Motivated by recent experimental studies of matter waves and optical beams in double-well potentials, we study the corresponding solutions of the nonlinear Schrödinger equation. Using a Galerkin-type approach, we obtain a detailed handle on the nonlinear solution branches of the problem, starting from the corresponding linear ones, and we predict the relevant bifurcations for both attractive and repulsive nonlinearities. The dynamics of the ensuing unstable solutions is also examined. The results illustrate the differences that arise between the steady states and the bifurcations emerging in symmetric and asymmetric double wells.

Introduction of optical gain in a disordered system results in enhanced fluctuations $[F_{\left(2\right)}=\mathrm{var}\left(\tilde{g}\right)∕{⟨\tilde{g}⟩}^2]$ of the dimensionless conductance $\tilde{g}$, similar to the effect of Anderson localization in a passive medium. Using numerical simulations we demonstrate that, despite such qualitative similarity, the whole distribution of the conductance of amplifying random media is drastically different from that of a passive system with the same value of $F_{\left(2\right)}$.

We calculate and analyze a thermodynamic limit of a multiscale molecular dynamics based scheme that we have developed previously for simulating shock waves. We validate and characterize the performance of the former scheme for several simple cases. Using model equations of state for chemical reactions and kinetics in a gas and a condensed phase explosive, we show that detonation wave profiles computed using the computational scheme are in good agreement with the steady state wave profiles of hydrodynamic direct numerical simulations. We also characterize the stability of the technique when applied to detonation waves and describe a technique for determining the detonation shock speed.

Transport of a Brownian particle moving along the axis of a three-dimensional asymmetric periodic tube is investigated in the presence of asymmetric unbiased forces. The reduction of the coordinates may involve not only the appearance of entropic barrier but also the effective diffusion coefficient. It is found that in the presence of entropic barrier, the asymmetry of the tube shape and the asymmetry of the unbiased forces are the two ways of inducing a net current. The current is a peaked function of temperature which indicates that the thermal noise may facilitate the transport even in the presence of entropic barrier. An optimized radius exists at the bottleneck at which the current takes its maximum value. Competition between the two opposite driving factors may induce current reversal.

The exact expression of the first moment and the signal-to-noise ratio (SNR) have been calculated for a linear system subject to an external periodic field as well as a multiplicative asymmetric dichotomous noise, by using the Shapiro-Loginov formula. It has been found that the amplitude of the output signal, and the SNR, respectively, exhibit two kinds of the phenomena of stochastic resonance: one is as the functions of the parameters of the asymmetric dichotomous noise, such as the noise strength $D$, and the parameter $k$ describing the asymmetric degree of the dichotomous noise; the other is as the function of the parameter of the input signal, such as the input signal frequency $\omega$.

A thermally reversible repulsive hard-sphere to sticky-sphere transition was studied in a model colloidal system over a wide volume fraction range. The static microstructure was obtained from high resolution small angle x-ray scattering, the colloid dynamics was probed by dynamic x-ray and light scattering, and supplementary mechanical properties were derived from bulk rheology. At low concentration, the system shows features of gas-liquid type phase separation. The bulk phase separation is presumably interrupted by a gelation transition at the intermediate volume fraction range. At high volume fractions, fluid-attractive glass and repulsive glass-attractive glass transitions are observed. It is shown that the volume fraction of the particles can be reliably deduced from the absolute scattered intensity. The static structure factor is modeled in terms of an attractive square-well potential, using the leading order series expansion of Percus-Yevick approximation. The ensemble-averaged intermediate scattering function shows different levels of frozen components in the attractive and repulsive glassy states. The observed static and dynamic behavior are consistent with the predictions of a mode-coupling theory and numerical simulations for a square-well attractive system.

We consider the feasibility of large-scale free-standing thin liquid film experiment in the space environment as a new realization to study two-dimensional hydrodynamics. We identify material and environmental criteria necessary to avoid freezing, evaporation, chemical degradation, and spontaneous collapse of the film. These criteria pose no obstacles to achieving films of kilometer scale and lifetime of many months, with attainable Reynolds number up to ${10}^7$. However, impacts from meteoroids pose a serious threat to the film, and require substantial shielding or unproven self-healing properties in the film. Current theoretical and experimental studies of two-dimensional turbulence are briefly reviewed. We also describe a specific candidate liquid for the film.

Nematic elastomers do not show the discontinuous, first-order, phase transition that the Landau-De Gennes mean field theory predicts for a quadrupolar ordering in three dimensions. We attribute this behavior to the presence of network crosslinks, which act as sources of quenched orientational disorder. We show that the addition of weak random anisotropy results in a singular renormalization of the Landau-De Gennes expression, adding an energy term proportional to the inverse quartic power of order parameter $Q$. This reduces the first-order discontinuity in $Q$. For sufficiently high disorder strength the jump disappears altogether and the phase transition becomes continuous, in some ways resembling the supercritical transitions in external field.

