We investigate bond- and site-percolation models on several two-dimensional lattices numerically, by means of transfer-matrix calculations and Monte Carlo simulations. The lattices include the square, triangular, honeycomb kagome, and diced lattices with nearest-neighbor bonds, and the square lattice with nearest- and next-nearest-neighbor bonds. Results are presented for the bond-percolation thresholds of the kagome and diced lattices, and the site-percolation thresholds of the square, honeycomb, and diced lattices. We also include the bond- and site-percolation thresholds for the square lattice with nearest- and next-nearest-neighbor bonds. We find that corrections to scaling behave according to the second temperature dimension $X_{t2}=4$ predicted by the Coulomb gas theory and the theory of conformal invariance. In several cases there is evidence for an additional term with the same exponent, but modified by a logarithmic factor. Only for the site-percolation problem on the triangular lattice does such a logarithmic term appear to be small or absent. The amplitude of the power-law correction associated with $X_{t2}=4$ is found to be dependent on the orientation of the lattice with respect to the cylindrical geometry of the finite systems.

In a number of stochastic systems the random forcing is represented as a dichotomous Markov noise. A common characteristic of these models is that the noise is usually supposed to be independent of the state of the forced dynamical system. However, there are several situations in which positive or negative feedback exist between the system and the random driver. This paper investigates a class of systems characterized by feedback between dichotomous Markov noise and the system’s dynamics. The effect of the feedback is accounted for through a state dependency in the transition rates of the dichotomous noise. We study noise-induced transitions in these systems, with special attention to the delicate problem of correctly defining the deterministic counterpart of the stochastic system. We find that (i) if in the absence of any feedback the dynamical system has a single deterministic stable point, the deterministic dynamics remain monostable when a negative feedback is introduced, while they may become bistable in the presence of a positive feedback. (ii) Noise may induce bistability in the presence of a null or negative feedback. (iii) Bistable deterministic dynamics, induced by the positive feedback, may be destroyed by the noise, which tends to stabilize the system around a new intermediate stable state between those of the deterministic dynamics.

In ferromagnetic spin models above the critical temperature $(T>T_{\mathrm{cr}})$ the partition function zeros accumulate at complex values of the magnetic field $\left(H_E\right)$ with a universal behavior for the density of zeros $\rho \left(H\right)\sim {\mid H-H_E\mid }^{\sigma }$. The critical exponent $\sigma$ is believed to be universal at each space dimension and it is related to the magnetic scaling exponent $y_h$ via $\sigma =(d-y_h)∕y_h$. In two dimensions we have $y_h=12∕5(\sigma =-1∕6)$ while $y_h=2(\sigma =-1∕2)$ in $d=1$. For the one-dimensional Blume-Capel and Blume-Emery-Griffiths models we show here, for different temperatures, that a value $y_h=3(\sigma =-2∕3)$ can emerge if we have a triple degeneracy of the transfer matrix eigenvalues.

We study aging in a colloidal suspension consisting of micron-sized particles in a liquid. This system is made glassy by increasing the particle concentration. We observe samples composed of particles of two sizes, with a size ratio of 1:2.1 and a volume fraction ratio 1:6, using fast laser scanning confocal microscopy. This technique yields real-time, three-dimensional movies deep inside the colloidal glass. Specifically, we look at how the size, motion, and structural organization of the particles relate to the overall aging of the glass. Particles move in spatially heterogeneous cooperative groups. These mobile regions tend to be richer in small particles, and these small particles facilitate the motion of nearby particles of both sizes.

Methods of biomedical optics employed to study and characterize biological media involve measuring light diffusely transmitted though tissue. In this paper we describe boundary conditions for the propagation of light at interfaces of biological media. Boundary conditions are derived for a generalized radiative transfer model that accounts for the propagation of light in biological media with a spatially varying refractive index and includes the effects of ray divergence. This work will improve the accuracy of the measurements in biomedical optics techniques and will enhance our understanding of tissue physiology.

We consider a three-dimensional model of active Brownian particles interacting via a Morse-type potential. The system exhibits two modes of motion: translation and a coherent rotation in a torus-shape structure. We observe noise-induced transitions in both directions between the two states. These occur at different noise intensities, thus leading to a hysteresis curve. For certain parameter regions, the system switches persistently between the states such that the center of mass alternates between ballistic and diffusive motion. The coherent rotation leads to a pronounced mean angular momentum that changes its direction diffusively. We derive an analytic expression for the diffusion of the angular momentum of one particle in an external harmonic potential and show that it is always faster than the stochastic switching of the direction of motion in the two-dimensional case.

Complex clustered networks are ubiquitous in natural and technological systems. Understanding the physics of the security of such networks in response to attacks is of significant value. We develop a model, based on physical analysis and numerical computations, for the key ingredients of load dynamics in typical clustered networks. With this understanding, an effective strategy is proposed for preventing cascading breakdown, one of the most disastrous events that can happen to a complex networked system.

