A study is presented on the effects of smooth nanoparticles on the structure and elastic moduli of a polymer matrix. Structural changes between the unfilled polymer matrix and the nanocomposite give rise to the formation of a glassy layer that surrounds the nanoparticles. Results for the effects of particle size and concentration on the local and overall mechanical properties of the polymer are consistent with experimental macroscopic observations. At the molecular level, it is found that dispersed, attractive nanoparticles alter the nonaffine displacement fields that arise in the polymer glass upon deformation, thereby rendering the nanocomposite glass less fragile.

In this paper we study a simple cascading process in a structured heterogeneous population, namely, a network composed of two loosely coupled communities. We demonstrate that under certain conditions the cascading dynamics in such a network has a two-tiered structure that characterizes activity spreading at different rates in the communities. We study the dynamics of the model using both simulations and an analytical approach based on annealed approximation and obtain good agreement between the two. Our results suggest that network modularity might have implications in various applications, such as epidemiology and viral marketing.

A variational method is developed to describe the dynamics of a Bose-Einstein condensate (BEC) trapped in an applied external potential consisting of both a harmonic and periodic component. Using this variational method, the BEC dynamics is shown to be well approximated by four coupled nonlinear differential equations, which describe the fundamental interactions in the system arising from the interplay of amplitude (width), chirp, center position, and center frequency. The simplified analytic theory allows for an efficient and convenient method for characterizing the experimental BEC behavior when localized condensates are generated. It further gives the critical strength ratio of harmonic to periodic potential necessary to support multiple stable lattice sites for the condensate and demonstrates that there can be an underlying chaotic behavior in the condensate system.

The phenomenon of length-scale competition in soliton-bearing equations perturbed by spatially dependent terms [A. Sánchez and A. R. Bishop, SIAM Rev. 40, 579 (1998)] is analyzed from a general viewpoint. We show that the perturbation gives rise to an effective potential for solitons, which consists of wells and barriers. We calculate the effect of these potential features on the solitons, establishing a direct relationship between the maxima, minima, and curvature of the potential with soliton deformations. When the typical wavelength of the perturbation is of the order of the soliton width, these deformations are seen to correspond to the excitation of shape modes and can lead to the dissipation of the soliton kinetic energy and, further, to the impossibility of soliton propagation. Thus, we demonstrate that the mechanism for length-scale competition is related to changes in the dynamics of the internal modes. We study different examples where the perturbation is introduced parametrically and nonparametrically to make it clear that our results apply to a wide class of equations.

We develop a method of solution for recently found integrable system of the reduced Maxwell-Bloch equations with two components of polarization and with an anisotropic dipole momentum by using the appropriate modification of the inverse scattering transform. The method is based on solution of the Riemann-Hilbert problem with taking into account symmetry properties of corresponding fundamental solutions. We show that these symmetries lead to some particular forms of the inverse scattering transform equations which may be used for finding as soliton-type as radiation-type solutions.

A method is proposed to construct an equilibrium density distribution function in the simulation of compressible flows at high Mach number by the lattice-Boltzmann method. In this method, the conventional Maxwellian distribution function is replaced by a circular function which is very simple and satisfies all needed statistical relations to recover the compressible Navier-Stokes equations. The circular function is then distributed to the lattice velocity directions by Lagrangian interpolation in such a way that all the needed statistical relations are exactly satisfied when the integral in the phase space is replaced by the summation in the context of the lattice-Boltzmann (LB) method. In this framework, the equilibrium distribution functions and the associated lattice velocity model can be derived naturally without assuming specific forms. Two LB models with adjustable specific heat ratio, respectively, for one-dimensional (1D) and two-dimensional (2D) compressible flows are shown in the paper. Some test cases of compressible flows with strong shock waves are simulated to validate the present approach. Excellent results are obtained. Note that in the simulation, the total variation diminishing (TVD) scheme was used to capture the discontinuity with coarse mesh.

We present a generalization of the often-used Crank-Nicolson (CN) method of obtaining numerical solutions of the time-dependent Schrödinger equation. The generalization yields numerical solutions accurate to order ${\left(\Delta x\right)}^{2r-1}$ in space and ${\left(\Delta t\right)}^{2M}$ in time for any positive integers $r$ and $M$, while CN employ $r=M=1$. We note dramatic improvement in the attainable precision (circa ten or greater orders of magnitude) along with several orders of magnitude reduction of computational time. The improved method is shown to lead to feasible studies of coherent-state oscillations with additional short-range interactions, wave-packet scattering, and long-time studies of decaying systems.

We analyze the structure of fluctuations near critical points and spinodals in mean-field and near-mean-field systems. Unlike systems that are non-mean-field, for which a fluctuation can be represented by a single cluster in a properly chosen percolation model, a fluctuation in mean-field and near-mean-field systems consists of a large number of clusters, which we term fundamental clusters. The structure of the latter and the way that they form fluctuations has important physical consequences for phenomena as diverse as nucleation in supercooled liquids, spinodal decomposition and continuous ordering, and the statistical distribution of earthquakes. The effects due to the fundamental clusters implies that they are physical objects and not only mathematical constructs.

