It is well established that the phase transition between survival and extinction in spreading models with short-range interactions is generically associated with the directed percolation (DP) universality class. In many realistic spreading processes, however, interactions are long ranged and well described by Lévy flights—i.e., by a probability distribution that decays in $d$ dimensions with distance $r$ as $r^{-d-\sigma }$. We employ the powerful methods of renormalized field theory to study DP with such long-range Lévy-flight spreading in some depth. Our results unambiguously corroborate earlier findings that there are four renormalization group fixed points corresponding to, respectively, short-range Gaussian, Lévy Gaussian, short-range, and Lévy DP and that there are four lines in the $(\sigma ,d)$ plane which separate the stability regions of these fixed points. When the stability line between short-range DP and Lévy DP is crossed, all critical exponents change continuously. We calculate the exponents describing Lévy DP to second order in an $\epsilon$ expansion, and we compare our analytical results to the results of existing numerical simulations. Furthermore, we calculate the leading logarithmic corrections for several dynamical observables.

The growth of ballistic aggregates on deterministic fractal substrates is studied by means of numerical simulations. First, we attempt the description of the evolving interface of the aggregates by applying the well-established Family-Vicsek dynamic scaling approach. Systematic deviations from that standard scaling law are observed, suggesting that significant scaling corrections have to be introduced in order to achieve a more accurate understanding of the behavior of the interface. Subsequently, we study the internal structure of the growing aggregates that can be rationalized in terms of the scaling behavior of frozen trees, i.e., structures inhibited for further growth, lying below the growing interface. It is shown that the rms height $\left(h_s\right)$ and width $\left(w_s\right)$ of the trees of size $s$ obey power laws of the form $h_s\propto s^{{\nu }_{\parallel }}$ and $w_s\propto s^{{\nu }_{\perp }}$, respectively. Also, the tree-size distribution $\left(n_s\right)$ behaves according to $n_s\sim s^{-\tau }$. Here, ${\nu }_{\parallel }$ and ${\nu }_{\perp }$ are the correlation length exponents in the directions parallel and perpendicular to the interface, respectively. Also, $\tau$ is a critical exponent. However, due to the interplay between the discrete scale invariance of the underlying fractal substrates and the dynamics of the growing process, all these power laws are modulated by logarithmic periodic oscillations. The fundamental scaling ratios, characteristic of these oscillations, can be linked to the (spatial) fundamental scaling ratio of the underlying fractal by means of relationships involving critical exponents. We argue that the interplay between the spatial discrete scale invariance of the fractal substrate and the dynamics of the physical process occurring in those media is a quite general phenomenon that leads to the observation of logarithmic-periodic modulations of physical observables.

We propose an alternative segregation mechanism where the species-dependent interactions are dynamically induced by the phase synchronization of beads. Based on this scenario, we report an alternative segregation among beads of different restitution coefficients by molecular dynamics simulations. Since the beads are of equal size and mass, this is not related to the Brazilian-nut effect, nor can it be explained by the depletion force. Instead, this phenomenon derives from the phase synchronization, a concept which helps us determine the criteria for segregation and the phase boundaries that agree excellently with the simulation results.

A field theoretical method for fluctuating hydrodynamics with preserved fluctuation-dissipation relations is reformulated. By assuming that the correlations including momentum are irrelevant in the long time region, we demonstrate that the equation obtained from the first-order perturbation is reduced to that for standard mode-coupling theory.

In the technique of microrheology, macroscopic rheological parameters as well as information about local structure are deduced from the behavior of microscopic probe particles under thermal or active forcing. Microrheology requires knowledge of the relation between macroscopic parameters and the force felt by a particle in response to displacements. We investigate this response function for a spherical particle using the two-fluid model, in which the gel is represented by a polymer network coupled to a surrounding solvent via a drag force. We obtain an analytic solution for the response function in the limit of small volume fraction of the polymer network, and neglecting inertial effects. We use no-slip boundary conditions for the solvent at the surface of the sphere. The boundary condition for the network at the surface of the sphere is a kinetic friction law, for which the tangential stress of the network is proportional to relative velocity of the network and the sphere. This boundary condition encompasses both no-slip and frictionless boundary conditions as limits. Far from the sphere there is no relative motion between the solvent and network due to the coupling between them. However, the different boundary conditions on the solvent and network tend to produce different far-field motions. We show that the far-field motion and the force on the sphere are controlled by the solvent boundary conditions at high frequency and by the network boundary conditions at low frequency. At low frequencies compression of the network can also affect the force on the sphere. We find the crossover frequencies at which the effects of sliding of the sphere past the polymer network and compression of the gel become important. The effects of sliding alone can lead to an underestimation of moduli by up to 33%, while the effects of compression alone can lead to an underestimation of moduli by up to 20%, and the effects of sliding and compression combined can lead to an underestimation of moduli by up to 43%.

