Tailored mixing inside individual droplets could be useful to ensure that reactions within microscopic discrete fluid volumes, which are used as microreactors in “digital microfluidic” applications, take place in a controlled fashion. In this paper we consider a translating spherical liquid drop to which we impose a time periodic rigid-body rotation. Such a rotation not only induces mixing via chaotic advection, which operates through the stretching and folding of material lines, but also offers the possibility of tuning the mixing by controlling the location and size of the mixing region. Tuned mixing is achieved by judiciously adjusting the amplitude and frequency of the rotation, which are determined by using a resonance condition and following the evolution of adiabatic invariants. As the size of the mixing region is increased, complete mixing within the drop is obtained.

The behavior of an atom in a nonuniform magnetic field is analyzed, as well as the motion of a classical magnetic dipole (a spinning charged ball) and a rotating charged ring. For the atom it is shown that, while the magnetic field does no work on the electron-orbital contribution to the magnetic moment (the source of translational kinetic energy being the internal energy of the atom), whether or not it does work on the electron-spin contribution to the magnetic moment depends on whether the electron has an intrinsic rotational kinetic energy associated with its spin. A rotational kinetic energy for the electron is shown to be consistent with the Dirac equation. If the electron does have a rotational kinetic energy, the acceleration of a silver atom in a Stern-Gerlach experiment or the emission of a photon from an electron spin flip can be explained without requiring the magnetic field to do work. For a constant magnetic field gradient along the $z$ axis, it is found that the classical objects oscillate in simple harmonic motion along the $z$ axis, the total kinetic energy—translational plus rotational—being a constant of the motion. For the charged ball, the change in rotational kinetic energy is associated only with a change in the precession frequency, the rotation rate about the figure axis remaining constant.

In their textbook, Suzuki and Varga [Stochastic Variational Approach to Quantum-Mechanical Few-Body Problems (Springer, Berlin, 1998)] present the stochastic variational method with the correlated Gaussian basis in a very exhaustive way. The matrix elements for central potentials are expressed in an elegant form, but the elements for spin-dependent operators, when treated, are given as very cumbersome expressions. We find many formulas for those elements. Their expressions are given in terms of the same geometrical functions that appear in the case of central potentials. These functions obtain therefore a universal status; this property is very useful for numerical applications.

The standard lattice Boltzmann equation (LBE) is inadequate for simulating gas flows with a large Knudsen number. In this paper we propose a generalized lattice Boltzmann equation with effective relaxation times based on a recently developed generalized Navier-Stokes constitution [Guo et al., Europhys Lett. 80, 24001 (2007)] for nonequilibrium flows. A kinetic boundary condition corresponding to a generalized second-order slip scheme is also designed for the model. The LBE model and the boundary condition are analyzed for a unidirectional flow, and it is found that in order to obtain the generalized Navier-Stokes equations, the relaxation times must be properly chosen and are related to the boundary condition. Numerical results show that the proposed method is able to capture the Knudsen layer phenomenon and can yield improved predictions in comparison with the standard lattice Boltzmann equation.

We study the Zhang model of sandpile on a one-dimensional chain of length $L$, where a random amount of energy is added at a randomly chosen site at each time step. We show that in spite of this randomness in the input energy, the probability distribution function of energy at a site in the steady state is sharply peaked, and the width of the peak decreases as $L^{-1/2}$ for large $L$. We discuss how the energy added at one time is distributed among different sites by topplings with time. We relate this distribution to the time-dependent probability distribution of the position of a marked grain in the one-dimensional Abelian model with discrete heights. We argue that in the large $L$ limit, the variance of energy at site $x$ has a scaling form $L^{-1}g\left(x/L\right)$, where $g\left(\xi \right)$ varies as $\text{ln}\left(1/\xi \right)$ for small $\xi$, which agrees very well with the results from numerical simulations.

We investigate the memory effects present in the asymptotic dynamics of a classical harmonic oscillator governed by a generalized Langevin equation. Using Laplace analysis together with Tauberian theorems we derive asymptotic expressions for the mean values, variances, and velocity autocorrelation function in terms of the long-time behavior of the memory kernel and the correlation function of the random force. The internal and external noise cases are analyzed. A simple criterion to determine if the diffusion process is normal or anomalous is established.

