We investigate analytically the anchoring of a nematic liquid crystal on a two-dimensionally grooved surface of arbitrary shape, induced by the elastic distortions of a liquid crystal adjacent to the surface. Our theoretical framework applied to a surface with square grooves reveals that such a surface can exhibit bistable anchoring, while a direct extension of a well-known theory of Berreman [Phys. Rev. Lett. 28, 1683 (1972)] results in no azimuthal anchoring in the so-called one-constant case ($K_1=K_2=K_3$, with $K_1$, $K_2$, and $K_3$ being the splay, twist, and bend elastic constants, respectively). We show under the assumption of $K_1=K_2=K$ that the direction of the bistable easy axes and the anchoring strength crucially depend on the ratios $K_3/K$ and $K_{24}/K$, where $K_{24}$ is the saddle-splay surface elastic constant. To demonstrate the applicability of our theory to general cases and to elucidate the effect of surface shape and the elastic constants on the properties of surface anchoring, we also consider several specific cases of interest; one-dimensional grooves of arbitrary shape, rhombic grooves, and surfaces possessing $2N$-fold symmetry, including hexagonal grooves, and show the following: (i) The rescaled anchoring energy $f\left(\varphi \right)/f\left(\pi /2\right)$ of one-dimensional grooves, with $\varphi$ being the angle between the director $\mathit{n}$ and the groove direction, is independent of the groove shape. (ii) Whether two diagonal axes of rhombic grooves can become easy axes depends sensitively on $K_3/K$, $K_{24}/K$ and the angle $\alpha$ between the grooves. The angle $\alpha$ yielding the maximum anchoring strength for given groove pitch and amplitude depends again on $K_3/K$ and $K_{24}/K$; in some cases $\alpha =0$ (one-dimensional grooves), and in other cases $\alpha \ne 0$, gives the maximum anchoring strength. Square grooves $\left(\alpha =\pi /2\right)$ do not necessarily exhibit the largest anchoring strength. (iii) A surface possessing $2N$-fold symmetry can yield $N$-stable azimuthal anchoring. However, when $K_1=K_2=K_3$ and $N\ge 3$, azimuthal anchoring is totally absent irrespective of the value of $K_{24}$. The direction of the easy axes depends on $K_3/K$, $K_{24}/K$, and whether $N$ is even or odd.

Characteristic versus critical features of earthquakes are studied on the basis of the Olami-Feder-Christensen model. It is found that the local recurrence-time distribution exhibits a sharp $\delta$-function-like peak corresponding to rhythmic recurrence of events with a fixed “period” uniquely determined by the transmission parameter of the model, together with a power-law-like tail corresponding to scale-free recurrence of events. The model exhibits phenomena closely resembling the asperity known in seismology.

We investigate the predictability of extreme events in time series. The focus of this work is to understand under which circumstances large events are better predictable than smaller events. Therefore we use a simple prediction algorithm based on precursory structures which are identified using the maximum likelihood principle. Using the receiver operator characteristic curve as a measure for the quality of predictions we find that the dependence on the event size is closely linked to the probability distribution function of the underlying stochastic process. We evaluate this dependence on the probability distribution function analytically and numerically. If we assume that the optimal precursory structures are used to make the predictions, we find that large increments are better predictable if the underlying stochastic process has a Gaussian probability distribution function, whereas larger increments are harder to predict if the underlying probability distribution function has a power-law tail. In the case of an exponential distribution function we find no significant dependence on the event size. Furthermore we compare these results with predictions of increments in correlated data, namely, velocity increments of a free jet flow. The velocity increments in the free jet flow are in dependence on the time scale either asymptotically Gaussian or asymptotically exponential distributed. The numerical results for predictions within free jet data are in good agreement with the previous analytical considerations for random numbers.

The one-sided Lévy-stable probability densities and the discrete-stable distributions form a doubly stochastic Poisson transform pair. This relationship facilitates the formulation of a class of continuous-stable stochastic processes.

We describe a lattice-gas model suitable for studying the generic effects of polydispersity on liquid-vapor phase equilibria. Using Monte Carlo simulation methods tailored for the accurate determination of phase behavior under conditions of fixed polydispersity, we trace the cloud and shadow curves for a particular Schulz distribution of the polydisperse attribute. Although polydispersity enters the model solely in terms of the strengths of the interparticle interactions, this is sufficient to induce the broad separation of cloud and shadow curves seen both in more realistic models and experiments.

