We consider the nonlinear Dirac equations (NLDE’s) in $1+1$ dimension with scalar-scalar self interaction $\frac{g^2}{k+1}{\left(\overline{\Psi }\Psi \right)}^{k+1}$, as well as a vector-vector self interaction $\frac{g^2}{k+1}{\left(\overline{\Psi }{\gamma }_{\mu }\Psi \overline{\Psi }{\gamma }^{\mu }\Psi \right)}^{1/2\left(k+1\right)}$. We find the exact analytic form for solitary waves for arbitrary $k$ and find that they are a generalization of the exact solutions for the nonlinear Schrödinger equation (NLSE) and reduce to these solutions in a well defined nonrelativistic limit. We perform the nonrelativistic reduction and find the $1/2m$ correction to the NLSE, valid when $\left|\omega -m\right|⪡2m$, where $\omega$ is the frequency of the solitary wave in the rest frame. We discuss the stability and blowup of solitary waves assuming the modified NLSE is valid and find that they should be stable for $k<2$.

We numerically examine dynamical heterogeneity in a highly supercooled three-dimensional liquid via molecular-dynamics simulations. To define the local dynamics, we consider two time intervals: ${\tau }_{\alpha }$ and ${\tau }_{\text{ngp}}$. ${\tau }_{\alpha }$ is the $\alpha$ relaxation time, and ${\tau }_{\text{ngp}}$ is the time at which non-Gaussian parameter of the Van Hove self-correlation function is maximized. We determine the lifetimes of the heterogeneous dynamics in these two different time intervals, ${\tau }_{\text{hetero}}\left({\tau }_{\alpha }\right)$ and ${\tau }_{\text{hetero}}\left({\tau }_{\text{ngp}}\right)$, by calculating the time correlation function of the particle dynamics, i.e., the four-point correlation function. We find that the difference between ${\tau }_{\text{hetero}}\left({\tau }_{\alpha }\right)$ and ${\tau }_{\text{hetero}}\left({\tau }_{\text{ngp}}\right)$ increases with decreasing temperature. At low temperatures, ${\tau }_{\text{hetero}}\left({\tau }_{\alpha }\right)$ is considerably larger than ${\tau }_{\alpha }$, while ${\tau }_{\text{hetero}}\left({\tau }_{\text{ngp}}\right)$ remains comparable to ${\tau }_{\alpha }$. Thus, the lifetime of the heterogeneous dynamics depends strongly on the time interval.

A two-dimensional directed stochastic sandpile model is studied both numerically and analytically. One of the known analytical approaches is extended by considering general stochastic toppling rules. The probability density distribution for the first-passage time of stochastic process described by a nonlinear Langevin equation with power-law dependence of the diffusion coefficient is obtained. Large-scale Monte Carlo simulations are performed with the aim to analyze statistical properties of the avalanches, such as the asymmetry between the initial and final stages, scaling of voids and the width of the thickest branch. Comparison with random walks description is drawn and different plausible scenarios for the avalanche evolution and the scaling exponents are suggested.

The dispersion relations of bulk modes propagating within a hexagonal close-packed structure of noncohesive monodisperse spherical elastic beads are derived. The contacts are modeled by two springs with stiffnesses given by the Hertz-Mindlin theory, one for normal interactions and one for transverse interactions. The existence of the transverse interaction requires to take into account the rotational degrees of freedom of the beads in the analysis. This leads to the prediction of translational modes and, due to the rotational degrees of freedom, of rotational modes and coupled rotational and translational modes. The study of the dispersion relations in a direction of high symmetry allows to identify the different modes and the influence of the rotational degrees of freedom on the bulk mode propagation. The evaluated dispersion relations provide guidelines for the experimental observation of rotational modes. Opportunities for controlling the dispersion laws of the modes by an external loading on the granular structure are discussed.

