The bistable Swift-Hohenberg equation possesses a variety of time-independent spatially localized solutions organized in the so-called snakes-and-ladders structure. This structure is a consequence of a phenomenon known as homoclinic snaking, and is in turn a consequence of spatial reversibility of the equation. We examine here the consequences of breaking spatial reversibility on the snakes-and-ladders structure. We find that the localized states now drift, and show that the snakes-and-ladders structure breaks up into a stack of isolas. We explore the evolution of this new structure with increasing reversibility breaking and study the dynamics of the system outside of the snaking region using a combination of numerical and analytical techniques.

We determine the full multipoint concentration statistics for superdiffusive transport in a steady stratified random velocity field. Using a Lagrangian approach, we derive explicit analytical expressions for the multipoint moments of concentration and specifically for the concentration variance, which is a measure for concentration uncertainty. The multipoint concentration moments are fully characterized by the Lagrangian mean velocity and by the one and two particle velocity correlations. While the relative variance at the center of mass of the mean concentration is constant, it increases exponentially with time and distance from the center of mass. This implies that small concentration values are particularly uncertain, which can pose a serious practical concern as these are typically the earliest and latest to arrive at a point.

Cascaded lattice Boltzmann method (cascaded-LBM) employs a class of collision operators aiming to stabilize computations and remove certain modeling artifacts for simulation of fluid flow on lattice grids with sizes arbitrarily larger than the smallest physical dissipation length scale [Geier et al., Phys. Rev. E 63, 066705 (2006)]. It achieves this and distinguishes from other collision operators, such as in the standard single or multiple relaxation-time approaches, by performing relaxation process due to collisions in terms of moments shifted by the local hydrodynamic fluid velocity, i.e., central moments, in an ascending order by order at different relaxation rates. In this paper, we propose and derive source terms in the cascaded-LBM to represent the effect of external or internal forces on the dynamics of fluid motion. This is essentially achieved by matching the continuous form of the central moments of the source or forcing terms with its discrete version. Different forms of continuous central moments of sources, including one that is obtained from a local Maxwellian, are considered in this regard. As a result, the forcing terms obtained in this formulation are Galilean invariant by construction. To alleviate lattice artifacts due to forcing terms in the emergent macroscopic fluid equations, they are proposed as temporally semi-implicit and second order, and the implicitness is subsequently effectively removed by means of a transformation to facilitate computation. It is shown that the impressed force field influences the cascaded collision process in the evolution of the transformed distribution function. The method of central moments along with the associated orthogonal properties of the moment basis completely determines the analytical expressions for the source terms as a function of the force and macroscopic velocity fields. In contrast to the existing forcing schemes, it is found that they involve higher-order terms in velocity space. It is shown that the proposed approach implies “generalization” of both local equilibrium and source terms in the usual lattice frame of reference, which depend on the ratio of the relaxation times of moments of different orders. An analysis by means of the Chapman-Enskog multiscale expansion shows that the cascaded-LBM with forcing terms is consistent with the Navier-Stokes equations. Computational experiments with canonical problems involving different types of forces demonstrate its accuracy.

An analytical and numerical analysis of the no-slip boundary condition at walls at rest for the lattice Boltzmann Bhatnagar-Gross-Krook method is performed. The main result of this analysis is an alternative formulation for the no-slip boundary condition at walls at rest. Numerical experiments assess the accuracy and stability of this formulation for Poiseuille and Womersley flows, flow over a backward facing step, and unsteady flow around a square cylinder. This no-slip boundary condition is compared analytically and numerically to the boundary conditions of Inamuro et al. [Phys. Fluids 7, 2928 (1995)] and Zou and He [Phys. Fluids 9, 1591 (1997)] and it is found that all three make use of the same mechanism for the off-diagonal element of the stress tensor. Mass conservation, however, is only assured by the present one. In addition, our analysis points out which mechanism lies behind the instabilities also observed by Lätt et al. [Phys. Rev. E 77, 056703 (2008)] for this kind of boundary conditions. We present a way to remove these instabilities, allowing one to reach relaxation frequencies considerably closer to 2.

