We consider a general system of coupled nonlinear diffusion equations that are characterized by having degenerate source terms and thereby not having isolated rest states. Using a general form of physically relevant source terms, we derive conditions that are required to trigger traveling waves when a stable uniform steady-state solution is perturbed by a highly localized disturbance. We show that the degeneracy in the source terms implies that traveling waves have a number of surprising properties that are not present for systems with nondegenerate source terms. We also show that such systems can lead to a pair of waves that initially propagate outwards from the disturbance, slow down, and reverse direction before ultimately colliding and annihilating each other.

The breakup of inwardly rotating spiral waves has been investigated in an oscillatory reaction-diffusion system near a Hopf bifurcation point. The breakup first occurred at the region far away from the core area, then gradually involved the whole medium by increasing the diffusion coefficient ratio between the two components of the oscillator system. With the approximation of the Complex-Ginzburg-Landau equation (CGLE), the criteria for the occurrence of the inwardly rotating spiral wave are examined theoretically. The analysis of the stability in the corresponding CGLE revealed that the breakup of the inward spiral wave was related to the Eckhaus instability.

Statistical mechanics is applied to lossy compression using multilayer perceptrons for unbiased Boolean messages. We utilize a treelike committee machine (committee tree) and treelike parity machine (parity tree) whose transfer functions are monotonic. For compression using a committee tree, a lower bound of achievable distortion becomes small as the number of hidden units $K$ increases. However, it cannot reach the Shannon bound even where $K\to \infty$. For a compression using a parity tree with $K\ge 2$ hidden units, the rate distortion function, which is known as the theoretical limit for compression, is derived where the code length becomes infinity.

Families of solitons in one- and two-dimensional (1D and 2D) Gross-Pitaevskii equations with the repulsive nonlinearity and a potential of the quasicrystallic type are constructed (in the 2D case, the potential corresponds to a fivefold optical lattice). Stable 1D solitons in the weak potential are explicitly found in three band gaps. These solitons are mobile, and they collide elastically. Many species of tightly bound 1D solitons are found in the strong potential, both stable and unstable (unstable ones transform themselves into asymmetric breathers). In the 2D model, families of both fundamental and vortical solitons are found and are shown to be stable.

The growth and ordering of striped domains has recently received renewed attention due in part to experimental studies in diblock copolymers and electroconvection. One surprising result has been the relatively slow dynamics associated with the growth of striped domains. One potential source of the slow dynamics is the pinning of defects in the periodic potential of the stripes. Of interest is whether or not external noise will have a significant impact on the domain ordering, perhaps by reducing the pinning and increasing the rate of ordering. In contrast, we present experiments using electroconvection in which we show that a particular type of external noise decreases the rate of domain ordering.

Fluid-fluid binodals of binary hard-sphere mixtures are computed from the recently proposed fundamental measure functional–mean spherical approximation closure of the two-component Ornstein-Zernike equation. The results, especially in the dense fluid region that was not accessible by previous theoretical methods, are compared with the corresponding ones for the one-component fluid of big spheres with effective potential obtained from the same closure. The general trends are those expected for hard-sphere potentials but small difference are detectable. The overall agreement found validates the equivalence of the two descriptions for size ratios $R=8.5$ or greater.

Conventional wisdom presumes that low-coordinated crystal ground states require directional interactions. Using our recently introduced optimization procedure to achieve self-assembly of targeted structures [M. C. Rechtsman et al., Phys. Rev. Lett. 95, 228301 (2005); Phys. Rev. E 73, 011406 (2006)], we present an isotropic pair potential $V\left(r\right)$ for a three-dimensional many-particle system whose classical ground state is the low-coordinated simple cubic lattice. This result is part of an ongoing pursuit by the authors to develop analytical and computational tools to solve statistical-mechanical inverse problems for the purpose of achieving targeted self-assembly. The purpose of these methods is to design interparticle interactions that cause self-assembly of technologically important target structures for applications in photonics, catalysis, separation, sensors, and electronics. We also show that standard approximate integral-equation theories of the liquid state that utilize pair correlation function information cannot be used in the reverse mode to predict the correct simple cubic potential. We report in passing optimized isotropic potentials that yield the body-centered-cubic and simple hexagonal lattices, which provide other examples of non-close-packed structures that can be assembled using isotropic pair interactions.

Surface freezing was detected and studied in mixtures of alcohol and alkane molecules, using surface tensiometry and surface-specific x-ray scattering methods. Considering that surface freezing in pure alkanes forms an ordered monolayer and in alcohols it forms an ordered bilayer, the length mismatch repulsion was minimized by varying the carbon number of the alkane component around $2n$, where $n$ is the carbon number of the alcohol molecule. A solutionlike behavior was found for all mixtures, where the ideal liquid mixture phase-separates upon freezing both in the bulk and the surface. The solid exhibits a herringbone crystalline phase below an alkane mole fraction ${\varphi }_t\approx 0.8$ and a rotator phase above it. The surface frozen film below ${\varphi }_t$ is an alkane monolayer exhibiting a next-nearest neighbor molecular tilt of a composition-dependent magnitude. Above ${\varphi }_t$, no diffraction peaks were observed. This could be explained by the intrinsically shorter-range order of the rotator phase and a possible proliferation of defects.