Evolution of the energy landscape during physical aging of glassy materials can be understood from the frequency and strain dependence of the shear modulus but the nonstationary nature of these systems frustrates investigation of their instantaneous underlying properties. Using a series of time-dependent measurements we systematically reconstruct the frequency and strain dependence as a function of age for a repulsive colloidal glass undergoing structural arrest. In this manner, we are able to unambiguously observe the structural relaxation time, which increases exponentially with sample age at short times. The yield stress varies logarithmically with time in the arrested state, consistent with recent simulation results, whereas the yield strain is nearly constant in this regime. Strikingly, the frequency dependence at fixed times can be rescaled onto a master curve, implying a simple connection between the aging of the system and the change in the frequency dependent modulus.

In its common implementation, the mode-coupling theory of the glass transition predicts the time evolution of the intermediate scattering functions in viscous liquids on the sole basis of the structural information encoded in two-point density correlations. We provide a critical test of this property and show that the theory fails to describe the strong differences of dynamical behavior seen in two model liquids characterized by very similar pair-correlation functions. Because we use “exact” static information provided by numerical simulations, our results are a direct indication that some important information about the dynamics of viscous liquids is not captured by pair correlations and is thus not described by the mode-coupling theory, even in the temperature regime where the theory is usually applied.

We use the annealed formulation of complex networks to study the dynamical behavior of disease spreading on both static and adaptive networked systems. This unifying approach relies on the annealed adjacency matrix, representing one network ensemble, and allows to solve the dynamical evolution of the whole network ensemble all at once. Our results accurately reproduce those obtained by extensive numerical simulations showing a large improvement with respect to the usual heterogeneous mean-field formulation. Moreover, by means of the annealed formulation we derive a new heterogeneous mean-field formulation that correctly reproduces the epidemic dynamics.

We study a continuous quasi-two-dimensional order-disorder phase transition that occurs in a simple model of a material that is inhomogeneously strained due to the presence of dislocation lines. Performing Monte Carlo simulations of different system sizes and using finite size scaling, we measure critical exponents describing the transition of $\beta =0.18\pm 0.02$, $\gamma =1.0\pm 0.1$, and $\alpha =0.10\pm 0.02$. Comparable exponents have been reported in a variety of physical systems. These systems undergo a range of different types of phase transitions, including structural transitions, exciton percolation, and magnetic ordering. In particular, similar exponents have been found to describe the development of magnetic order at the onset of the pseudogap transition in high-temperature superconductors. Their common universal critical exponents suggest that the essential physics of the transition in all of these physical systems is the same as in our simple model. We argue that the nature of the transition in our model is related to surface transitions although our model has no free surface.

We present a study of the dynamics of confined suspensions whose dimensionality is intermediate between quasi-one-dimensional and quasi-two-dimensional (q2D) using microfluidic channels of various widths. The crossover between the two limiting behaviors is found to occur to different extent for different dynamic correlations between a pair of particles. In particular, the transverse coupling diffusion coefficient of particle pairs significantly deviates from its q2D form even in surprisingly wide channels.

Glass transition constitutes one of main problems of condensed matter physics and material engineering that remains unsolved. The common acceptance of the Vogel-Fulcher-Tammann (VFT) equation for portraying the primary relaxation time or shear viscosity indicated a possible phase transition, hidden below the glass transition temperature $\left(T_g\right)$. Recently Hecksher et al. [Nat. Phys. 4, 737 (2008)] delivered strong empirical arguments that VFT description lacks a direct experimental basis and thus theories not predicting a dynamic divergence should be focused on. We present clear evidence for a superiority of critical-like divergent equation $\tau \left(T\right)={\tau }_0{\left(T-T_C\right)}^{-\varphi }$ and $T_C<T_g$ for liquid crystalline (LC) glass formers and orientationally disordered crystals (ODIC). Such dependence was already known for spin-glasslike systems and the dynamical scaling model, although the latter was hardly explored so far. The pressure-related behavior is also discussed. Results obtained support arguments for the suggested direct link between critical phenomena and vitrification [Tanaka et al., Nat. Mater. 9, 324 (2010)]. LCs and ODICs may be considered as simple experimental model systems for the structural glass formers group.

