Anisotropy-driven dynamics in vibrated granular rods

Toothpicks and toy football players: frictional jamming causes locomotion of vertically-shaken assymetrical objects

In a vertically shaken vertical round container of loosely packed nominally-vertical toothpicks, the toothpicks gradually progress horizontally. This vibration-driven locomotion is also observed in vintage toy football players on a buzzing support and in modern "Bristlebots" which are toothbrush heads with phone buzzers mounted on top. In all these cases the objects, or their lower parts, have a slant. With experiments, simulations and simple approximate theories we identify the two key features of horizontal locomotion induced by vertical-shaking: (1) the normal-force dependence of Coulomb friction and (2) some, but not too much, rotational freedom of the slanted legs. The small rotational freedom causes the necessary elastic or inertial anisotropy of the slanted objects. On the other hand, the slight restriction on rotation keeps the orientation near this anisotropic state. A key factor is jamming, which tends to occur if the slant of the principal axes of the contact mass or stiffness matrix is less than the friction angle ϕf ≡ arctan(μ), where μ is the friction coefficient.

Stochastic motion under nonlinear friction representing shear thinning

We study stochastic motion under a nonlinear frictional force that levels off with increasing velocity. Specifically, our frictional force is of the so-called Coulomb-tanh type. At small speed, it increases approximately linearly with velocity, while at large speed, it approaches a constant magnitude, similarly to solid (dry, Coulomb) friction. In one spatial dimension, a formal analogy between the associated Fokker-Planck equation and the Schrödinger equation for a quantum mechanical oscillator in a nonharmonic Pöschl-Teller potential is revealed. Then, the stationary velocity statistics can be treated analytically. From such analytical considerations, we determine associated diffusion coefficients, which we confirm by agent-based simulations. Moreover, from such simulations and from numerically solving the associated Fokker-Planck equation, we find that the spatial distribution function, starting from an initial Gaussian shape, develops tails that appear exponential at intermediate timescales. At small magnitudes of stochastic driving, the velocity distribution resembles the case of linear friction, while at large magnitudes, it rather approaches the case of solid (dry, Coulomb) friction. The same is true for diffusion coefficients. In a certain sense thus interpolating between the two extreme cases of linear friction and solid (dry, Coulomb) friction, our approach should be useful to describe several cases of practical relevance. For instance, a reduced increase in friction with increasing relative speed is typical of shear-thinning behavior. Therefore, driven motion in shear-thinning environments is one specific example to which our description may be applied.

Density-mediated spin correlations drive edge-to-bulk flow transition in active chiral matter

We demonstrate that edge currents develop in active chiral matter due to boundary shielding over a wide range of densities corresponding to a gas, fluid, and crystal. The system is composed of spinning disk-shaped grains with chirally arranged tilted legs confined in a circular vibrating chamber. The edge currents are shown to increasingly drive circulating bulk flows with area fraction as percolating clusters develop due to increasing spin-coupling between neighbors mediated by frictional contacts. Edge currents are observed even in the dilute limit. While, at low area fraction, the average flux vanishes except within a distance that is of the order of a particle diameter of the boundary, the penetration depth grows with increasing area fraction until a solid-body rotation is achieved corresponding to the highest packing, where the particles are fully caged with hexagonal order and spin in phase with the entire packing. A coarse-grained model, based on the increased collisional interlocking of the particles with area fraction and the emergence of order, captures the observed flow fields.

Tunable collective dynamics of ellipsoidal Quincke particles

Collective behaviors in active systems become dramatically complicated in the presence of chirality. In this study, we show that ellipsoidal Quincke particles driven by an electric field exhibit flexible and tunable chirality because of the tilting of the spinning axis. As the tilting torque decreases with the increase of angular speed, the motion of individual particles transforms from localized circle motion to global rolling. However, because of the anisotropic shape and the resulting anisotropic polar interactions, it is dynamically easier for ellipsoids to bind and form rotating structures rather than to align their velocities. In dense systems, the suppression of velocity aligning produces transient dense clusters which produce dynamic heterogeneity. The formation and dissociation of dense clusters prohibit the emergence of large-scale collective motions and limit the amplitude of density fluctuations. These findings demonstrate that collective dynamics and thus the scale of density fluctuations in active systems with tunable chirality can be well controlled. This has potential applications in exploring disordered hyperuniform states.

