Motivated by recent experiments, we present a study of the dynamics of cracks in thin sheets. While the equations of elasticity for thin plates are well known, there remains the question of path selection for a propagating crack. We invoke a generalization of the principle of local symmetry to provide a criterion for path selection and demonstrate qualitative agreement with the experimental findings. The nature of the singularity at the crack tip is studied with and without the interference of nonlinear terms.

In this paper, we consider spatial networks that realize a balance between an infrastructure cost (the cost of wire needed to connect the network in space) and communication efficiency, measured by average shortest path length. A global optimization procedure yields network topologies in which this balance is optimized. These are compared with network topologies generated by a competitive process in which each node strives to optimize its own cost-communication balance. Three phases are observed in globally optimal configurations for different cost-communication trade offs: (i) regular small worlds, (ii) starlike networks, and (iii) trees with a center of interconnected hubs. In the latter regime, i.e., for very expensive wire, power laws in the link length distributions $P\left(w\right)\propto w^{-\alpha }$ are found, which can be explained by a hierarchical organization of the networks. In contrast, in the local optimization process the presence of sharp transitions between different network regimes depends on the dimension of the underlying space. Whereas for $d=\infty$ sharp transitions between fully connected networks, regular small worlds, and highly cliquish periphery-core networks are found, for $d=1$ sharp transitions are absent and the power law behavior in the link length distribution persists over a much wider range of link cost parameters. The measured power law exponents are in agreement with the hypothesis that the locally optimized networks consist of multiple overlapping suboptimal hierarchical trees.

In this work, we introduce two models of Feynman’s ratchet and pawl system. Molecular dynamics is carried out to simulate the two designs for Feynman’s ratchet and pawl systems followed by a Langevin dynamics simulation of the reduced system. We find that the ratchet will rotate as designed when the temperature of the pawl chamber is lower than that of the ratchet chamber, which is consistent with the second law of thermodynamics. Different parameters and configurations are tested, and the results show that the efficiency of the ratchet depends on the applied torque. We find further that efficiencies of the Feynman’s ratchet and pawl systems depend greatly on the details of the systems.

The phase behavior of hard superballs is examined using molecular dynamics within a deformable periodic simulation box. A superball’s interior is defined by the inequality ${\left|x\right|}^{2q}+{\left|y\right|}^{2q}+{\left|z\right|}^{2q}\le 1$, which provides a versatile family of convex particles $\left(q\ge 0.5\right)$ with cubelike and octahedronlike shapes as well as concave particles $\left(q<0.5\right)$ with octahedronlike shapes. Here, we consider the convex case with a deformation parameter $q$ between the sphere point $\left(q=1\right)$ and the cube $\left(q=\infty \right)$. We find that the asphericity plays a significant role in the extent of cubatic ordering of both the liquid and crystal phases. Calculation of the first few virial coefficients shows that superballs that are visually similar to cubes can have low-density equations of state closer to spheres than to cubes. Dense liquids of superballs display cubatic orientational order that extends over several particle lengths only for large $q$. Along the ordered, high-density equation of state, superballs with $1<q<3$ exhibit clear evidence of a phase transition from a crystal state to a state with reduced long-ranged orientational order upon the reduction of density. For $q\ge 3$, long-ranged orientational order persists until the melting transition. The width of the apparent coexistence region between the liquid and ordered, high-density phase decreases with $q$ up to $q=4.0$. The structures of the high-density phases are examined using certain order parameters, distribution functions, and orientational correlation functions. We also find that a fixed simulation cell induces artificial phase transitions that are out of equilibrium. Current fabrication techniques allow for the synthesis of colloidal superballs and thus the phase behavior of such systems can be investigated experimentally.

