The probability that a random walker returns to its origin for large times scales as $t^{-\overline{d}∕2}$, where $\overline{d}$ is the spectral dimension. We calculate $\overline{d}$ for a class of tree structures using a renormalization technique on an infinite continued fraction. We consider a wide range of homogeneous networks based on replacing the branches of a self-similar tree with arbitrary fractals and composite fractals. We also consider a new class of inhomogeneous hyperbranched trees.

We consider the persistence phenomenon in advected passive scalar equation in one dimension. The velocity field is random with the $⟨v(k,\omega )v(-k,-\omega )⟩\sim {\mid k\mid }^{-(2+\alpha )}$. In the presence of the nonlinearity the complete Green’s function becomes $G^{-1}=-i\omega +D{k}^2+\Sigma$. We determine $\Sigma$ self-consistently from the correlation function which gives $\Sigma \sim k^{\beta }$, with $\beta =(1-\alpha )∕2$. The effect of the nonlinear term in the equation in the $\mathcal{O}\left({ϵ}^2\right)$ is to replace the diffusion term due to molecular viscosity by an effective term of the form ${\Sigma }_0{k}^{\beta }$. The stationary correlator for this system is ${\left[\mathrm{sech}(T∕2)\right]}^{1∕\beta }$. Using the self-consistent theory we have determined the relation between $\beta$ and $\alpha$. Finally, the independent interval approximation (IIA) method is used to determine the persistent exponent.

Recent progress in nanoscale materials science has pushed the basic research on physical effects while going to such length scales. Particularly, the study of size effects on heat conductivity has attracted much theoretical and experimental interest. In this work we explore the possibilities to study such kinds of phenomena through the quantization of the dynamic equations of a class of nonlinear heat transport macroscopic models. We develop a quantization procedure to search the energy level structure of one-dimensional nanosystems. The procedure may have applications in other fields of the physics of mesoscopic systems.

It is known that there are thermodynamic states for which the Gaussian-core fluid displays anomalous properties such as expansion upon isobaric cooling (density anomaly) and increased single-particle mobility upon isothermal compression (self-diffusivity anomaly). Here, we investigate how temperature and density affect its short-range translational structural order, as characterized by the two-body excess entropy. We find that there is a wide range of conditions for which the short-range translational order of the Gaussian-core fluid decreases upon isothermal compression (structural order anomaly). As we show, the origin of the structural anomaly is qualitatively similar to that of other anomalous fluids (e.g., water or colloids with short-range attractions) and is connected to how compression affects static correlations at different length scales. Interestingly, we find that the self-diffusivity of the Gaussian-core fluid obeys a scaling relationship with the two-body excess entropy that is very similar to the one observed for a variety of simple liquids. One consequence of this relationship is that the state points for which structural, self-diffusivity, and density anomalies of the Gaussian-core fluid occur appear as cascading regions on the temperature-density plane; a phenomenon observed earlier for models of waterlike fluids. There are, however, key differences between the anomalies of Gaussian-core and waterlike fluids, and we discuss how those can be qualitatively understood by considering the respective interparticle potentials of these models. Finally, we note that the self-diffusivity of the Gaussian-core fluid obeys different scaling laws depending on whether the two-body or total excess entropy is considered. This finding, which deserves more comprehensive future study, appears to underscore the significance of higher-body correlations for the behavior of fluids with bounded interactions.

We present experimental results obtained with a two-dimensional silo discharging under gravity through an orifice at the flat bottom. High-speed measurements provide enough time resolution to detect every single bead that goes out and this allows the measurement of the flow rate in short-time windows. Two different regimes are clearly distinguished: one for large orifices, which can be described by Gaussian fluctuations, and another for small orifices, in which extreme events appear. The frontier between those two regimes coincides with the outlet size below which jamming events are frequent. Moreover, it is shown that the power spectrum of the flow-rate oscillations is not dominated by any particular frequency.

Null transmission ellipsometer was employed to study the surface-induced tilt transition of an antiferroelectric liquid crystal compound in the bulk smectic-$A$ $\left(\text{Sm}A\right)$ temperature range. The data of the surface tilt transition can be fitted with an extended mean-field model. These results indicate the transition is near a mean-field tricritical point. The surface layers’ tilt directions are found to be in different planes several degrees above the bulk $\text{Sm}A\text{-Sm}{C}_{\alpha }^{\ast }$ transition.

Complicated patterns showing various spatial scales have been obtained in the past by coupling Turing systems in such a way that the scales of the independent systems resonate. This produces superimposed patterns with different length scales. Here we propose a model consisting of two identical reaction-diffusion systems coupled together in such a way that one of them produces a simple Turing pattern of spots or stripes, and the other traveling wave fronts that eventually become stationary. The basic idea is to assume that one of the systems becomes fixed after some time and serves as a source of morphogens for the other system. This mechanism produces patterns very similar to the pigmentation patterns observed in different species of stingrays and other fishes. The biological mechanisms that support the realization of this model are discussed.

