLE, the scale-dependent Lyapunov exponent (SDLE). We show that with our algorithm, the SDLE can be accurately computed from short noisy time series and readily classify various types of motions, including truly low-dimensional chaos, noisy chaos, noise-induced chaos, random $1∕f^{\alpha }$ and $\alpha$-stable Levy processes, stochastic oscillations, and complex motions with chaotic behavior on small scales but diffusive behavior on large scales. To our knowledge, no other measures are able to accurately characterize all these different types of motions. Based on the distinctive behaviors of the SDLE for different types of motions, we propose a scheme to distinguish chaos from noise.

While fundamental-mode discrete solitons have been demonstrated with both self-focusing and defocusing nonlinearity, high-order-mode localized states in waveguide lattices have been studied thus far only for the self-focusing case. In this paper, the existence and stability regimes of dipole, quadrupole, and vortex soliton structures in two-dimensional lattices induced with a defocusing nonlinearity are examined by the theoretical and numerical analysis of a generic envelope nonlinear lattice model. In particular, we find that the stability of such high-order-mode solitons is quite different from that with self-focusing nonlinearity. As a simple example, a dipole (“twisted”) mode soliton with adjacent excited sites which may be stable in the focusing case becomes unstable in the defocusing regime. Our results may be relevant to other two-dimensional defocusing periodic nonlinear systems such as Bose-Einstein condensates with a positive scattering length trapped in optical lattices.

The design of appropriate multifractal analysis algorithms, able to correctly characterize the scaling properties of multifractal systems from experimental, discretized data, is a major challenge in the study of such scale invariant systems. In the recent years, a growing interest for the application of the microcanonical formalism has taken place, as it allows a precise localization of the fractal components as well as a statistical characterization of the system. In this paper, we deal with the specific problems arising when systems that are strictly monofractal are analyzed using some standard microcanonical multifractal methods. We discuss the adaptations of these methods needed to give an appropriate treatment of monofractal systems.

Brownian trajectory simulation methods are employed to fully establish the non-Gaussian fluctuation effects predicted by our nonlinear Langevin equation theory of single particle activated dynamics in glassy hard-sphere fluids. The consequences of stochastic mobility fluctuations associated with the space-time complexities of the transient localization and barrier hopping processes have been determined. The incoherent dynamic structure factor was computed for a range of wave vectors and becomes of an increasingly non-Gaussian form for volume fractions beyond the (naive) ideal mode coupling theory (MCT) transition. The non-Gaussian parameter (NGP) amplitude increases markedly with volume fraction and is well described by a power law in the maximum restoring force of the nonequilibrium free energy profile. The time scale associated with the NGP peak becomes much smaller than the $\alpha$ relaxation time for systems characterized by significant entropic barriers. An alternate non-Gaussian parameter that probes the long time $\alpha$ relaxation process displays a different shape, peak intensity, and time scale of its maximum. However, a strong correspondence between the classic and alternate NGP amplitudes is predicted which suggests a deep connection between the early and final stages of cage escape. Strong space-time decoupling emerges at high volume fractions as indicated by a nondiffusive wave vector dependence of the relaxation time and growth of the translation-relaxation decoupling parameter. Displacement distributions exhibit non-Gaussian behavior at intermediate times, evolving into a strongly bimodal form with slow and fast subpopulations at high volume fractions. Qualitative and semiquantitative comparisons of the theoretical results with colloid experiments, ideal MCT, and multiple simulation studies are presented.

We present a method of visualizing topological defects arising in numerical simulations of liquid crystals. The method is based on scientific visualization techniques developed to visualize second-rank tensor fields, yielding information not only on the local structure of the field but also on the continuity of these structures. We show how these techniques can be used to first locate topological defects in fluid simulations of nematic liquid crystals where the locations are not known a priori and then study the properties of these defects including the core structure. We apply these techniques to simulation data obtained by previous authors who studied a rapid quench and subsequent equilibration of a Gay-Berne nematic. The quench produces a large number of disclination loops which we locate and track with the visualization methods. We show that the cores of the disclination lines have a biaxial region and the loops themselves are of a hybrid wedge-twist variety.

The dynamics of unidirectionally propagating pulses in a two-dimensional uniform excitable reaction-diffusion medium is investigated. It is shown that under weak diffusion coupling between medium points such a pulse can evolve into a pair of counter-rotating spirals (spiral pair). We analyze the drift of such a pair and examine the collisions between several drifting pairs. It is demonstrated that collisions can result in a special type of reflection or, alternatively, in new types of complex stationary spiral structures. A possible application of these findings for the diagnosis of cardiac arrhythmias is suggested.

Cavitation bubbles are generated in water by low-energy femtosecond laser pulses in the presence of an ultrasonic field. Bubble dynamics and cavitation luminescence are investigated by CCD photography and photomultiplier measurements in dependence on the phase of the acoustic cycle at which the bubbles are generated. The experimental results demonstrate that the initially small laser-generated bubbles can be expanded significantly by the sound field and that weak cavitation luminescence can be observed in two small intervals of the seeding phase. The luminescence yield sensitively depends on the degree of sphericity of bubble collapse.

We introduce a pattern-formation model based on a symmetric system of three linearly coupled cubic-quintic complex Ginzburg-Landau equations, which form a triangular configuration. This is the simplest model of a multicore fiber laser. We identify stability regions for various types of localized patterns possible in this setting, which include stationary and breathing triangular vortices.