Finite-size effects are investigated in the Gaussian model with isotropic and anisotropic short-range interactions in film geometry with nonperiodic boundary conditions (bc) above, at, and below the bulk critical temperature $T_{\mathrm{c}}$. We have obtained exact results for the free energy and the Casimir force for antiperiodic, Neumann, Dirichlet, and Neumann-Dirichlet mixed bc in $1<d<4$ dimensions. For the Casimir force, finite-size scaling is found to be valid for all bc. For the free energy, finite-size scaling is valid in $1<d<3$ and $3<d<4$ dimensions for antiperiodic, Neumann, and Dirichlet bc, but logarithmic deviations from finite-size scaling exist in $d=3$ dimensions for Neumann and Dirichlet bc. This is explained in terms of the borderline dimension $d^{*}=3$, where the critical exponent $1-\alpha -\nu =(d-3)∕2$ of the Gaussian surface energy density vanishes. For Neumann-Dirichlet bc, finite-size scaling is strongly violated above $T_{\mathrm{c}}$ for $1<d<4$ because of a cancelation of the leading scaling terms. For antiperiodic, Dirichlet, and Neumann-Dirichlet bc, a finite film critical temperature $T_{\mathrm{c},\text{film}}\left(L\right)<T_{\mathrm{c}}$ exists at finite film thickness $L$. Our results include an exact description of the dimensional crossover between the $d$-dimensional finite-size critical behavior near bulk $T_{\mathrm{c}}$ and the $(d-1)$-dimensional critical behavior near $T_{\mathrm{c},\text{film}}\left(L\right)$. This dimensional crossover is illustrated for the critical behavior of the specific heat. Particular attention is paid to an appropriate representation of the free energy in the region $T_{\mathrm{c},\text{film}}\left(L\right)⩽T⩽T_{\mathrm{c}}$. For $2<d<4$, the Gaussian results are renormalized and reformulated as one-loop contributions of the ${\phi }^4$ field theory at fixed dimension $d$ and are then compared with the $\epsilon =4-d$ expansion results at $\epsilon =1$ as well as with $d=3$ Monte Carlo data. For $d=2$, the Gaussian results for the Casimir force scaling function are compared with those for the Ising model with periodic, antiperiodic, and free bc; unexpected exact relations are found between the Gaussian and Ising scaling functions. For both the $d$-dimensional Gaussian model and the two-dimensional Ising model it is shown that anisotropic couplings imply nonuniversal scaling functions of the Casimir force that depend explicitly on microscopic couplings. Our Gaussian results provide the basis for the investigation of finite-size effects of the mean spherical model in film geometry with nonperiodic bc above, at, and below the bulk critical temperature.

The definition of a nonequilibrium temperature through generalized fluctuation-dissipation relations relies on the independence of the fluctuation-dissipation temperature from the observable considered. We argue that this observable independence is deeply related to the uniformity of the phase-space probability distribution on the hypersurfaces of constant energy. This property is shown explicitly on three different stochastic models, where observable dependence of the fluctuation-dissipation temperature arises only when the uniformity of the phase-space distribution is broken. The first model is an energy transport model on a ring, with biased local transfer rules. In the second model, defined on a fully connected geometry, energy is exchanged with two heat baths at different temperatures, breaking the uniformity of the phase-space distribution. Finally, in the last model, the system is connected to a zero temperature reservoir, and preserves the uniformity of the phase-space distribution in the relaxation regime, leading to an observable-independent temperature.

This paper presents a simulation study of the applicability of the Rosenfeld entropy scaling to the systems which cannot be approximated by the effective hard spheres. Three systems are studied: the Herzian spheres, the Gauss core model, and a soft repulsive shoulder potential. These systems demonstrate diffusion anomalies at low temperatures: the diffusion coefficient increases with increasing density or pressure. It is shown that for the first two systems belonging to a class of bounded potentials, the Rosenfeld scaling formula is valid only in the infinite-temperature limit where there are no anomalies. For the soft repulsive shoulder potential, the scaling formula is valid already at sufficiently low temperatures, however, out of the anomaly range.

We report the synchronization behavior in an ensemble of identical limit cycle oscillators coupled to a mass-spring load via a force relation. We consider the effect of periodic parametric modulation on the final states of the system. Two types of external parametric excitations are investigated numerically: periodic modulation of the stiffness of the inertial oscillator and periodic excitation of the frequency of the self-oscillatory element. We show that the synchronization scenarios are not only ruled by the choice of parameters of the excitation force but also depend on the initial collective state in the ensemble. We give detailed analysis of entrainment behavior for initially synchronous or antiphase synchronous states as well as for disorganized states.