Nucleation is considered for nonzero rate of change of the dimensionless barrier $B$, which is characterized by a finite, slowly varying “nonstationary index” $\mathcal{N}\propto -\left(dB/dt\right)$. In the standard adiabatic approximation it is assumed that after the start of nucleation an $\mathcal{N}$-independent nucleation flux is established instantaneously, with a quasi-steady-state (QSS) value determined by the current barrier $B\left(t\right)$. Those assumptions, however, can be justified only in the strict limit $\mathcal{N}\to 0$, and otherwise both transient nucleation at small times, and subsequent deviations from QSS are essential. Earlier results for the non-QSS transient flux are refined and generalized to account for arbitrary relations between the rates of the change of the barrier and of the critical size, and for a variable $\mathcal{N}\left(t\right)$. The $\mathcal{N}$-dependent transient distributions of growing nuclei and their numbers also are obtained. The treatment is mostly based on matched asymptotic (singular perturbation) analysis of the Becker-Döring equation (BDE), and involves comparison with exact numerics. General results are specified within the continuous Zeldovich-Frenkel approximation to BDE, with a large fixed critical size and a barrier which either increases $\left(\mathcal{N}<0\right)$ or decays $\left(\mathcal{N}>0\right)$ with time. In such cases growth can be described exactly, allowing to extend the nucleation solution to arbitrary sizes without additional approximations. Resulting distributions $f\left(r,t\right)$ are monotonic in size $r$ for $\mathcal{N}\ge 0$, with a diverging total number of particles $\rho$ as $t\to \infty$. For $\mathcal{N}<0$ distributions acquire an asymmetric bell shape with a finite $\rho$, which is exponentially small compared to ${\rho }^{QSS}$.

Experiments on the formation of spiraling hexagons (350–1000 nm in width) from a solution of nanoparticles are presented. Transmission electron microscopy images of the reaction products of chemically synthesized cadmium nanocrystals indicate that the birth of the hexagons proceeds without assistance from static screw or edge dislocations, that is, they spiral without constraints provided by an underlying substrate. Instead, the growth mechanism relies on a dynamical dislocation identified as a dense aggregate of small nanocrystals that intersects the spiraling hexagon at the crystal surface. This nanocrystal bundle, which we term the “feeder,” also appears to release nanocrystals into the spiral during the growth process.

It has recently been observed that on application of an alternating electric field, the B1 liquid-crystalline phase may be induced to form a switchable phase. This induced phase has been shown to have an almost thresholdless dielectric response. In this paper we examine this $E$-field-induced transition as a function of temperature both with and without a ferroelectric liquid-crystal dopant. Although the aim of this experiment was to enhance and stabilize the field-induced phase on addition of dopant, we find that the opposite is the case. The ferroelectric dopant actually increases the threshold $E$-fields required for transformation to the switching phase and at just $5\text{wt}\mathrm{\%}$ dopant a smectic-$A$ phase is formed. Addition of the dopant also acts against the low-field switching and significantly speeds relaxation back to the B1 phase on field removal. In addition we find that the field-induced phase experiences a slower crystallization. We use polarized optical microscopy, differential scanning calorimetry, and x-ray scattering experiments to characterize the physical properties of the mixtures.

