Eutectic dendrites forming in a model ternary system have been studied using the phase-field theory. The eutectic and one-phase dendrites have similar forms, and the tip radius scales with the interface free energy as for one-phase dendrites. The steady-state eutectic patterns, appearing on these two-phase dendrites, include concentric rings and single- to multiarm spirals from which the fluctuations choose; a stochastic phenomenon characterized by a peaked probability distribution. The number of spiral arms correlates with the tip radius and the kinetic anisotropy.

The spatial structure and statistical properties of polymer interfaces grown by vapor deposition polymerization have been studied in the context of fluid turbulence. The extended self-similarity present in the correlation functions of the polymer interface uncovers two types of multiscaling for different spatial scales. The relative $q$th-order scaling exponents and the probability distribution functions of the height gradient display intermittent height fluctuations, which lead to spatial multiscaling analogous to the velocity fluctuations in turbulent fluid.

DNA conformation plays an important role in a host of cellular processes. Despite the central importance of DNA conformation, there is not yet a general-purpose calculator for conformational statistics that is designed for the scientific community. Here we describe a public tool we developed for calculating an important class of conformational statistics: the end-to-end probability density of finding a locus of the DNA polymer at a given displacement and orientation relative to a second locus on the same polymer. As a demonstration, we propose a cyclization experiment and use our calculator to show that this experiment could measure the energy of DNA bending as a direct function of bend angle in the poorly understood high-bending regime. Our tool is available as both an online calculator and a downloadable program at http://mtshasta.phys.washington.edu/wormulator/.

The metapopulation framework is adopted in a wide array of disciplines to describe systems of well separated yet connected subpopulations. The subgroups or patches are often represented as nodes in a network whose links represent the migration routes among them. The connections have been so far mostly considered as static, but in general evolve in time. Here we address this case by investigating simple contagion processes on time-varying metapopulation networks. We focus on the SIR process and determine analytically the mobility threshold for the onset of an epidemic spreading in the framework of activity-driven network models. We find profound differences from the case of static networks. The threshold is entirely described by the dynamical parameters defining the average number of instantaneously migrating individuals and does not depend on the properties of the static network representation. Remarkably, the diffusion and contagion processes are slower in time-varying graphs than in their aggregated static counterparts, the mobility threshold being even two orders of magnitude larger in the first case. The presented results confirm the importance of considering the time-varying nature of complex networks.

By applying a stepwise overlimiting voltage to a nanoslot system in equilibrium, it is possible to follow the time evolution of the electroconvective instability vortex array via the depletion dynamics or, alternatively, by following dielectrophoretically trapped particles at the stagnation points of each of the hydrodynamic vortex pairs. Particles are advected to the stagnation point by the hydrodynamics, where they are trapped by a short-range dielectrophoretic force. It is experimentally confirmed that the wavelength selection process occurs at the diffusive time scale and that the wavelength selection mechanism, started by the Rubinstein and Zaltzman electroconvective instability is eventually determined by the system lateral geometry and dictated by Dukhin's electro-osmosis of the second kind. The steady-state case was numerically studied by solving the fully coupled electroconvective problem, confirming that the vortex stagnation point is indeed of a converging type. The particle's planar (two-dimensional) equations of motion are solved after adding a dielectrophoretic force that accounts for quasi-three-dimensional effects. It is shown that this force can account for trapping at the nanoslot interface.

Including the effect of thermal fluctuations in traditional computational fluid dynamics requires developing numerical techniques for solving the stochastic partial differential equations of fluctuating hydrodynamics. These Langevin equations possess a special fluctuation-dissipation structure that needs to be preserved by spatio-temporal discretizations in order for the computed solution to reproduce the correct long-time behavior. In particular, numerical solutions should approximate the Gibbs-Boltzmann equilibrium distribution, and ideally this will hold even for large time step sizes. We describe finite-volume spatial discretizations for the fluctuating Burgers and fluctuating incompressible Navier-Stokes equations that obey a discrete fluctuation-dissipation balance principle just like the continuum equations. We develop implicit-explicit predictor-corrector temporal integrators for the resulting stochastic method-of-lines discretization. These stochastic Runge-Kutta schemes treat diffusion implicitly and advection explicitly, are weakly second-order accurate for additive noise for small time steps, and give a good approximation to the equilibrium distribution even for very strong fluctuations. Numerical results demonstrate that a midpoint predictor-corrector scheme is very robust over a broad range of time step sizes.