We report a systematic experimental study of the orientation and the flow strength of the large-scale circulation (LSC) in water-filled cylindrical Rayleigh-Bénard convection cells with aspect ratios 2.3, 1, and 0.5 by both direct velocity measurement and the indirect multithermal-probe measurement. Unlike its weak effect in the system’s global heat transport, the aspect ratio $\Gamma$ is found to play an important role in the dynamics of the azimuthal motion of the LSC. It is found that in larger $\Gamma$ geometries the azimuthal motion of the LSC’s vertical plane is confined in smaller azimuthal region than that in smaller $\Gamma$ geometries. The twisting motion between top and bottom parts of the LSC observed in the $\Gamma =1$ geometry is found to be absent in the $\Gamma =1∕2$ case. It is found that in the $\Gamma =1∕2$ geometry the orientational change $\mid \Delta \varphi \mid$ through a reorientation has an exponential distribution, in contrast to the power-law distribution for the $\Gamma =1$ case. Despite the difference in orientational change, the occurrence of the reorientations is a Poisson process in both geometries. Using the conditional average of the time interval between adjacent cessations or reversals on the rebound flow strength, we demonstrate the possibility to empirically predict when the next cessation or reversal will most likely occur if the rebound flow strength of the preceding cessation or reversal is given.

To achieve propulsion at low Reynolds number, a swimmer must deform in a way that is not invariant under time-reversal symmetry; this result is known as the scallop theorem. However, there is no many-scallop theorem. We demonstrate here that two active particles undergoing reciprocal deformations can swim collectively; moreover, polar particles also experience effective long-range interactions. These results are derived for a minimal dimers model, and generalized to more complex geometries on the basis of symmetry and scaling arguments. We explain how such cooperative locomotion can be realized experimentally by shaking a collection of soft particles with a homogeneous external field.

We study the continuous absorbing-state phase transition in the contact process on the Voronoi-Delaunay lattice. The Voronoi construction is a natural way to introduce quenched coordination disorder in lattice models. We simulate the disordered system using the quasistationary simulation method and determine its critical exponents and moment ratios. Our results suggest that the critical behavior of the disordered system is unchanged with respect to that on a regular lattice, i.e., that of directed percolation.

A combination of recent numerical and theoretical advances are applied to analyze the scaling behavior of the site-diluted Ising model in two dimensions, paying special attention to the implications for multiplicative logarithmic corrections. The analysis focuses primarily on the odd sector of the model (i.e., that associated with magnetic exponents), and in particular on its Lee-Yang zeros, which are determined to high accuracy. Scaling relations are used to connect to the even (thermal) sector, and a first analysis of the density of zeros yields information on the specific heat and its corrections. The analysis is fully supportive of the strong scaling hypothesis and of the scaling relations for logarithmic corrections.

A jumping process, defined in terms of the Lévi distributed jumping size and the Poissonian, position-dependent waiting time with the algebraic jumping rate, is discussed on the assumption that parameters of both distributions are themselves random variables which are determined from given probability distributions. The fractional equation for the distributed Lévy order parameter $\mu$ is derived and solved. The solution is of the form of a combination of the Fox functions and simple scaling is lacking. The problem of accelerated diffusion is also discussed. The case of the distributed waiting time parameter $\theta$ is similarly solved and the solution offers a possibility to manage processes which are characterized by more general forms of the jumping rate, not only algebraic. Moreover, we mention a possibility that the parameters $\mu$ and $\theta$ are mutually dependent.

The nonequilibrium collective behavior of self-propelled colloidal rods in a confining channel is studied using Brownian dynamics simulations and dynamical density functional theory. We observe an aggregation process in which rods self-organize into transiently jammed clusters at the channel walls. In the early stage of the process, fast-growing hedgehoglike clusters are formed which are largely immobile. At later stages, most of these clusters dissolve and mobilize into nematized aggregates sliding past the walls.

We present a nonequilibrium statistical mechanics description of rank abundance relations (RAR) in random community models of ecology. Specifically, we study a multispecies replicator system with quenched random interaction matrices. We here consider symmetric interactions as well as asymmetric and antisymmetric cases. RARs are obtained analytically via a generating functional analysis, describing fixed-point states of the system in terms of a small set of order parameters, and in dependence on the symmetry or otherwise of interactions and on the productivity of the community. Our work is an extension of Tokita [Phys. Rev. Lett. 93, 178102 (2004)], where the case of symmetric interactions was considered within an equilibrium setup. The species abundance distribution in our model come out as truncated normal distributions or transformations thereof and, in some case, are similar to left-skewed distributions observed in ecology. We also discuss the interaction structure of the resulting food-web of stable species at stationarity, cases of heterogeneous cooperation pressures as well as effects of finite system size and of higher-order interactions.