Tilt-induced polar order and topological defects in growing bacterial populations

Rod-shaped bacteria, such as Escherichia coli, commonly live forming mounded colonies. They initially grow two-dimensionally on a surface and finally achieve three-dimensional growth. While it was recently reported that three-dimensional growth is promoted by topological defects of winding number +1/2 in populations of motile bacteria, how cellular alignment plays a role in nonmotile cases is largely unknown. Here, we investigate the relevance of topological defects in colony formation processes of nonmotile E. coli populations, and found that both ±1/2 topological defects contribute to the three-dimensional growth. Analyzing the cell flow in the bottom layer of the colony, we observe that +1/2 defects attract cells and -1/2 defects repel cells, in agreement with previous studies on motile cells, in the initial stage of the colony growth. However, later, cells gradually flow toward -1/2 defects as well, exhibiting a sharp contrast to the existing knowledge. By investigating three-dimensional cell orientations by confocal microscopy, we find that vertical tilting of cells is promoted near the defects. Crucially, this leads to the emergence of a polar order in the otherwise nematic two-dimensional cell orientation. We extend the theory of active nematics by incorporating this polar order and the vertical tilting, which successfully explains the influx toward -1/2 defects in terms of a polarity-induced force. Our work reveals that three-dimensional cell orientations may result in qualitative changes in properties of active nematics, especially those of topological defects, which may be generically relevant in active matter systems driven by cellular growth instead of self-propulsion.

Statistics for an object actively driven by spontaneous symmetry breaking into reversible directions

Propulsion of otherwise passive objects is achieved by mechanisms of active driving. We concentrate on cases in which the direction of active drive is subject to spontaneous symmetry breaking. In our case, this direction will be maintained until a large enough impulse by an additional stochastic force reverses it. Examples may be provided by self-propelled droplets, gliding bacteria stochastically reversing their propulsion direction, or nonpolar vibrated hoppers. The magnitude of active forcing is regarded as constant, and we include the effect of inertial contributions. Interestingly, this situation can formally be mapped to stochastic motion under (dry, solid) Coulomb friction, however, with a negative friction parameter. Diffusion coefficients are calculated by formal mapping to the situation of a quantum-mechanical harmonic oscillator exposed to an additional repulsive delta-potential. Results comprise a ditched or double-peaked velocity distribution and spatial statistics showing outward propagating maxima when starting from initially concentrated arrangements.

Statistical mechanics of active Ornstein-Uhlenbeck particles

We study the statistical properties of active Ornstein-Uhlenbeck particles (AOUPs). In this simplest of models, the Gaussian white noise of overdamped Brownian colloids is replaced by a Gaussian colored noise. This suffices to grant this system the hallmark properties of active matter, while still allowing for analytical progress. We study in detail the steady-state distribution of AOUPs in the small persistence time limit and for spatially varying activity. At the collective level, we show AOUPs to experience motility-induced phase separation both in the presence of pairwise forces or due to quorum-sensing interactions. We characterize both the instability mechanism leading to phase separation and the resulting phase coexistence. We probe how, in the stationary state, AOUPs depart from their thermal equilibrium limit by investigating the emergence of ratchet currents and entropy production. In the small persistence time limit, we show how fluctuation-dissipation relations are recovered. Finally, we discuss how the emerging properties of AOUPs can be characterized from the dynamics of their collective modes.

Mechanically driven active and passive grains as models for egress dynamics

Passages of people or cattle through narrow entrances or exits occur in manifold situations. They are difficult to study experimentally, because one has to carefully separate objective, physical parameters from subjective, individual motivations, manners and temperament. Mechanically excited physical model systems can help to discriminate some of these classes of parameters. We characterize active and passive particles of equal shape and mass on a vibrating plate and study their bottleneck passage dynamics. They show fundamentally different scaling behavior.

