We provide an analytical expression for the mean frequency shift in phase oscillators as a function of the standard deviation, $\sigma$ and the autocorrelation time, $\tau$ of small random perturbations. We show that the frequency shift is negative and proportional to ${\sigma }^2$. Its absolute value increases monotonically with $\tau$, approaching an asymptote determined by the ${\text{L}}^2$-norm of the phase-response curve. We validate our theoretical predictions with computer simulations and discuss their implications for the design of electronic oscillators and for the encoding of information in biological neural networks.

Mixing of shear thinning polymeric fluids in long channels with patterned boundary conditions is studied through molecular dynamics simulations. Patterned wettability was shown to induce spatially varying slip lengths at the channel walls which in turn induce mixing in the fluid. To quantify the amount of mixing for different wave lengths of patterns, transverse velocity profiles were evaluated. The transverse velocity profiles from the molecular dynamics simulations were then compared with predictions from continuum modeling and good quantitative agreement was found. Offsetting the pattern was shown to produce better mixing in the center of the channel. Transverse flow is found to increase when the radius of gyration of the chains is smaller than the pattern length. We also implement an oscillating (time dependent) body force and find that the transverse flow increases significantly. However, we do not find an increase in transverse flow with frequency of the oscillation as predicted from continuum modeling and we postulate reasons for this behavior.

The problem of the reconstruction of a flow of conducting incompressible fluid generating a given magnetic mode is considered. We use the magnetic induction equation to derive ordinary differential equations along the magnetic field lines, which gives an opportunity to determine the generating flow if additional data is provided on a two-dimensional manifold transversal to magnetic field lines, and show that an arbitrary solenoidal vector field cannot be a neutral magnetic mode sustained by any flow of conducting fluid, unless a scalar consistency condition is satisfied at every point within the fluid volume.

The survival probability of immobile targets annihilated by a population of random walkers on inhomogeneous discrete structures, such as disordered solids, glasses, fractals, polymer networks, and gels, is analytically investigated. It is shown that, while it cannot in general be related to the number of distinct visited points as in the case of homogeneous lattices, in the case of bounded coordination numbers its asymptotic behavior at large times can still be expressed in terms of the spectral dimension $\tilde{d}$ and its exact analytical expression is given. The results show that the asymptotic survival probability is site-independent of recurrent structures $\left(\tilde{d}\le 2\right)$, while on transient structures $\left(\tilde{d}>2\right)$ it can strongly depend on the target position, and such dependence is explicitly calculated.

The process of dimerization, in which two monomers bind to each other and form a dimer, is common in nature. This process can be modeled using rate equations, from which the average copy numbers of the reacting monomers and of the product dimers can then be obtained. However, the rate equations apply only when these copy numbers are large. In the limit of small copy numbers the system becomes dominated by fluctuations, which are not accounted for by the rate equations. In this limit one must use stochastic methods such as direct integration of the master equation or Monte Carlo simulations. These methods are computationally intensive and rarely succumb to analytical solutions. Here we use the recently introduced moment equations which provide a highly simplified stochastic treatment of the dimerization process. Using this approach, we obtain an analytical solution for the copy numbers and reaction rates both under steady-state conditions and in the time-dependent case. We analyze three different dimerization processes: dimerization without dissociation, dimerization with dissociation, and heterodimer formation. To validate the results we compare them with the results obtained from the master equation in the stochastic limit and with those obtained from the rate equations in the deterministic limit. Potential applications of the results in different physical contexts are discussed.

Single-file diffusion of multiple strongly interacting particles in a one-dimensional pore is described within a general analytical framework. The theory accounts for nonequilibrium conditions, explicit particle-particle interactions, external potential acting on the particles and the fluctuations of the number of particles due to their exchange with external equilibrium reservoirs. It is shown that the problem can be reduced to a closed hierarchical set of partial differential equations of increasing dimensionality, which can be solved numerically. Our framework allows computing any macroscopic characteristic of multiparticle diffusion in the pore. It is shown that the pore occupancy probabilities and the current are rational functions of external concentrations in the steady state. The theory is tested on a simplified model of the narrow rigid pore inspired by the selectivity filter of biological ion channel. Perspectives and limitations of the theory are discussed.

We study critical-point finite-size effects on the behavior of susceptibility of a film placed in the Earth’s gravitational field. The fluid-fluid and substrate-fluid interactions are characterized by van der Waals type power-law tails, and the boundary conditions are consistent with bounding surfaces that strongly prefer the liquid phase of the system. Specific predictions are made with respect to the behavior of ${}^3\text{H}\text{e}$ and ${}^4\text{H}\text{e}$ films in the vicinity of their respective liquid-gas critical points. We find that for all film thicknesses of current experimental interest the combination of van der Waals interactions and gravity leads to substantial deviations from the behavior predicted by models in which all interatomic forces are very short ranged and gravity is absent. In the case of a completely short-ranged system exact mean-field analytical expressions are derived, within the continuum approach, for the behavior of both the local and the total susceptibilities.