We present results of an extensive numerical study of the dynamics of networks of integrate-and-fire neurons connected randomly through inhibitory interactions. We first consider delayed interactions with infinitely fast rise and decay. Depending on the parameters, the network displays transients which are short or exponentially long in the network size. At the end of these transients, the dynamics settle on a periodic attractor. If the number of connections per neuron is large $\left(\sim 1000\right)$, this attractor is a cluster state with a short period. In contrast, if the number of connections per neuron is small $\left(\sim 100\right)$, the attractor has complex dynamics and very long period. During the long transients the neurons fire in a highly irregular manner. They can be viewed as quasistationary states in which, depending on the coupling strength, the pattern of activity is asynchronous or displays population oscillations. In the first case, the average firing rates and the variability of the single-neuron activity are well described by a mean-field theory valid in the thermodynamic limit. Bifurcations of the long transient dynamics from asynchronous to synchronous activity are also well predicted by this theory. The transient dynamics display features reminiscent of stable chaos. In particular, despite being linearly stable, the trajectories of the transient dynamics are destabilized by finite perturbations as small as $O\left(1/N\right)$. We further show that stable chaos is also observed for postsynaptic currents with finite decay time. However, we report in this type of network that chaotic dynamics characterized by positive Lyapunov exponents can also be observed. We show in fact that chaos occurs when the decay time of the synaptic currents is long compared to the synaptic delay, provided that the network is sufficiently large.

We report on the experimental evidence of noise amplification in an erbium-doped fiber laser in the vicinity of saddle-node, period-doubling, and crisis bifurcations. We demonstrate this interesting phenomenon by analyzing the laser bifurcation diagrams and power spectra. Numerical simulations on the base of an advanced laser model display good agreement with the experimental results.

Porous materials having multiple scales of porosity afford the opportunity to combine the high surface area and functionality of nanopores with the superior charge/discharge characteristics of wider transport channels. However, the relative volume fractions assigned to nanopores and transport channels must be thoughtfully balanced because the introduction of transport channels reduces the volume available for nanopore functionality. In the present paper, the optimal balance between nanopore capacity and system response time is achieved by adjusting the aperture and spacing of a family of transport channels that provide access to adjacent nanopores during recharge/discharge cycles of materials intended for storage of gas or electric charge. A diffusive transport model is used to describe alternative processes of viscous gas flow, Knudsen gas flow, and ion diffusion or electromigration. The coupled transport equations for the nanopores and transport channels are linearized and solved analytically for a periodic variation in external gas pressure, ion concentration, or electric potential using a separation-of-variables approach in the complex domain. Optimization of these solutions yields closed-form expressions for channel apertures and spacing that provide maximum discharge of gas or electric charge for a fixed system volume and a desired discharge time.

Important features associated with the segregration of particles in turbulent flow are investigated by considering the statistical distribution (phase-space number density) of particles subject to the combined effects of straining flow and stochastic forcing. A Fokker-Planck model is used to obtain results for the phase-space distributions of particles that are entrained into straining flow fields. The analysis shows that, in marked contrast to the zero strain case, nonsingular steady-state distributions are generated, and also confirms that the diffusional effect resulting from stochastic forcing is sufficient to offset the otherwise singular distributions that would result from the indefinite accumulation of particles along stagnation lines. The influence of particle inertia (Stokes number) on the form of the resulting distributions is considered and several significant results are observed. The influence of strain rate on the attenuation of particle kinetic stresses is quantified and explained. The development of large third-order velocity moments is observed for Stokes numbers above a critical value. The mechanism underlying this phenomenon is seen to be a generic feature of particle transport in flows where vortex structures induce local counterflows of particles. The system therefore provides an ideal test for closure models for third-order moments of particle velocities, and here the standard Chapman-Enskog approximation is assessed.

In the present work experimental, numerical, and theoretical investigations of a normal drop impact onto a liquid film of finite thickness are presented. The dynamics of drop impact on liquid surfaces, the shape of the cavity, the formation and propagation of a capillary wave in the crater, and the residual film thickness on the rigid wall are determined and analyzed. The shape of the crater within the film and the uprising liquid sheet formed upon the impact are observed using a high-speed video system. The effects of various influencing parameters such as drop impact velocity, liquid film thickness and physical properties of the liquids, including viscosity and surface tension, on the time evolution of the crater formation are investigated. Complementary to experiments the direct numerical simulations of the phenomena are performed using an advanced free-surface capturing model based on a two-fluid formulation of the classical volume-of-fluid (VOF) model in the framework of the finite volume numerical method. In this model an additional convective term is introduced into the transport equation for phase fraction, contributing decisively to a sharper interface resolution. Furthermore, an analytical model for the penetration depth of the crater is developed accounting for the liquid inertia, viscosity, gravity, and surface tension. The model agrees well with the experiments at the early times of penetration far from the wall if the impact velocity is high. Finally, a scaling analysis of the residual film thickness on the wall is conducted demonstrating a good agreement with the numerical predictions.

