An exact lattice Langevin equation is derived for the ballistic deposition model of surface growth. The continuum limit of this equation is dominated by the Kardar-Parisi-Zhang (KPZ) equation at all length and time scales. For a one-dimensional substrate the solution of the exact lattice Langevin equation yields the KPZ scaling exponents without any extrapolation. For a two-dimensional substrate the scaling exponents are different from those found from computer simulations. This discrepancy is discussed in relation to analytic approaches to the KPZ equation in higher dimensions.

The diffusion of a pulse of small grains in an horizontal rotating drum is studied through discrete elements methods simulations. We present a theoretical analysis of the diffusion process in a one-dimensional confined space in order to elucidate the effect of the confining end-plate of the drum. We then show that the diffusion is neither subdiffusive nor superdiffusive but normal. This is demonstrated by rescaling the concentration profiles obtained at various stages and by studying the time evolution of the mean squared deviation. Finally we study the self-diffusion of both large and small grains, and we show that it is normal and that the diffusion coefficient is independent of the grain size.

We present a network model for the study of melting and liquid structure in two dimensions, the first in which the presence and energy of topological defects (dislocations and disclinations) and of geometrical defects (elemental voids) can be independently controlled. Interparticle interaction is via harmonic springs and control is achieved by Monte Carlo moves which springs can either be orientationally “flipped” between particles to generate topological defects, or can be “popped” in force-free shape, to generate geometrical defects. With the geometrical defects suppressed the transition to the liquid phase occurs via disclination unbinding, as described by the Kosterlitz-Thouless-Halperin-Nelson-Young model and found in soft potential two-dimensional (2D) systems, such as the dipole-dipole potential [H. H. von Grünberg et al., Phys. Rev. Lett. 93, 255703 (2004)]. By contrast, with topological defects suppressed, a disordering transition, the Glaser-Clark condensation of geometrical defects [M. A. Glaser and N. A. Clark, Adv. Chem. Phys. 83, 543 (1993); M. A. Glaser et al., Springer Proceedings in Physics: Dynamics and Patterns in Complex Fluids (Springer-Verlag, Berlin, 1990), Vol. 52, p. 141], produces a state that accurately characterizes the local liquid structure and first-order melting observed in hard-potential 2D systems, such as hard disk and the Weeks-Chandler-Andersen (WCA) potentials (M. A. Glaser and co-workers, see above). Thus both the geometrical and topological defect systems play a role in melting. The present work introduces a system in which the relative roles of topological and geometrical defects and their interactions can be explored. We perform Monte Carlo simulations of this model in the isobaric-isothermal ensemble, and present the phase diagram as well as various thermodynamic, statistical, and structural quantities as a function of the relative populations of geometrical and topological defects. The model exhibits a rich phase behavior including hexagonal and square crystals, expanded crystal, dodecagonal quasicrystal, and isotropic liquid phases. In this system the geometrical defects effectively control the melting, reducing the solid-liquid transition temperature by a factor of $\sim 3$ relative to the topological-only case. The local structure of the dense liquid has been investigated and the results are compared to that from simulations of WCA systems.

A polyimide coated substrate is treated so that vertical liquid crystal alignment $(\theta =0)$ obtains over the temperature range $T_{\mathrm{NA}}<T<T_a$, where $T_{\mathrm{NA}}$ is the nematic-smectic-$A$ transition temperature. When the cell is filled with a chiral liquid crystal whose helical pitch is unwound (surface stabilized), application of an in-plane electric field for $T_{\mathrm{NA}}<T<T_a$ induces a nonzero polar tilt $\theta \propto E$ of the liquid crystal director at the surface, where the tilted orientation propagates elastically into the bulk. On heating toward $T_a$, this surface electroclinic response becomes large, corresponding to the onset of a surface tilt transition at $T_a$ from $\theta =0$ to nonzero $\theta$.

The influence of confinement (droplet size) and liquid crystal orientational order (smectic-$A$ and nematic) on the interfacial polarization field effects [Maxwell-Wagner-Sillars (MWS) effect] existing in liquid-crystal-droplets–polymer systems is investigated by broadband dielectric spectroscopy and a forward transmittance measurement technique. A relaxation process observed in the low frequency domain of the dielectric spectrum has been associated with a MWS effect for both micron-size and submicron-size droplets. Using electro-optical measurements and numerical simulations of the field inside droplets, it is shown that a depolarization field takes place in the same frequency range as that determined by dielectric spectroscopy. Differential scanning calorimetry measurements allowed to estimate the phase-separated liquid crystal [4,$4^{\prime }$-octylcyanobiphenyl (8CB)] fraction, which was found in the range of 55% for both micron-size and submicron-size droplets. X-ray diffraction experiments showed that smectic 8CB confined to micron-size cavities adopt bulklike properties, i.e., a partial bilayer structure, whereas in submicron-size droplets the layer spacing of the smectic phase is increased due to the strong bending deformations induced by the high curvature of the cavity walls.