We have investigated by numerical simulation the density segregation of fine equal-sized bronze and glass particles subject to vertical vibrations. The model was found to be capable of predicting the two main segregation forms (“bronze on top” and “sandwich”) in roughly the same regions of the phase diagram as was found experimentally by Burtally et al. We investigated the effects of pressure air forcing, friction and restitution of kinetic energy in collisions, and box size on the segregation behavior. We find that next to the interstitial air friction also has a large influence on the formation of the sandwich structure.

A lattice Boltzmann model was used to simulate the accelerated transport of dense inert particles in low Reynolds number flows in smooth- and rough-walled narrow channels. The simulations showed that, after an initial transient, an initially immobile particle migrated faster than the average fluid velocity. The sensitivity of the particle residence time to wall roughness increased with decreasing Reynolds numbers. The relationship between the exit position and residence time of a particle was sensitive to the release position, flow strength, and the wall roughness. A particle with a density 5% larger than the density of the fluid migrated to an equilibrium position between the centerline and the wall for the slowest flow rates in rough-walled channels, displaying the Segre-Silberberg effect that a rigid neutrally buoyant spherical particle exhibits in small Reynolds number flows. However, a particle that was 35% denser than the density of the fluid drifted to the centerline in the slowest flows due to the gravitational settling effect. The difference in the residence time of the less-dense and dense particles was most sensitive to the surface roughness at the smallest Reynolds number investigated.

Shear localization in an aqueous clay suspension of Laponite is investigated through dynamic light scattering, which provides access both to the dynamics of the system (homodyne mode) and to the local velocity profile (heterodyne mode). When shear bands form, a relaxation of the dynamics typical of a gel phase is observed in both bands soon after the flow stops. Periodic oscillations of the flow behavior, typical of a stick-slip phenomenon, are also observed when shear localization occurs. Both results are discussed in the light of various theoretical models for soft glassy gels.

The melting of a self-organized system composed of classical particles confined in a two-dimensional parabolic trap and interacting through a potential with a short-range attractive part and a long-range repulsive potential is studied. Different behaviors of the melting temperature are found depending on the strength $\left(B\right)$ of the attractive part of the interparticle potential. The melting of a system consisting of small bubbles takes place through a two-step melting process. A reentrant behavior and a thermally induced structural phase transition are observed in a small region of the $\left(B,\kappa \right)$ space. A hysteresis effect in the configuration of the particles is observed as a function of temperature. This is a consequence of the presence of a potential barrier between different configurations of the system.

Classically, there have been two different ways to obtain mean-field theories for liquid crystals. One is based on short-range repulsive steric forces and the other on long-range attractive dispersion forces. In the former approach, it is the anisotropic shape of the molecules that leads to the anisotropic interaction, and in the latter it is the anisotropy of the molecular polarizability. In real molecules both causes of anisotropy can be expected to contribute to the effective interaction, and so it is desirable to assess the combined effect of anisotropic long-range attraction and short-range repulsion. Here we present an avenue to this end. We start from dispersion forces interactions and combine them with hard-core repulsions in a formal theory, whose crucial element is the steric tensor, a fourth-rank tensor depending on the anisotropy of the interacting molecules. This tensor can be determined analytically for a special class of molecular shapes.

We employ dissipative particle dynamics (DPD) to examine the effects of composition and interaction parameter on the resulting phase behavior of $ABC$ star copolymers. Here, we assume that the interaction parameters among the three components are equal. When the three components have comparable volume fractions, our DPD results illustrate that the unique formation of various types of three-phase separated polygonal cylinders is mainly dominated by the composition but not influenced by the interaction parameter. In contrast, when two of the three components are minor, the resulting morphology type is greatly influenced by the interaction parameter. Generally speaking, with an increase in the interaction parameter, the two minority components first act like one component and the system forms a one-length-scale ordered microstructure. Then a further segregation between the two minority components within the large-length-scale phase can be induced as the interaction parameter keeps increasing. In general, our DPD results, a systematic study of the morphological transition behavior obtained by varying the interaction parameter and composition, bridge the gap between the previous theoretical results in the strong and weak segregation regimes via Monte Carlo and two-dimensional self-consistent mean-field methods, respectively.