We study the quasiperiodically driven Hénon and Standard maps in the weak dissipative limit. In the absence of forcing, there are a large number of coexisting periodic attractors. Although chaotic attractors can also be found, these typically have vanishingly small basins of attraction. Quasiperiodic forcing reduces the multistability in the system, and as the bifurcation parameter is varied, strange nonchaotic attractors (SNAs) are created. The attractor basin for SNAs appears to be the largest among those of all coexisting attractors at such a transition.

Absorbing phase transitions in the conserved threshold transfer process (CTTP) on diluted lattices, i.e., on infinite percolation networks, were studied in two dimensions by Monte Carlo simulations. Lattice dilution was found to be irrelevant to the critical behavior for the concentration of diluted sites $x$ far less than the critical concentration $x_c$ ($=1-p_c$, with $p_c$ being the percolation threshold), whereas for $x$ close to $x_c$ the critical exponents slightly deviated from those of the pure CTTP and new critical exponents were established at $x_c$. The results for $x<x_c$ were consistent with the recent results for the conserved lattice gas (CLG) model, whereas the results for $x=x_c$ were different from those of the CLG model, for which nonuniversal power-law behavior was observed [Lee, Phys. Rev. E 84, 041123 (2011)]. On the other hand, the critical exponents of the CTTP on a percolation backbone were similar to those on an infinite network and similar to those of the CLG model on a backbone network. The diluted CTTP in three and four dimensions was also investigated, and the critical exponents were similar to those in two dimensions; i.e., the critical exponents were independent of or weakly dependent on the dimensionality. A comment on the recent claim of the absence of a Manna class was also presented.

We elucidate the effects of chiral quenched disorder on the scaling properties of pure systems by considering a reduced model that is a variant of the quenched disordered cubic anisotropic $\text{O}\left(N\right)$ model near its second-order phase transition. A generic short-ranged Gaussian disorder distribution is considered. For distributions not invariant under spatial inversion (hence chiral), the scaling exponents are found to depend continuously on a model parameter that describes the extent of inversion symmetry breaking. Experimental and phenomenological implications of our results are discussed.

We investigate the critical properties of the $d=3$ random-field Ising model with a Gaussian field distribution at zero temperature. By implementing suitable graph-theoretical algorithms, we perform a large-scale numerical simulation of the model for a vast range of values of the disorder strength $h$ and system sizes $\mathcal{V}=L\times L\times L$, with $L\le 156$. Using the sample-to-sample fluctuations of various quantities and proper finite-size scaling techniques we estimate with high accuracy the critical disorder strength $h_{\mathrm{c}}$ and the correlation length exponent $\nu$. Additional simulations in the area of the estimated critical-field strength and relevant scaling analysis of the bond energy suggest bounds for the specific heat critical exponent $\alpha$ and the violation of the hyperscaling exponent $\theta$. Finally, a data collapse analysis of the order parameter and disconnected susceptibility provides accurate estimates for the critical exponent ratios $\beta /\nu$ and $\overline{\gamma }/\nu$, respectively.

Relativistic transport phenomena are important from both a theoretical and practical point of view. Accordingly, hydrodynamics of relativistic gas has been extensively studied theoretically. Here we introduce a three-dimensional canonical model of hard-sphere relativistic gas which allows us to impose appropriate temperature gradient along a given direction maintaining the system in a nonequilibrium steady state. We use such a numerical laboratory to study the appropriateness of the so-called first order (Chapman-Enskog) relativistic hydrodynamics by calculating various transport coefficients. Our numerical results are consistent with predictions of such a theory for a wide range of temperatures. Our results are somewhat surprising since such linear theories are not consistent with the fundamental assumption of the special theory of relativity ($v\le c$). We therefore seek to explain such results by studying the appropriateness of diffusive transport in the relativistic gas, comparing our results with that of a classical gas. We find that the relativistic correction (constraint) in the hydrodynamic limit amounts to small negligible corrections, thus indicating the validity of the linear approximation in near equilibrium transport phenomena.

