We present a numerical study of the anomalies in transport and diffusion of overdamped Brownian particles in totally disordered potential landscapes in one and in two dimensions. We characterize and analyze the effects of three different disordered potentials. The anomalous regimes are characterized by the time exponents that exhibit the statistical moments of the ensemble of particle trajectories. The anomaly in the transport is always of the subtransport type, but diffusion presents a greater variety of anomalies: Both subdiffusion and superdiffusion are possible. In two dimensions we present a mixed anomaly: subdiffusion in the direction perpendicular to the force and superdiffusion in the parallel direction.

The adoption of the formalism of fractional calculus is an elegant way to simulate either subdiffusion or superdiffusion from within a renewal perspective where the occurrence of an event at a given time $t$ does not have any memory of the events occurring at earlier times. We illustrate a physical model to assign infinite memory to renewal anomalous diffusion and we find (i) a condition where the simultaneous action of a renewal and a memory source of subdiffusion generates localization and (ii) a condition where they make subdiffusion weaker and superdiffusion emerge. We argue that our approach may provide important contributions to the current search to distinguish the renewal from the memory source of subdiffusion.

We numerically examine colloidal particles driven over a muffin tin substrate. Previous studies of this model identified a variety of commensurate and incommensurate static phases in which topological defects can form domain walls, ordered stripes, superlattices, or disordered patchy regimes as a function of the filling fraction. Here, we show that the addition of an external drive to these static phases can produce distinct dynamical responses. At incommensurate fillings the flow occurs in the form of localized pulses or solitons correlated with topological defect structures. Transitions between different modes of motion can occur as a function of increasing drive. We measure the average particle velocity for specific ranges of external drive and show that changes in the velocity response correlate with changes in the topological defect arrangements. We also demonstrate that in the different dynamic phases, the particles have distinct trajectories and velocity distributions. Dynamic transitions between ordered and disordered flows exhibit hysteresis, while in strongly disordered regimes there is no hysteresis and the velocity-force curves are smooth. When stripe patterns are present, transport can occur at an angle to the driving direction.

A fundamental problem in the physics of amorphous materials is understanding the transition from reversible to irreversible plastic behavior and its connection to yield. Currently, continuum material modeling relies on phenomenological yield thresholds, however in many cases the transition from elastic to plastic behavior is gradual, which makes it difficult to identify an exact yield criterion. Here we show that under periodic shear, amorphous solids undergo a transition from repetitive, predictable behavior to chaotic, irregular behavior as a function of the strain amplitude. In both the periodic and chaotic regimes, localized particle rearrangements are observed. We associate the point of transition from repetitive to chaotic behavior with the yield strain and suggest that at least for oscillatory shear, yield in amorphous solids is a result of a “transition to chaos.”

The Onsager theory is known to be inaccurate in its prediction of the critical transition density for small aspect ratio hard rods. In this paper we generalize the Onsager theory in two dimensions by taking into account the short-range order as well as the higher-order virial coefficients, up to the fourth order. By carrying out molecular dynamics (MD) simulations on “molecules” comprising linked hard disks with an aspect ratio $\ell$ ranging from 5 to 13, we show that the generalized theory is much improved as compared to the traditional theory, with its predictions of the transition density agreeing well with the simulation results. This indicates the importance of short-range order considerations (in conjunction with steric repulsion) for molecules with $\ell \le 10$, a group which includes the most commonly encountered thermotropic liquid crystals. MD simulations further yield evidence for hexagonal order for molecules with $\ell \le 8$, indicating an intermediate hexagonal phase before solidifying at higher densities.

A numerical gradient flow procedure was devised to characterize minimal energy shapes of fusion pores connecting two parallel planar bilayer membranes. Pore energy, composed of splay, tilt, and stretching, was obtained by modeling each bilayer as two monolayers and treating each monolayer of a bilayer membrane as a freely deformable surface described with a mean lipid orientation field. Voids between the two monolayers were prevented by a steric penalty formulation. Pore shapes were assumed to possess both axial and reflectional symmetry. For fixed pore radius and bilayer separation, the gradient flow procedure was applied to initially toroidal pore shapes. Using initially elliptical pore shapes yielded the same final shape. The resulting minimal pore shapes and energies were analyzed as a function of pore dimension and lipid composition. Previous studies either assumed or confined pore shapes, thereby tacitly supplying an unspecified amount of energy to maintain shape. The shapes derived in the present study were outputs of calculations and an externally provided energy was not supplied. Our procedure therefore yielded energy minima significantly lower than those reported in prior studies. The membrane of minimal energy pores bowed outward near the pore lumen, yielding a pore length that exceeded the distance between the two fusing membranes.

