We study the ensemble statistics of the particle density in a random medium whose mean transport dynamics describes a continuous time random walk. Starting from a Langevin equation for the particle motion in a single disorder realization, we derive evolution equations for the $n$-point moments of concentration by coarse graining and ensemble averaging the microscale transport problem. The governing equations describe multidimensional continuous time random walks whose waiting time distribution is given in terms of the disorder distribution. We find that the concentration is not self-averaging even for normal mean behavior. The relative concentration variance for anomalous is larger than for normal mean behavior. These results may have some impact on risk and extreme value analysis in stochastic dynamic systems.

A uniform electric field is used for cleaning drops of the particles they often carry on their surface. In a first step, particles migrate to either the drop’s poles or equator. This is due to the presence of an electrostatic force for which an analytical expression is derived. In a second step, particles concentrated near the poles are released into the ambient liquid via tip streaming, and those near the equator are removed by stretching the drop and breaking it into several droplets. In the latter case, particles are all concentrated in a small middle daughter droplet.

A problem of current interest is the estimation of spatially distributed processes at locations where measurements are missing. Linear interpolation methods rely on the Gaussian assumption, which is often unrealistic in practice, or normalizing transformations, which are successful only for mild deviations from the Gaussian behavior. We propose to address the problem of missing value estimation on two-dimensional grids by means of spatial classification methods based on spin (Ising, Potts, and clock) models. The “spin” variables provide an interval discretization of the process values, and the spatial correlations are captured in terms of interactions between the spins. The spins at the unmeasured locations are classified by means of the “energy matching” principle: the correlation energy of the entire grid (including prediction sites) is estimated from the sample-based correlations. We investigate the performance of the spin classifiers in terms of computational speed, misclassification rate, class histogram, and spatial correlations reproduction using simulated realizations of spatial random fields, real rainfall data, and a digital test image. We also compare the spin-based methods with standard classifiers such as the $k$-nearest neighbor, the fuzzy $k$-nearest neighbor, and the support vector machine. We find that the spin-based classifiers provide competitive choices.

The total entropy production fluctuations are studied in some exactly solvable models. For these systems, the detailed fluctuation theorem holds even in the transient state, provided initially that the system is prepared in thermal equilibrium. The nature of entropy production during the relaxation of a system to equilibrium is analyzed. The averaged entropy production over a finite time interval gives a better bound for the average work performed on the system than that obtained from the well-known Jarzynski equality. Moreover, the average entropy production as a quantifier for information theoretic nature of irreversibility for finite time nonequilibrium processes is discussed.

A recent theoretical model developed by Imparato et al. [Phys. Rev. E 76, 050101(R) (2007)] of the experimentally measured heat and work effects produced by the thermal fluctuations of single micron-sized polystyrene beads in stationary and moving optical traps has proved to be quite successful in rationalizing the observed experimental data. The model, based on the overdamped Brownian dynamics of a particle in a harmonic potential that moves at a constant speed under a time-dependent force, is used to obtain an approximate expression for the distribution of the heat dissipated by the particle at long times. In this paper, we generalize the above model to consider particle dynamics in the presence of colored noise, without passing to the overdamped limit, as a way of modeling experimental situations in which the fluctuations of the medium exhibit long-lived temporal correlations, of the kind characteristic of polymeric solutions, for instance, or of similar viscoelastic fluids. Although we have not been able to find an expression for the heat distribution itself, we do obtain exact expressions for its mean and variance, both for the static and for the moving trap cases. These moments are valid for arbitrary times and they also hold in the inertial regime, but they reduce exactly to the results of Imparato et al. in appropriate limits.

For some nonlinear systems the performance can improve with an increasing noise level. Such noise-induced improvement in static nonlinearities is of great interest for practical applications since many systems can be modeled in that way (e.g., sensors, quantizers, limiters, etc.). We present experimental evidence that noise-induced performance improvement occurs in those systems as a consequence of discretization in time with the achievable signal-to-noise ratio (SNR) gain increasing with decreasing ratio of input noise bandwidth and total measurement bandwidth. By modifying the input noise bandwidth, noise-induced improvement with SNR gain larger than unity is demonstrated in a system where it was not previously thought possible. Our experimental results bring closer two different theoretical models for the same class of nonlinearities and shed light on the behavior of static nonlinear discrete-time systems.

