The state-space-split method for solving the Fokker-Planck-Kolmogorov equations in high dimensions is extended to solving the generalized Fokker-Planck-Kolmogorov equations in high dimensions for stochastic dynamical systems with a polynomial type of nonlinearity and excited by Poissonian white noise. The probabilistic solution of the motion of the stretched Euler-Bernoulli beam with cubic nonlinearity and excited by uniformly distributed Poissonian white noise is analyzed with the presented solution procedure. The numerical analysis shows that the results obtained with the state-space-split method together with the exponential polynomial closure method are close to those obtained with the Monte Carlo simulation when the relative value of the basic system relaxation time and the mean arrival time of the Poissonian impulse is in some limited range.

The percolation aspect of random sequential adsorption of extended objects on a triangular lattice is studied by means of Monte Carlo simulations. The depositing objects are formed by self-avoiding lattice steps on the lattice. Jamming coverage ${\theta }_{\mathrm{jam}}$, percolation threshold ${\theta }_p^{*}$, and their ratio ${\theta }_p^{*}/{\theta }_{\mathrm{jam}}$ are determined for objects of various shapes and sizes. We find that the percolation threshold ${\theta }_p^{*}$ may decrease or increase with the object size, depending on the local geometry of the objects. We demonstrate that for various objects of the same length, the threshold ${\theta }_p^{*}$ of more compact shapes exceeds the ${\theta }_p^{*}$ of elongated ones. In addition, we study polydisperse mixtures in which the size of line segments making up the mixture gradually increases with the number of components. It is found that the percolation threshold decreases, while the jamming coverage increases, with the number of components in the mixture.

Experimental and computational model systems composed of frictionless particles in a fixed geometry have a finite number of distinct mechanically stable (MS) packings. The frequency of occurrence for each MS packing is highly variable and depends strongly on preparation protocol. Despite intense work, it is extremely difficult to predict a priori the MS packing probabilities. We describe a novel computational method for calculating the volume and other geometrical properties of the “basin of attraction” for each MS packing. The basin of attraction for an MS packing contains all initial conditions in configuration space that map to that MS packing using a given preparation protocol. We find that the basin is a highly complex structure. For a compressive-quench-from-zero-density protocol, we show the existence of a small core volume of the basin around each MS packing for which all points map to that MS packing. However, in contrast to previous studies for supercooled liquids, glasses, and over-compressed jammed systems, we find that the MS packing probabilities are very weakly correlated with this core volume. Instead, MS packing probabilities obtained from compression protocols that use initially dilute configurations and do not allow particle overlaps (i.e., those relevant to granular media) are determined by complex geometric features of the basin of attraction that are distant from the MS packing. In particular, we find that the shape of the average basin profile function $S\left(l\right)$, which gives the probability for a point on a hyperspherical shell a distance $l$ from a given MS packing to map back to that packing, can be described by a $\Gamma$ distribution with a peak that increases as the system size increases and as the quench rate decreases. We find a simple model which predicts $S\left(l\right)$ for the extreme cases of very slow and fast quench rates.

We combine the shear-transformation-zone (STZ) theory of amorphous plasticity with Edwards’ statistical theory of granular materials to describe shear flow in a disordered system of thermalized hard spheres. The equations of motion for this system are developed within a statistical thermodynamic framework analogous to that which has been used in the analysis of molecular glasses. For hard spheres, the system volume $V$ replaces the internal energy $U$ as a function of entropy $S$ in conventional statistical mechanics. In place of the effective temperature, the compactivity $X=\partial V/\partial S$ characterizes the internal state of disorder. We derive the STZ equations of motion for a granular material accordingly, and predict the strain rate as a function of the ratio of the shear stress to the pressure for different values of a dimensionless, temperature-like variable near a jamming transition. We use a simplified version of our theory to interpret numerical simulations by Haxton, Schmiedeberg, and Liu, and in this way are able to obtain useful insights about internal rate factors and relations between jamming and glass transitions.