Determinants of the switching speed of the genetic toggle switch remain unknown. Analysis shows that the decay rate of proteins predominantly sets the speed limit, but its modification introduces a trade-off between increased speed and decreased bistability. Incorporating protein-modifying enzymes into the switch gives extra degrees of freedom to address this trade-off. The condition for bistability when such enzymes are incorporated is derived. Under this condition, speed increases with the maximal rate of gene expression.

A concept of neighborhood in complex networks is addressed based on the criterion of the minimal number of steps to reach other vertices. This amounts to, starting from a given network $R_1$, generating a family of networks $R_{\mathcal{l}},\mathcal{l}=2,3,\dots$ such that, the vertices that are $\mathcal{l}$ steps apart in the original $R_1$, are only 1 step apart in $R_{\mathcal{l}}$. The higher order networks are generated using Boolean operations among the adjacency matrices $M_{\mathcal{l}}$ that represent $R_{\mathcal{l}}$. The families originated by the well known linear and the Erdös-Renyi networks are found to be invariant, in the sense that the spectra of $M_{\mathcal{l}}$ are the same, up to finite size effects. A further family originated from small world network is identified.

Node betweenness has been studied recently by a number of authors, but until now less attention has been paid to edge betweenness. In this paper, we present an exact analytic study of edge betweenness in evolving scale-free and non-scale-free trees. We aim at the probability distribution of edge betweenness under the condition that a local property, the in-degree of the “younger” node of a randomly selected edge, is known. En route to the conditional distribution of edge betweenness the exact joint distribution of cluster size and in-degree, and its one-dimensional marginal distributions have been presented in the paper as well. From the derived probability distributions the expectation values of different quantities have been calculated. Our results provide an exact solution not only for infinite, but for finite networks as well.

We propose a mean-field theory for the localization of damage in a quasistatic fuse model on a cylinder. Depending on the quenched disorder distribution of the fuse thresholds, we show analytically that the system can either stay in a percolation regime up to breakdown, or start at some imposed current, to localize starting from the smallest scale (lattice spacing), or instead go to a diffuse localization regime where damage starts to concentrate in bands of width scaling as the width of the system, but remains diffuse at smaller scales. Depending on the nature of the quenched disorder on the fuse thresholds, we derive analytically the phase diagram of the system separating these regimes and the current levels for the onset of these possible localizations. We compare these predictions to numerical results.

We present the analysis of the first passage time problem on a finite interval for the generalized Wiener process that is driven by Lévy stable noises. The complexity of the first passage time statistics (mean first passage time, cumulative first passage time distribution) is elucidated together with a discussion of the proper setup of corresponding boundary conditions that correctly yield the statistics of first passages for these non-Gaussian noises. The validity of the method is tested numerically and compared against analytical formulas when the stability index $\alpha$ approaches $2$, recovering in this limit the standard results for the Fokker-Planck dynamics driven by Gaussian white noise.

We present results from a systematic direct-numerical simulation study of pressure fluctuations in an unforced, incompressible, homogeneous, and isotropic three-dimensional turbulent fluid. At cascade completion, isosurfaces of low pressure are found to be organized as slender filaments, whereas the predominant isostructures appear sheetlike. We exhibit several results, including plots of probability distributions of the spatial pressure difference, the pressure-gradient norm, and the eigenvalues of the pressure-Hessian tensor. Plots of the temporal evolution of the mean pressure-gradient norm, and the mean eigenvalues of the pressure-Hessian tensor are also exhibited. We find the statistically preferred orientations between the eigenvectors of the pressure-Hessian tensor, the pressure gradient, the eigenvectors of the strain-rate tensor, the vorticity, and the velocity. Statistical properties of the nonlocal part of the pressure-Hessian tensor are also exhibited. We present numerical tests (in the viscous case) of some conjectures of Ohkitani [Phys. Fluids A 5, 2570 (1993)] and Ohkitani and Kishiba [Phys. Fluids 7, 411 (1995)] concerning the pressure-Hessian and the strain-rate tensors, for the unforced, incompressible, three-dimensional Euler equations.

We make an experimental characterization of the effect that static disorder has on the shape of a normal Saffman-Taylor finger. We find that static noise induces a small amplitude and long wavelength instability on the sides of the finger. Fluctuations on the finger sides have a dominant wavelength, indicating that the system acts as a selective amplifier of static noise. The dominant wavelength does not seem to be very sensitive to the intensity of static noise present in the system. On the other hand, at a given flow rate, rms fluctuations of the finger width, decrease with decreasing intensity of static noise. This might explain why the sides of the fingers are flat for typical Saffman-Taylor experiments. Comparison with previous numerical studies of the effect that temporal noise has on the Saffman-Taylor finger, leads to conclude that the effect of temporal noise and static noise are similar. The behavior of fluctuations of the finger width found in our experiments, is qualitatively similar to one recently reported, in the sense that, the magnitude of the width fluctuations decays as a power law of the capillary number, at low flow rates, and increases with capillary number for larger flow rates.