A universal pattern emerging from the analysis of the distribution of relaxation times in the isotropic phase of liquid crystalline $n$-alkylcyanobiphenyls $\left(n\text{CB}\right)$ from 4CB to 14CB is presented. The increase of the length of $n\text{CB}$ molecules causes the high-frequency (short-time) branch of the primary relaxation loss curve to approach the form $\epsilon \left(f>f_{peak}\right)\sim {\omega }^{-n}$, with $n\to 1/2$ in frequency $\left(\omega =2\pi f\right)$ or $\sim \sqrt{t}$ in time on cooling toward the isotropic-mesophase “clearing” phase-transition temperature $\left(T^C\right)$. Recently, such behavior was suggested as a hypothetical universal pattern for diverse glass forming organic liquids on approaching the glass temperature [A. I. Nielsen, T. Christensen, B. Jakobsen, K. Niss, N. B. Olsen, R. Richert, and J. C. Dyre, J. Chem. Phys. 130, 154508 (2009)]. The isotropic phase of rodlike liquid crystalline compounds is considered to be an important experimental model system for studying glassy dynamics since it constitutes a link with the model fluid of hard ellipsoids of revolution.

We use a Potts model community detection algorithm to accurately and quantitatively evaluate the hierarchical or multiresolution structure of a graph. Our multiresolution algorithm calculates correlations among multiple copies (“replicas”) of the same graph over a range of resolutions. Significant multiresolution structures are identified by strongly correlated replicas. The average normalized mutual information, the variation in information, and other measures, in principle, give a quantitative estimate of the “best” resolutions and indicate the relative strength of the structures in the graph. Because the method is based on information comparisons, it can, in principle, be used with any community detection model that can examine multiple resolutions. Our approach may be extended to other optimization problems. As a local measure, our Potts model avoids the “resolution limit” that affects other popular models. With this model, our community detection algorithm has an accuracy that ranks among the best of currently available methods. Using it, we can examine graphs over $40\times {10}^6$ nodes and more than $1\times {10}^9$ edges. We further report that the multiresolution variant of our algorithm can solve systems of at least $200000$ nodes and $10\times {10}^6$ edges on a single processor with exceptionally high accuracy. For typical cases, we find a superlinear scaling $O\left(L^{1.3}\right)$ for community detection and $O\left(L^{1.3}\text{log}N\right)$ for the multiresolution algorithm, where $L$ is the number of edges and $N$ is the number of nodes in the system.

Quantitative characterization of interaction between processes from time series is often required in different fields of natural science including geophysics and biophysics. Typically, one estimates “short-term” influences, e.g., the widely used Granger causality is defined via one-step-ahead predictions. Such an approach does not reveal how strongly the “long-term” behavior of one process under study is affected by the others. To overcome this problem, we introduce the concept of long-term causality, which extends the concept of Granger causality. The long-term causality is estimated from data via empirical modeling and analysis of model dynamics under different conditions. Apart from mathematical examples, we apply both approaches to find out how strongly the global surface temperature (GST) is affected by variations in carbon dioxide atmospheric content, solar activity, and volcanic activity during the last 150 years. Influences of all the three factors on GST are detected with the Granger causality. However, the long-term causality shows that the rise in GST during the last decades can be explained only if the anthropogenic factor $\left({\text{CO}}_2\right)$ is taken into account in a model.

A thin plate accelerated or decelerated in a free-molecular gas at rest by a constant external force is considered. The force is in the direction perpendicular to the plate. In this situation, the plate velocity approaches its final constant velocity as time goes on. It is shown numerically that, under the diffuse-reflection boundary condition, the difference between the plate velocity and its final value decreases in proportion to an inverse power of time. This agrees with the previous theoretical result obtained under the assumption that the initial plate velocity is sufficiently close to the final one.

A simple lattice Boltzmann model for numerical simulation of fluid flow and heat transfer inside a rotating disk-cylinder configuration, which is of fundamental interest and practical importance in science as well as in engineering, is proposed in this paper. Unlike existing lattice Boltzmann models for such flows, which were based on “primitive-variable” Navier-Stokes equations, the target macroscopic equations of the present model for the flow field are vorticity–stream function equations, inspired by our recent work designed for nonrotating flows [S. Chen, J. Tölke, and M. Krafczyk, Phys. Rev. E 79, 016704 (2009); S. Chen, J. Tölke, S. Geller, and M. Krafczyk, Phys. Rev. E 78, 046703 (2008)]. The flow field and the temperature field both are solved by the D2Q5 model. Compared with the previous models, the present model is more efficient, more stable, and much simpler. It was found that, even though with a relatively low grid resolution, the present model can still work well when the Grashof number is very high. The advantages of the present model are validated by numerical experiments.

A study of the gas dynamics of a dilute collection of the inelastically colliding hard spheres is presented. When diffusive processes are neglected the gas density blows up in a finite time. The blowup is the mathematical expression for one of the possible mechanisms for cluster formation in dissipative gases. The way diffusive processes smoothen the singularity has been studied. Exact localized states of the gas when heat diffusion balances nonlinear cooling are obtained. The presented results generalize previous findings for planar flows.