We use a high level virial expansion to investigate the properties of the isotropic and nematic phases of the hard spheroid fluid. We use the Monte Carlo techniques described previously to calculate the virial coefficients up to seventh order and we represent the dependence of these coefficients on particle orientations via a spherical harmonic expansion. The expansion coefficients are determined using Lebedev quadrature which carries out the angular integration required exactly. For fairly spherical spheroids ($1/3<a/b<3$, where $a$ and $b$ are the semiaxes of the spheroid) the virial expansion appears to be convergent in both the isotropic and nematic phases up to high density. For more aspherical particles, the predictions varied in a more erratic fashion upon increasing the number of virials. The use of a finite angular basis set to represent the orientationally dependent virial coefficients could be a contributory cause to this.

A detailed stochastic model of single-gene autoregulation is established and its solutions are explored when mRNA dynamics is fast compared with protein dynamics and in the opposite regime. The model includes all the sources of randomness that are intrinsic to the autoregulation process and it considers both transcriptional and post-transcriptional regulation. The time-scale separation allows the derivation of analytic expressions for the equilibrium distributions of protein and mRNA. These distributions are generally well described in the continuous approximation, which is used to discuss the qualitative features of the protein equilibrium distributions as a function of the biological parameters in the fast mRNA regime. The performance of the time-scale approximation is assessed by comparison with simulations of the full stochastic system, and a good quantitative agreement is found for a wide range of parameter values. We show that either unimodal or bimodal equilibrium protein distributions can arise, and we discuss the autoregulation mechanisms associated with bimodality.

We investigate hydrodynamic interactions between two three-sphere swimmers analytically and numerically. Hydrodynamic forces exerted on the swimmers as well as the translational and angular velocities of them are obtained in the far field regime. We demonstrate that the active term of the translational velocity is along the intrinsic direction of swimming $\left(\mathbf{n}\right)$ and has no component along the direction of relative positions of swimmers $\left(\widehat{\mathbf{r}}\right)$ as reported in previous papers. Using numerical simulations we investigate the long-time swimming paths of swimmers in two general situations of swimming in the same and opposite directions. The former reveals four swimming states for symmetric swimmers—attractive, repulsive, parallel, and oscillatory—and only three swimming states for asymmetric swimmers—attractive, repulsive, and contracting-oscillatory, confirming that the expanding-oscillatory state reported in previous papers is not stable. The latter shows that there are rotative bound states in hydrodynamic scattering of the swimmers.

We derive the critical line of the O($n$) loop model on the martini lattice as a function of the loop weight $n$ basing on the critical points on the honeycomb lattice conjectured by Nienhuis [Phys. Rev. Lett. 49, 1062 (1982)]. In the limit $n\to 0$ we prove the connective constant $\mu =1.7505645579\cdots$ of self-avoiding walks on the martini lattice. A finite-size scaling analysis based on transfer matrix calculations is also performed. The numerical results coincide with the theoretical predictions with a very high accuracy. Using similar numerical methods, we also study the O($n$) loop model on the 3-12 lattice. We obtain similarly precise agreement with the critical points given by Batchelor [J. Stat. Phys. 92, 1203 (1998)].

Maximum-likelihood exponent maps have been studied as a technique to increase the understanding and improve the fit of power-law exponents to experimental and numerical simulation data, especially when they exhibit both upper and lower cutoffs. The use of the technique is tested by analyzing seismological data, acoustic emission data, and avalanches in numerical simulations of the three-dimensional random field Ising model. In the different examples we discuss the nature of the deviations observed in the exponent maps and some relevant conclusions are drawn for the physics behind each phenomenon.

We study the properties of complex networks embedded in a Euclidean space of communicability distances. The communicability distance between two nodes is defined as the difference between the weighted sum of walks self-returning to the nodes and the weighted sum of walks going from one node to the other. We give some indications that the communicability distance identifies the least crowded routes in networks where simultaneous submission of packages is taking place. We define an index $Q$ based on communicability and shortest path distances, which allows reinterpreting the “small-world” phenomenon as the region of minimum $Q$ in the Watts-Strogatz model. It also allows the classification and analysis of networks with different efficiency of spatial uses. Consequently, the communicability distance displays unique features for the analysis of complex networks in different scenarios.