We consider the interaction of multivalent counterions with spherical polyelectrolyte brushes (SPBs). SPBs result if linear polyelectrolyte chains (contour length 60 nm) are densely grafted to colloidal spheres of 116 nm in diameter. When dispersed in water the surface layer, consisting of chains of the strong polyelectrolyte poly(styrene sulfonic acid), will swell. Recent work [Mei et al., Phys. Rev. Lett. 97, 158301 (2006)] has demonstrated that spherical polyelectrolyte brushes undergo a collapse in the presence of a mixture of monovalent and multivalent counterions. The collapse crossover could be well described by a mean-field approach. Here we demonstrate that the application of a mean-field approach is well founded by simulation results done with molecular dynamics (MD). MD simulations show that over a wide range of multivalent counterion concentration the effects of ion correlation and fluctuations can be neglected. Higher-valent counterions are shown to interact strongly with the polyelectrolyte chains of the SPBs and thus exhibit a much reduced osmotic activity in the system. This reduction is the driving force for the collapse.

We investigate in detail the model of a trophic web proposed by Amaral and Meyer [Phys. Rev. Lett. 82, 652 (1999)]. We focus on small-size systems that are relevant for real biological food webs and for which the fluctuations play an important role. We show, using Monte Carlo simulations, that such webs can be nonviable, leading to extinction of all species in small and/or weakly coupled systems. Estimations of the extinction times and survival chances are also given. We show that before the extinction the fraction of highly connected species (“omnivores”) is increasing. Viable food webs exhibit a pyramidal structure, where the density of occupied niches is higher at lower trophic levels, and moreover the occupations of adjacent levels are closely correlated. We also demonstrate that the distribution of the lengths of food chains has an exponential character and changes weakly with the parameters of the model. On the contrary, the distribution of avalanche sizes of the extinct species depends strongly on the connectedness of the web. For rather loosely connected systems, we recover the power-law type of behavior with the same exponent as found in earlier studies, while for densely connected webs the distribution is not of a power-law type.

We study the spatiotemporal properties of coherent states (peaks, holes, and fronts) in a bistable activator-inhibitor system that exhibits biochemical saturated autocatalysis, and in which fronts do not preserve spatial parity symmetry. Using the Gierer-Meinhardt prototype model, we find the conditions in which two distinct pinning regions are formed. The first pinning type is known in the context of variational systems while the second is structurally different due to the presence of a heteroclinic bifurcation between two uniform states. The bifurcation also separates the parameter regions of counterpropagating fronts, leading in turn to the growth or contraction of activator domains. These phenomena expand the range of pattern formation theory and its biomedical applications: activator domain retraction suggests potential therapeutic strategies for patterned pathologies, such as cardiovascular calcification.

The way cooperation organizes dynamically strongly depends on the topology of the underlying interaction network. We study this dependence using heterogeneous scale-free networks with different levels of (a) degree-degree correlations and (b) enhanced clustering, where the number of neighbors of connected nodes are correlated and the number of closed triangles are enhanced, respectively. Using these networks, we analyze a finite population analog of the evolutionary replicator dynamics of the prisoner’s dilemma, the latter being a two-player game with two strategies, defection and cooperation, whose payoff matrix favors defection. Both topological features significantly change the dynamics with respect to the one observed for fully randomized scale-free networks and can strongly facilitate cooperation even for a large temptation to defect, and should hence be considered as important factors in the evolution and sustainment of cooperation.

We demonstrate that the dynamics toward and within the Feigenbaum attractor combine to form a $q$-deformed statistical-mechanical construction. The rate at which ensemble trajectories converge to the attractor (and to the repellor) is described by a $q$ entropy obtained from a partition function generated by summing distances between neighboring positions of the attractor. The values of the $q$ indices involved are given by the unimodal map universal constants, while the thermodynamic structure is closely related to that formerly developed for multifractals. As an essential component in our demonstration we expose, in great detail, the features of the dynamics of trajectories that either evolve toward the Feigenbaum attractor or are captured by its matching repellor. The dynamical properties of the family of periodic superstable cycles in unimodal maps are seen to be key ingredients for the comprehension of the discrete scale invariance features present at the period-doubling transition to chaos. Elements in our analysis are the following. (i) The preimages of the attractor and repellor of each of the supercycles appear entrenched into a fractal hierarchical structure of increasing complexity as period doubling develops. (ii) The limiting form of this rank structure results in an infinite number of families of well-defined phase-space gaps in the positions of the Feigenbaum attractor or of its repellor. (iii) The gaps in each of these families can be ordered with decreasing width in accordance with power laws and are seen to appear sequentially in the dynamics generated by uniform distributions of initial conditions. (iv) The power law with log-periodic modulation associated with the rate of approach of trajectories toward the attractor (and to the repellor) is explained in terms of the progression of gap formation. (v) The relationship between the law of rate of convergence to the attractor and the inexhaustible hierarchy feature of the preimage structure is elucidated. (vi) A "mean field" evaluation of the atypical partition function, a thermodynamic interpretation of the time evolution process, and a crossover to ordinary exponential statistics are given. We make clear the dynamical origin of the anomalous thermodynamic framework existing at the Feigenbaum attractor.