Liquid drops can be kept from touching a plane solid surface by a gas stream entering from underneath, as it is observed for water drops on a heated plate, kept aloft by a stream of water vapor. We investigate the limit of small flow rates, for which the size of the gap between the drop and the substrate becomes very small, to obtain a full analytical description of stationary drop states and their stability. Above a critical drop radius no stationary drops can exist, below the critical radius two solutions coexist. However, only the solution with the smaller gap width is stable, the other is unstable. We compare to experimental data and use boundary integral simulations to show that unstable drops develop a gas “chimney” that breaks the drop in its middle.

We present a hierarchical transform that can be applied to Laplace-like differential equations such as Darcy’s equation for single-phase flow in a porous medium. A finite-difference discretization scheme is used to set the equation in the form of an eigenvalue problem. Within the formalism suggested, the pressure field is decomposed into an average value and fluctuations of different kinds and at different scales. The application of the transform to the equation allows us to calculate the unknown pressure with a varying level of detail. A procedure is suggested to localize important features in the pressure field based only on the fine-scale permeability, and hence we develop a form of adaptive coarse graining. The formalism and method are described and demonstrated using two synthetic toy problems.

This work investigates the scaling properties of fluctuations in the flux of individual agents with respect to their average flux in an interbank network. The analyzed data provide information on daily values of $f_i^{\text{asset}}$, the credit provided by bank $i$ in the interbank network, and $f_i^{\text{liab}}$, the credit received by bank $i$ from the other banks of the network. The investigation focuses on the scaling properties of the fluctuations in the raw data $f_i^{\text{asset}}$, $f_i^{\text{liab}}$, and $f_{R,i}^{\text{ext}}\left(t\right)=f_i^{\text{asset}}-f_i^{\text{liab}}$, as well as on similar properties internal and external fluctuations $f_i^{\text{int}}$ and $f_i^{\text{ext}}$, which are derived according to a recently proposed methodology [M. Argollo de Menezes and A. L. Barabasi, Phys. Rev. Lett. 93, 068701 (2004)]. Finally, a “rolling sampling” approach is introduced in order to deal with the nonstationarity of the fluxes. The results suggest that exponents are time varying, hinting that the considered interbank network is changing with time.

The mean time for a Brownian particle to find a small target inside a narrow domain is a key parameter for many chemical reactions occurring in cellular microstructures. Although current estimations are given for a large class of domains, they cannot be used for narrow domains often encountered in cellular biology, such as the synaptic cleft, narrow compartments in the outer segment of vertebrate photoreceptors, or neuron-glia contact. We compute here the mean time for a Brownian particle to hit a small target placed on the surface of a narrow cylinder. We then use this result to estimate the rate constant of cyclic-GMP (cGMP) hydrolysis by the activated enzyme phosphodiesterase (PDE) in the narrow microdomains that build up the outer segment of a rod photoreceptor. By controlling the cGMP concentration, PDE activity is at the basis of the early photoresponse chemical reaction cascade. Our approach allows us to compute the cGMP rate constant as a function of biophysical parameters.

We investigate the selective forces that promote the emergence of modularity in nature. We demonstrate the spontaneous emergence of modularity in a population of individuals that evolve in a changing environment. We show that the level of modularity correlates with the rapidity and severity of environmental change. The modularity arises as a synergistic response to the noise in the environment in the presence of horizontal gene transfer. We suggest that the hierarchical structure observed in the natural world may be a broken symmetry state, which generically results from evolution in a changing environment. To support our results, we analyze experimental protein interaction data and show that protein interaction networks became increasingly modular as evolution proceeded over the last four billion years. We also discuss a method to determine the divergence time of a protein.

We report long-duration real-time observations of the dynamics of hexagonal (rodlike) directional-solidification patterns in bulk samples of a transparent eutectic alloy. A slight forward curvature of the isotherms induces a slow dilatation of the growth pattern at constant solidification rate and triggers the rod-splitting instability. At long times, the rod-splitting frequency exactly balances the dilatation driven by the curved isotherms. The growth pattern is then disordered and nonstationary but has a sharply selected mean spacing. Well-ordered growth patterns can be grown using time-dependent solidification rates.