Social media have provided plentiful evidence of their capacity for information diffusion. Fads and rumors but also social unrest and riots travel fast and affect large fractions of the population participating in online social networks (OSNs). This has spurred much research regarding the mechanisms that underlie social contagion, and also who (if any) can unleash system-wide information dissemination. Access to real data, both regarding topology—the network of friendships—and dynamics—the actual way in which OSNs users interact, is crucial to decipher how the former facilitates the latter's success, understood as efficiency in information spreading. With the quantitative analysis that stems from complex network theory, we discuss who (and why) has privileged spreading capabilities when it comes to information diffusion. This is done considering the evolution of an episode of political protest which took place in Spain, spanning one month in 2011.

Sequence motifs are words of nucleotides in DNA with biological functions, e.g., gene regulation. Identification of such words proceeds through rejection of Markov models on the expected motif frequency along the genome. Additional biological information can be extracted from the correlation structure among patterns of motif occurrences. In this paper a log-linear multivariate intensity Poisson model is estimated via expectation maximization on a set of motifs along the genome of E. coli K12. The proposed approach allows for excitatory as well as inhibitory interactions among motifs and between motifs and other genomic features like gene occurrences. Our findings confirm previous stylized facts about such types of interactions and shed new light on genome-maintenance functions of some particular motifs. We expect these methods to be applicable to a wider set of genomic features.

The four-fifths law for third-order longitudinal moments is examined, using direct numerical simulation (DNS) data on three-dimensional (3D) forced incompressible magnetohydrodynamic (MHD) turbulence without a uniformly imposed magnetic field in a periodic box. The magnetic Prandtl number is set to one, and the number of grid points is ${512}^3$. A generalized Kármán-Howarth-Kolmogorov equation for second-order velocity moments in isotropic MHD turbulence is extended to anisotropic MHD turbulence by means of a spherical average over the direction of $\mathit{r}$. Here, $\mathit{r}$ is a separation vector. The viscous, forcing, anisotropic and nonstationary terms in the generalized equation are quantified. It is found that the influence of the anisotropic terms on the four-fifths law is negligible at small scales, compared to that of the viscous term. However, the influence of the directional anisotropy, which is measured by the departure of the third-order moments in a particular direction of $\mathit{r}$ from the spherically averaged ones, on the four-fifths law is suggested to be substantial, at least in the case studied here.

The influence of a magnetic field on the dynamics of the flow of a ferrofluid in the gap between two concentric, independently rotating cylinders is investigated numerically. The Navier-Stokes equations are solved using a hybrid finite difference and Galerkin method. We show that the frequently used assumption that the internal magnetic field within a ferrofluid is equal to the external applied field is only a leading-order approximation. By accounting for the ferrofluid's magnetic susceptibility, we show that a uniform externally imposed magnetic field is modified by the presence of the ferrofluid within the annulus. The modification to the magnetic field has an $r^{-2}$ radial dependence and a magnitude that scales with the susceptibility. For ferrofluids typically used in laboratory experiments of the type simulated in this paper, the modification to the imposed magnetic field can be substantial. This has significant consequences on the structure and stability of the basic states, as well as on the bifurcating solutions.

A stationary principle is developed for dynamical systems by formulating the concept of mixed convolved action, which is written in terms of displacement and force variables, using temporal convolutions and fractional derivatives. The classical canonical single-degree-of-freedom dynamical system is considered as an initial application. With this new formulation, a single real scalar functional provides the governing differential equations, along with all the pertinent initial conditions, as the Euler-Lagrange equations emanating from the stationarity of the mixed convolved action. Both conservative and non-conservative processes can be considered within a common framework, thus resolving a long-standing limitation of variational approaches for dynamical systems. Several results in fractional calculus also are developed.