Elasticity and strength of individual beta-sheet protein domains govern key biological functions and the mechanical properties of biopolymers including spider silk, amyloids, and muscle fibers. The worm-like-chain (WLC) model is commonly used to describe the entropic elasticity of polypeptides and other biomolecules. However, force spectroscopy experiments have shown pronounced deviations from the ideal WLC behavior, leading to controversial views about the appropriate elastic description of proteins at nanoscale. Here we report a simple model that explains the physical mechanism that leads to the breakdown of the WLC idealization in experiments by using only two generic parameters of the protein domain, the H-bond energy and the protein backbone’s persistence length. We show that a rupture initiation condition characterized by the free energy release rate of H-bonds characterizes the limit of WLC entropic elasticity of beta-sheet protein domains and the onset of rupture. Our findings reveal that strength and elasticity are coupled and cannot be treated separately. The predictions of the model are compared with atomic force microscopy experiments of protein rupture.

A mechanics model describing how a cell membrane with diffusive mobile receptors wraps around a ligand-coated cylindrical or spherical particle has been recently developed to model the role of particle size in receptor-mediated endocytosis. The results show that particles in the size range of tens to hundreds of nanometers can enter cells even in the absence of clathrin or caveolin coats. Here we report further progress on modeling the effects of size and shape in diffusion, interaction, and absorption of finite-sized colloidal particles near a partially absorbing sphere. Our analysis indicates that, from the diffusion and interaction point of view, there exists an optimal hydrodynamic size of particles, typically in the nanometer regime, for the maximum rate of particle absorption. Such optimal size arises as a result of balance between the diffusion constant of the particles and the interaction energy between the particles and the absorbing sphere relative to the thermal energy. Particles with a smaller hydrodynamic radius have larger diffusion constant but weaker interaction with the sphere while larger particles have smaller diffusion constant but stronger interaction with the sphere. Since the hydrodynamic radius is also determined by the particle shape, an optimal hydrodynamic radius implies an optimal size as well as an optimal aspect ratio for a nonspherical particle. These results show broad agreement with experimental observations and may have general implications on interaction between nanoparticles and animal cells.

A class of cubic networks composed of a regular one-dimensional lattice and a set of long-range links is introduced. Networks parametrized by a positive integer $k$ are constructed by starting from a one-dimensional lattice and iteratively connecting each site of degree 2 with a $k\text{th}$ neighboring site of degree 2. Specifying the way pairs of sites to be connected are selected, various random and regular networks are defined, all of which have a power-law edge-length distribution of the form $P_{>}\left(l\right)\sim l^{-s}$ with the marginal exponent $s=1$. In all these networks, lengths of shortest paths grow as a power of the distance and random walk is superdiffusive. Applying a renormalization group method, the corresponding shortest-path dimensions and random-walk dimensions are calculated exactly for $k=1$ networks and for $k=2$ regular networks; in other cases, they are estimated by numerical methods. Although, $s=1$ holds for all representatives of this class, the above quantities are found to depend on the details of the structure of networks controlled by $k$ and other parameters.