Material with the phase sequence $\mathrm{Sm}A\text{−}\mathrm{Sm}AP\text{−}\mathrm{Sm}CP$ is studied as an example of a system in which the spontaneous electric polarization and the molecular tilt develop independently at the $\mathrm{Sm}A\text{−}\mathrm{Sm}AP$ and the $\mathrm{Sm}AP\text{−}\mathrm{Sm}CP$ phase transition, respectively. The temperature dependence of the spontaneous electric polarization clearly shows a strong coupling between the polarization and tilt. The system exhibits also very strong precritical polarization and tilt fluctuations. Experimental observations are explained within the theoretical model.

In this paper, we present ${}^{13}\mathrm{C}$ and ${}^1\mathrm{H}$ NMR investigations of 2, 3, 6, 7, 10, 11-hexahexyl-thiotriphenylene (HHTT). The measurements were carried out under both static and magic-angle spinning conditions. The phase diagram of HHTT is $K\leftrightarrow H\leftrightarrow D_{hd}\leftrightarrow I$, where $H$ is a helical phase and $D_{hd}$ is a columnar liquid crystal. The motivation was to characterize the molecular order and dynamics and to investigate differences at the molecular level between the two mesophases: $H$ and $D_{hd}$. It is shown that $D_{hd}$ is a conventional columnar liquid crystal, where the molecular core undergoes fast rotation about the symmetry axis. The orientational order in this mesophase is lower and the temperature dependence of the order parameter is steeper than in other triphenylene-based compounds. On the other hand, in the helical phase the core, similarly to the solid phase, is essentially rigid. The difference between the solid and helical phases is mainly manifested in an increased mobility of the aliphatic chains observed in the latter phase. In addition, the sample exhibits thermal history effects, which are observed in the different behavior upon cooling and heating.

We present a detailed numerical study of a microscopic artificial swimmer realized recently by Dreyfus et al. in experiments [Dreyfus et al., Nature 437, 862 (2005)]. It consists of an elastic filament composed of superparamagnetic particles that are linked together by DNA strands. Attached to a load particle, the resulting swimmer is actuated by an oscillating external magnetic field so that it performs a nonreciprocal motion in order to move forward. We model the superparamagnetic filament by a bead-spring configuration that resists bending like a rigid rod and whose beads experience friction with the surrounding fluid and hydrodynamic interactions with each other. We show that, aside from finite-size effects, its dynamics is governed by the dimensionless sperm number, the magnitude of the magnetic field, and the angular amplitude of the field’s oscillating direction. Then we study the mean velocity and the efficiency of the swimmer as a function of these parameters and the size of the load particle. In particular, we clarify that the real velocity of the swimmer is influenced by two main factors, namely the shape of the beating filament (determined by the sperm number and the magnetic-field strength) and the oscillation frequency. Furthermore, the load size influences the performance of the swimmer and has to be chosen as a compromise between the largest swimming velocity and the best efficiency. Finally, we demonstrate that the direction of the swimming velocity changes in a symmetry-breaking transition when the angular amplitude of the field’s oscillating direction is increased, in agreement with experiments.

We investigate the possible quasiperiodic property in the sequences of Alu repeats, one of typical noncoding DNA sequences. We calculated the quasiperiods of the right and left monomers of Alu repeats of different families with quasiperiodic matrix algorithm. It is interesting that the right monomers of all families show significant quasiperiod 8 in their sequences while the left monomers show quasiperiods 8 or 5. Our results indicate that there exist common quasiperiods in most Alu repeats. This may be helpful to further explore possible functions of Alu repeats.

We consider a Markovian jumping process which is defined in terms of the jump-size distribution and the waiting-time distribution with a position-dependent frequency, in the diffusion limit. We assume the power-law form for the frequency. For small steps, we derive the Fokker-Planck equation and show the presence of the normal diffusion, subdiffusion, and superdiffusion. For the Lévy distribution of the step size, we construct a fractional equation, which possesses a variable coefficient, and solve it in the diffusion limit. Then we calculate fractional moments and define the fractional diffusion coefficient as a natural extension to the cases with the divergent variance. We also solve the master equation numerically and demonstrate that there are deviations from the Lévy stable distribution for large wave numbers.

We study the phase diagram and the commensurate-incommensurate transitions in a phase field model of a two-dimensional crystal lattice in the presence of an external pinning potential. The model allows for both elastic and plastic deformations and provides a continuum description of lattice systems, such as for adsorbed atomic layers or two-dimensional vortex lattices. Analytically, a mode expansion analysis is used to determine the ground states and the commensurate-incommensurate transitions in the model as a function of the strength of the pinning potential and the lattice mismatch parameter. Numerical minimization of the corresponding free energy shows reasonable agreement with the analytical predictions and provides details on the topological defects in the transition region. We find that for small mismatch the transition is of first order, and it remains so for the largest values of mismatch studied here. Our results are consistent with results of simulations for atomistic models of adsorbed overlayers.