We study the critical behavior of a frustrated Blume-Capel (BC) antiferromagnet on a triangular lattice by Monte Carlo simulations. For a reduced single-ion anisotropy strength $-1.47\lesssim \Delta <0$, we find two phase transitions. The low-temperature phase is characterized by the antiferromagnetic long-range ordering (LRO) on two sublattices with the third one remaining in a nonmagnetic state. At higher temperatures, there is a critical region of the Berezinskii-Kosterlitz-Thouless (BKT) type with a power-law decaying spin-correlation function. For $-1.5\le \Delta \lesssim -1.47$, there is only one phase transition from the LRO to the paramagnetic region and the transition is of first order. The presence of the BKT phase in the current frustrated BC model is a new feature not observed in its nonfrustrated counterparts. The values of the decay exponent $\eta$ of the BKT phase corresponding to upper and lower temperatures appear to be consistent with the theoretical predictions for the six-state clock model.

The phase angle between the imposed ac electric field and the oscillations in particle height is the key parameter governing the sign of interparticle force during two-dimensional directed assembly. The phase angle depends on a number of experimental parameters, including the frequency of the electric field and dispersing electrolytes. The origin of electrolyte dependence in this phase angle has been a mystery for a decade. Electrolyte dependence arises from polarization of the particle's diffuse layer, which affects the dynamic electrophoretic mobility of the particle. A full description of the magnitude and phase angle of the dynamic electrophoretic mobility was incorporated into a nonlinear integro-differential equation of motion for a 5.7 μm diameter particle suspended in 0.15 mM KOH, KCl, NaHCO${}_3$, NH${}_4$OH, and NaOH at frequencies between 5 and 1000 Hz. Integration of the equation revealed that the phase angles for a particle in KOH, NH${}_4$OH, and NaOH were smaller than the phase angles calculated for a particle in KCl and NaHCO${}_3$, which is consistent with previously published experiments. Although the phase angles for each electrolyte are spread over only ∼1°, the results cluster around 90°, which is the crucial boundary between particle aggregation (>90°) and separation (<90°). A family of curves of the oscillation in particle height collapsed to a master curve when the amplitude of motion was scaled with the product of the dynamic electrophoretic mobility and electric field strength. These results constitute the first a priori prediction of electrolyte-dependent motion of a particle near an electrode during ac polarization.

We present a statistical mechanics analysis of the finite-size elasticity of model polymers, consisting of domains that can exhibit transitions between more than one stable state at large applied force. The constant-force (Gibbs) and constant-displacement (Helmholtz) formulations of single-molecule stretching experiments are shown to converge in the thermodynamic limit. Monte Carlo simulations of continuous three-dimensional polymers of variable length are carried out, based on this formulation. We demonstrate that the experimental force-extension curves for short and long polymers are described by a unique universal model, despite the differences in chemistry and rate-dependence of transition forces.

Molecular spiders are synthetic molecular motors based on DNA nanotechnology. While natural molecular motors have evolved towards very high efficiency, it remains a major challenge to develop efficient designs for man-made molecular motors. Inspired by biological motor proteins such as kinesin and myosin, molecular spiders comprise a body and several legs. The legs walk on a lattice that is coated with substrate which can be cleaved catalytically. We propose a molecular spider design in which $n$ spiders form a team. Our theoretical considerations show that coupling several spiders together alters the dynamics of the resulting team significantly. Although spiders operate at a scale where diffusion is dominant, spider teams can be tuned to behave nearly ballistic, which results in fast and predictable motion. Based on the separation of time scales of substrate and product dwell times, we develop a theory which utilizes equivalence classes to coarse-grain the microstate space. In addition, we calculate diffusion coefficients of the spider teams, employing a mapping of an $n$-spider team to an $n$-dimensional random walker on a confined lattice. We validate these results with Monte Carlo simulations and predict optimal parameters of the molecular spider team architecture which makes their motion most directed and maximally predictable.

We consider the problem of inferring meaningful spatial information in networks from incomplete information on the connection intensity between the nodes of the network. We consider two spatially distributed networks: a population migration flow network within the US, and a network of mobile phone calls between cities in Belgium. For both networks we use the eigenvectors of the Laplacian matrix constructed from the link intensities to obtain informative visualizations and capture natural geographical subdivisions. We observe that some low-order eigenvectors localize very well and seem to reveal small geographically cohesive regions that match remarkably well with political and administrative boundaries. We discuss possible explanations for this observation by describing diffusion maps and localized eigenfunctions. In addition, we discuss a possible connection with the weighted graph cut problem, and provide numerical evidence supporting the idea that lower-order eigenvectors point out local cuts in the network. However, we do not provide a formal and rigorous justification for our observations.