In this paper, critical global connectivity of mobile ad hoc networks (MANETs) is investigated. We model the two-dimensional plane on which nodes move randomly with a triangular lattice. Demanding the best communication of the network, we account the global connectivity $\eta$ as a function of occupancy $\sigma$ of sites in the lattice by mobile nodes. Critical phenomena of the connectivity for different transmission ranges $r$ are revealed by numerical simulations, and these results fit well to the analysis based on the assumption of homogeneous mixing. Scaling behavior of the connectivity is found as $\eta \sim f\left(R^{\beta }\sigma \right)$, where $R=(r-r_0)∕r_0$, $r_0$ is the length unit of the triangular lattice, and $\beta$ is the scaling index in the universal function $f\left(x\right)$. The model serves as a sort of geometric distance-dependent site percolation on dynamic complex networks. Moreover, near each critical ${\sigma }_c\left(r\right)$ corresponding to certain transmission range $r$, there exists a cutoff degree $k_c$ below which the clustering coefficient of such self-organized networks keeps a constant while the averaged nearest-neighbor degree exhibits a unique linear variation with the degree $k$, which may be useful to the designation of real MANETs.

It is unlikely that a dissipative reaction-diffusion system exhibits static domains composed of different pattern elements. So far as we know, there is only one exception in the energy conserving system: the generalized Swift-Hohenberg (GSH) system [M’F. Hilali et al., Phys. Rev. E 51, 2049 (1995)]. Our paper reports that both a spot-domain and a line-domain coexist in a dissipative reaction-diffusion system with the reversible Gray-Scott dynamics. The system has the features that a local perturbation induces the self-rearrangement of the pattern elements and/or self-replication of spots. These features and controllability of the pattern are different from those in the GSH system.

We identify an unconventional route to the creation of a strange nonchaotic attractor (SNA) in a quasiperiodically forced electronic circuit with a nonsinusoidal (square wave) force as one of the quasiperiodic forces through numerical and experimental studies. We find that bubbles appear in the strands of the quasiperiodic attractor due to the instability induced by the additional square-wave-type force. The bubbles then enlarge and get increasingly wrinkled as a function of the control parameter. Finally, the bubbles get extremely wrinkled (while the remaining parts of the strands of the torus remain largely unaffected) resulting in the creation of the SNA; we term this the bubbling route to the SNA. We characterize and confirm this creation from both experimental and numerical data using maximal Lyapunov exponents and their variance, Poincaré maps, Fourier amplitude spectra, and spectral distribution functions. We also strongly confirm the creation of a SNA via the bubbling route by the distribution of the finite-time Lyapunov exponents.

We propose a scheme for studying thin liquid films on a solid substrate using a phase field model. For a van der Waals fluid—far from criticality—the most natural phase field function is the fluid density. The theoretical description is based on the Navier-Stokes equation with extra phase field terms and the continuity equation. In this model free of interface conditions, the contact angle can be controlled through the boundary conditions for the density field at the solid walls [L. M. Pismen and Y. Pomeav, Phys. Rev. E 62, 2480 (2000)]. We investigate the stability of a thin liquid film on a flat homogeneous solid support with variable wettability. For almost hydrophobic surfaces, the liquid film breaks up and transitions from a flat film to drops occur. Finally, we report on two-dimensional numerical simulations for static liquid drops resting on a flat horizontal solid support and for drops sliding down on inclined substrates under gravity effects.

We study the time evolution of two correlated electrons of opposite spin in an anharmonic lattice chain. The electrons are described quantum mechanically by the Hubbard model while the lattice is treated classically. The lattice units are coupled via Morse-Toda potentials. Interaction between the lattice and the electrons arises due to the dependence of the electron transfer-matrix element on the distance between neighboring lattice units. Localized configurations comprising a paired electron and a pair of lattice deformation solitons are constructed such that an associated energy functional is minimized. We investigate long-lived, stable pairing features. It is demonstrated that traveling pairs of lattice solitons serve as carriers for the paired electrons realizing coherent transport of the two correlated electrons. We also observe dynamical narrowing of the states, that is, starting from an initial double-peak profile of the electron probability distribution, a single-peak profile is adopted going along with enhancement of localization of the paired electrons. Interestingly, a parameter regime is identified for which supersonic transport of paired electrons is achieved.

We present an elastic model for B-form DNA with variable radius to study the elastic twist-stretch coupling of stretched DNA. In this model, only two strain variables as well as the changes in the energy of the hydrogen and covalent bonds of DNA during the deformation are considered. It is found that, depending on the elastic constants, the correlation between twisting and stretching of a helical molecule can be positive or negative. It is shown that for the suitable elastic constants in the model, the twist-stretch coupling of DNA behaves nonmonotonically, and contrary to intuition, the DNA twisting and stretching are positively correlated for small distortions. This result is entirely consistent with recent experimental results.