Equilibrium and nonequilibrium growth phenomena, e.g., surface growth, generically yields self-affine distributions. Analysis of statistical properties of these distributions appears essential in understanding statistical mechanics of underlying phenomena. Here, we analyze scaling properties of the cumulative distribution of iso-height loops (i.e., contour lines) of rough self-affine surfaces in terms of loop area and system size. Inspired by the Coulomb gas methods, we find the generating function of the area of the loops. Interestingly, we find that, after sorting loops with respect to their perimeters, Zipf-like scaling relations hold for ranked loops. Numerical simulations are also provided in order to demonstrate the proposed scaling relations.

Using a generalized Landau theory involving orientational, layering, tilt, and biaxial order parameters we analyze the smectic-$A^{\ast }$ and smectic-$C^{\ast }$ (Sm-$A^{\ast }$–Sm-$C^{\ast }$) transitions, showing that a combination of small orientational order and large layering order leads to Sm-$A^{\ast }$–Sm-$C^{\ast }$ transitions that are either continuous and close to tricriticality or first order. The model predicts that in such systems the increase in birefringence upon entry to the Sm-$C^{\ast }$ phase will be especially rapid. It also predicts that the change in layer spacing at the Sm-$A^{\ast }$–Sm-$C^{\ast }$ transition will be proportional to the orientational order. These are two hallmarks of Sm-$A^{\ast }$–Sm-$C^{\ast }$ transitions in de Vries materials. We analyze the electroclinic effect in the Sm-$A^{\ast }$ phase and show that as a result of the zero-field Sm-$A^{\ast }$–Sm-$C^{\ast }$ transition being either continuous and close to tricriticality or first order (i.e., for systems with a combination of weak orientational order and strong layering order), the electroclinic response of the tilt will be unusually strong. Additionally, we investigate the associated electrically induced change in birefringence and layer spacing, demonstrating de Vries behavior for each, i.e., an unusually large increase in birefringence and an unusually small layer contraction. Both the induced changes in birefringence and layer spacing are shown to scale quadratically with the induced tilt angle.

In their recent paper [Phys. Rev. E 79, 031115 (2009)], Ning Jia and Shoufeng Ma used some Markov chain arguments for the analytical description of inflow in the deterministic Nagel-Schreckenberg model with open boundaries. In this context, they considered two different mechanisms of injecting vehicles: the standard injection rule and a popular expanded injection rule. While the results for the first one seem to be correct, simulations show that the inflow formula in case of the expanded injection rule yields only approximate results. Therefore, this comment provides the exact formula also in this case and explains the shortcoming in the derivation of Jia and Ma.

The phenomenon of coherence resonance is investigated in an unijunction transistor relaxation oscillator and quantified by estimating the normal variance (NV). Depending on the measuring points, two types of NV curves have been obtained. We have observed that the degradations in coherency at higher noise amplitudes in our system is probably the result of direct interference of coherent oscillations and the stochastic perturbation. Degradation of coherency may be minimal if this direct interference of noise and coherent oscillations is eliminated.

We report a deformation phenomenon occurring at the interface between two immiscible liquids. The two liquids are in a cylindrical container and set into motion by a rotating lid positioned above the interface. The upper liquid is more viscous than the lower and the difference in densities between the liquids is fairly small. As the rotational speed of the lid $\left(\Omega \right)$ is increased in small increments, the center of the interface rises and the height increases. Depending on $\Omega$, the topology of the interface changes drastically. The shape of the interface depends strongly on $\Omega$ and also the volume fraction of the liquids. Some dynamical behavior of the interface accompanies the changes in topology: interfacial waves, the generation of droplets, and interfacial instability.

Numerical simulation of rotating convection in plane layers with free slip boundaries show that the convective flows can be classified according to a quantity constructed from the Reynolds, Prandtl, and Ekman numbers. Three different flow regimes appear: laminar flow close to the onset of convection, turbulent flow in which the heat flow approaches the heat flow of nonrotating convection, and an intermediate regime in which the heat flow scales according to a power law independent of thermal diffusivity and kinematic viscosity.

We study the biased random-walk process in random uncorrelated networks with arbitrary degree distributions. In our model, the bias is defined by the preferential transition probability, which, in recent years, has been commonly used to study the efficiency of different routing protocols in communication networks. We derive exact expressions for the stationary occupation probability and for the mean transit time between two nodes. The effect of the cyclic search on transit times is also explored. Results presented in this paper provide the basis for a theoretical treatment of transport-related problems in complex networks, including quantitative estimation of the critical value of the packet generation rate.

We solve the first-passage problem for the Heston random diffusion model. We obtain exact analytical expressions for the survival and the hitting probabilities to a given level of return. We study several asymptotic behaviors and obtain approximate forms of these probabilities which prove, among other interesting properties, the nonexistence of a mean-first-passage time. One significant result is the evidence of extreme deviations—which implies a high risk of default—when certain dimensionless parameter, related to the strength of the volatility fluctuations, increases. We confront the model with empirical daily data and we observe that it is able to capture a very broad domain of the hitting probability. We believe that this may provide an effective tool for risk control which can be readily applicable to real markets both for portfolio management and trading strategies.