A dynamical phase transition from reversible to irreversible behavior occurs when particle suspensions are subjected to uniform oscillatory shear, even in the Stokes flow limit. We consider a more general situation with nonuniform strain (e.g., oscillatory channel flow), which is observed to exhibit markedly different dynamics. The onset of irreversibility is delayed, and occurs simultaneously across the entire channel. This behavior is only partially explained by self-organization and shear-induced migration. The onset is accompanied by long-range correlated particle motion even at the channel center, where the strain is negligible; this motion prevents the system from evolving into a reversible state.

The asymptotic behaviour of a distribution function $P\left(X\right)$ for $X$ clusters is investigated in aggregation-chipping processes, where aggregation and chipping off of a finite unit of size less than $L$ take place simultaneously. Numerical simulations show that above a certain threshold ${⟨X⟩}_c$ of an average cluster size, the system exhibits partial condensation where one condensed cluster coexists with a universal power-law distribution with the exponent $-5/2$. The critical value ${⟨X⟩}_c$ is calculated and turns out to increase monotonously with $L$. The $z$-transform technique is used to analyze the case $L=2$ in detail. Obtained results agree well with numerical ones. Finally, universality of the asymptotic power law is discussed for general cases. It becomes evident that universality holds as long as the size of chipped off unit is finite.

The phase field crystal (PFC) model captures the elastic and topological properties of crystals with a single scalar field at small undercooling. At large undercooling, new foamlike behavior emerges. We characterize this foam phase of the PFC equation and propose a modified PFC equation that may be used for the simulation of foam dynamics. This minimal model reproduces von Neumann’s rule for two-dimensional dry foams and Lifshitz-Slyozov coarsening for wet foams. We also measure the coordination number distribution and find that its second moment is larger than previously reported experimental and theoretical studies of soap froths, a finding that we attribute to the wetness of the foam increasing with time.

We study the statistical properties of the recurrence intervals $\tau$ between successive trading volumes exceeding a certain threshold $q$. The recurrence interval analysis is carried out for the 20 liquid Chinese stocks covering a period from January 2000 to May 2009, and two Chinese indices from January 2003 to April 2009. Similar to the recurrence interval distribution of the price returns, the tail of the recurrence interval distribution of the trading volumes follows a power-law scaling, and the results are verified by the goodness-of-fit tests using the Kolmogorov-Smirnov (KS) statistic, the weighted KS statistic and the Cramér–von Mises criterion. The measurements of the conditional probability distribution and the detrended fluctuation function show that both short-term and long-term memory effects exist in the recurrence intervals between trading volumes. We further study the relationship between trading volumes and price returns based on the recurrence interval analysis method. It is found that large trading volumes are more likely to occur following large price returns, and the comovement between trading volumes and price returns is more pronounced for large trading volumes.

Bred vectors are a type of finite perturbation used in prediction studies of atmospheric models that exhibit spatially extended chaos. We study the structure, spatial correlations, and the growth rates of logarithmic bred vectors (which are constructed by using a given norm). We find that, after a suitable transformation, logarithmic bred vectors are roughly piecewise copies of the leading Lyapunov vector. This fact allows us to deduce a scaling law for the bred vector growth rate as a function of its amplitude. In addition, we relate growth rates with the spectrum of Lyapunov exponents corresponding to the most expanding directions. We illustrate our results with simulations of the Lorenz 1996 model.

In this work, we study the one-dimensional stabilized Kuramoto Sivashinsky equation with additive uncorrelated stochastic noise. The Eckhaus stable band of the deterministic equation collapses to a narrow region near the center of the band. This is consistent with the behavior of the phase diffusion constants of these states. Some connections to the phenomenon of state selection in driven out of equilibrium systems are made.

Classically, the net action of nonlinear turbulent processes is interpreted as either a direct or inverse cascade. However, in nonuniform/shear flows the dominant process is a nonlinear redistribution over wave number angle of perturbation spatial Fourier harmonics. We call this process a nonlinear transverse redistribution (NTR). This phenomenon is demonstrated for a simple two-dimensional constant shear (non-normal) flow by numerically simulating the nonlinear dynamics of coherent and stochastic vortical perturbations in the flow. NTR is a general feature of nonlinear processes that should manifest itself in nonuniform engineering, environmental, and astrophysical flows. The conventional characterization of turbulence in terms of direct and inverse cascades, which ignores NTR, appears to be misleading for shear flow turbulence. We focus on the action of nonlinear processes on the spectral energy. NTR redistributes perturbations over different quadrants of the wave number plane and the interplay of this nonlinear redistribution with linear phenomena becomes intricate: it can realize either positive or negative feedback. In the case of positive feedback, it repopulates the quadrants in wave number space where the shear flow induces linear transient growth.