In many experiments, the aim is to deduce an underlying multisubstate on-off kinetic scheme (KS) from the statistical properties of a two-state trajectory. However, a two-state trajectory that is generated from an on-off KS contains only partial information about the KS, and so, in many cases, more than one KS can be associated with the data. We recently showed that the optimal way to solve this problem is to use canonical forms of reduced dimensions (RDs). RD forms are on-off networks with connections only between substates of different states, where the connections can have nonexponential waiting time probability density functions (WT-PDFs). In theory, only a single RD form can be associated with the data. To utilize RD forms in the analysis of the data, a RD form should be associated with the data. Here, we give a toolbox for building a RD form from a finite time, noiseless, two-state trajectory. The methods in the toolbox are based on known statistical methods in data analysis, combined with statistical methods and numerical algorithms designed specifically for the current problem. Our toolbox is self-contained—it builds a mechanism based only on the information it extracts from the data, and its implementation is fast (analyzing a ${10}^6\text{cycle}$ trajectory from a 30-parameter mechanism takes a couple of hours on a PC with a $2.66\mathrm{GHz}$ processor). The toolbox is automated and is freely available for academic research upon electronic request.

We investigate the spatiotemporal properties of a lattice of chaotic maps whose coupling connections are rewired to random sites with probability $p$. Keeping $p$ constant, we change the random links at different frequencies in order to discern the effect (if any) of the time dependence of the links. We observe two different regimes in this network: (i) when the network is rewired slowly, namely, when the random connections are quite static, the dynamics of the network is spatiotemporally chaotic and (ii) when these random links are switched around fast, namely, the network is rewired frequently, one obtains a spatiotemporal fixed point over a large range of coupling strengths. We provide evidence of a sharp transition from a globally attracting spatiotemporal fixed point to spatiotemporal chaos as the rewiring frequency is decreased. Thus, in addition to geometrical properties such as the fraction of random links in the network, dynamical information on the time dependence of these links is crucial in determining the spatiotemporal properties of complex dynamical networks.

The issue of the thermalization of an isolated quantum system is addressed by considering a perfect gas confined by gravity and initially trapped above a certain height. As we are interested in the behavior of truly isolated systems, we assume the gas is in a pure state of macroscopically well-defined energy. We show that, in general, for single-particle distributions, such a state is strictly equivalent to the microcanonical mixed state at the same energy. We derive an expression for the time-dependent gas density that depends on the initial gas state only via a few thermodynamic parameters. Though we consider noninteracting particles, the density relaxes into an asymptotic profile, but it is not the thermal equilibrium one determined by the gas energy and particle number.

It is well known that statistical mechanics systems exhibit subtle behavior in high dimensions. In this paper, we show that certain natural soft-core models, such as the Gaussian core model, have unexpectedly complex ground states even in relatively low dimensions. Specifically, we disprove a conjecture of Torquato and Stillinger, who predicted that dilute ground states of the Gaussian core model in dimensions 2 through 8 would be Bravais lattices. We show that in dimensions 5 and 7, there are in fact lower-energy non-Bravais lattices. (The nearest three-dimensional analog is the hexagonal close-packing, but it has higher energy than the face-centered cubic lattice.) We believe these phenomena are in fact quite widespread, and we relate them to decorrelation in high dimensions.

We report an unusual behavior of a ferroelectric $\mathrm{smectic}\text{−}C{P}_{\mathrm{F}}$ film formed by bent-shaped molecules. The ground state of the $c$-director in such film is not uniform but forms a striped pattern with alternating bend deformation. We found that the sense of the alternating bend is not related to an alternating handedness defined by the mutual orientation of the tilt ($c$ director) and the bow ($p$ director) of the molecules. Despite its similarity to a previously described twist-bend instability [J. Pang and N. A. Clark, Phys. Rev. Lett. 73, 2332 (1994)], this pattern cannot be explained in terms of spontaneous chiral symmetry breaking with continuous variation of the chirality order parameter, since the synclinic order of the polar molecules predefines the chirality of the film. We discuss possible models describing the spontaneous formation of an ambidextrous bend pattern of the $c$ director.