We study bond percolations on hierarchical scale-free networks with the open bond probability of the shortcuts $\tilde{p}$ and that of the ordinary bonds $p$. The system has a critical phase in which the percolating probability $P$ takes an intermediate value $0<P<1$. Using generating function approach, we calculate the fractal exponent $\psi$ of the root clusters to show that $\psi$ varies continuously with $\tilde{p}$ in the critical phase. We confirm numerically that the distribution $n_s$ of cluster size $s$ in the critical phase obeys a power law $n_s\propto s^{-\tau }$, where $\tau$ satisfies the scaling relation $\tau =1+{\psi }^{-1}$. In addition the critical exponent $\beta \left(\tilde{p}\right)$ of the order parameter varies as $\tilde{p}$, from $\beta \simeq 0.164694$ at $\tilde{p}=0$ to infinity at $\tilde{p}={\tilde{p}}_c=5/32$.

The fractal Weyl law connects the asymptotic level number with the fractal dimension of the chaotic repeller. We provide the first test for the fractal Weyl law for a three-dimensional open scattering system. For the four-sphere billiard, we investigate the chaotic repeller and discuss the semiclassical quantization of the system by the method of cycle expansion with symmetry decomposition. We test the fractal Weyl law for various symmetry subspaces and sphere-to-sphere separations.

We study how the clustering coefficient influences the evolution of cooperation in scale-free public goods games. In games played by groups of individuals, triangle loops provide stronger support for mutual cooperation to resist invasion of selfish behavior than that in the absence of such loops, so that diffusion of cooperative behavior is relatively promoted. The feedback reciprocity mechanism of triangle plays a key role in facilitating cooperation in high clustered networks.

It is well known that optical force fields are not conservative. This has important consequences for the thermal motion of optically trapped dielectric spheres. In particular, the spheres do not reach thermodynamic equilibrium. Instead, a steady state is achieved in which the stochastic trajectory contains an underlying deterministic bias toward cyclic motion, and the energy of the sphere deviates from that implied by the equipartition theorem. Such effects are second order and only observed at low trap powers when the sphere is able to explore regions of the trap beyond the linear regime. Analogous effects may be expected for particles of less than spherical symmetry. However, in this case the effects are first order and depend on the linear term in the optical force field. As such they are not suppressed by increases in beam power, although the frequency and amplitude of the cyclic motion will be affected by it. In this paper, we present an analysis of the first-order nonconservative behavior of nonspherical particles in optical traps. The analysis is supported by optical force calculations and Brownian dynamics simulations of dielectric microrods held vertically in Gaussian optical traps.

A method based on multicanonical Monte Carlo is applied to the calculation of large deviations in the largest eigenvalue of random matrices. The method is successfully tested with the Gaussian orthogonal ensemble, sparse random matrices, and matrices whose components are subject to uniform density. Specifically, the probability that all eigenvalues of a matrix are negative is estimated in these cases down to the values of $\sim {10}^{-200}$, a region where simple random sampling is ineffective. The method can be applied to any ensemble of matrices and used for sampling rare events characterized by any statistics.

Mapping of the diffusion equation in a channel of varying cross section onto the longitudinal coordinate is already a well studied procedure for a slowly changing radius. We examine here the mapping of diffusion in a channel with abrupt change of diameter. In two dimensions, our considerations are based on solution of the exactly solvable geometry with abruptly doubled width at $x=0$. We verify the surmise of Berezhkovskii et al. [J. Chem. Phys. 131, 224110 (2009)] that one-dimensional diffusion behaves as free in such channels everywhere except at the point of change, which looks like a local trap for the particles. Applying the method of “sewing” of solutions, we show that this picture is valid also for three-dimensional symmetric channels.