We derive the generalized master equation for reaction-diffusion on networks from an underlying stochastic process, the continuous time random walk (CTRW). The nontrivial incorporation of the reaction process into the CTRW is achieved by splitting the derivation into two stages. The reactions are treated as birth-death processes and the first stage of the derivation is at the single particle level, taking into account the death process, while the second stage considers an ensemble of these particles including the birth process. Using this model we have investigated different types of pattern formation across the vertices on a range of networks. Importantly, the CTRW defines the Laplacian operator on the network in a non-ad hoc manner and the pattern formation depends on the structure of this Laplacian. Here we focus attention on CTRWs with exponential waiting times for two cases: one in which the rate parameter is constant for all vertices and the other where the rate parameter is proportional to the vertex degree. This results in nonsymmetric and symmetric CTRW Laplacians, respectively. In the case of symmetric Laplacians, pattern formation follows from the Turing instability. However in nonsymmetric Laplacians, pattern formation may be possible with or without a Turing instability.

We study the occurrence of modulational instabilities in lattices with nonlocal power-law hoppings and interactions. Choosing as a case study the discrete nonlinear Schrödinger equation, we consider one-dimensional chains with power-law decaying interactions (with exponent $\alpha$) and hoppings (with exponent $\beta$): An extensive energy is obtained for $\alpha ,\beta >1$. We show that the effect of power-law interactions is that of shifting the onset of the modulational instabilities region for $\alpha >1$. At a critical value of the interaction strength, the modulational stable region shrinks to zero. Similar results are found for effectively short-range nonlocal hoppings ($\beta >2$): At variance, for longer-ranged hoppings ($1<\beta <2$) there is no longer any modulational stability. The hopping instability arises for $q=0$ perturbations, thus the system is most sensitive to the perturbations of the order of the system size. We also discuss the stability regions in the presence of the interplay between competing interactions - (e.g., attractive local and repulsive nonlocal interactions). We find that noncompeting nonlocal interactions give rise to a modulational instability emerging for a perturbing wave vector $q=\pi$ while competing nonlocal interactions may induce a modulational instability for a perturbing wave vector $0<q<\pi$. Since for $\alpha >1$ and $\beta >2$ the effects are similar to the effect produced on the stability phase diagram by finite range interactions and/or hoppings, we conclude that the modulational instability is ``genuinely'' long-ranged for $1<\beta <2$ nonlocal hoppings.

Turbulent Taylor-Couette flow between counter-rotating cylinders develops intermittently fluctuating boundary layers for sufficient counter-rotation. We demonstrate the phenomenon in direct numerical simulations for radius ratios $\eta =0.5$ and 0.71 and propose a theoretical model for the critical value in the rotation ratio. Numerical results as well as experiments show that the onset of this intermittency coincides with the maximum in torque. The variations in torque correlate with the variations in mean Taylor vortex flow, which is first enhanced for weak counter-rotation and then is reduced as intermittency sets in. To support the model, we compare it to numerical results, to experiments at higher Reynolds numbers, and to Wendt's data.

The dynamics of dissipative soft-sphere gases obeys Newton's equations of motion, which are commonly solved numerically by (force-based) Molecular Dynamics (MD) schemes. With the assumption of instantaneous, pairwise collisions, the simulation can be accelerated considerably using event-driven MD, where the coefficient of restitution is derived from the interaction force between particles. Recently it was shown, however, that this approach may fail dramatically, that is, the obtained trajectories deviate significantly from the ones predicted by Newton's equations. In this paper, we generalize the concept of the coefficient of restitution and derive a numerical scheme which, in the case of dilute systems and frictionless interaction, allows us to perform highly efficient event-driven MD simulations even for noninstantaneous collisions. We show that the particle trajectories predicted by our scheme agree perfectly with the corresponding (force-based) MD, except for a short transient period whose duration corresponds to the duration of the contact. Thus, the new algorithm solves Newton's equations of motion like force-based MD while preserving the advantages of event-driven simulations.