Numerical propagation of the Schrödinger equation is the bottleneck in many quantum optimal control computations. For a quantum system of $N$ states with an electric-field–dipole interaction, the use of a propagation toolkit introduced in a prior work yields an $O\left(N\right)$ reduction in floating-point operations per wave function propagation. A concatenation scheme for the toolkit method is introduced, and a scaling analysis shows a significant additional reduction in computational cost. The method exploits the fact that the same sequences of discretized control field values are often repeated many times in a control simulation. The concatenated toolkit is benchmarked against the standard toolkit in a numerical simulation.

We have used x-ray reflectivity to study the coupling of surface and interface layering in a molecularly thin normal liquid [tetrakis(2-ethylhexoxy)silane (TEHOS)], as a function of temperature and film thickness. The best fits to the data were obtained with an electron density model that consists of a uniform density component superimposed upon molecular-scale density oscillations (layers). The two types of layer profiles were observed to vary with temperature from $187–286\mathrm{K}$. The amount of material in the molecular layers increases as that in the uniform density layer decreases, with the onset of liquid-to-layered transition occurring at a total film thickness of $\sim 40\mathrm{\AA }$ (about twice the bulk correlation length of TEHOS).

The effect of sidewall nucleation on nanowire morphology is studied theoretically. The model provides a semiquantitative description of nanowire radius as a function of its length and the distance from the surface. It is demonstrated that the wire shape critically depends on the diffusion flux of adatoms from the substrate and on the rate of direct impingement to the sidewalls. At high diffusion flux the wire shape is cylindrical. A decrease of diffusion from the surface leads to the onset of nucleation on the sidewalls resulting in the lateral extension and in the reduction of wire length. The wire shape changes from cylindrical to conical, because the supersaturation of adatoms driving the nucleation is higher at the wire foot than at the top. It is shown that the shape modification becomes pronounced at low growth temperatures. Theoretical results are used to model the experimentally observed shapes of GaAs and GaP wires, grown by Au-assisted molecular beam epitaxy at different temperatures.

We report the results of a synchrotron x-ray reflectivity study of bulk liquid-crystal surfaces that are coated by thin wetting films of an immiscible liquid. The liquid-crystal subphase consisted of the nematic or isotropic phase of 4-octyl-$4^{\prime }$-cyanobiphenyl (8CB), and the wetting film was formed by the fluorocarbon perfluoromethylcyclohexane (PFMC), a volatile liquid. The thickness of the wetting film was controlled by the temperature difference $\Delta T_{\mu }$ between the sample and a reservoir of bulk PFMC, contained within the sealed sample cell. Phase information on the interfacial electron density profiles has been extracted from the interference between the scattering from the PFMC-vapor interface and the surface-induced smectic order of the 8CB subphase. The liquid-crystal side of the nematic-liquid (8CB-PFMC) interface is characterized by a density oscillation whose period corresponds to the smectic layer spacing and whose amplitude decays exponentially toward the underlying nematic subphase. The decay length $\xi$ of the smectic amplitude is independent of the PFMC film thickness but increases as the nematic–smectic-$A$ transition temperature $T_{\text{NA}}$ is approached, in agreement with the longitudinal correlation length ${\xi }_{\parallel }\propto {\left(T-T_{\text{NA}}\right)}^{-0.7}$ for the smectic fluctuations in the bulk nematic. The results indicate that the homeotropic orientation of the 8CB molecules is preferred at the 8CB-PFMC interface and that the observed temperature dependence of the smectic layer growth is consistent with the critical adsorption mechanism. The observed $\Delta T_{\mu }$ dependence of the PFMC film thickness, $L\propto {\left(\Delta T_{\mu }\right)}^{-1/3}$, implies that PFMC completely wets the 8CB surface and is dominated by the nonretarded dispersion interactions between hydro- and fluorocarbons. The complete wetting behavior of PFMC is nearly independent of the degree of interfacial smectic order in the subphase.