We study the directional-ordering transition in the two-dimensional classical and quantum compass models on the square lattice by means of Monte Carlo simulations. An improved algorithm is presented which builds on the Wolff cluster algorithm in one-dimensional subspaces of the configuration space. This improvement allows us to study classical systems up to $L=512$. Based on this algorithm, we give evidence for the presence of strongly anomalous scaling for periodic boundary conditions which is much worse than anticipated before. We propose and study alternative boundary conditions for the compass model which do not make use of extended configuration spaces and show that they completely remove the problem with finite-size scaling. In the last part, we apply these boundary conditions to the quantum problem and present a considerably improved estimate for the critical temperature which should be of interest for future studies on the compass model. Our investigation identifies a strong one-dimensional magnetic ordering tendency with a large correlation length as the cause of the unusual scaling and moreover allows for a precise quantification of the anomalous length scale involved.

We measure the two-point density correlations and Voronoi cell distributions of cyclically sheared granular spheres obtained with a fluorescence technique and compare them with random packing of frictionless spheres. We find that the radial distribution function $g\left(r\right)$ is captured by the Percus-Yevick equation for initial volume fraction $\varphi =0.59$. However, small but systematic deviations are observed because of the splitting of the second peak as $\varphi$ is increased toward random close packing. The distribution of the Voronoi free volumes deviates from postulated $\Gamma$ distributions, and the orientational order metric $Q_6$ shows local order but no long range order. Overall, these measures show significant similarity of random packing of granular and frictionless spheres, but some systematic differences as well.

Nematic solids respond strongly to changes in ambient heat or light, significantly differently parallel and perpendicular to the director. This phenomenon is well characterized for uniform director fields but not for defect textures. We analyze the elastic ground states of a nematic glass in the membrane approximation as a function of temperature for some disclination defects with an eye toward reversibly inducing three-dimensional shapes from flat sheets of material, at the nanoscale all the way to macroscopic objects, including nondevelopable surfaces. The latter offers a paradigm to actuation via switchable stretch in thin systems.

We study time-dependent correlation functions of classical and quantum gases using methods of equilibrium statistical mechanics for systems of uniform as well as nonuniform densities. The basis for our approach is the path integral formalism of quantum mechanical systems. With this approach the statistical mechanics of a quantum mechanical system becomes the equivalent of a classical polymer problem in four dimensions where imaginary time is the fourth dimension. Several nontrivial results for quantum systems have been obtained earlier by this analogy. Here, we will focus upon the presence of a time-dependent electromagnetic pair interaction where the electromagnetic vector potential that depends upon currents, will be present. Thus both density and current correlations are needed to evaluate the influence of this interaction. Then we utilize that densities and currents can be expressed by polarizations by which the ionic fluid can be regarded as a dielectric one for which a nonlocal susceptibility is found. This nonlocality has as a consequence that we find no contribution from a possible transverse electric zero-frequency mode for the Casimir force between metallic plates. Further, we establish expressions for a leading correction to ab initio calculations for the energies of the quantized electrons of molecules where now retardation effects also are taken into account.

We consider the infinite-range spin glass in which the spins have $m>1$ components (a vector spin glass). Applying a magnetic field which is random in direction, there is a de Almeida-Thouless (AT) line below which the “replica symmetric” solution is unstable, just as for the Ising $\left(m=1\right)$ case. We calculate the location of this AT line for Gaussian random fields for arbitrary $m$ and verify our results by numerical simulations for $m=3$.

The density relaxation phenomenon is modeled using both Monte Carlo and discrete element simulations to investigate the effects of regular taps applied to a vessel having a planar floor filled with monodisperse spheres. Results suggest the existence of a critical tap intensity which produces a maximum bulk solids fraction. We find that the mechanism responsible for the relaxation phenomenon is an evolving ordered packing structure propagating upwards from the plane floor.

We consider the finite generation-time effect in virus evolution models, introducing differential equations with delay. The suggested approach more adequately describes the evolution in case of growing populations than the popular models of population genetics, especially for the viruses with large number of offspring during one life cycle. Now the mean fitness, as a coefficient for exponential population growth, could not be defined via instant characteristics of the model. For the constant population size the finite generation-time does not affect mean fitness in the steady state. The growing virus population is characterized by different fitness than the population with a constant size.