The tension generated by randomly distributed myosin minifilaments in an actin gel is evaluated using a rigorous theorem relating the surface forces acting on the gel to the forces exerted by the myosins. The maximum tension generated per myosin depends strongly on the lengths of the myosin minifilaments and the actin filaments. The result is used to place an upper bound on the tension that can be generated during cytokinesis. It is found that actomyosin contraction by itself generates too little force for ring contraction during cytokinesis unless the actin filaments are tightly crosslinked into inextensible units much longer than a single actin filament.

The ground state of an externally confined one-component Yukawa plasma is derived analytically. In particular, the radial density profile is computed. The results agree very well with computer simulations of three-dimensional spherical Coulomb crystals. We conclude in presenting an exact equation for the density distribution for a confinement potential of arbitrary geometry.

We present results for the high-temperature series expansions of the susceptibility and free energy of the $q$-state Potts model on a $D$-dimensional hypercubic lattice ${\mathbb{Z}}^D$ for arbitrary values of $q$. The series are up to order 20 for dimension $D\le 3$, order 19 for $D\le 5$, and up to order 17 for arbitrary $D$. Using the $q\to 1$ limit of these series, we estimate the percolation threshold $p_c$ and critical exponent $\gamma$ for bond percolation in different dimensions. We also extend the $1∕D$ expansion of the critical coupling for arbitrary values of $q$ up to order $D^{-9}$.

We confine charged spheres in cells with the smallest dimension along the direction of gravity $\widehat{g}$. The particles are density mismatched with the surrounding medium and sediment along $\widehat{g}$ with typical Péclet numbers of $\mathrm{Pe}\approx {10}^{-3}$. After a certain time, we find that the number of particles $N$ increases near both upper and lower plates until a characteristic time $\tau$ is reached; above this time $N$ plateaus. We attribute the observed phenomenology to collective particle motions driven by gravity and mediated by hydrodynamic interactions; these could yield formation of swirls made of particles with correlated velocities that could eventually drive the particles towards the upper plate. The characteristic time for these migrations scales with plate-to-plate separation $L_z$ as $\tau \sim L_z^{1.2}$, exactly as the characteristic decay time of velocity fluctuations in sedimentation processes [S. Y. Tee et al., Phys. Rev. Lett. 89, 054501 (2002)], despite that in these experiments the smallest cell dimension is perpendicular to $\widehat{g}$ and $7<\mathrm{Pe}<50$. In the absence of gravitational field, the observed particle migrations disappear, emphasizing the key role played by gravity in these experiments.

Two models exist for the orientational distribution of the long molecular axes in smectic-$A$ liquid crystals: the conventional unimodal distribution and the “cone-shaped” de Vries distribution. The de Vries hypothesis provides a plausible picture of how, at a molecular level, a first-order $\mathrm{Sm}\text{−}A$ to $\mathrm{Sm}\text{−}{C}^{*}$ transition may occur, especially if there is little or no concomitant shrinkage of the layer spacing. This work investigates two materials with such transitions: C7 and TSiKN65. The azimuthal distribution of in-layer directors is probed using IR and polarized Raman spectroscopy, which allows us to obtain orientational order parameters. In C7, we observe a discontinuous change in the order parameter, the magnitude of which is small compared with the corresponding change in the in-layer director tilt angle $\Theta$. Assuming that the smectic-$A$ liquid crystal is of the de Vries type, we calculate the $\Theta$ required to reproduce the apparent order parameter ${⟨P_2⟩}_{\mathrm{app}}$, obtained from IR, by using the true order parameter $⟨P_2⟩$, obtained from polarized Raman scattering. The results indicate that, for C7, the tilt angle so calculated is much smaller than that in the $\mathrm{Sm}\text{−}{C}^{*}$ angle and hence de Vries behavior may not be the appropriate explanation in this case. Conversely, we find that TSiKN65 shows a different behavior to C7, which can be explained in terms of the de Vries concept. Thus, we conclude that either type of distribution may exist in $\mathrm{Sm}\text{−}A$ phases which undergo a first-order transition to the $\mathrm{Sm}\text{−}{C}^{*}$ phase. We also discuss the changes in the smectic layer spacing and the orientational order parameters across the $\mathrm{Sm}\text{−}A\text{−}\mathrm{Sm}\text{−}{C}^{*}$ phase transition, together with changes in birefringence with applied electric field.