Using videomicroscopy data of a two-dimensional colloidal system the bond-order correlation function $G_6$ is calculated and used to determine both the orientational correlation length ${\xi }_6$ in the liquid phase and the modulus of orientational stiffness, Frank’s constant $F_A$, in the hexatic phase. The latter is an anisotropic fluid phase between the crystalline and the isotropic liquid phase. $F_A$ is found to be finite within the hexatic phase, takes the value $72∕\pi$ at the $\text{hexatic}\leftrightarrow \text{isotropic}$ liquid phase transition, and diverges at the $\text{hexatic}\leftrightarrow \text{crystal}$ transition as predicted by the Kosterlitz-Thouless-Halperin-Nelson-Young theory. This is a quantitative test of the mechanism of breaking the orientational symmetry by disclination unbinding.

Within the Landau–de Gennes phenomenological theory, we study the influence of an applied electric field with average strength $E_a$ on the position of a nematic line defect with topological charge $M=\pm \frac{1}{2}$ in a hybrid cell. We explore the biaxial structure of the defect core and we describe its expulsion from the cell upon increasing $E_a$. We show that prior to the expulsion the defect core displays dramatic changes for strong enough surface anchorings. At a critical value of $E_a$, the core broadens and merges into a surface layer with a large biaxiality. This transition corresponds to the reconstruction of the nematic order already observed in the bulk in response to an applied electric field. A similar order reconstruction could take place even in the absence of defects, but at a higher threshold.

One of the most intriguing features of the immune system is regulation: a limited response when perturbed repeatedly. We propose a minimal network model for immune regulation in a lymphocyte network containing two types of elements: B lymphocytes and ligands that bind to their receptors. Effective interactions between B cells, mediated by other components of the immune system can be excitatory or inhibitory. In our model, B cell clones and ligand species are represented by nodes, and interactions by links. We expect that, as in many complex systems, the connectivity distribution is broad, motivating study of the model on a scale-free network; for comparison we study the same dynamics on a random graph. We characterize the dynamics of the model and its response to perturbations. Our model reproduces several key features of immune system dynamics: regulation (saturation of response), and more rapid response upon repeated perturbation with the same agents. Our results suggest that a scale-free network of interactions contributes to the regulation and dynamics of the immune system.

Motivated by recent studies of models of particle and heat quantum pumps, we study similar simple classical models and examine the possibility of heat pumping. Unlike many of the usual ratchet models of molecular engines, the models we study do not have particle transport. We consider a two-spin system and a coupled oscillator system which exchange heat with multiple heat reservoirs and which are acted upon by periodic forces. The simplicity of our models allows accurate numerical and exact solutions and unambiguous interpretation of results. We demonstrate that while both our models seem to be built on similar principles, one is able to function as a heat pump (or engine) while the other is not.

We present the first measurements of the temperature dependent extension of single stranded DNA. At forces between 2 and $10\mathrm{pN}$ the extension increases with temperature. This increase in extension is consistent with the disruption of hairpins, and a simple theory that includes hairpin formation shows good agreement with the data at these low forces. In contrast, at forces above $10\mathrm{pN}$ and temperatures higher than $40°\mathrm{C}$, the extension decreases rapidly with temperature in a manner not consistent with predictions.

We investigate the critical behavior of a model with two coupled critical densities, one of which is diffusive. The model simulates the propagation of an epidemic process in a population, which uses the underlying lattice to leave a track of the recent disease history. We determine the critical density of the population above which the system reaches an active stationary state with a finite density of active particles. We also perform a scaling analysis to determine the order parameter, the correlation length, and critical relaxation exponents. We show that the model does not belong to the usual directed percolation universality class and is compatible with the class of directed percolation with diffusive and conserved fields.

We study the statistical properties of wave scattering in a disordered waveguide. The statistical properties of a “building block” of length $\delta L$ are derived from a potential model and used to find the evolution with length of the expectation value of physical quantities. In the potential model the scattering units consist of thin potential slices, idealized as delta slices, perpendicular to the longitudinal direction of the waveguide; the variation of the potential in the transverse direction may be arbitrary. The sets of parameters defining a given slice are taken to be statistically independent from those of any other slice and identically distributed. In the dense-weak-scattering limit, in which the potential slices are very weak and their linear density is very large, so that the resulting mean free paths are fixed, the corresponding statistical properties of the full waveguide depend only on the mean free paths and on no other property of the slice distribution. The universality that arises demonstrates the existence of a generalized central-limit theorem. Our final result is a diffusion equation in the space of transfer matrices of our system, which describes the evolution with the length $L$ of the disordered waveguide of the transport properties of interest. In contrast to earlier publications, in the present analysis the energy of the incident particle is fully taken into account. For one propagating mode, $N=1$, we have been able to solve the diffusion equation for a number of particular observables, and the solution is in excellent agreement with the results of microscopic calculations. In general, we have not succeeded in finding a solution of the diffusion equation. We have thus developed a numerical simulation, to be called “random walk in the transfer matrix space,” in which the universal statistical properties of a “building block” are first implemented numerically, and then the various building blocks are combined to find the statistical properties of the full waveguide. The reported results thus obtained (in which use was made of a “short-wavelength approximation”) are in very good agreement with those arising from truly microscopic calculations, for both bulk and surface disorder. Since the paper has a clear pedagogical aim, we have included, for the benefit of experts and nonexperts, a number of appendixes that contain the more involved calculations.