Liquid-liquid phase separation leading to complex coacervation in a ternary system (oppositely charged polyion and macroion in a solvent) is discussed within the framework of a statistical thermodynamics model. The polyion and the macroion in the ternary system interact to form soluble aggregates (complexes) in the solvent, which undergoes liquid-liquid phase separation. Four necessary conditions are shown to drive the phase separation: (i) ${\left({\sigma }_{23}\right)}^{3}r/{\Phi }_{23c}\ge \left(\frac{64}{9{\alpha }^2}\right){\left({\chi }_{23}{\Phi }_3\right)}^2$, (ii) $r\ge {\left[\frac{64{\left({\chi }_{23}{\Phi }_3\right)}^2}{9{\alpha }^2{\sigma }_{23}^3}\right]}^{1/2}$, (iii) ${\chi }_{23}\ge \frac{\left(2{\chi }_{231}-1\right)}{{\Phi }_{23c}{\Phi }_3}$, and (iv) ${\left({\sigma }_{23}\right)}^2/\sqrt{I}\ge \frac{8}{3\alpha }\left(2{\chi }_{231}-1\right)$ (where ${\sigma }_{23}$ is the surface charge on the complex formed due to binding of the polyelectrolyte and macroion, ${\Phi }_{23c}$ is the critical volume fraction of the complex, ${\chi }_{23}$ is the Flory interaction parameter between polyelectrolyte and macroion, ${\chi }_{231}$ is the same between solvent and the complex, ${\Phi }_3$ is the volume fraction of the macroions, $I$ is the ionic strength of the solution, $\alpha$ is electrostatic interaction parameter and $r$ is typically of the order of molecular weight of the polyions). It has been shown that coacervation always requires a hydrated medium. In the case of a colloidal macroion and polyelectrolyte coacervation, molecular weight of polyelectrolyte must satisfy the condition $r\ge {10}^3\text{Da}$ to exhibit liquid-liquid phase separation. This model has been successfully applied to study the coacervation phenomenon observed in aqueous Laponite (macroion)–gelatin (polyion) system where it was found that the coacervate volume fraction, $\delta {\Phi }_{23}\sim {\chi }_{231}^2$ (where $\delta {\Phi }_{23}$ is the volume fraction of coacervates formed during phase separation). The free energy and entropy of this process have been evaluated, and a free-energy landscape has been drawn for this system that maps the pathway leading to phase separation.

In many applications one is interested in finding a simplified model which captures the essential dynamical behavior of a real life process. If the essential dynamics can be assumed to be (approximately) memoryless then a reasonable choice for a model is a Markov model whose parameters are estimated by means of Bayesian inference from an observed time series. We propose an efficient Monte Carlo Markov chain framework to assess the uncertainty of the Markov model and related observables. The derived Gibbs sampler allows for sampling distributions of transition matrices subject to reversibility and/or sparsity constraints. The performance of the suggested sampling scheme is demonstrated and discussed for a variety of model examples. The uncertainty analysis of functions of the Markov model under investigation is discussed in application to the identification of conformations of the trialanine molecule via Robust Perron Cluster Analysis $\left(\text{PCCA}+\right)$.

We study a modified model of the Kardar-Paris-Zhang equation with quenched disorder, in which the driving force decreases as the interface rises up. A critical state is self-organized, and the anomalous scaling law with roughness exponent $\alpha \sim 0.63$ is numerically obtained.

We develop a geometric framework to study the structure and function of complex networks. We assume that hyperbolic geometry underlies these networks, and we show that with this assumption, heterogeneous degree distributions and strong clustering in complex networks emerge naturally as simple reflections of the negative curvature and metric property of the underlying hyperbolic geometry. Conversely, we show that if a network has some metric structure, and if the network degree distribution is heterogeneous, then the network has an effective hyperbolic geometry underneath. We then establish a mapping between our geometric framework and statistical mechanics of complex networks. This mapping interprets edges in a network as noninteracting fermions whose energies are hyperbolic distances between nodes, while the auxiliary fields coupled to edges are linear functions of these energies or distances. The geometric network ensemble subsumes the standard configuration model and classical random graphs as two limiting cases with degenerate geometric structures. Finally, we show that targeted transport processes without global topology knowledge, made possible by our geometric framework, are maximally efficient, according to all efficiency measures, in networks with strongest heterogeneity and clustering, and that this efficiency is remarkably robust with respect to even catastrophic disturbances and damages to the network structure.