We study linear arrays of different number of quartic oscillators shaped in the form of a ring when Gaussian noise (temperature) is added. Frustration is introduced through periodic boundary conditions and repulsive, directional interactions between neighboring oscillators. We show that these systems have similar dynamic properties than the arrays of fluxgates magnetometers. We find that there is a critical number of oscillators separating the regimes arising for systems with few and many oscillators and show that they reach an optimum ordering for a nonvanishing temperature. We also find that they have a relaxation process with an infinite mean life that is typical of glassy systems.

We study the domain geometry during spinodal decomposition of a 50:50 binary mixture in two dimensions. Extending arguments developed to treat nonconserved coarsening, we obtain approximate analytic results for the distribution of domain areas and perimeters during the dynamics. The main approximation is to regard the interfaces separating domains as moving independently. While this is true in the nonconserved case, it is not in the conserved one. Our results can therefore be considered as a “first-order” approximation for the distributions. In contrast to the celebrated Lifshitz-Slyozov-Wagner distribution of structures of the minority phase in the limit of very small concentration, the distribution of domain areas in the 50:50 case does not have a cutoff. Large structures (areas or perimeters) retain the morphology of a percolative or critical initial condition, for quenches from high temperatures or the critical point, respectively. The corresponding distributions are described by a $c{A}^{-\tau }$ tail, where $c$ and $\tau$ are exactly known. With increasing time, small structures tend to have a spherical shape with a smooth surface before evaporating by diffusion. In this regime, the number density of domains with area $A$ scales as $A^{1/2}$, as in the Lifshitz-Slyozov-Wagner theory. The threshold between the small and large regimes is determined by the characteristic area $A\sim t^{2/3}$. Finally, we study the relation between perimeters and areas and the distribution of boundary lengths, finding results that are consistent with the ones summarized above. We test our predictions with Monte Carlo simulations of the two-dimensional Ising model.

We develop a finite-temperature quantized version of density-functional theory of atomic and molecular liquids (QLDFT). Following the Kohn-Sham partitioning of the free energy, we introduce a noninteracting reference fluid of particles obeying the Maxwell-Boltzmann statistics. The kinetic and potential energy of the reference fluid are evaluated exactly. All remaining contributions, including interactions between fluid particles and corrections due to the appropriate quantum statistics are subsumed by an excess (in electronic DFT called exchange-correlation) functional. Two variants to approximate the excess functional are presented: the simplest local-interaction expression (LIE-0) avoids the direct calculation of interparticle interactions and includes them in the excess functional, which is parametrized to reproduce experimental equation of state of normal hydrogen. The more sophisticated LIE-1 approximation is based on the weighted local-density approximation and includes the explicit interparticle interaction potential as well as the local approximation of the excess functional, the latter being weighted by the average over a spherical environment to include nonlocal effects in an approximate way. We apply LIE-0 and LIE-1 to two benchmark systems, bulk fluid hydrogen and hydrogen in a slit pore, and compare it with classical molecular-dynamics simulations employing the same potential. Both functionals produce similar results for direct quantum effects in adsorption free energy. At the same time, LIE-1 also yields a reasonable description of the fluid structure and classical packing effects, which are not reproduced by LIE-0. The source code of our implementation of the LIE-QLDFT is distributed under the GNU public license and is included as a supporting material.

Using differential scanning calorimetry, the transition enthalpies and temperatures for the bulk smectic-isotropic phase transition have been measured for a series of liquid-crystal compounds. For five compounds, those values were used as parameters in a microscopic mean-field model to predict the temperature dependence of the difference in free-energy density between a sample of material in a free-standing smectic film and that in the bulk. The model predicts a weak temperature dependence below the bulk clearing point and a pronounced monotonic increase with temperature above the transition temperature. The compounds used in this study were chosen specifically because they were also the subject of a previous independent experimental study [M. Veum et al., Phys. Rev. E 74, 011703 (2006)] that demonstrated a sudden monotonic increase in the free-standing film tension with temperature, which is qualitatively consistent with the predictions of the above-mentioned mean-field model. This study presents a direct and quantitative comparison between the predictions of the mean-field model and the results from previous tension experiments.

The dynamics of light particles in chaotic oscillating cellular flows is investigated both analytically and numerically by means of Monte Carlo simulations. At level of linear analysis (in the oscillation amplitude) we determined how the known fixed points relative to the stationary cellular flow deform into closed stable trajectories. Once the latter have been analytically determined, we numerically show that they possess the dynamical role of attracting all asymptotic trajectories for a wide range of parameters values. The robustness of the attracting trajectories is tested by adding a white-noise contribution to the particle equation of motion. As a result, attracting trajectories persist up to a critical Péclet number above which an average rising velocity sets in. Possible implications of our results on the long-standing problem related to the explanation of the observed oceanic plankton patchiness will be also discussed.