The aim of this paper is to contribute to the understanding of and to model the processes controlling the amplitude of the wind of Rayleigh-Bénard convection. We analyze results from direct simulation of an $L/H=4$ aspect-ratio domain with periodic sidewalls at $\text{Ra}=\left\{{10}^5,{10}^6,{10}^7,{10}^8\right\}$ and at $\text{Pr}=1$ by decomposing independent realizations into wind and fluctuations. It is shown that, deep inside the thermal boundary layer, horizontal heat fluxes exceed the average vertical heat flux by a factor of 3 due to the interaction between the wind and the mean temperature field. These large horizontal heat fluxes are responsible for spatial temperature differences that drive the wind by creating pressure gradients. The wall fluxes and turbulent mixing in the bulk provide damping. Using the direct numerical simulation results to parametrize the unclosed terms, a simple model capturing the essential processes governing the wind structure is derived. The model consists of two coupled differential equations for wind velocity and temperature amplitude. The equations indicate that the formation of a wind structure is inevitable due to the positive feedback resulting from the interaction between the wind and temperature field. Furthermore, the wind velocity is largely determined by the turbulence in the bulk rather than by the wall-shear stress. The model reproduces the Ra dependence of wind Reynolds number and temperature amplitude.

The scaling of the kinematic boundary layer thickness ${\lambda }_u$ and the friction factor $C_f$ at the top and bottom walls of Rayleigh-Bénard convection is studied by direct numerical simulation (DNS). By a detailed analysis of the friction factor, a new parameterisation for $C_f$ and ${\lambda }_u$ is proposed. The simulations were made of an $L/H=4$ aspect-ratio domain with periodic lateral boundary conditions at $\text{Ra}=\left\{{10}^5,{10}^6,{10}^7,{10}^8\right\}$ and $\text{Pr}=1$. The continuous spectrum, as well as significant forcing due to Reynolds stresses, clearly indicates a turbulent character of the boundary layer, while viscous effects cannot be neglected, judging from the scaling of classical integral boundary layer parameters with Reynolds number. Using a conceptual wind model, we find that the friction factor $C_f$ should scale proportionally to the thermal boundary layer thickness as $C_f\propto {\lambda }_{\Theta }/H$, while the kinetic boundary layer thickness ${\lambda }_u$ scales inversely proportionally to the thermal boundary layer thickness and wind Reynolds number ${\lambda }_u/H\propto {({\lambda }_{\Theta }/H)}^{-1}{\text{Re}}^{-1}$. The predicted trends for $C_f$ and ${\lambda }_u$ are in agreement with DNS results.

Pattern selection near the onset of convection in a cylindrical container heated from below is investigated numerically for a water-ethanol mixture, with parameter values and boundary conditions relevant to experiments. The Boussinesq three-dimensional equations for binary fluid convection are simulated for cylinders of aspect ratio $\Gamma =11$ and $10.5$ ($\Gamma \equiv R/d$, where $R$ is the radius of the cell and $d$ its height). The onset of convection occurs via a subcritical Hopf bifurcation in which the critical mode is strongly influenced by small variations of the aspect ratio of the cell. During the linear regime, an $m=1$ azimuthal mode consisting of radially traveling waves grows in amplitude in the $\Gamma =11$ cell, while an $m=0$ azimuthal mode is selected in the $\Gamma =10.5$ cylinder. As convection evolves, simulations for subcritical and supercritical Rayleigh numbers reveal differences in the dynamics. Very close to the critical value, convection is erratic and focuses along one or more diameters of the cell; growths and collapses of the convection amplitude take place, but convection eventually dies away for subcritical values and persists for slightly supercritical values. For larger supercritical values, convection grows progressively in amplitude, and patterns consist of traveling-wave regions of convection initially focused near the cell center, though expanding slowly until a large-amplitude state is reached. Depending on the reduced Rayleigh number, the final state can be a nonsteady state filling the cell or a disordered confined state.