Correlation in complex networks follows a linear relation between the degree of a node and the total degrees of its neighbors for six different classes of real networks. This general linear relation is an extension of the Aboav-Weaire law in two-dimensional cellular structures and provides a simple and different perspective on the correlation in complex networks, which is complementary to an existing description using Pearson correlation coefficients and a power law fit. Analytical expression for this linear relation for three standard models of complex networks: the Erdos-Renyi, Watts-Strogatz, and Barabasi-Albert networks is provided. The slope and intercept of this linear relation are described by a single parameter $a$ together with the first and second moment of the degree distribution of the network. The assortivity of the network can be related to the sign of the intercept.

A renormalization group theory for the nonlinear and nonequilibrium responses to linearly varying external probes is formulated for the critical dynamics of a time-dependent Ginzburg-Landau equation with a scalar nonconserved order parameter. A series of nonequilibrium dynamic critical exponents and their scaling relations characterizing the time-dependent probes and the nonequilibrium hysteresis induced are derived analytically and systematically and agree well with numerical results. The three-dimensional dynamic critical exponent $z$ is accordingly determined as $z=2.036\left(11\right)$. These show that the linearly varying external fields can be applied to probe criticality effectively.

Using event-driven molecular dynamics we study one- and two-dimensional ballistic annihilation. We estimate exponents $\xi$ and $\gamma$, which describe the long-time decay of the number of particles $\left[n\right(t)\sim t^{-\xi }]$ and of their typical velocity $\left[v\right(t)\sim t^{-\gamma }]$. To a good accuracy our results confirm the scaling relation $\xi +\gamma =1$. In the two-dimensional case our results are in good agreement with those obtained from Boltzmann kinetic theory.

We develop a general criterion about coarsening for some classes of nonlinear evolution equations describing one-dimensional pattern-forming systems. This criterion allows one to discriminate between the situation where a coarsening process takes place and the one where the wavelength is fixed in the course of time. An intermediate scenario may occur, namely “interrupted coarsening.” The power of the criterion on which a brief account has been given [Politi and Misbah, Phys. Rev. Lett. 92, 090601 (2004)], and which we extend here to more general equations, lies in the fact that the statement about the occurrence of coarsening, or selection of a length scale, can be made by only inspecting the behavior of the branch of steady state periodic solutions. The criterion states that coarsening occurs if ${\lambda }^{\prime }\left(A\right)>0$ while a length scale selection prevails if ${\lambda }^{\prime }\left(A\right)<0$, where $\lambda$ is the wavelength of the pattern and $A$ is the amplitude of the profile (prime refers to differentiation). This criterion is established thanks to the analysis of the phase diffusion equation of the pattern. We connect the phase diffusion coefficient $D\left(\lambda \right)$ (which carries a kinetic information) to ${\lambda }^{\prime }\left(A\right)$, which refers to a pure steady state property. The relationship between kinetics and the behavior of the branch of steady state solutions is established fully analytically. Another important and new result which emerges here is that the exploitation of the phase diffusion coefficient enables us to determine in a rather straightforward manner the dynamical coarsening exponent. Our calculation, based on the idea that $\mid D\left(\lambda \right)\mid \sim {\lambda }^2∕t$, is exemplified on several nonlinear equations, showing that the exact exponent is captured. We are not aware of another method that so systematically provides the coarsening exponent. Contrary to many situations where the one-dimensional character has proven essential for the derivation of the coarsening exponent, this idea can be used, in principle, at any dimension. Some speculations about the extension of the present results are outlined.

We study the persistent random walk of photons on a one-dimensional lattice of random transmittances. Transmittances at different sites are assumed independent, distributed according to a given probability density $f\left(t\right)$. Depending on the behavior of $f\left(t\right)$ near $t=0$, diffusive and subdiffusive transports are predicted by the disorder expansion of the mean square-displacement and the effective medium approximation. Monte Carlo simulations confirm the anomalous diffusion of photons. To observe photon subdiffusion experimentally, we suggest a dielectric film stack for realization of a distribution $f\left(t\right)$.

A phase-field method is applied to the modeling of flow and breakup of droplets in a T-shaped junction in the hydrodynamic regime where capillary and viscous stresses dominate over inertial forces, which is characteristic of microfluidic devices. The transport equations are solved numerically in the three-dimensional geometry, and the dependence of the droplet breakup on the flow rates, surface tension and viscosities of the two components is investigated in detail. The model reproduces quite accurately the phase diagram observed in experiments performed with immiscible fluids. The critical capillary number for droplet breakup depends on the viscosity contrast, with a trend which is analogous to that observed for individual isolated droplets in hyperbolic flow.