We have used a Linkam CSS 450 stage equipped with an optical microscope to study online the evolution of the morphology and structure of polyamide 6 (PA6)–poly(styrene-ethylene-butadiene-styrene) (SEBS)–polypropylene (PP) ternary blends under shear flow. At a high shear rate $\left(60{\text{s}}^{-1}\right)$, large drops of PA6 and SEBS broke up to form small randomly distributed PA6 droplets covered by SEBS in the PP matrix. These droplets aggregated into a well-ordered structure, i.e., parallel clusters, after decreasing the shear rate abruptly to within the range of $1.5–3.5{\text{s}}^{-1}$. Interestingly, these parallel clusters were aligned perpendicular to the direction of shear flow. An important feature of these parallel clusters is that they can be maintained for a long period of time (several tens of minutes) at the given shear rate or after cessation of flow; this feature allows us to freeze the structure by decreasing the temperature.

The structures of the peptides and their assembly are largely modulated by the environment. To discover the physical principles governing the structural modulations of peptides by the environment would be useful for many applications. As the typical examples, the structures of three kinds of ionic-complementary EAK16-family peptides under various environmental conditions are studied with simulations in this work. A model with intermediate resolution is used, in which both the backbone hydrogen bonds and electrostatic interactions are explicitly considered. The thermodynamics of these peptides (including the free energy and heat capacity) are described for various strengths of the electrostatic interactions which reflect the variation of environment. With these results, the phase diagrams of these peptides related to the temperature and the strength of electrostatic interactions are presented and compared. Based on the differences in the phase structures of the peptide, the different aggregation behaviors are explained based on the monomeric structural features of the peptides. Through the analysis on the stability of various secondary structures of these peptides, it is demonstrated that the charge pattern is the basic reason of the different responses of the EAK16-family peptides to the environmental changes. These results provide some examples and insights for the principles of structural selection by environment and may be helpful for further analysis and designs of peptide systems.

We report on a self-emerging chimera state in a homogeneous chain of nonlocally and nonlinearly coupled oscillators. This chimera, i.e., a state with coexisting regions of complete and partial synchrony, emerges via a supercritical bifurcation from a homogeneous state. We develop a theory of chimera based on the Ott-Antonsen equations for the local complex order parameter. Applying a numerical linear stability analysis, we also describe the instability of the chimera and transition to phase turbulence with persistent patches of synchrony.

The fossil record has been interpreted as exhibiting a gradual decrease in the extinction rate. We use the individual based Tangled Nature model of evolutionary ecology to study the mechanisms behind this kind of nonstationary macrodynamics. We demonstrate that the long time aging in the system (manifested as a slowing down of the rate of large jumps, or quakes, that the system undergoes) is related to decreasing fluctuations in the offspring probability. The scenario is reminiscent of relaxation in a quenched spin glass but purely dynamical in nature since no energy barriers can be defined.

We investigate anchored scroll-wave filaments in an excitable medium whose diffusivity matrix, including its determinant, is spatially nonuniform. The study is motivated by cardiological applications where scroll-wave behavior in the presence of diffusivity gradients is believed to play an important role in the development of severe arrhythmias. A diffusivity gradient is expected to make the filament drift, unless drift is prevented (“frustrated”) by anchoring to localized defects in the propagation medium. The resulting stationary filament is a geodesic curve, as demonstrated here in the case of a nonzero but constant gradient. That is, the diffusivity matrix has a determinant that varies in space, in contrast to what was assumed in earlier work. Here, we show that the filament shape results from a metric tensor of the form $\left(\text{det}D\right)D^{-1}$, where $D$ is the diffusivity tensor. The filament’s shape is solely determined by the diffusivity tensor and is independent of the equation’s reaction terms. We derive the analytic solution for the filament and determine conditions for the existence of that solution. The theory is in excellent agreement with numerical simulations.

