We show that some boundary conditions assumed at a thin membrane may result in normal diffusion not being the stochastic Markov process. We consider boundary conditions defined in terms of the Laplace transform in which there is a linear combination of probabilities and probability fluxes defined on both membrane surfaces. The coefficients of the combination may depend on the Laplace transform parameter. Such boundary conditions are most commonly used when considering diffusion in a membrane system unless collective or nonlocal processes in particles diffusion occur. We find Bachelier-Smoluchowski-Chapmann-Kolmogorov (BSCK) equation in terms of the Laplace transform and we derive the criterion to check whether the boundary conditions lead to fundamental solutions of diffusion equation satisfying this equation. If the BSCK equation is not met, then the Markov property is broken. When a probability flux is continuous at the membrane, the general forms of the boundary conditions for which the fundamental solutions meet the BSCK equation are derived. A measure of broken of semi-group property is also proposed. The relation of this measure to the non-Markovian property measure is discussed.

We present numerical findings on the behavior of the athermal nonequilibrium random-field Ising model of spins at the thin striplike $L_1\times L_2\times L_3$ cubic lattices with $L_1<L_2<L_3$. Changing of system sizes highly influences the evolution and shape of avalanches. The smallest avalanches [classified as three-dimension- (3D) like] are unaffected by the system boundaries, the larger are sandwiched between the top and bottom system faces so are 2D-like, while the largest are extended over the system lateral cross section and propagate along the length $L_3$ like in 1D systems. Such a structure of avalanches causes double power-law distributions of their size, duration, and energy with larger effective critical exponent corresponding to 3D-like and smaller to 2D-like avalanches. The distributions scale with thickness $L_1$ and are collapsible following the proposed scaling predictions which, together with the distributions' shape, might be important for analysis of the Barkhausen noise experimental data for striplike samples. Finally, the impact of system size on external field that triggers the largest avalanche for a given disorder is presented and discussed.

We present an extensive Markov chain Monte Carlo study of the finite-size scaling behavior of the Fortuin-Kasteleyn Ising model on five-dimensional hypercubic lattices with periodic boundary conditions. We observe that physical quantities, which include the contribution of the largest cluster, exhibit complete graph asymptotics. However, for quantities where the contribution of the largest cluster is removed, we observe that the scaling behavior is mainly controlled by the Gaussian fixed point. Our results therefore suggest that both scaling predictions, i.e., the complete graph and the Gaussian fixed point asymptotics, are needed to provide a complete description for the five-dimensional finite-size scaling behavior on the torus.

We study the low-temperature out-of-equilibrium Monte Carlo dynamics of the disordered Ising $p$-spin Model with $p=3$ and a small number of spin variables. We focus on sequences of configurations that are stable against single spin flips obtained by instantaneous gradient descent from persistent ones. We analyze the statistics of energy gaps, energy barriers, and trapping times on subsequences such that the overlap between consecutive configurations does not overcome a threshold. We compare our results to the predictions of various trap models finding the best agreement with the step model when the $p$-spin configurations are constrained to be uncorrelated.

It is shown that angular stiffness in the hexagonal lattice model plays a significant role in the geometrical nonlinear terms in the equations of the continuum limit. A geometrically nonlinear discrete model is formulated for the hexagonal lattice by considering the interaction of two sublattices. An asymptotic procedure is developed in order to obtain the nonlinear coupled equations of motion in the continuum limit of the discrete model. An interaction of longitudinal and shear plane strain waves is studied by using the solutions of the obtained equations.

We present experimental and numerical results for the parametric fluctuation properties in the spectra of classically chaotic quantum graphs with unitary or symplectic symmetry. A level dynamics is realized by changing the lengths of a few bonds parametrically. The long-range correlations in the spectra reveal at a fixed parameter value deviations from those expected for generic chaotic systems with corresponding universality class. They originate from modes which are confined to individual bonds or explore only a fraction of the quantum graph. Similarly, discrepancies are observed in the avoided-crossing distribution, velocity correlation function, and the curvature distribution of the level dynamics which also may be attributed to such localized modes. We demonstrate that these may be easily identified by inspecting the level dynamics and consequently their nonuniversal contributions to the parametric spectral properties may be diminished considerably. This is corroborated by numerical studies.