The boundary behavior of axisymmetric microswimming squirmers is theoretically explored within an inertialess Newtonian fluid for a no-slip interface and also a free surface in the small capillary number limit, preventing leading-order surface deformation. Such squirmers are commonly presented as abridged models of ciliates, colonial algae, and Janus particles and we investigate the case of low-mode axisymmetric tangential surface deformations with, in addition, the consideration of a rotlet dipole to represent torque-motor swimmers such as flagellated bacteria. The resulting boundary dynamics reduces to a phase plane in the angle of attack and distance from the boundary, with a simplifying time-reversal duality. Stable swimming adjacent to a no-slip boundary is demonstrated via the presence of stable fixed points and, more generally, all types of fixed points as well as stable and unstable limit cycles occur adjacent to a no-slip boundary with variations in the tangential deformations. Nonetheless, there are constraints on swimmer behavior—for instance, swimmers characterized as pushers are never observed to exhibit stable limit cycles. All such generalities for no-slip boundaries are consistent with observations and more geometrically faithful simulations to date, suggesting the tangential squirmer is a relatively simple framework to enable predications and classifications for the complexities associated with axisymmetric boundary swimming. However, in the presence of a free surface, with asymptotically small capillary number, and thus negligible leading-order surface deformation, no stable surface swimming is predicted across the parameter space considered. While this is in contrast to experimental observations, for example, the free-surface accumulation of sterlet sperm, extensive surfactants are present, most likely invalidating the low capillary number assumption. In turn, this suggests the necessity of surface deformation for stable free-surface three-dimensional finite-size microswimming, as previously highlighted in a two-dimensional mathematical study of singularity swimmers [Crowdy et al., J. Fluid Mech. 681, 24 (2011)].

This paper provides time-dependent expressions for the expected degree distribution of a given network that is subject to growth. We consider both uniform attachment, where incoming nodes form links to existing nodes selected uniformly at random, and preferential attachment, where probabilities are assigned proportional to the degrees of the existing nodes. We consider the cases of single and multiple links being formed by each newly introduced node. The initial conditions are arbitrary, that is, the solution depends on the degree distribution of the initial graph which is the substrate of the growth. Previous work in the literature focuses on the asymptotic state, that is, when the number of nodes added to the initial graph tends to infinity, rendering the effect of the initial graph negligible. Our contribution provides a solution for the expected degree distribution as a function of time, for arbitrary initial condition. Previous results match our results in the asymptotic limit. The results are discrete in the degree domain and continuous in the time domain, where the addition of new nodes to the graph are approximated by a continuous arrival rate.

Epidemics have so far been mostly studied in undirected networks. However, many real-world networks, such as the online social network Twitter and the world wide web, on which information, emotion, or malware spreads, are directed networks, composed of both unidirectional links and bidirectional links. We define the directionality $\xi$ as the percentage of unidirectional links. The epidemic threshold ${\tau }_c$ for the susceptible-infected-susceptible (SIS) epidemic is lower bounded by $1/{\lambda }_1$ in directed networks, where ${\lambda }_1$, also called the spectral radius, is the largest eigenvalue of the adjacency matrix. In this work, we propose two algorithms to generate directed networks with a given directionality $\xi$. The effect of $\xi$ on the spectral radius ${\lambda }_1$, principal eigenvector $x_1$, spectral gap $({\lambda }_1-\left|{\lambda }_2\right|),$ and algebraic connectivity ${\mu }_{N-1}$ is studied. Important findings are that the spectral radius ${\lambda }_1$ decreases with the directionality $\xi$, whereas the spectral gap and the algebraic connectivity increase with the directionality $\xi$. The extent of the decrease of the spectral radius depends on both the degree distribution and the degree-degree correlation ${\rho }_D$. Hence, in directed networks, the epidemic threshold is larger and a random walk converges to its steady state faster than that in undirected networks with the same degree distribution.