A powerful and well-established tool for free-energy estimation is Bennett’s acceptance ratio method. Central properties of this estimator, which employs samples of work values of a forward and its time-reversed process, are known: for given sets of measured work values, it results in the best estimate of the free-energy difference in the large sample limit. Here we state and prove a further characteristic of the acceptance ratio method: the convexity of its mean-square error. As a two-sided estimator, it depends on the ratio of the numbers of forward and reverse work values used. Convexity of its mean-square error immediately implies that there exists a unique optimal ratio for which the error becomes minimal. Further, it yields insight into the relation of the acceptance ratio method and estimators based on the Jarzynski equation. As an application, we study the performance of a dynamic strategy of sampling forward and reverse work values.

Based on the Langevin description of the continuous time random walk (CTRW), we consider a generalization of CTRW in which the waiting times between the subsequent jumps are correlated. We discuss the cases of exponential and slowly decaying persistent power-law correlations between the waiting times as two generic examples and obtain the corresponding mean squared displacements as functions of time. In the case of exponential-type correlations the (sub)diffusion at short times is slower than in the absence of correlations. At long times the behavior of the mean squared displacement is the same as in uncorrelated CTRW. For power-law correlations we find subdiffusion characterized by the same exponent at all times, which appears to be smaller than the one in uncorrelated CTRW. Interestingly, in the limiting case of an extremely long power-law correlations, the (sub)diffusion exponent does not tend to zero, but is bounded from below by the subdiffusion exponent corresponding to a short-time behavior in the case of exponential correlations.

We analyze confining mechanisms for Lévy flights. When they evolve in suitable external potentials their variance may exist and show signatures of a superdiffusive transport. Two classes of stochastic jump-type processes are considered: those driven by Langevin equation with Lévy noise and those, named topological Lévy processes (occurring in systems with topological complexity such as folded polymers or complex networks), whose Langevin representation is unknown and possibly nonexistent. Our major finding is that both above classes of processes stay in affinity and may share common stationary probability density, even if their detailed dynamical behavior look different. This near-equilibrium observation seems to be generic to a broad class of Lévy noise-driven processes, such as e.g., superdiffusion on folded polymers, geophysical flows, and even climatic changes.

We propose a polymer growth model, in which propagating radicals can grow through propagation processes or annihilate through termination (disproportionation or combination) processes. Considering a simple case in which the propagation and termination rates of each polymer chain are both independent of its length, we then investigate analytically the kinetics of the model by means of the rate-equation approach. The propagating radicals will be exhausted eventually and only the inert polymers (the termination products of propagating radicals) can survive in the end. Moreover, the size distribution of propagating radicals can always take the form of the Poisson distribution at a given time, while that of inert polymers is dependent strongly on the details of the reaction-rate kernels. For the case in which the propagation rate constant $J_1$ is less than the termination rate constant $J_2$, the size distribution of inert polymers can always take a power-law form $c_k\left(t\right)\sim k^{-2-J_1/\left(J_2-J_1\right)}$, in the region of $t⪢1$ and $k⪢1$. For the $J_1>J_2$ case, the kinetic evolution of inert polymers is very complex and $c_k\left(t\right)$ can take one of the three forms: monotone decreasing, single peak (Poisson-like distribution), and double peak. For the special $J_1=J_2$ case, $c_k\left(t\right)$ exhibits an exponential decay in size.

We consider an Ising model where longitudinal components of every pair of spins have antiferromagnetic interaction of the same magnitude. When subjected to a transverse magnetic field at zero temperature, the system undergoes a phase transition of second order to an ordered phase and if the temperature is now increased, there is another phase transition to disordered phase. We provide derivation of these features by perturbative treatment up to the second order and argue that the results are nontrivial and not derivable from the known results about related models.