We have studied the effects of quenched random disorder created by dispersed aerosil nanoparticle gels on the nematic to smectic-$A$ (N-$\text{Sm}A$) and smectic-$A$ to reentrant nematic ($\text{Sm}A$-RN) phase transitions of thermotropic liquid-crystal mixtures of hexyloxycyanobiphenyl (6OCB) and octyloxycyanobiphenyl (8OCB). These effects are probed using high-resolution synchrotron x-ray diffraction techniques. We find that the reentrant characteristics of the system are largely unchanged by the presence of the aerosil gel network. By comparing measurements of the smectic static structure amplitude for this 8OCB-$\text{6OCB}+\text{aerosil}$ system with those for butyloxybenzilidene-octylaniline $\left(4\text{O}.8\right)+\text{aerosil}$ gels, we find that the short-range smectic order in the smectic-$A$ phase is significantly weaker in the reentrant system. This result is consistent with the behavior seen in pure 8OCB-6OCB mixtures. The strength of the smectic ordering decreases progressively as the 6OCB concentration is increased. Detailed line shape analysis shows that the high- and low-temperature nematic phases (N and RN) are similar to each other.

Molecular dynamics (MD) simulations are used to study the interaction of a zwitterionic palmitoyl-oleoyl-phosphatidylcholine (POPC) bilayer with the cationic antimicrobial peptide bovine lactoferricin (LFCinB) in a 100 mM NaCl solution at 310 K. The interaction of LFCinB with POPC is used as a model system for studying the details of membrane-peptide interactions, with the peptide selected because of its antimicrobial nature. Seventy-two 3 ns MD simulations, with six orientations of LFCinB at 12 different distances from a POPC membrane, are carried out to determine the potential of mean force (PMF) or free energy profile for the peptide as a function of the distance between LFCinB and the membrane surface. To calculate the PMF for this relatively large system a new variant of constrained MD and thermodynamic integration is developed. A simplified method for relating the PMF to the LFCinB-membrane binding free energy is described and used to predict a free energy of adsorption (or binding) of $-1.05\pm 0.39\text{kcal}/\text{mol}$, and corresponding maximum binding force of about 20 pN, for LFCinB-POPC. The contributions of the ions-LFCinB and the water-LFCinB interactions to the PMF are discussed. The method developed will be a useful starting point for future work simulating peptides interacting with charged membranes and interactions involved in the penetration of membranes, features necessary to understand in order to rationally design peptides as potential alternatives to traditional antibiotics.

We analytically investigate the interspike interval (ISI) density, the Fano factor, and the coefficient of variation of a leaky integrate-and-fire neuron model driven by exponentially correlated Gaussian noise with a large correlation time $\tau$. We find a burstinglike behavior of the spike train, which is revealed by a dominant peak of the ISI density at small intraburst intervals and a slow power-law decay of long interburst intervals. The large, power-law distributed ISIs give rise to a coefficient of variation which diverges as $\sqrt{\tau }$. This leads to the paradoxical effect that ISI correlations, as expressed by the serial correlation coefficient, vanish for large correlation times. This is in contrast to findings of previous works on a simpler neuron model where the effect of noise correlations appeared in higher-order statistical measures.

The asymptotic mean number of distinct sites visited by a subdiffusive continuous-time random walker in two dimensions seems not to have been explicitly calculated anywhere in the literature. This number has been calculated for other dimensions for only one specific asymptotic behavior of the waiting time distribution between steps. We present an explicit derivation for two cases in all integer dimensions so as to formally complete a tableau of results. In this tableau we include the dominant as well as subdominant contributions in all integer dimensions. Other quantities that can be calculated from the mean number of distinct sites visited are also discussed.

In previous papers, we found that the mean square displacement $⟨u^2⟩$ in polystyrene thin films decreased with film thickness and discussed two possible causes for the decrease. One is the hard interface layer between the polymer thin film and the substrate, and the other is the spatial confinement effect of polymer chains. In order to disclose the cause of the decrease of $⟨u^2⟩$ with film thickness, we studied the molecular weight $M_w$ dependence of $⟨u^2⟩$ in polystyrene thin films. We found that $⟨u^2⟩$ was independent of $M_w$ within experimental error, concluding that the interface effect is a main reason for the decrease of $⟨u^2⟩$ with film thickness.