In this work, we present an effective discrete Edwards-Wilkinson equation aimed to describe the single-file diffusion process. The key physical properties of the system are captured defining an effective elasticity, which is proportional to the single particle diffusion coefficient and to the inverse squared mean separation between particles. The effective equation gives a description of single-file diffusion using the global roughness of the system of particles, which presents three characteristic regimes, namely, normal diffusion, subdiffusion, and saturation, separated by two crossover times. We show how these regimes scale with the parameters of the original system. Additional repulsive interaction terms are also considered and we analyze how the crossover times depend on the intensity of the additional terms. Finally, we show that the roughness distribution can be well characterized by the Edwards-Wilkinson universal form for the different single-file diffusion processes studied here.

We present a fundamental theory for the kinetics of systems of classical particles. The theory represents a unification of kinetic theory, Brownian motion, and field theory. It is self-consistent and is the dynamic generalization of the functional theory of fluids in equilibrium. This gives one a powerful tool for investigating the existence of ergodic-nonergodic transitions near the liquid-glass transition.

We describe a technique that allows the exact analytical computation of the mean first passage time (MFPT) for infinite families of trees using their recursive properties. The method is based in the relationship between the MFPT and the eigenvalues of the Laplacian matrix of the trees but avoids their explicit computation. We apply this technique to find the MFPT for a family of generalized deterministic recursive trees. The method, however, can be adapted to other self-similar tree families.

A striking feature of the marine ecosystem is the regularity in its size spectrum: the abundance of organisms as a function of their weight approximately follows a power law over almost ten orders of magnitude. We interpret this as evidence that the population dynamics in the ocean is approximately scale-invariant. We use this invariance in the construction and solution of a size-structured dynamical population model. Starting from a Markov model encoding the basic processes of predation, reproduction, maintenance respiration, and intrinsic mortality, we derive a partial integro-differential equation describing the dependence of abundance on weight and time. Our model represents an extension of the jump-growth model and hence also of earlier models based on the McKendrick-von Foerster equation. The model is scale-invariant provided the rate functions of the stochastic processes have certain scaling properties. We determine the steady-state power-law solution, whose exponent is determined by the relative scaling between the rates of the density-dependent processes (predation) and the rates of the density-independent processes (reproduction, maintenance, and mortality). We study the stability of the steady-state against small perturbations and find that inclusion of maintenance respiration and reproduction in the model has a strong stabilizing effect. Furthermore, the steady state is unstable against a change in the overall population density unless the reproduction rate exceeds a certain threshold.

We develop an interacting particle model to simulate the life cycle of myxobacteria, which consists of two main stages—the swarming stage and the development (fruiting body formation) stage. As experiments have shown that the phase transition from swarming to development stage is triggered by starvation, we incorporate into the simulation a system of ordinary differential equations (ODEs) called the dynamic energy budget, which controls the uptake and use of energy by individuals. This inclusion successfully automates the phase transition in our simulation. Only one parameter, namely, the food density, controls the entire simulation of the life cycle.

Using multiple nonlinear techniques, we revealed the existence of chaos in the spontaneous activity of neuronal networks in vitro. The spatial-temporal dynamics of these networks indicated that emergent transition between chaotic behavior and superburst occurred periodically in low-frequency oscillations. An analysis of network-wide activity indicated that chaos was synchronized among different sites. Moreover, we found that the degree of chaos increased as the number of active sites in the network increased during long-term development (over three months in vitro). The chaotic behavior of the dissociated networks had similar spatial-temporal characteristics (rapid transition, periodicity, and synchronization) as the intact brain; however, the degree of chaos depended on the number of active sites at the mesoscopic level. This work could provide insight into neural coding and neurocybernetics.

Hwang in [Phys. Rev. E 80, 042103 (2009)] suggested that the two transitions of the six-state clock model on the square lattice are not of the Kosterlitz-Thouless type. Here we show from simulations that at the upper transition, the helicity modulus does make a discontinuous jump to zero. This gives strong evidence for a Kosterlitz-Thouless transition.