We undertake a systematic numerical exploration of self-organized states in a deterministic model of interacting self-propelled particles in two dimensions. In the process, we identify various types of collective motion, namely, disordered swarms, rings, and droplets. We construct a “phase diagram,” which summarizes our results as it delineates phase transitions (all discontinuous) between disordered swarms and vortical flocks on one hand and bound vortical flocks and expanding formations on the other. One of the transition lines is found to have a close analogy with the temperature-driven gas-liquid transition, in finite clusters with the same interparticle potential. Furthermore, we report on a type of flocking which takes place in the presence of a uniform external driver. Altogether, our results set a rather firm stage for experimental refinement and/or falsification of this class of models.

A corrected second-order slip boundary condition is proposed to solve the Navier-Stokes equations for fluid flows confined in parallel-plate nanochannels. Compared with the classical second-order slip boundary condition proposed by Beskok and Karniadakis, the corrected slip boundary condition is not only dependent on the Knudsen number and the tangential momentum accommodation coefficient, but also dependent on the relative position of the slip surface in the Knudsen layer. For the fluid flows in slip-flow regime with the Knudsen number less than 0.3, Couette cell is investigated using molecular-dynamics simulations to verify Newtonian flow behaviors by examining the constitutive relationship between shear stress and strain rate. By comparing the velocity profiles of Poiseuille flows predicted from the Navier-Stokes equations with the corrected slip boundary condition with that from molecular-dynamics simulations, it is found that the flow behaviors in our models can be effectively captured.

Besides the well-known effect of autocorrelations in time series of Monte Carlo simulation data resulting from the underlying Markov process, using the same data pool for computing various estimates entails additional cross correlations. This effect, if not properly taken into account, leads to systematically wrong error estimates for combined quantities. Using a straightforward recipe of data analysis employing the jackknife or similar resampling techniques, such problems can be avoided. In addition, a covariance analysis allows for the formulation of optimal estimators with often significantly reduced variance as compared to more conventional averages.

Particle diffusion in a two-dimensional curved surface embedded in $R_3$ is considered. In addition to the usual diffusion flow, we find a flow with an explicit curvature dependence. Diffusion equation is obtained in $ϵ$ (thickness of surface) expansion. As an example, the surface of elliptic cylinder is considered, and curvature-dependent diffusion coefficient is calculated.

Mechanical properties of structural protein materials are crucial for our understanding of biological processes and disease states. Through utilization of molecular simulation based on stress wave tracking, we investigate mechanical energy transfer processes in fibrous beta-sheet-rich proteins that consist of highly ordered hydrogen bond (H-bond) networks. By investigating four model proteins including two morphologies of amyloids, beta solenoids, and silk beta-sheet nanocrystals, we find that all beta-sheet-rich protein fibrils provide outstanding elastic moduli, where the silk nanocrystal reaches the highest value of $\approx 40\text{GPa}$. However, their capacities to dissipate mechanical energy differ significantly and are controlled strongly by the underlying molecular structure of H-bond network. Notably, silk beta-sheet nanocrystals feature a ten times higher energy damping coefficient than others, owing to flexible intrastrand motions in the transverse directions. The results demonstrate a unique feature of silk nanocrystals, their capacity to simultaneously provide extreme stiffness and energy dissipation capacity. Our results could help one to explain the remarkable properties of silks from an atomistic and molecular perspective, in particular its great toughness and energy dissipation capacity, and may enable the design of multifunctional nanomaterials with outstanding stiffness, strength, and impact resistance.

It is known that the relaxation of the shape fluctuations of nearly spherical lipid vesicles is accompanied by a lateral displacement of the monolayers, comprising their bilayers. In this work a dissipation mechanism of the mechanical energy stored in the fluctuation is revealed that concerns the viscous friction of the flow in the liquid around the vesicle caused by this displacement. The time correlation functions of each of the vesicle’s fluctuation modes are calculated as a function of the mechanical and rheological properties of the system which are the tension of the vesicle bilayer, its bending elasticities at free and blocked flip-flop, the viscosities of the liquids bathing the bilayer, the friction coefficient between the two monolayers, as well as the vesicle’s dimensions: its bilayer thickness and radius. The correlations of the shape fluctuations of nearly spherical emulsion droplets are also calculated for different viscosities of the liquid inside and outside the droplet.