The structure of a network can significantly influence the properties of the dynamical processes that take place on them. While many studies have been paid to this influence, much less attention has been devoted to the interplay and feedback mechanisms between dynamical processes and network topology on adaptive networks. Adaptive rewiring of links can happen in real life systems such as acquaintance networks, where people are more likely to maintain a social connection if their views and values are similar. In our study, we consider different variants of a model for consensus formation. Our investigations reveal that the adaptation of the network topology fosters cluster formation by enhancing communication between agents of similar opinion, although it also promotes the division of these clusters. The temporal behavior is also strongly affected by adaptivity: while, on static networks, it is influenced by percolation properties, on adaptive networks, both the early and late time evolutions of the system are determined by the rewiring process. The investigation of a variant of the model reveals that the scenarios of transitions between consensus and polarized states are more robust on adaptive networks.

A combined experimental and numerical study of the boundary layer in a 4:1 aspect-ratio Rayleigh-Bénard cell over a four-decade range of Rayleigh numbers has been undertaken aimed at gaining a better insight into the character of the boundary layers. The experiments involved the simultaneous laser Doppler anemometry measurements of fluid velocity at two locations, i.e., in the boundary layer and far away from it in the bulk, for Rayleigh numbers varying between $1.6\times {10}^7$ and $2.4\times {10}^9$. In parallel, direct numerical simulations have been performed for the same configuration for Rayleigh numbers between $7.0\times {10}^4$ and $7.7\times {10}^7$. The temperature and velocity probability density functions and the power spectra of the horizontal velocity fluctuations measured in the boundary layer and in the bulk flow are found to be practically identical. Except for the smallest Rayleigh numbers, the spectra in the boundary layer and in the bulk central region are continuous and have a wide range of active scales. This indicates that both the bulk and the boundary layers are turbulent in the Ra number range considered. However, molecular effects can still be observed and the boundary layer does not behave like a classical shear-driven turbulent boundary layer.

We study the effects of the existence of another type of agents, called spies, in the minority game (MG). Unlike the normal agents in the MG, the spies do not carry any strategy. Instead, they decide their action by scouting some normal agents and take the minority action of the spied group. For a few spies and when there is useful information in the normal agents’ actions, the spies can avoid the crowd effect of the normal agents and win more readily. When information becomes less useful and when more spies are present, the spies’ crowd effect hurts the success rate of the spies themselves, and the normal agents could have a higher success rate than the spies. More spies actually assist more normal agents to win, as the spies also provide more winning quotas. This leads to a nonmonotonic behavior in the total success rate of the population as a function of the fraction of spies.

The problem of diffusion in a time-dependent (and generally inhomogeneous) external field is considered on the basis of a generalized master equation with two times, introduced by Trigger and co-authors [S. A. Trigger, G. J. F. van Heijst, and P. P. J. M. Schram, Physica A 347, 77 (2005); J. Phys.: Conf. Ser. 11, 37 (2005)]. We consider the case of the quasi-Fokker-Planck approximation, when the probability transition function for diffusion (PTD function) does not possess a long tail in coordinate space and can be expanded as a function of instantaneous displacements. The more complicated case of long tails in the PTD will be discussed separately. We also discuss diffusion on the basis of hydrodynamic and kinetic equations and show the validity of the phenomenological approach. A type of “collision” integral is introduced for the description of diffusion in a system of particles, which can transfer from a moving state to the rest state (with some waiting time distribution). The solution of the appropriate kinetic equation in the external field also confirms the phenomenological approach of the generalized master equation.

We found that in transition arrays of complex atomic spectra, the strengths of electric-dipolar lines obey Benford’s law, which means that their significant digits follow a logarithmic distribution favoring the smallest values. This indicates that atomic processes result from the superposition of uncorrelated probability laws and that the occurrence of digits reflects the constraints induced by the selection rules. Furthermore, Benford’ law can be a useful test of theoretical spectroscopic models. Its applicability to the statistics of electric-dipolar lines can be understood in the framework of random matrix theory and is consistent with the Porter-Thomas law.