Phenomena following a rapid mechanical quench of liquid ${}^4\mathrm{He}$ from its normal to its superfluid phase are reported and discussed. The mechanical expansion apparatus is an improved version of that described previously. It uses a double-cell geometry to effect a partial separation of the sample from the convolutions of the bellows that form the outer wall of the cell. Consistent with earlier work, no evidence is found for the production of quantized vortices via the Kibble-Zurek (KZ) mechanism. Although the expansion is complete within $15\mathrm{ms}$, the second-sound velocity and attenuation continue to increase for a further $\sim 60\mathrm{ms}$; correspondingly the temperature decreases. Subsequently, the temperature rises again toward its final value as the second-sound velocity and attenuation decrease. It is shown that this unexpected behavior is apparently associated with a large-amplitude second-sound oscillation produced by the expansion, and it is suggested that the observed vortices are created by the normal fluid–superfluid counterflow that constitutes the second-sound wave. If production of large-amplitude second sound is inherent to the mechanical expansion of liquid ${}^4\mathrm{He}$ through the superfluid transition, as appears to be the case for final temperatures more than $3\mathrm{mK}$ from the $\lambda$ transition, the phenomenon sets a lower bound on the density of KZ vortices that can be detected in this type of experiment.

Diffuse reflection boundary conditions are introduced in a thermal lattice Boltzmann model to allow for variable fluid density and temperature along the walls. The capability of this model to capture the main characteristics of the thermal transpiration phenomenon in a box at nonvanishing Knudsen numbers is demonstrated. The thermal creep velocity is found to be proportional to the temperature gradient imposed at the wall, whereas the accuracy of the simulation results are found to be of first or second order, depending on the numerical scheme.

We study an autocatalytic reaction-diffusion scheme, the Gray-Scott model, when the mixing processes do not homogenize the reactants. Starting from the master equation, we derive the resulting coupled, nonlinear, stochastic partial differential equations that rigorously include the spatiotemporal fluctuations resulting from the interplay between the reaction and mixing processes. The fields are complex and depend on correlated complex noise terms. We implement a method to solve for these complex fields numerically and extract accurate information about the system evolution and stationary states under different mixing regimes. Through this example, we show how the reaction-induced fluctuations interact with the temporal nonlinearities, leading to results that differ significantly from the mean-field (perfectly mixed) approach. This procedure can be applied to an arbitrary nonlinear reaction diffusion scheme.

The single-file diffusion of water in the straight channels of two different crystalline microporous aluminosilicates (zeolites bikitaite and Li-ABW) was studied by comparing the results of molecular dynamics computer simulations with the predictions of anomalous diffusion theory modeled by using fractional diffusion equations. At high coverage, the agreement is reasonably good, in particular for sufficiently large displacements and sufficiently long times. At low coverage, interesting phenomena appear in the simulation results, such as multimodal propagators, which could be interpreted on the basis of fractional Fokker-Planck equations. The results are discussed also in view of different theories that have been proposed to model the single-file diffusion process.

An initially well mixed granular material composed of two distinct subclasses of particles, small and large or light and heavy, segregates radially into stable lobed patterns when rotated in various quasi-two-dimensional, regular polygonal tumblers. The patterns are highly sensitive to the time-periodic flow, which in turn depends critically on the fill fraction and container shape. Simulations of a simple model reproduce the segregation patterns observed in experiment. Kolmogorov-Arnol’d-Moser (KAM) regions in Poincaré plots of the velocity field used to model the flow attract smaller (denser) particles and their spatial symmetries mirror those of the segregation patterns, suggesting that competition between the driving forces for radial segregation (percolation and buoyancy) and those for chaotic mixing plays a key role.

We present results from an extensive experimental investigation into granular segregation of a shallow binary mixture in which particles are driven by frictional interactions with the surface of a vibrating horizontal tray. Three distinct phases of the mixture are established viz. binary gas (unsegregated), segregation liquid, and segregation crystal. Their ranges of existence are mapped out as a function of the system’s primary control parameters using a number of measures based on Voronoi tessellation. We study the associated transitions and show that segregation can be suppressed as the total filling fraction of the granular layer, $C$, is decreased below a critical value, $C_c$, or if the dimensionless acceleration of the driving, $\gamma$, is increased above a value ${\gamma }_c$.

Based in a linearized hydrodynamic model, we study the relaxation dynamics of hexatic-$B$ liquid-crystal films in two distinct cases. First, we investigate the hexatic order relaxing through a purely diffusive process for which the particles velocity field is assumed to be negligible. In this case, the asymptotic relaxation of deformations in the hexatic order presents a pronounced dependence on the boundary conditions. We found that a surface tilted order is enough to drive the slowest relaxation mode from an acoustic to an optic character. Second, we study the viscous case which is characterized by the coupling between the particles velocity field and the bond orientational order. In this case, underdamped modes with oscillatory-exponential relaxation are obtained on a narrow range of wave vectors. Further, the viscous relaxation of surface and inner layers deformations display distinct scaling behaviors under a surface tilted order.