Vertically aligned carbon nanotube (VA-CNT) arrays were grown on several chromium (Cr)-coated glass substrates using a plasma-enhanced chemical vapor deposition system. The CNTs were $2\mu \mathrm{m}$ long and had a site density of $2\times {10}^9\mathrm{c}{\mathrm{m}}^{-2}$ on the substrates. Two VA-CNT slides on Cr glass substrates were put together to design a homeotropic electro-optic liquid crystal (LC) device. A negative dielectric anisotropic LC was used in the device. The $\pi -\pi$ stacking interaction between the LC and the VA-CNTs allows the LC material to align homeotropically in the cell. When an external electric field was applied using the transparent conducting Cr layers, the LC achieves a planar orientation above a threshold field. These results successfully demonstrate the optical, electro-optical operations, and the field-induced dynamic response of a homeotropic LC device employing the VA-CNT arrays as the homeotropic-alignment agent. This study significantly advances the range and understanding of nanostructured surfaces that provide vertical alignment of LCs.

We study the mobility of objects embedded in an immersed granular packing and subjected to cyclic loadings. With this aim, we conducted uplift experiments whereby a horizontal plate is embedded in the packing and subjected to a vertical cyclic force oscillating between zero and a maximum amplitude. Tests performed at different cyclic force frequencies and amplitudes evidence the development of three mobility regimes whereby the plate stays virtually immobile, moves up steadily, or slowly creeps upwards. Results show that steady plate uplift can occur at lower force magnitudes when the frequency is increased. We propose an interpretation of this frequency-weakening behavior based on force relaxation experiments and on the analysis of the mobility response of theoretical viscoelastoplastic mechanical analog. These results and analysis point out inherent differences in mobility response between steady and cyclic loadings in immersed granular materials.

Using extensive nonequilibrium molecular dynamics simulations, we investigate a glass-forming binary Lennard-Jones mixture under shear. Both supercooled liquids and glasses are considered. Our focus is on the characterization of inhomogeneous flow patterns such as shear bands that appear as a transient response to the external shear. For the supercooled liquids, we analyze the crossover from Newtonian to non-Newtonian behavior with increasing shear rate $\dot{\gamma }$. Above a critical shear rate ${\dot{\gamma }}_c$ where a non-Newtonian response sets in, the transient dynamics are associated with the occurrence of short-lived vertical shear bands, i.e., bands of high mobility that form perpendicular to the flow direction. In the glass states, long-lived horizontal shear bands, i.e., bands of high mobility parallel to the flow direction, are observed in addition to vertical ones. The systems with shear bands are characterized in terms of mobility maps, stress-strain relations, mean-squared displacements, and (local) potential energies. The initial formation of a horizontal shear band provides an efficient stress release, corresponds to a local minimum of the potential energy, and is followed by a slow broadening of the band towards the homogeneously flowing fluid in the steady state. Whether a horizontal or a vertical shear band forms cannot be predicted from the initial undeformed sample. Furthermore, we show that with increasing system size, the probability for the occurrence of horizontal shear bands increases.

In this work, we consider the stability of a spherical shell under combined loading from a uniform external pressure and a homogenous natural curvature. Nonmechanical stimuli, such as one that tends to modify the rest curvature of an elastic body, are prevalent in a wide range of natural and engineered systems, and may occur due to thermal expansion, changes in pH, differential swelling, and differential growth. Here we investigate how the presence of both an evolving natural curvature and an external pressure modifies the stability of a complete spherical shell. We show that due to a mechanical analogy between pressure and curvature, positive natural curvatures can severely destabilize a thin shell, while negative natural curvatures can strengthen the shell against buckling, providing the possibility to design shells that buckle at or above the theoretical limit for pressure alone, i.e., a strengthening factor. These results extend directly from the classical analysis of the stability of shells under pressure, and highlight the important role that nonmechanical stimuli can have on modifying the membrane state of stress in a thin shell.