We present an analytic study of traveling fronts, localized colonies, and extended patterns arising from a reaction-diffusion equation which incorporates simultaneously two features: the well-known Allee effect and spatially nonlocal competition interactions. The former is an essential ingredient of most systems in population dynamics and involves extinction at low densities, growth at higher densities, and saturation at still higher densities. The latter feature is also highly relevant, particularly to biological systems, and goes beyond the unrealistic assumption of zero-range interactions. We show via exact analytic methods that the combination of the two features yields a rich diversity of phenomena and permits an understanding of a variety of issues including spontaneous appearance of colonies.

This paper demonstrates the existence of self-thermophoresis, a phenomenon whereby a virtual thermophoretic force arising from a temperature gradient in a quiescent single-component liquid or gas acts upon an individual molecule of that fluid in much the same manner as a “real” thermophoretic force acts upon a macroscopic, non-Brownian body immersed in that same fluid. In turn, self-thermophoresis acting in concert with Brownian self-diffusion gives rise to the phenomenon of thermal self-diffusion in single-component fluids. The latter furnishes quantitative explanations of both thermophoresis in pure fluids and thermal diffusion in binary mixtures (the latter composed of a dilute solution of a physicochemically inert solute whose molecules are large compared with those of the solvent continuum). Explicitly, the self-thermophoretic theory furnishes a simple expression for both the thermophoretic velocity $\mathbf{U}$ of a macroscopic body in a single-component fluid subjected to a temperature gradient $\mathbf{\nabla }T$, and the intimately related binary thermal diffusion coefficient $D_T$ for a two-component colloidal or macromolecular mixture. The predicted expressions $\mathbf{U}=-D_T\mathbf{\nabla }T\equiv -\beta D_S\mathbf{\nabla }T$ and $D_T=\beta D_S$ (with $\beta$ and $D_S$ the pure solvent’s respective thermal expansion and isothermal self-diffusion coefficients) are each noted to accord reasonably well with experimental data for both liquids and gases. The likely source of systematic deviations of the predicted values of $D_T$ from these data is discussed. This appears to be the first successful thermodiffusion theory applicable to both liquids and gases, a not insignificant achievement considering that the respective thermal diffusivities and thermophoretic velocities of these two classes of fluids differ by as much as six orders of magnitude.

The normal modes of a three-dimensional Yukawa plasma in an isotropic harmonic confinement are investigated by solving the linearized cold fluid equations. The eigenmodes are found analytically and expressed in terms of hypergeometric functions. It is found that the mode frequencies solely depend on the dimensionless plasma parameter $\xi =\kappa R$, where $R$ is the plasma radius and $\kappa$ is the inverse screening length. The eigenfrequencies increase monotonically with $\xi$ and saturate in the limit $\xi \to \infty$. Compared with the results in the Coulomb limit [D. H. E. Dubin, Phys. Rev. Lett. 66, 2076 (1991)], we find an additional class of modes characterized by the number $n$ which determines the number of radial nodes in the perturbed potential. These modes originate from the degenerate bulk modes of the Coulomb system. Analytical formulas for the eigenfrequencies are derived for limiting cases.

Using symmetry analysis we systematically present a higher-dimensional similarity transformation reducing the $\left(3+1\right)$-dimensional inhomogeneous nonlinear Schrödinger (NLS) equation with variable coefficients and parabolic potential to the $\left(1+1\right)$-dimensional NLS equation with constant coefficients. This transformation allows us to relate certain class of localized exact solutions of the $\left(3+1\right)$-dimensional case to the variety of solutions of integrable NLS equation of the $\left(1+1\right)$-dimensional case. As an example, we illustrated our technique using two lowest-order rational solutions of the NLS equation as seeding functions to obtain rogue wavelike solutions localized in three dimensions that have complicated evolution in time including interactions between two time-dependent rogue wave solutions. The obtained three-dimensional rogue wavelike solutions may raise the possibility of relative experiments and potential applications in nonlinear optics and Bose-Einstein condensates.