We study correlation functions of the Calogero-Sutherland model in the whole range of the interaction parameter. Using the replica method we obtain analytical expressions for the long-distance asymptotics of the one-body density matrix in addition to the previously derived asymptotics of the pair-distribution function [D.M. Gangardt and A. Kamenev, Nucl. Phys. B 610, 578 (2001)]. The leading analytic and nonanalytic terms in the short-distance expansion of the one-body density matrix are discussed. Numerical results for these correlation functions are obtained using Monte Carlo techniques for all distances. The momentum distribution and static structure factor are calculated. The potential and kinetic energies are obtained using the Hellmann-Feynman theorem. Perfect agreement is found between the analytical expressions and numerical data. These results allow for the description of physical regimes of the Calogero-Sutherland model. The zero temperature phase diagram is found to be of a crossover type and includes quasicondensation, quasicrystallization and quasisupersolid regimes.

Dense suspensions of small strongly interacting particles are complex systems that are rarely understood on the microscopic level. We investigate properties of dense suspensions and sediments of small spherical ${\mathrm{Al}}_2{\mathrm{O}}_3$ particles in a shear cell by means of a combined molecular-dynamics and stochastic rotation dynamics simulation. We study structuring effects and the dependence of the suspension’s viscosity on the shear rate and shear thinning for systems of varying salt concentration and $p\mathrm{H}$ value. To show the agreement of our results with experimental data, the relation between the bulk $p\mathrm{H}$ value and surface charge of spherical colloidal particles is modeled by Debye-Hückel theory in conjunction with a $2\mathrm{pK}$ charge regulation model.

This paper is the fourth in a series exploring the physical consequences of the solidity of highly viscous liquids. It is argued that the two basic characteristics of a flow event (a jump between two energy minima in configuration space) are the local density change and the sum of all particle displacements. Based on this it is proposed that density fluctuations are described by a time-dependent Ginzburg-Landau equation with rates in $\mathbf{k}$ space of the form ${\Gamma }_0+D{k}^2$ with $D⪢{\Gamma }_0{a}^2$ where $a$ is the average intermolecular distance. The inequality expresses a long-wavelength dominance of the dynamics which implies that the Hamiltonian (free energy) may be taken to be ultralocal. As an illustration of the theory the case with the simplest nontrivial Hamiltonian is solved to second order in the Gaussian approximation, where it predicts an asymmetric frequency dependence of the isothermal bulk modulus with Debye behavior at low frequencies and an ${\omega }^{-1∕2}$ decay of the loss at high frequencies. Finally, a general formalism for the description of viscous liquid dynamics, which supplements the density dynamics by including stress fields, a potential energy field, and molecular orientational fields, is proposed.

We present a detailed computer simulation and integral equation study of the phase behavior of a nematogenic system composed of hard spheres with embedded three-dimensional Maier-Saupe spins. For this well-known system, we map the gas-liquid equilibrium, which is coupled to a first-order isotropic-nematic transition. The anisotropic integral equation theory is found to yield excellent agreement with the simulation data within the fluid regime. Additionally, we determine the fluid-solid equilibrium transition by means of computer simulation.

The principal features of the volumetric as well as the viscoelastic response of mechanically stimulated glasses can be summarized as follows: (i) the time-aging time shift factors contract upon increasing the probe stress (i.e., the stress apparently modifies the volume recovery kinetics), (ii) the volume recovery baseline remains unaltered (i.e., the underlying structure of the stimulated glass remains unchanged). Here we present a series of numerically simulated results concerning the responses of glassy polycarbonate that simultaneously fulfill these apparent contradictions. The problem was tackled coupling a modified Kalroush, Aklonis, Hotchinson, Ramos equation with the constitutive law for linear viscoelasticity within the domain of the reduced time. It was argued that the relaxation times under isobaric conditions depend on the temperature, the dimensionless volume, and the isotropic components of the stress tensor. Simulations are obtained with a minimum of experimental ($PVT$ and linear viscoelastic) data inputs. Different loading protocols consisting of complex combinations and/or sequences of large and small mechanical stimuli were tested. Volumetric as well as viscoelastic behavior are systematically reported. A tentative explanation of the origin of the time-aging time contraction was finally proposed while some additional features concerning the volumetric response emerged.

The linear stability analysis of dielectric liquid placed in uniform electric field with respect to perturbations of density of an initially uniform state was carried out. The electric field increases the instability increment for the stratification along the field and decreases it for the transversal stratification. Thus, anisotropic separation into liquid and vapor phases is possible in high electric fields for a liquid that is initially in unstable state, as well as in metastable or stable states. Computer simulations of electrohydrodynamics confirm the theoretical calculations. It is important that new regions of low density phase appear as narrow cylindrical channels oriented along the field. This mechanism of generation of gaseous phase in locally high electric field can play a key role in processes of inception and ultrafast propagation of streamers during breakdown of liquid dielectrics in nanosecond range.