We present constant-pressure Monte Carlo simulations of nuclear magnetic resonance (NMR) spectral parameters, nuclear magnetic shielding relative to the free atom as well as nuclear quadrupole coupling, for atomic xenon dissolved in a model thermotropic liquid crystal. The solvent is described by Gay-Berne (GB) molecules with parametrization $\kappa =4.4$, ${\kappa }^{\prime }=20.0$, and $\mu =\nu =1$. The reduced pressure of $P^{\star }=2.0$ is used. Previous simulations of a pure GB system with this parametrization have shown that upon lowering the temperature, the model exhibits isotropic, nematic, smectic-$A$, and smectic-$B$/molecular crystal phases. We introduce spherical xenon solutes and adjust the energy and length scales of the $\mathrm{GB}\text{−}\mathrm{Xe}$ interaction to those of the GB-GB interaction. This is done through first principles quantum chemical calculations carried out for a dimer of model mesogens as well as the mesogen-xenon complex. We preparametrize quantum chemically the Xe nuclear shielding and quadrupole coupling tensors when interacting with the model mesogen, and use the parametrization in a pairwise additive fashion in the analysis of the simulation. We present the temperature evolution of ${}^{129∕131}\mathrm{Xe}$ shielding and ${}^{131}\mathrm{Xe}$ quadrupole coupling in the different phases of the GB model. From the simulations, separate isotropic and anisotropic contributions to the experimentally available total shielding can be obtained. At the experimentally relevant concentration, the presence of the xenon atoms does not significantly affect the phase behavior as compared to the pure GB model. The simulations reproduce many of the characteristic experimental features of Xe NMR in real thermotropic LCs: Discontinuity in the value or trends of the shielding and quadrupole coupling at the nematic-isotropic and smectic-$A$–nematic phase transitions, nonlinear shift evolution in the nematic phase reflecting the behavior of the orientational order parameter, and decreasing shift in the smectic-$A$ phase. The last observation is due to the preference of the xenon solutes to occupy the interlayer space where the density of the medium is reduced as compared to the layers. There are systematic deviations, however, in the magnitude of the shielding and its discontinuities, as well as the distribution of the solutes in the translationally ordered smectic-$A$ phase, between the simulation and experiment. These deficiencies are believed to result from the lack of flexibility of the GB model.

Transport of molecular motors, stimulated by interactions with specific links between consecutive binding sites (called “bridges”), is investigated theoretically by analyzing discrete-state stochastic “burnt-bridge” models. When an unbiased diffusing particle crosses the bridge, the link can be destroyed (“burned”) with a probability $p$, creating a biased directed motion for the particle. It is shown that for probability of burning $p=1$ the system can be mapped into a one-dimensional single-particle hopping model along the periodic infinite lattice that allows one to calculate exactly all dynamic properties. For the general case of $p<1$ a theoretical method is developed and dynamic properties are computed explicitly. Discrete-time and continuous-time dynamics for periodic distribution of bridges and different burning dynamics are analyzed and compared. Analytical predictions are supported by extensive Monte Carlo computer simulations. Theoretical results are applied for analysis of the experiments on collagenase motor proteins.

Transversely finite sound beams traveling in fluid media with negligible viscous damping and scattering have seven quantities constant along the length of the beam. The simplest of these is the cycle-averaged momentum per unit length of the beam. The seven constant quantities are proved to be invariants from conservation laws. An angular momentum flux density tensor is introduced, in the formulation of conservation of angular momentum. Examples of the invariants are given, for approximate solutions of the wave equation (Gaussian beams), and for a set of exact solutions (generalized Bessel beams).

We study a model of interacting random walkers that proposes a simple mechanism for the emergence of cooperation in a group of individuals. Each individual, represented by a Brownian particle, experiences an interaction produced by the local unbalance in the spatial distribution of the other individuals. This interaction results in a nonlinear velocity driving the particle trajectories in the direction of the nearest more crowded region; the competition among different aggregating centers generates nontrivial dynamical regimes. Our simulations show that for sufficiently low randomness the system evolves through a coalescence behavior characterized by clusters of particles growing with a power law in time. In addition, the typical scaling properties of the general theory of stochastic aggregation processes are verified.