Langevin dynamics simulations are used to study the effect of shear on a two-dimensional colloidal crystal (with implicit solvent) confined by structured parallel walls. When walls are sheared very slowly, only two or three crystalline layers next to the walls move along with them, while the inner layers of the crystal are only slightly tilted. At higher shear velocities, this inner part of the crystal breaks into several pieces with different orientations. The velocity profile across the slit is reminiscent of shear banding in flowing soft materials, where liquid and solid regions coexist; the difference, however, is that in the latter case the solid regions are glassy while here they are crystalline. At even higher shear velocities, the effect of the shearing becomes smaller again. Also the effective temperature near the walls (deduced from the velocity distributions of the particles) decreases again when the wall velocity gets very large. When the walls are placed closer together, thereby introducing an incommensurability between the periodicity of the confined crystal and the walls, a structure containing a soliton staircase arises in simulations without shear. Introducing shear increases the disorder in these systems until no solitons are visible anymore. Instead, similar structures like in the case without mismatch result. At high shear rates, configurations where the incommensurability of the crystalline structure is compensated by the creation of holes become relevant.

Nematic molecules confined in an asymmetric π cell and subjected to strong electric fields exhibit textural distortions involving nematic order variations, described by the Landau-de Gennes Q-tensor theory. We investigated the evolution of order variations as function of the applied electric pulse amplitude and of the nematic surface pretilt anchoring angles by implementing a Q-tensor model with a moving mesh finite element method. The proposed technique is able to clearly distinguish the bulk and the surface order reconstruction which occur in the cell.

We consider the main transition in single-component membranes using computer simulations of the Pink model [D. A. Pink et al., Biochemistry 19, 349 (1980)]. We first show that the accepted parameters of the Pink model yield a main transition temperature that is systematically below experimental values. This resolves an issue that was first pointed out by Corvera and co-workers [Phys. Rev. E 47, 696 (1993)]. In order to yield the correct transition temperature, the strength of the van der Waals coupling in the Pink model must be increased; by using finite-size scaling, a set of optimal values is proposed. We also provide finite-size scaling evidence that the Pink model belongs to the universality class of the two-dimensional Ising model. This finding holds irrespective of the number of conformational states. Finally, we address the main transition in the presence of quenched disorder, which may arise in situations where the membrane is deposited on a rough support. In this case, we observe a stable multidomain structure of gel and fluid domains, and the absence of a sharp transition in the thermodynamic limit.

The World Trade Web (WTW), which models the international transactions among countries, is a fundamental tool for studying the economics of trade flows, their evolution over time, and their implications for a number of phenomena, including the propagation of economic shocks among countries. In this respect, the possible existence of communities is a key point, because it would imply that countries are organized in groups of preferential partners. In this paper, we use four approaches to analyze communities in the WTW between 1962 and 2008, based, respectively, on modularity optimization, cluster analysis, stability functions, and persistence probabilities. Overall, the four methods agree in finding no evidence of significant partitions. A few weak communities emerge from the analysis, but they do not represent secluded groups of countries, as intercommunity linkages are also strong, supporting the view of a truly globalized trading system.

In this paper, we investigated a density-dependent reaction-diffusion equation, $u_t={\left(u^m\right)}_{xx}+u-u^m$. This equation is known as the extension of the Fisher or Kolmogoroff-Petrovsky-Piscounoff equation, which is widely used in population dynamics, combustion theory, and plasma physics. By employing a suitable transformation, this equation was mapped to the anomalous diffusion equation where the nonlinear reaction term was eliminated. Due to its simpler form, some exact self-similar solutions with compact support have been obtained. The solutions, evolving from an initial state, converge to the usual traveling wave at a certain transition time. Hence, the connection between the self-similar solution and the traveling wave solution is quite clear from these results. Moreover, the solutions were found in a manner that propagates either to the right or to the left. Furthermore, the two solutions form a symmetric solution, expanding in both directions. Applications to spatiotemporal pattern formation in biological populations is discussed.