We study multifractal properties of wave functions for a one-parameter family of quantum maps displaying the whole range of spectral statistics intermediate between integrable and chaotic statistics. We perform extensive numerical computations and provide analytical arguments showing that the generalized fractal dimensions are directly related to the parameter of the underlying classical map, and thus to other properties such as spectral statistics. Our results could be relevant for Anderson and quantum Hall transitions, where wave functions also show multifractality.

We study the Laplacian operator of an uncorrelated random network and, as an application, consider hopping processes (diffusion, random walks, signal propagation, etc.) on networks. We develop a strict approach to these problems. We derive an exact closed set of integral equations, which provide the averages of the Laplacian operator’s resolvent. This enables us to describe the propagation of a signal and random walks on the network. We show that the determining parameter in this problem is the minimum degree $q_m$ of vertices in the network and that the high-degree part of the degree distribution is not that essential. The position of the lower edge of the Laplacian spectrum ${\lambda }_c$ appears to be the same as in the regular Bethe lattice with the coordination number $q_m$. Namely, ${\lambda }_c>0$ if $q_m>2$, and ${\lambda }_c=0$ if $q_m\le 2$. In both of these cases the density of eigenvalues $\rho \left(\lambda \right)\to 0$ as $\lambda \to {\lambda }_c+0$, but the limiting behaviors near ${\lambda }_c$ are very different. In terms of a distance from a starting vertex, the hopping propagator is a steady moving Gaussian, broadening with time. This picture qualitatively coincides with that for a regular Bethe lattice. Our analytical results include the spectral density $\rho \left(\lambda \right)$ near ${\lambda }_c$ and the long-time asymptotics of the autocorrelator and the propagator.

We report on an experimental study of long normal Saffman-Taylor fingers subject to periodic forcing. The sides of the finger develop a low amplitude, long wavelength instability. We discuss the finger response in stationary and nonstationary situations, as well as the dynamics towards the stationary states. The response frequency of the instability increases with forcing frequency at low forcing frequencies, while, remarkably, it becomes independent of forcing frequency at large forcing frequencies. This implies a process of wavelength selection. These observations are in good agreement with previous numerical results reported in [Ledesma-Aguilar et al., Phys. Rev. E 71, 016312 (2005)]. We also study the average value of the finger width, and its fluctuations, as a function of forcing frequency. The average finger width is always smaller than the width of the steady-state finger. Fluctuations have a nonmonotonic behavior with a maximum at a particular frequency.

A model for autonomous motion of the contact line of the oil-water interface along a solid surface is proposed. The present model is inspired by the spontaneous wave generation and the contact line motion of an oil-water interface composed of cationic surfactant and oil-soluble anions. The motion is created through wetting by an adsorption of surfactant followed by an autocatalytic process with a chemical reaction and also dewetting due to the desorption of the remaining monolayer. The wetting process is accelerated by the contact line motion itself through the convection-enhanced transport of reaction constituents, which is autocatalytic in nature. These processes are expressed by nonlinear time-evolution equations for the velocity and the amplitude. Following the model, the autonomous motion is essentially excitable, and hence the spatiotemporal pattern is of the noise-induced type. The present model explains well the diverse features of the experimental results with a few parameters.

The existence and stability of discrete breathers (DBs) in one- and two-dimensional magnetic metamaterials (MMs), which consist of periodic arrangements (arrays) of split-ring resonators (SRRs), are investigated numerically. We consider different configurations of the SRR arrays, which are related to the relative orientation of the SRRs in the MM, in both one and two spatial dimensions. In the latter case we also consider anisotropic MMs. Using standard numerical methods we construct several types of linearly stable breather excitation in both Hamiltonian and dissipative MMs (dissipative breathers). The study of stability in both cases is performed using standard Floquet analysis. In both cases we find that the increase of dimensionality from one to two spatial dimensions does not destroy the DBs, which may also exist in the case of moderate anisotropy (in two dimensions). In dissipative MMs, the dynamics is governed by a power balance between the mainly Ohmic dissipation and driving by an alternating magnetic field. In that case it is demonstrated that DB excitation locally alters the magnetic response of MMs from paramagnetic to diamagnetic. Moreover, when the frequency of the applied field approaches the SRR resonance frequency, the magnetic response of the MM in the region of the DB excitation may even become negative (extremely diamagnetic).