Generic Hamiltonian systems have a mixed phase space, where classically disjoint regions of regular and chaotic motion coexist. We present an iterative method to construct an integrable approximation $H_{\mathrm{reg}}$, which resembles the regular dynamics of a given mixed system $H$ and extends it into the chaotic region. The method is based on the construction of an integrable approximation in action representation which is then improved in phase space by iterative applications of canonical transformations. This method works for strongly perturbed systems and arbitrary degrees of freedom. We apply it to the standard map and the cosine billiard.

Transient patterns in a bistable ring of bidirectionally coupled sigmoidal neurons were studied. When the system had a pair of spatially uniform steady solutions, the instability of unstable spatially nonuniform steady solutions decreased exponentially with the number of neurons because of the symmetry of the system. As a result, transient spatially nonuniform patterns showed dynamical metastability: Their duration increased exponentially with the number of neurons and the duration of randomly generated patterns obeyed a power-law distribution. However, these metastable dynamical patterns were easily stabilized in the presence of small variations in coupling strength. Metastable rotating waves and their pinning in the presence of asymmetry in the direction of coupling and the disappearance of metastable dynamical patterns due to asymmetry in the output function of a neuron were also examined. Further, in a two-dimensional array of neurons with nearest-neighbor coupling, intrinsically one-dimensional patterns were dominant in transients, and self-excitation in these neurons affected the metastable dynamical patterns.

The statistics of velocity fluctuations of turbulent Taylor-Couette flow are examined. The rotation rates of the inner and outer cylinders are varied while keeping the Taylor number fixed to $1.49\times {10}^{12}$ [$O\left(\text{Re}\right)={10}^6$]. The azimuthal velocity component of the flow is measured using laser Doppler anemometry. For each experiment $5\times {10}^6$ data points are acquired and carefully analyzed. Using extended self-similarity [Benzi et al., Phys. Rev. E 48, R29 (1993)] the longitudinal structure function exponents are extracted and are found to weakly depend on the ratio of the rotation rates. For the case where only the inner cylinder rotates the results are in good agreement with results measured by Lewis and Swinney [Phys. Rev. E 59, 5457 (1999)] using hot-film anemometry. The power spectra show clear $-5/3$ scaling for the intermediate angular velocity ratios $-{\omega }_o/{\omega }_i\in \{0.6,0.8,1.0\}$, roughly $-5/3$ scaling for $-{\omega }_o/{\omega }_i\in \{0.2,0.3,0.4,2.0\}$, and no clear scaling law can be found for $-{\omega }_0/{\omega }_i=0$ (inner cylinder rotation only); the local scaling exponent of the spectra has a strong frequency dependence. We relate these observations to the shape of the probability density function of the azimuthal velocity and the presence of a neutral line.

This work discusses the role of gravity on the Faraday instability, and the differences one can expect to observe in a low-gravity experiment when compared to an earth-based system. These differences are discussed in the context of the viscous linear theory for laterally infinite systems, and a surprising result of the analysis is the existence of a crossover frequency where an interface in low gravity switches from being less to more stable than an earth-based system. We propose this crossover exists in all Faraday systems, and the frequency at which it occurs is shown to be strongly influenced by layer height. In presenting these results physical explanations are provided for the behavior of the predicted forcing amplitude thresholds and wave number selection.

We employ a weakly nonlinear perturbative scheme to investigate the stretch flow of a non-Newtonian fluid confined in Hele-Shaw cell for which the upper plate is lifted. A generalized Darcy's law is utilized to model interfacial fingering formation in both the weak shear-thinning and weak shear-thickening limits. Within this context, we analyze how the interfacial finger shapes and the nonlinear competition dynamics among fingers are affected by the non-Newtonian nature of the stretched fluid.