Recently, Dean and Lefèvre [Phys. Rev. Lett. 90, 198301 (2003)] developed a method for testing the statistical mechanical theory of granular packings proposed by Edwards and co-workers [Physica A 157, 1080 (1989); Phys. Rev. E 58, 4758 (1998)]. The method relies on the prediction that the ratio of two overlapping volume histograms should be exponential in volume. We extend the method by showing that one can also calculate the entropy of the packing and also that the method can yield false positive results when the histograms are Gaussians with nearly identical variances. We then apply the method to simulations and experiments of granular compaction. The distribution of global volumes (the volume of the entire packing) is nearly Gaussian and it is difficult to conclude if the theory is valid. On the other hand, the distribution of Voronoï volumes clearly obeys the theoretical prediction.

The characteristics of steady granular flow in quasi-two-dimensional rotating tumblers have been thoroughly investigated and are fairly well understood. However, unsteady time-varying flow has not been studied in detail. The linear response of granular flow in quasi-two-dimensional rotating tumblers is presented for periodic forcing protocols via sinusoidal variation in the rotational speed of the tumbler and for step changes in rotational speed. Variations in the tumbler radius, particle size, and forcing frequency are explored. Similarities to steady flow include the fastest flow occurring at the free surface of the flowing layer and an instantaneous approximately linear velocity profile through the depth. The flowing layer depth varies by 2–5 particle diameters between minimum and maximum rotation rates. However, unsteady forcing also causes the flow to exhibit dynamic properties. For periodic rotational speeds, the phase lag of the flowing layer depth increases linearly with increasing input forcing frequency up to nearly 2.0 rad over 0–20 cycles per tumbler revolution. The amplitude responses of the velocity and shear rate show a resonance behavior unique to the system level parameters. The phase lag of all flow properties appears to be related to the number of particle contacts from the edge of the rotating tumbler. Characterization via step changes in rotational speed shows dynamic properties of overshoot (up to 35%) and rise times on the order of 0.2–0.7 s. The results suggest that the unsteady granular flow analysis may be beneficial for characterizing the “flowability” and “rheology” of granular materials based on particle size, moisture content, or other properties.

We study the phase behavior of a classical system of particles interacting through a strictly convex soft-repulsive potential which, at variance with the pairwise softened repulsions considered so far in the literature, lacks a region of downward or zero curvature. Nonetheless, such interaction is characterized by two length scales, owing to the presence of a range of interparticle distances where the repulsive force increases, for decreasing distance, much more slowly than in the adjacent regions. We investigate, using extensive Monte Carlo simulations combined with accurate free-energy calculations, the phase diagram of the system under consideration. We find that the model exhibits a fluid-solid coexistence line with multiple re-entrant regions, an extremely rich solid polymorphism with solid-solid transitions, and waterlike anomalies. In spite of the isotropic nature of the interparticle potential, we find that among the crystal structures in which the system can exist, there are also a number of non-Bravais lattices, such as $cI16$ and diamond.

A rigorous theory of liquid-crystal transitions is developed starting from the Liouville equation. The starting point is an all-atom description and a set of order-parameter field variables that are shown to evolve slowly via Newton’s equations. The separation of time scales between that of atomic collision or vibrations and the order-parameter fields enables the derivation of rigorous equations for stochastic order-parameter field dynamics. When the fields provide a measure of the spatial profile of the probability of molecular position, orientation, and internal structure, a theory of liquid-crystal transitions emerges. The theory uses the all-atom/continuum approach developed earlier to obtain a functional generalization of the Smoluchowski equation wherein key atomic details are embedded. The equivalent nonlocal Langevin equations are derived, and the computational aspects are discussed. The theory enables simulations that are much less computationally intensive than molecular dynamics and thus does not require oversimplification of the system’s constituent components. The equations obtained do not include factors that require calibration and can thus be applicable to various phase transitions which overcomes the limitations of phenomenological field models. The relation of the theory to phenomenological descriptions of nematic and smectic phase transitions, and the possible existence of other types of transitions involving intermolecular structural parameters are discussed.