The output of a linear filter driven by a randomly polarized square wave, when viewed backward in time, is shown to exhibit determinism at all times when embedded in a three-dimensional state space. Combined with previous results establishing exponential divergence equivalent to a positive Lyapunov exponent, this result rigorously shows that such reverse-time synthesized waveforms appear equally to have been produced by a deterministic chaotic system.

To examine the anchoring energy of a surface with one-dimensional grooves of sinusoidal shape, we carry out numerical calculation of the Frank elastic energy of a nematic cell composed of such a grooved surface and a flat surface. We evaluate the anchoring energy of the grooved surface by carefully eliminating the contribution from a uniform twist deformation in the bulk. When $qA\lesssim 0.2$, with $q$ and $A$ being the wave number and the amplitude of the surface groove, we find that the azimuthal-angle dependence of the calculated anchoring energy agrees perfectly with our previous analytical result under the assumption of $qA⪡1$ [Fukuda et al., Phys. Rev. Lett. 98, 187803 (2007); 99, 139902(E) (2007)]. Even when $qA\simeq 0.6$ or 1, we observe an unexpectedly good agreement between the calculated and the analytical anchoring energies, indicating the wide applicability of the analytical anchoring energy in spite of the assumption of $qA⪡1$ in its derivation.

We have shown that the steady state probability distribution function of a diffusion-coalescence system on a one-dimensional lattice of length $L$ with reflecting boundaries can be written in terms of a superposition of double-shock structures which perform biased random walks on the lattice while repelling each other. The shocks can enter into the system and leave it from the boundaries. Depending on the microscopic reaction rates, the system is known to have two different phases. We have found that the mean distance between the shock positions is of order $L$ in one phase while it is of order 1 in the other phase.

Adaptive populations such as those in financial markets and distributed control can be modeled by the Minority Game. We consider how their dynamics depends on the agents’ initial preferences of strategies, when the agents use linear or quadratic payoff functions to evaluate their strategies. We find that the fluctuations of the population making certain decisions (the volatility) depends on the diversity of the distribution of the initial preferences of strategies. When the diversity decreases, more agents tend to adapt their strategies together. In systems with linear payoffs, this results in dynamical transitions from vanishing volatility to a nonvanishing one. For low signal dimensions, the dynamical transitions for the different signals do not take place at the same critical diversity. Rather, a cascade of dynamical transitions takes place when the diversity is reduced. In contrast, no phase transitions are found in systems with the quadratic payoffs. Instead, a basin boundary of attraction separates two groups of samples in the space of the agents’ decisions. Initial states inside this boundary converge to small volatility, while those outside diverge to a large one. Furthermore, when the preference distribution becomes more polarized, the dynamics becomes more erratic. All the above results are supported by good agreement between simulations and theory.

We study the nonlinear elastic response of a two-dimensional material to a localized boundary force, with the particular goal of understanding the differences observed between isotropic granular materials and those with hexagonal anisotropy. Corrections to the classical Boussinesq result for the stresses in an infinite half space of a linear, isotropic material are developed in a power series in inverse distance from the point of application of the force. The breakdown of continuum theory on scales of order of the grain size is modeled with phenomenological parameters characterizing the strengths of induced multipoles near the point of application of the external force. We find that the data of Geng et al. [Phys. Rev. Lett. 87, 035506 (2001)] on isotropic and hexagonal packings of photoelastic grains can be fitted within this framework. Fitting the hexagonal packings requires a choice of elastic coefficients with hexagonal anisotropy stronger than that of a simple ball-and-spring model. For both the isotropic and hexagonal cases, induced dipole and quadrupole terms produce propagation of stresses away from the vertical direction over short distances. The scale over which such propagation occurs is significantly enhanced by the nonlinearities that generate hexagonal anisotropy.

Numerical simulations of assemblies of grains under cyclic loading exhibit “granular ratcheting:” a small net deformation occurs with each cycle, leading to a linear accumulation of deformation with cycle number. We show that this is due to a curious property of the most frequently used models of the particle-particle interaction: namely, that the potential energy stored in contacts is path dependent. There exist closed paths that change the stored energy, even if the particles remain in contact and do not slide. An alternative method for calculating the tangential force removes granular ratcheting.