Both cooperation and migration are ubiquitous in human society and animal world. In this Rapid Communication, we propose an aspiration-induced migration in which individuals will migrate to new sites provided that their payoffs are below some aspiration level. It is found that moderate aspiration level can best favor cooperative behavior. In particular, moderate aspiration level enables cooperator clusters to maintain and expand whereas induces defector clusters to disintegrate, thus promoting the diffusion of cooperation among population. Our results provide insights into understanding the role played by migration in the emergence of cooperative behavior.

We show that the statistics of spreads in real order books is characterized by an intrinsic asymmetry due to discreteness effects for even or odd values of the spread. An analysis of data from the New York Stock Exchange (NYSE) order book points out that traders’ strategies contribute to this asymmetry. We also investigate this phenomenon in the framework of a microscopic model and, by introducing a nonuniform deposition mechanism for limit orders, we are able to quantitatively reproduce the asymmetry found in the experimental data. Simulations of our model also show a realistic dynamics with a sort of intermittent behavior characterized by long periods in which the order book is compact and liquid interrupted by volatile configurations. The order placement strategies produce a nontrivial behavior of the spread relaxation dynamics which is similar to the one observed in real markets.

We analyze and exploit some scaling properties of the affinity propagation (AP) clustering algorithm proposed by Frey and Dueck [Science 315, 972 (2007)]. Following a divide and conquer strategy we setup an exact renormalization-based approach to address the question of clustering consistency, in particular, how many cluster are present in a given data set. We first observe that the divide and conquer strategy, used on a large data set hierarchically reduces the complexity $\mathcal{O}\left(N^2\right)$ to $\mathcal{O}\left(N^{\left(h+2\right)/\left(h+1\right)}\right)$, for a data set of size $N$ and a depth $h$ of the hierarchical strategy. For a data set embedded in a $d$-dimensional space, we show that this is obtained without notably damaging the precision except in dimension $d=2$. In fact, for $d$ larger than 2 the relative loss in precision scales such as $N^{\left(2-d\right)/\left(h+1\right)d}$. Finally, under some conditions we observe that there is a value $s^{\ast }$ of the penalty coefficient, a free parameter used to fix the number of clusters, which separates a fragmentation phase (for $s<s^{\ast }$) from a coalescent one (for $s>s^{\ast }$) of the underlying hidden cluster structure. At this precise point holds a self-similarity property which can be exploited by the hierarchical strategy to actually locate its position, as a result of an exact decimation procedure. From this observation, a strategy based on AP can be defined to find out how many clusters are present in a given data set.

The problem of nonlinear Hamiltonian oscillators is one of the classical questions in physics. When an analytic solution is not possible, one can resort to obtaining a numerical solution or using perturbation theory around the linear problem. We apply the Fourier series expansion to find approximate solutions to the oscillator position as a function of time as well as the period-amplitude relationship. We compare our results with other recent approaches such as variational methods or heuristic approximations, in particular the Ren-He’s method. Based on its application to the Duffing oscillator, the nonlinear pendulum and the eardrum equation, it is shown that the Fourier series expansion method is the most accurate.

Rotating spiral waves are a form of self-organization observed in spatially extended systems of physical, chemical, and biological nature. In the presence of a small perturbation, the spiral wave’s center of rotation and fiducial phase may change over time, i.e., the spiral wave drifts. In linear approximation, the velocity of the drift is proportional to the convolution of the perturbation with the spiral’s response functions, which are the eigenfunctions of the adjoint linearized operator corresponding to the critical eigenvalues $\lambda =0,\pm i\omega$. Here, we demonstrate that the response functions give quantitatively accurate prediction of the drift velocities due to a variety of perturbations: a time dependent, periodic perturbation (inducing resonant drift); a rotational symmetry-breaking perturbation (inducing electrophoretic drift); and a translational symmetry-breaking perturbation (inhomogeneity induced drift) including drift due to a gradient, stepwise, and localized inhomogeneity. We predict the drift velocities using the response functions in FitzHugh-Nagumo and Barkley models, and compare them with the velocities obtained in direct numerical simulations. In all cases good quantitative agreement is demonstrated.