Dispersions of hydrophilic (Aeroxide Alu C) and hydrophobic (Aeroxide Alu C 805) pyrogenic alumina $\left({\text{Al}}_2{\text{O}}_3\right)$ in liquid crystal $4^{\prime }$-n-pentyl-4-cyanobiphenyl (5CB) were investigated with deuteron nuclear magnetic resonance. The disorder effects of ${\text{Al}}_2{\text{O}}_3$ particles on the orientational order of liquid-crystal media and on the field-induced director configuration were studied as a function of alumina density in samples prepared by zero-field cooling and in-field cooling procedures. The order parameters and their variation with alumina density suggest a stronger disordering effect from the nonpolar surface of Alu C 805 particles. For dispersions of hydrophobic Alu C 805 experiments involving in-field cooling from the isotropic phase indicate that the director of “disordered” domains can be aligned, though not perfectly, by the field-aided annealing process. But the same in-field cooling procedure has shown rather limited alignment effects for hydrophilic Alu C/5CB samples. The more robust network of hydrophilic gel possibly coupled with weak liquid-crystal-network interactions could be responsible for the observed behavior. Spectra recorded during in-field cooling and within the isotropic-nematic coexistence region reveal the augmentation of the disorder strength during the transition and illustrate the effect of field-aided annealing. The stability of the aligned states as revealed by deuteron NMR is described. The results are discussed in comparison with previous studies of aerosil dispersions in alkylcyanobiphenyl.

High-resolution optical coherence tomography is used to study the onset of a large-scale convective motion in free-standing thin films of adjustable thickness containing suspensions of swimming aerobic bacteria. Clear evidence is found that beyond a threshold film thickness there exists a transition from quasi-two-dimensional collective swimming to three-dimensional turbulent behavior. The latter state, qualitatively different from bioconvection in dilute bacterial suspensions, is characterized by enhanced diffusivities of oxygen and bacteria. These results emphasize the impact of self-organized bacterial locomotion on the onset of three-dimensional dynamics, and suggest key ingredients necessary to extend standard models of bioconvection to incorporate effects of large-scale collective motion.

The self-field and the associated plasma response of an externally injected charged particle in plasmas are investigated within a self-consistent formalism that extends a formula used in plasma wake-field acceleration. The response of plasmas to such a self-field is also investigated through this self-consistent theory. This self-consistent theory leads to a correct formula of plasma wake field.

Analytical results are derived for the bond percolation threshold and the size of the giant connected component in a class of random networks with nonzero clustering. The network’s degree distribution and clustering spectrum may be prescribed and theoretical results match well with numerical simulations on both synthetic and real-world networks.

This paper explores the physics of the electromagnetic diffraction radiation of a subwavelength holes array excited by a set of evanescent waves generated by a line charge of electron beam moving parallel to the array. Activated by a uniformly moving line charge, numerous physical phenomena occur such as the diffraction radiation on both sides of the array as well as the electromagnetic penetration or transmission below or above the cut-off through the holes. As a result the subwavelength holes array becomes a radiation array. Making use of the integral equation with relevant Green’s functions, an analytical theory for such a radiation system is built up. The results of the numerical calculations based on the theory agree well with that obtained by the computer simulation. The relation among the effective surface plasmon wave, the electromagnetic penetration or transmission of the holes and the diffraction radiation is revealed. The energy dependence of and the influence of the hole thickness on the diffraction radiation and the electromagnetic penetration or transmission are investigated in detail. Therefore, a distinct diffraction radiation phenomenon is discovered.

We demonstrate the power of a recently proposed approximation scheme for the nonperturbative renormalization group that gives access to correlation functions over their full momentum range. We solve numerically the leading-order flow equations obtained within this scheme and compute the two-point functions of the $O\left(N\right)$ theories at criticality, in two and three dimensions. Excellent results are obtained for both universal and nonuniversal quantities at modest numerical cost.

According to the Einstein relation the ratio between independent measurements of fluctuations (e.g., diffusivity) and the response to a weak external field (e.g., mobility) is equal to the thermal temperature when the system is kept close to thermal equilibrium. For strongly disordered systems, which are not self-averaging this ratio is a random variable and hence in this sense the Einstein relation is not valid. Thus effective temperatures found using the fluctuation dissipation ratio are at least in some cases stochastic variables. This scenario is tested with the quenched trap model. An average over an ensemble of systems yields an averaged effective temperature, which is compared with results obtained from the mean-field continuous-time random-walk model.