Motivated by a beautiful demonstration of the Faraday and the Lenz laws in which a small neodymium magnet falls slowly through a conducting nonferromagnetic tube, we consider the dynamics of a magnet falling coaxially through a superconducting pipe. Unlike the case of normal conducting pipes, in which the magnet quickly reaches the terminal velocity, inside a superconducting tube the magnet falls freely. On the other hand, to enter the pipe the magnet must overcome a large electromagnetic energy barrier. For sufficiently strong magnets, the barrier is so large that the magnet will not be able to penetrate it and will be levitated over the mouth of the pipe. We calculate the work that must done to force the magnet to enter a superconducting tube. The calculations show that superconducting pipes are very efficient at screening magnetic fields. For example, the magnetic field of a dipole at the center of a short pipe of radius $a$ and length $L\gtrsim a$ decays, in the axial direction, with a characteristic length $\xi \approx 0.26a$. The efficient screening of the magnetic field might be useful for shielding highly sensitive superconducting quantum interference devices. Finally, the motion of the magnet through a superconducting pipe is compared and contrasted to the flow of ions through a trans-membrane channel.

We argue that structural rearrangements experienced by an assembly of hard disks under increasing disk density are accompanied by the mutual caging of each disk by its three alternating Voronoi nearest neighbors. This caging becomes effective at a packing fraction $\eta =\pi \sqrt{3}∕8\approx 0.680$ when the average gap width between neighboring disks in the system shrinks to about 15% of the disk diameter. The freezing occurs when the fraction of caged disks is about 40%.

The Betti numbers are fundamental topological quantities that describe the $k$-dimensional connectivity of an object: ${\beta }_0$ is the number of connected components and ${\beta }_k$ effectively counts the number of $k$-dimensional holes. Although they are appealing natural descriptors of shape, the higher-order Betti numbers are more difficult to compute than other measures and so have not previously been studied per se in the context of stochastic geometry or statistical physics. As a mathematically tractable model, we consider the expected Betti numbers per unit volume of Poisson-centered spheres with radius $\alpha$. We present results from simulations and derive analytic expressions for the low intensity, small radius limits of Betti numbers in one, two, and three dimensions. The algorithms and analysis depend on alpha shapes, a construction from computational geometry that deserves to be more widely known in the physics community.

We exactly solve the ferromagnetic spin-$\frac{1}{2}$ Ising model on the Bethe lattice in the presence of an external magnetic field by means of the equations of motion method within the Green’s function formalism. In particular, such an approach is applied to an isomorphic model of localized Fermi particles interacting via an intersite Coulomb interaction. A complete set of eigenoperators is found together with the corresponding eigenvalues. The Green’s functions and the correlation functions are written in terms of a finite set of parameters to be self-consistently determined. A procedure is developed that allows us to exactly fix the unknown parameters in the case of a Bethe lattice with any coordination number $z$. Nonlocal correlation functions up to four points are also provided together with a study of the relevant thermodynamic quantities.

We study the statics and dynamics of an elastic manifold in a disordered medium with quenched defects correlated as $\sim r^{-a}$ for large separation $r$. We derive the functional renormalization-group equations to one-loop order, which allow us to describe the universal properties of the system in equilibrium and at the depinning transition. Using a double $\epsilon =4-d$ and $\delta =4-a$ expansion we compute the fixed points characterizing different universality classes and analyze their regions of stability. The long-range disorder-correlator remains analytic but generates short-range disorder whose correlator exhibits the usual cusp. The critical exponents and universal amplitudes are computed to first order in $\epsilon$ and $\delta$ at the fixed points. At depinning, a velocity-versus-force exponent $\beta$ larger than unity can occur. We discuss possible realizations using extended defects.

Quasirigidity means that one builds a theory for assemblies of grains under a slowly changing external load by using the deformation of those grains as a small parameter. Is quasirigidity a complete theory for these granular assemblies? Does it provide unique predictions of the assembly’s behavior, or must some other process be invoked to decide between several possibilities? We provide evidence that quasirigidity is a complete theory by showing that two possible sources of indeterminacy do not exist for the case of disk-shaped grains. One possible source of indeterminacy arises from zero-frequency modes present in the packing. This problem can be solved by considering the conditions required to obtain force equilibrium. A second possible source of indeterminacy is the necessity to choose the status (sliding or nonsliding) at each contact. We show that only one choice is permitted, if contacts slide only when required by Coulomb friction.

We investigate clogging of microchannels at the single-pore level using microfluidic devices as model porous media. The process of clogging is studied at low volume fractions and high flow rates, a technologically important regime. We show that clogging is independent of particle flow rate and volume fraction, indicating that collective effects do not play an important role. Instead, the average number of particles that can pass through a pore before it clogs scales with the ratio of pore to particle size. We present a simple model that accounts for the data.

We present a combined experimental and numerical investigation of the coiling of a liquid “rope” falling on a solid surface, focusing on three little-explored aspects of the phenomenon: The time dependence of “inertio-gravitational” coiling, the systematic dependence of the radii of the coil and the rope on the experimental parameters, and the “secondary buckling” of the columnar structure generated by high-frequency coiling. Inertio-gravitational coiling is characterized by oscillations between states with different frequencies, and we present experimental observations of four distinct branches of such states in the frequency-fall height space. The transitions between coexisting states have no characteristic period, may take place with or without a change in the sense of rotation, and usually (but not always) occur via an intermediate “figure of eight” state. We present extensive laboratory measurements of the radii of the coil and of the rope within it, and show that they agree well with the predictions of a “slender-rope” numerical model. Finally, we use dimensional analysis to reveal a systematic variation of the critical column height for secondary buckling as a function of (dimensionless) flow rate and surface tension parameters.