Direct numerical simulation and stereoscopic particle image velocimetry of turbulent convection are used to gather spatial data for the calculation of structure functions. We wish to add to the ongoing discussion in the literature whether temperature acts as an active or passive scalar in turbulent convection, with consequences for structure-function scaling. The simulation results show direct confirmation of the scalings derived by Bolgiano and Obukhov for turbulence with an active scalar for both velocity and temperature statistics. The active-scalar range shifts to larger scales when the forcing parameter (Rayleigh number) is increased. Furthermore, a close inspection of local turbulent length scales (Kolmogorov and Bolgiano lengths) confirms conjectures from earlier studies that the oft-used global averages are not suited for the interpretation of structure functions. In the experiment, a characterization of the domain-filling large-scale circulation of confined convection is carried out for comparison with other studies. The measured velocity fields are also used to calculate velocity structure functions, further confirming the Bolgiano-Obukhov scalings when interpreted with the local turbulent length scales found in the simulations. An extended self-similarity analysis shows that the relative scalings are different for the Kolmogorov and Bolgiano-Obukhov regimes.

Duration-controlled amplified spontaneous emission with an intensity of ${10}^{13}\mathrm{W}∕{\mathrm{cm}}^2$ is used to convert a $7.5\text{−}\mu \mathrm{m}$-thick polyimide foil into a near-critical plasma, in which the $p$-polarized, $45\text{−}\mathrm{fs}$, ${10}^{19}\text{−}\mathrm{W}∕{\mathrm{cm}}^2$ laser pulse generates $3.8\text{−}\mathrm{MeV}$ protons, emitted at some angle between the target normal and the laser propagation direction of 45°. Particle-in-cell simulations reveal that the efficient proton acceleration is due to the generation of a quasistatic magnetic field on the target rear side with magnetic pressure inducing and sustaining a charge separation electrostatic field.

The combination of continuum many-body quantum physics and Monte Carlo methods provide a powerful and well established approach to first principles calculations for large systems. Replacing the exact solution of the problem with a statistical estimate requires a measure of the random error in the estimate for it to be useful. Such a measure of confidence is usually provided by assuming the central limit theorem to hold true. In what follows it is demonstrated that, for the most popular implementation of the variational Monte Carlo method, the central limit theorem has limited validity, or is invalid and must be replaced by a generalized central limit theorem. Estimates of the total energy and the variance of the local energy are examined in detail, and shown to exhibit uncontrolled statistical errors through an explicit derivation of the distribution of the random error. Several examples are given of estimated quantities for which the central limit theorem is not valid. The approach used is generally applicable to characterizing the random error of estimates, and to quantum Monte Carlo methods beyond variational Monte Carlo.

Expectation values of physical quantities may accurately be obtained by the evaluation of integrals within many-body quantum mechanics, and these multidimensional integrals may be estimated using Monte Carlo methods. In a previous publication it has been shown that for the simplest, most commonly applied strategy in continuum quantum Monte Carlo, the random error in the resulting estimates is not well controlled. At best the central limit theorem is valid in its weakest form, and at worst it is invalid and replaced by an alternative generalized central limit theorem and non-normal random error. In both cases the random error is not controlled. Here we consider a new “residual sampling strategy” that reintroduces the central limit theorem in its strongest form, and provides full control of the random error in estimates. Estimates of the total energy and the variance of the local energy within variational Monte Carlo are considered in detail, and the approach presented may be generalized to expectation values of other operators, and to other variants of the quantum Monte Carlo method.

The recently developed tensor renormalization-group (TRG) method provides a highly precise technique for deriving thermodynamic and critical properties of lattice Hamiltonians. The TRG is a local coarse-graining transformation, with the elements of the tensor at each lattice site playing the part of the interactions that undergo the renormalization-group flows. These tensor flows are directly related to the phase diagram structure of the infinite system, with each phase flowing to a distinct surface of fixed points. Fixed-point analysis and summation along the flows give the critical exponents, as well as thermodynamic functions along the entire temperature range. Thus, for the ferromagnetic triangular lattice Ising model, the free energy is calculated to better than ${10}^{-5}$ along the entire temperature range. Unlike previous position-space renormalization-group methods, the truncation (of the tensor index range $D$) in this general method converges under straightforward and systematic improvements. Our best results are easily obtained with $D=24$, corresponding to 4624-dimensional renormalization-group flows.