For quantum particles, it is well known that static disorder would lead to Anderson localization (AL) while dynamic (evolving) disorder would destroy AL and facilitate the transport. In this article, we study the transport behavior of a quantum particle in weak dynamic disorder. Based on Wigner representation, we obtain the radiative transfer equation (a linear Boltzmann equation) in the weak dynamic disorder limit, which could lead to not only all the existing transport behaviors in the literature but also new transport behaviors (for example, Lévy flight in momentum space). Furthermore, for dimensions greater than one, though we can formally derive the diffusive transport approximation, we argue that this diffusive transport is not physical but the nondiffusive transport should persist forever. This provides a possible resolution for the long-standing puzzle whether diffusive or nondiffusive transport would prevail in the long time limit. Our result would have major implications for the hypertransport of light, matter wave dynamics in disordered media, and directed polymer problems.

We study a one-dimensional random walk with memory. The behavior of the walker is modified with respect to the simple symmetric random walk only when he or she is at the maximum distance ever reached from his or her starting point (home). In this case, having the choice to move farther or to move closer, the walker decides with different probabilities. If the probability of a forward step is higher then the probability of a backward step, the walker is bold, otherwise he or she is timorous. We investigate the asymptotic properties of this bold-timorous random walk, showing that the scaling behavior varies continuously from subdiffusive (timorous) to superdiffusive (bold). The scaling exponents are fully determined with a new mathematical approach based on a decomposition of the dynamics in active journeys (the walker is at the maximum distance) and lazy journeys (the walker is not at the maximum distance).

A new connection between maximum-power Curzon-Ahlborn thermal cycles and maximum-work reversible cycles is proposed. This linkage is built through a mapping between the exponents of a class of heat transfer laws and the exponents of a family of heat capacities depending on temperature. This connection leads to the recovery of known results and to a wide and interesting set of results for a class of thermal cycles. Among other results it was found that it is possible to use analytically closed expressions for maximum-work efficiencies to calculate good approaches to maximum-power efficiencies. Behind the proposed connection is an interpretation of endoreversibility hypothesis. Additionally, we suggest that certain reversible maximum-work cycles depending on working substance can be used as reversible landmarks for FTT maximum-power cycles, which also depend on working substance properties.

It is well known that density inhomogeneities at the solid-liquid interface can have a strong effect on the velocity profile of a nanoconfined fluid in planar Poiseuille flow. However, it is difficult to control the density inhomogeneities induced by solid walls, making this type of system unsuitable for a comprehensive study of the effect on density inhomogeneity on nanofluidic flow. In this paper, we employ an external force compatible with periodic boundary conditions to induce the density variation, which greatly simplifies the problem when compared to flow in nonperiodic nanoconfined systems. Using the sinusoidal transverse force method to produce shearing velocity profiles and the sinusoidal longitudinal force method to produce inhomogeneous density profiles, we are able to observe the interactions between the two property inhomogeneities at the level of individual Fourier components. This gives us a method for direct measurement of the coupling between the density and velocity fields and allows us to introduce various feedback control mechanisms which customize fluid behavior in individual Fourier components. We briefly discuss the role of temperature inhomogeneity and consider whether local thermal expansion due to nonuniform viscous heating is sufficient to account for shear-induced density inhomogeneities. We also consider the local Newtonian constitutive relation relating the shear stress to the velocity gradient and show that the local model breaks down for sufficiently large density inhomogeneities over atomic length scales.

It has recently been shown that the computing abilities of Boltzmann machines, or Ising spin-glass models, can be implemented by chaotic billiard dynamics without any use of random numbers. In this paper, we further numerically investigate the capabilities of the chaotic billiard dynamics as a deterministic alternative to random Monte Carlo methods by applying it to classical spin models in statistical physics. First, we verify that the billiard dynamics can yield samples that converge to the true distribution of the Ising model on a small lattice, and we show that it appears to have the same convergence rate as random Monte Carlo sampling. Second, we apply the billiard dynamics to finite-size scaling analysis of the critical behavior of the Ising model and show that the phase-transition point and the critical exponents are correctly obtained. Third, we extend the billiard dynamics to spins that take more than two states and show that it can be applied successfully to the Potts model. We also discuss the possibility of extensions to continuous-valued models such as the $XY$ model.