In a systematic study of shock wave propagating in crystalline polyethylenes using molecular dynamics method and the electron force field (eFF) potential, we show that microscopic structure of shock front is significantly affected by the anisotropy of long carbon chain and the bond breaking and recombination dynamics. However, macroscopic properties measured in Hugoniot experiments, such as compression ratio and shock velocity, are not sensitive to carbon chain anisotropy and bond dynamics. Our work also display that hydrogen molecules are formed when the piston speed is in the region between 10 km/s and 30 km/s. However, carbon-hydrogen pair distribution function does not display an indication of carbon-hydrogen phase segregation.

A new phenomenon called imposed ordered structures has been of particular interest to the dusty plasma community within the past several years. These structures are a new type of pattern formation within a dusty plasma in which the dust particles become fixed to a background confining potential whose spatial structure is determined by some conducting element present in the plasma. In previous works, this element has typically been a conducting wire mesh. One of the unanswered questions is whether the dust particles become trapped beneath the conducting surface of the mesh or in the gaps (holes) between the wires. This work makes use of a new electrode whose shape is designed to mimic that of wire mesh, but with much larger dimensions to facilitate in situ diagnostic measurements. Observations of the dust show that particles become confined to the regions beneath the holes of this new electrode. Measurements of the dust particle velocities allow for a determination of the kinetic energy within the dust cloud which show that the particles are in an energetic steady state. Comparisons of spatial profiles of the particle velocities also show that the dust particles typically reside in areas of increased plasma glow, possibly being trapped by plasma filaments.

In this paper, we first present a unified framework of multiple-relaxation-time lattice Boltzmann (MRT-LB) method for the Navier-Stokes and nonlinear convection-diffusion equations where a block-lower-triangular-relaxation matrix and an auxiliary source distribution function are introduced. We then conduct a comparison of the four popular analysis methods (Chapman-Enskog analysis, Maxwell iteration, direct Taylor expansion, and recurrence equations approaches) that have been used to obtain the macroscopic Navier-Stokes and nonlinear convection-diffusion equations from the MRT-LB method and show that from mathematical point of view, these four analysis methods can give the same equations at the second-order of expansion parameters. Finally, we give some elements that are needed in the implementation of the MRT-LB method and also find that some available LB models can be obtained from this MRT-LB method.

Dynamic critical behavior in superfluid systems is considered in the presence of external stirring and advecting processes. The latter are generated by means of the Gaussian random velocity ensemble with white-noise character in time variable and self-similar spatial dependence. The main focus of this work is to analyze an effect of compressible modes on the critical behavior. The model is formulated through stochastic Langevin equations, which are then recast into the Janssen-De Dominicis response formalism. Employing the field-theoretic perturbative renormalization group method we analyze large-scale properties of the model. Explicit calculations are performed to the leading one-loop approximation in the double $(\varepsilon ,y)$ expansion scheme, where $\varepsilon$ is a deviation from the upper critical dimension $d_c=4$ and $y$ describes a scaling property of the velocity ensemble. Altogether five distinct universality classes are expected to be macroscopically observable. In contrast to the incompressible case, we find that compressibility leads to an enhancement and stabilization of nontrivial asymptotic regimes.

Considering symbolic and numerical random sequences in the framework of the additive Markov chain approach, we establish a relation between their correlation functions and conditional entropies. We express the entropy by means of the two-point probability distribution functions and then evaluate the entropy for the numerical random chain in terms of the correlation function. We show that such approximation gives a satisfactory result only for special types of random sequences. In general case the conditional entropy of numerical sequences obtained in the two-point distribution function approach is lower. We derive the conditional entropy of the additive Markov chain as a sum of the Kullback–Leibler mutual information and give an example of random sequence with the exactly zero correlation function and the nonzero correlations.