We present and compare different numerical schemes for the integration of the variational equations of autonomous Hamiltonian systems whose kinetic energy is quadratic in the generalized momenta and whose potential is a function of the generalized positions. We apply these techniques to Hamiltonian systems of various degrees of freedom and investigate their efficiency in accurately reproducing well-known properties of chaos indicators such as the Lyapunov characteristic exponents and the generalized alignment indices. We find that the best numerical performance is exhibited by the “tangent map method,” a scheme based on symplectic integration techniques which proves to be optimal in speed and accuracy. According to this method, a symplectic integrator is used to approximate the solution of the Hamilton equations of motion by the repeated action of a symplectic map $S$, while the corresponding tangent map $TS$ is used for the integration of the variational equations. A simple and systematic technique to construct $TS$ is also presented.

We show that in a microfluidic network with low Reynolds numbers, a system can be irreversible due to hysteresis effects. We simulated a network of pipes that was used in a recent experiment. The network consists of one loop connected to input and output pipes. A train of droplets enters the system at a uniform rate, but the droplets may leave the system in a periodic or even a chaotic pattern. The output pattern depends on the time interval between incoming droplets as well as the network geometry. For some parameters, the system is not reversible.

We have studied finite $N$-body $D$-dimensional nonextensive ideal gases and harmonic oscillators, by using the maximum-entropy methods with the $q$ and normal averages ($q$: the entropic index). The validity range, specific heat and Tsallis entropy obtained by the two average methods are compared. Validity ranges of the $q$- and normal averages are $0<q<q_U$ and $q>q_L$, respectively, where $q_U=1+{\left(\eta DN\right)}^{-1}$, $q_L=1-{\left(\eta DN+1\right)}^{-1}$ and $\eta =1/2$ $\left(\eta =1\right)$ for ideal gases (harmonic oscillators). The energy and specific heat in the $q$ and normal averages coincide with those in the Boltzmann-Gibbs statistics, although this coincidence does not hold for the fluctuation of energy. The Tsallis entropy for $N\left|q-1\right|⪢1$ obtained by the $q$ average is quite different from that derived by the normal average, despite a fairly good agreement of the two results for $\left|q-1\right|⪡1$. It has been pointed out that first-principles approaches previously proposed in the superstatistics yield additive $N$-body entropy $\left(S^{\left(N\right)}=N{S}^{\left(1\right)}\right)$ which is in contrast with the nonadditive Tsallis entropy.

With the Monte Carlo methods, we systematically investigate the short-time dynamics of domain-wall motion in the two-dimensional random-field Ising model with a driving field (DRFIM). We accurately determine the depinning transition field and critical exponents. Through two different definitions of the domain interface, we examine the dynamics of overhangs and islands. At the depinning transition, the dynamic effect of overhangs and islands reaches maximum. We argue that this should be an important mechanism leading the DRFIM model to a different universality class from the Edwards-Wilkinson equation with quenched disorder.

Relatively general techniques for computing mean first-passage time (MFPT) of random walks on networks with a specific property are very useful since a universal method for calculating MFPT on general graphs is not available because of their complexity and diversity. In this paper, we present techniques for explicitly determining the partial mean first-passage time (PMFPT), i.e., the average of MFPTs to a given target averaged over all possible starting positions, and the entire mean first-passage time (EMFPT), which is the average of MFPTs over all pairs of nodes on regular treelike fractals. We describe the processes with a family of regular fractals with treelike structure. The proposed fractals include the $T$ fractal and the Peano basin fractal as their special cases. We provide a formula for MFPT between two directly connected nodes in general trees on the basis of which we derive an exact expression for PMFPT to the central node in the fractals. Moreover, we give a technique for calculating EMFPT, which is based on the relationship between characteristic polynomials of the fractals at different generations and avoids the computation of eigenvalues of the characteristic polynomials. Making use of the proposed methods, we obtain analytically the closed-form solutions to PMFPT and EMFPT on the fractals and show how they scale with the number of nodes. In addition, to exhibit the generality of our methods, we also apply them to the Vicsek fractals and the iterative scale-free fractal tree and recover the results previously obtained.