This article elucidates and analyzes the fundamental underlying structure of the renormalization group (RG) approach as it applies to the solution of any differential equation involving multiple scales. The amplitude equation derived through the elimination of secular terms arising from a naive perturbation expansion of the solution to these equations by the RG approach [L.-Y. Chen, N. Goldenfeld, and Y. Oono, Phys. Rev. E 54, 376 (1996)] is reduced to an algebraic equation which is expressed in terms of the Thiele semi-invariants or cumulants of the eliminant sequence ${\left\{Z_i\right\}}_{i=1}^{\infty }$. Its use is illustrated through the solution of both linear and nonlinear perturbation problems and certain results from the literature are recovered as special cases. The fundamental structure that emerges from the application of the RG approach is not the amplitude equation but the aforementioned algebraic equation.

We investigate experimentally the pattern formation process during injection of air in a noncohesive granular material confined in a linear Hele-Shaw cell. We characterize the features and dynamics of this pattern formation on the basis of fast image analysis and sensitive pressure measurements. Behaviors are classified using two parameters—injection pressure and plate opening—and four hydrodynamic regimes are defined. For some regions of the parameter space, flows of air and grains are shown to be strongly coupled and instable, and lead to channelization within the granular material with obvious large-scale permeability variations.

Many analytical and numerical works have been devoted to the prediction of macroscopic effective transport properties in particulate media. Usually, structure and properties of macroscopic balance and constitutive equations are stated a priori. In this paper, the upscaling of the transient diffusion equations in concentrated particulate media with possible particle-particle interfacial barriers, highly conductive particles, poorly conductive matrix, and temperature-dependent physical properties is revisited using the homogenization method based on multiple scale asymptotic expansions. This method uses no a priori assumptions on the physics at the macroscale. For the considered physics and microstructures and depending on the order of magnitude of dimensionless Biot and Fourier numbers, it is shown that some situations cannot be homogenized. For other situations, three different macroscopic models are identified, depending on the quality of particle-particle contacts. They are one-phase media, following the standard heat equation and Fourier’s law. Calculations of the effective conductivity tensor and heat capacity are proved to be uncoupled. Linear and steady state continuous localization problems must be solved on representative elementary volumes to compute the effective conductivity tensors for the two first models. For the third model, i.e., for highly resistive contacts, the localization problem becomes simpler and discrete whatever the shape of particles. In paper II [Vassal et al., Phys. Rev. E 77, 011303 (2008)], diffusion through networks of slender, wavy, entangled, and oriented fibers is considered. Discrete localization problems can then be obtained for all models, as well as semianalytical or fully analytical expressions of the corresponding effective conductivity tensors.

In paper I [Vassal et al., Phys. Rev. E77, 011302 (2008)] of this contribution, the effective diffusion properties of particulate media with highly conductive particles and particle-particle interfacial barriers have been investigated with the homogenization method with multiple scale asymptotic expansions. Three different macroscopic models have been proposed depending on the quality of contacts between particles. However, depending on the nature and the geometry of particles contained in representative elementary volumes of the considered media, localization problems to be solved to compute the effective conductivity of the two first models can rapidly become cumbersome, time and memory consuming. In this second paper, the above problem is simplified and applied to networks made of slender, wavy and entangled fibers. For these types of media, discrete formulations of localization problems for all macroscopic models can be obtained leading to very efficient numerical calculations. Semianalytical expressions of the effective conductivity tensors are also proposed under simplifying assumptions. The case of straight monodisperse and homogeneously distributed slender fibers with a circular cross section is further explored. Compact semianalytical and analytical estimations are obtained when fiber-fiber contacts are perfect or very poor. Moreover, two discrete element codes have been developed and used to solve localization problems on representative elementary volumes for the same types of contacts. Numerical results underline the significant roles of the fiber content, the orientation of fibers as well as the relative position and orientation of contacting fibers on the effective conductivity tensors. Semianalytical and analytical predictions are discussed and compared with numerical results.

The interplay of surface ordering and elasticity can be studied on the example of a liquid crystal confined to a cavity. We present a computer simulation study of a liquid of hard spherocylinders in a hard spherical cavity. With increasing density, first a uniaxial surface film forms and then a biaxial surface film, which eventually fills the entire cavity. We studied how the surface order, the adsorption, and the shape of the director field depend on the curvature of the wall. We find that orientational ordering at a curved wall is stronger than at a flat wall, while adsorption is weaker. For densities above the isotropic-nematic transition, we always find bipolar configurations.