Using the properties of random Möbius transformations, we investigate the statistical properties of the reflection coefficient in a random chain of lossy scatterers. We explicitly determine the support of the distribution and the condition for coherent perfect absorption to be possible. We show that at its boundaries the distribution has Lifshits-like tails, which we evaluate. We also obtain the extent of penetration of incoming waves into the medium via the Lyapunov exponent. Our results agree well when compared to numerical simulations in a specific random system.

In the thermodynamics of nanoscopic systems, the relation between classical and quantum mechanical description is of particular importance. To scrutinize this correspondence we study an anharmonic oscillator driven by a periodic external force with slowly varying amplitude both classically and within the framework of quantum mechanics. The energy change of the oscillator induced by the driving is closely related to the probability distribution of work for the system. With the amplitude $\lambda \left(t\right)$ of the drive increasing from zero to a maximum ${\lambda }_{\max }$ and then going back to zero again, the initial and final Hamiltonian coincide. The main quantity of interest is then the probability density $P\left(E_f\right|E_i)$ for transitions from initial energy $E_i$ to final energy $E_f$. In the classical case nondiagonal transitions with $E_f\ne E_i$ mainly arise due to the mechanism of separatrix crossing. We show that approximate analytical results within the pendulum approximation are in accordance with numerical simulations. In the quantum case numerically exact results are complemented with analytical arguments employing Floquet theory. For both the classical and quantum case we provide an intuitive explanation for the periodic variation of $P\left(E_f\right|E_i)$ with the maximal amplitude ${\lambda }_{\max }$ of the driving.

The frustrated $q$-state Potts model is solved exactly on a hierarchical lattice, yielding chaos under rescaling, namely, the signature of a spin-glass phase, as previously seen for the Ising $(q=2)$ model. However, the ground-state entropy introduced by the $(q>2)$-state antiferromagnetic Potts bond induces an escape from chaos as multiplicity $q$ increases. The frustration versus multiplicity phase diagram has a reentrant (as a function of frustration) chaotic phase.

We develop linear stability analysis (LSA) to quantitatively predict the dynamics of a perturbed plane wave in nonlinear systems. We take a nonintegrable fiber model with purely fourth-order dispersion as an example to demonstrate this method's effectiveness. For a Gaussian-type initial perturbation with cosine-type modulation on a plane wave, its propagation velocities, periodicity, and localization are predicted successfully, and the range of application is discussed. Importantly, the modulation-instability-induced growth of localized perturbation is proved different from the one of purely periodic perturbation and requires the modification of gain value for more accurate prediction. The method offers a needful supplement and improvement for LSA and paves a way to study the dynamics of a perturbed plane wave in more practical nonlinear systems.

We derive methods for estimating the topology of the stationary probability current ${\vec{j}}_s$ of the two-species Fokker-Planck equation (FPE) without the need to solve the FPE. These methods are chosen such that they become exact in certain limits, such as infinite system size or vanishing coupling between species in the diffusion matrix. The methods make predictions about the fixed points of ${\vec{j}}_s$ and their relation to extrema of the stationary probability distribution and to fixed points of the convective field, which is related to the deterministic drift of the system. Furthermore, they predict the rotation sense of ${\vec{j}}_s$ around extrema of the stationary probability distribution. Even though these methods cannot be proven to be valid away from extrema, the boundary lines between regions with different rotation senses are obtained with surprising accuracy. We illustrate and test these methods, using simple reaction systems with only one coupling term between the two species as well as a few generic reaction networks taken from the literature. We use it also to investigate the shape of nonphysical probability currents occurring in reaction systems with detailed balance due to the approximations involved in deriving the Fokker-Planck equation.