Heterogeneity of cardiac electrical properties can lead to heart rhythm disorders. Numerical studies have shown that stimuli chosen to maximize dynamic heterogeneity terminate wave propagation. However, experimental investigations suggest that similar sequences induce fragmentation of the wave fronts, rather than complete wave block. In this paper we show that an insulating boundary in an otherwise homogeneous medium can disrupt dynamically induced wave block by breaking a symmetry in the spatial pattern of action potential duration, leading to unidirectional block and reentrant activation.

We have developed a technique that circumvents the process of elimination of secular terms [L.-Y. Chen, N. Goldenfeld, and Y. Oono, Phys. Rev. E 54, 376 (1996)] and reproduces the uniformly valid approximations, amplitude equations, and first integrals. The technique is based on a rearrangement of secular terms and their grouping into the secular series that multiplies the constants of the asymptotic expansion. We illustrate the technique by deriving amplitude equations for standard nonlinear oscillator and boundary-layer problems.

The extreme value statistics of systems possessing a two-hump probability density of the relevant variable, in which the left peak is more pronounced than the right one, is studied. It is shown that systems of this type display a nontrivial transient behavior in the form of anomalous fluctuations around the mean, for certain (finite) ranges of observational time windows. The results are illustrated on independent identically distributed random variables, systems possessing two locally stable states and subjected to additive white noise, and dynamical systems in the regime of deterministic chaos.

Low amplitude, high frequency vibrations can induce in fluids under weightlessness behaviors that resemble those induced by gravity. Supercritical fluids (above their gas-liquid critical point) are used in the space industry and also display universal behavior. They are particularly sensitive to gravity effects. When submitted to vibration (typically $0.1\text{to}0.5\mathrm{mm}$ amplitude, $10\text{to}50\mathrm{Hz}$ frequency), a Rayleigh-Bénard-like instability is observed in experiments with ${\mathrm{H}}_2$ and $\mathrm{C}{\mathrm{O}}_2$ under weightlessness. The thermal boundary layer created during a temperature change displays periodic fingering perpendicular to the vibration direction. A systematic two-dimensional numerical study by the finite volume method is performed in $\mathrm{C}{\mathrm{O}}_2$ that shows that the fingering pattern is due to a thermovibrational instability, characterized by a vibrational Rayleigh number. The simulation and a simplified dimensional analysis show that the fingering wavelength and the vibrational Rayleigh number decrease as a power law with the distance in temperature to the critical point. However, due to the oversimplification of the analysis, the exponent in the simulation is found to be somewhat different than in the theoretical approach, calling for a more complete investigation of the problem.

We present a comprehensive, theoretical treatment of the orientational response to external stress of active, contractile cells embedded in a gel-like elastic medium. The theory includes both the forces that arise from the deformation of the matrix as well as forces due to the internal regulation of the stress fibers and focal adhesions of the cell. We calculate the time-dependent response of both the magnitude and the direction of the elastic dipole that characterizes the active forces exerted by the cell, for various situations. For static or quasistatic external stress, cells orient parallel to the stress while for high frequency dynamic external stress, cells orient nearly perpendicular. Both numerical and analytical calculations of these effects are presented. In addition we predict the relaxation time for the cellular response for both slowly and rapidly varying external stresses; several characteristic scaling regimes for the relaxation time as a function of applied frequency are predicted. We also treat the case of cells for which the regulation of the stress fibers and focal adhesions is controlled by strain (instead of stress) and show that the predicted dependence of the cellular orientation on the Poisson ratio of the matrix can differentiate strain vs stress regulation of cellular response.

Evaporation of droplets or liquid films lying on a substrate induces internal viscous flow, which affects the transport of suspended particles and, thus, the final deposit profile in numerous applications. In this work, the problem of Stokes flow inside a two-dimensional droplet, representing the cross section of an evaporating liquid line lying on a flat surface, is considered. The stream function formulation is adopted, leading to the biharmonic equation in bipolar coordinates. A solution in closed form is obtained for any contact angle in $(0,\pi )$ and is, thus, valid for both hydrophilic and hydrophobic substrates. The solution can be used with any type of evaporation mechanism, including diffusion, convection, or kinetically controlled modes. Both pinned and depinned contact lines are considered. For the boundary conditions to be compatible at the contact lines, the Navier slip boundary condition is applied on the substrate. Numerical results are presented for kinetically and diffusion controlled evaporation. For pinned contact lines, the flow inside the evaporating liquid line is directed towards the edges, thus, promoting the coffee stain phenomenon. In the case of depinned contact lines and contact angle less than $\pi ∕2$, the flow is directed towards the center of the droplet, whereas, for strongly hydrophobic substrates it is directed outwards.