We formulate a spectral graph-partitioning algorithm that uses the two leading eigenvectors of the matrix corresponding to a selected quality function to split a network into three communities in a single step. In so doing, we extend the recursive bipartitioning methods developed by Newman [M. E. J. Newman, Proc. Natl. Acad. Sci. U.S.A. 103, 8577 (2006); Phys. Rev. E 74, 036104 (2006)] to allow one to consider the best available two-way and three-way divisions at each recursive step. We illustrate the method using simple “bucket brigade” examples and then apply the algorithm to examine the community structures of the coauthorship graph of network scientists and of U. S. Congressional networks inferred from roll call voting similarities.

Oscillating electric potentials are applied to interdigitated arrays of cylindrical electrodes above and below a stationary conducting viscous fluid. The phases of these potentials are chosen to produce a longitudinal traveling wave that traps high-mobility ions and partially traps intermediate-mobility ions in periodic and narrowband chaotic attractors with average velocities that are commensurate with the wave speed. Stable periodic attractors have periods up to 101 times the wave period. Incommensurate broadband chaotic attractors are described by one-dimensional iterated contact-angle return maps, which feature self-similar nested sequences that converge geometrically at unstable trapped orbits. Sequences of singular angles and sequences of step transitions are characterized by distinct convergence factors. A criterion for allowed interelectrode orbits is developed. Experiments are suggested to evaluate the applicability of the theory to microfluidic separations.

The heat exchanged upon isothermal (0.5–200 K) stretching of aluminum and gold nanowires has been calculated by means of molecular dynamics. Atoms at fixed positions with velocities randomly distributed according to Maxwell distribution were taken as initial conditions. The results clearly reveal the presence of non-Gaussian (exponential) tails in the heat probability distribution function at low temperatures, both in gold and aluminum. As temperature is raised, tails rapidly disappear.

Instabilities and pattern formation is the rule in nonequilibrium systems. Selection of a persistent length scale or coarsening (increase in the length scale with time) are the two major alternatives. When and under which conditions one dynamics prevails over the other is a long-standing problem, particularly beyond one dimension. It is shown that the challenge can be defied in two dimensions, using the concept of phase diffusion equation. We find that coarsening is related to the $\lambda$ dependence of a suitable phase diffusion coefficient, $D_{11}\left(\lambda \right)$, depending on lattice symmetry and conservation laws. These results are exemplified analytically on prototypical nonlinear equations.

We demonstrate that demographic noise can induce persistent spatial pattern formation and temporal oscillations in the Levin-Segel predator-prey model for plankton-herbivore population dynamics. Although the model exhibits a Turing instability in mean-field theory, demographic noise greatly enlarges the region of parameter space where pattern formation occurs. To distinguish between patterns generated by fluctuations and those present at the mean-field level in real ecosystems, we calculate the power spectrum in the noise-driven case and predict the presence of fat tails not present in the mean-field case. These results may account for the prevalence of large-scale ecological patterns, beyond that expected from traditional nonstochastic approaches.

We study pulse propagation in one-dimensional tapered chains of spherical granules. Analytic results for the pulse velocity and other pulse features are obtained using a binary collision approximation. Comparisons with numerical results show that the binary collision approximation provides quantitatively accurate analytic results for these chains.

We investigate numerically static and dynamic properties of cholesteric blue phases. Our study is based on a Landau–de Gennes theory describing the orientational order of a liquid crystal in terms of a second-rank tensor. To find the shape and size of the unit cell conforming to the minimum of the free energy, we let the geometrical parameters characterizing the unit cell relax in the course of the time evolution via a simple relaxational equation. We investigate the effect of an electric field on the structure of cholesteric blue phases. We study how the deformation of the unit cell in response to the electric field $\mathit{E}$ depends on the strength and direction of the electric field and the original structure of cholesteric blue phases. Our results qualitatively agree with the experimental findings. Although in a weak field, the strain tensor is proportional to $E^2$ as previously argued, for a moderate field the distortion is no longer proportional to $E^2$ and can be even nonmonotonic with respect to $E^2$. Furthermore, we investigate the kinetic processes of the deformation, rearrangement, and extinction of disclination lines under a strong electric field. We show that the kinetics of disclination lines is highly complicated and sensitively depends on the initial structure of blue phases, the direction of the electric field, and the sign of dielectric anisotropy ${ϵ}_{\text{a}}$. In most cases, a strong field aligns the liquid crystals in a uniform (positive ${ϵ}_{\text{a}}$) or helical (negative ${ϵ}_{\text{a}}$) manner without disclination lines. However, for negative ${ϵ}_{\text{a}}$ and the direction of the electric field parallel to the body diagonal of the unit cell, disclination lines do not disappear and form a two-dimensional hexagonal lattice.

We present an approach to calculating the quantum resonances and resonance wave functions of chaotic scattering systems, based on the construction of states localized on classical periodic orbits and adapted to the dynamics. Typically only a few such states are necessary for constructing a resonance. Using only short orbits (with periods up to the Ehrenfest time), we obtain approximations to the longest-living states, avoiding computation of the background of short living states. This makes our approach considerably more efficient than previous ones. The number of long-lived states produced within our formulation is in agreement with the fractal Weyl law conjectured recently in this setting. We confirm the accuracy of the approximations using the open quantum baker map as an example.