Resonant phenomena and mechanism in vibrated granular systems

We were motivated to perform this research by the investigation of Brownian motors in excited granular materials converting the chaotic motion of granules into the oriented motion of motors. We conducted experimental studies to explore the horizontal motion of granules in vertically vibrated annular granular systems, including mixed and pure granular systems with an asymmetrical periodic structure on the bottom. The variations of the horizontal granular flow caused by the height, vibrating parameters, and mixing ratio were described in detail. Our results revealed considerable changes in the horizontal flow of different granular systems. Most importantly, resonance was induced in the horizontal granular flow by the vertical vibration; that is, the horizontal flow reached its maximum at specific vibrating parameters. A collisional model of rigid objects was constructed to probe the flowing resonances in these granular systems and provided a qualitative agreement with the experimental results obtained. We conclude that when a flowing resonance occurs, the granular system oscillates horizontally with a natural frequency under periodic external excitation. The frequency matching between the external excitation and the horizontal oscillation is responsible for the flowing resonance. Our results could improve the current understanding of the dynamic properties of granular systems under external excitation.

Short granular chain under vibration: Spontaneous switching of states

We study experimentally a short chain of N(≤8) loosely connected spheres bouncing against a horizontal surface that vibrates sinusoidally at intensity Γ. Distinct states are identified: a base state of uniform bouncing in-sync with the substrate prevails at low values of Γ, whereas increasing Γ can induce transitions to two excited states with appreciable storage of energy around one or both ends of the chain. We find that, in a transitional window of Γ, the chain can even switch spontaneously among states, resolving the mystery why different modes of motion can be initiated at the same position in our previous work along a gradient of vibration [Phys. Rev. Lett. 112, 058001 (2014)]. Preliminary interpretations on the parametric dependences and the optimal frequency window for seeing these transitions are offered, based on the microscopic and statistical evidence in our experiments up to date.

Bacterial social interactions drive the emergence of differential spatial colony structures

Background

Social interactions have been increasingly recognized as one of the major factors that contribute to the dynamics and function of bacterial communities. To understand their functional roles and enable the design of robust synthetic consortia, one fundamental step is to determine the relationship between the social interactions of individuals and the spatiotemporal structures of communities.

Results

We present a systematic computational survey on this relationship for two-species communities by developing and utilizing a hybrid computational framework that combines discrete element techniques with reaction-diffusion equations. We found that deleterious interactions cause an increased variance in relative abundance, a drastic decrease in surviving lineages, and a rough expanding front. In contrast, beneficial interactions contribute to a reduced variance in relative abundance, an enhancement in lineage number, and a smooth expanding front. We also found that mutualism promotes spatial homogeneity and population robustness while competition increases spatial segregation and population fluctuations. To examine the generality of these findings, a large set of initial conditions with varying density and species abundance was tested and analyzed. In addition, a simplified mathematical model was developed to provide an analytical interpretation of the findings.

Conclusions

This work advances our fundamental understanding of bacterial social interactions and population structures and, simultaneously, benefits synthetic biology for facilitated engineering of artificial microbial consortia.

Electronic supplementary material

The online version of this article (doi:10.1186/s12918-015-0188-5) contains supplementary material, which is available to authorized users.

Ordering in granular-rod monolayers driven far from thermodynamic equilibrium

The orientational order in vertically agitated granular-rod monolayers is investigated experimentally and compared quantitatively with equilibrium Monte Carlo simulations and density functional theory. At sufficiently high number density, short rods form a tetratic state and long rods form a uniaxial nematic state. The length-to-width ratio at which the order changes from tetratic to uniaxial is around 7.3 in both experiments and simulations. This agreement illustrates the universal aspects of the ordering of rod-shaped particles across equilibrium and nonequilibrium systems. Moreover, the assembly of granular rods into ordered states is found to be independent of the agitation frequency and strength, suggesting that the detailed nature of energy injection into such a nonequilibrium system does not play a crucial role.

Symmetry of interactions rules in incompletely connected random replicator ecosystems

The evolution of an incompletely connected system of species with speciation and extinction is investigated in terms of random replicators. It is found that evolving random replicator systems with speciation do become large and complex, depending on speciation parameters. Antisymmetric interactions result in large systems, whereas systems with symmetric interactions remain small. A co-dominating feature is within-species interaction pressure: large within-species interaction increases species diversity. Average fitness evolves in all systems, however symmetry and connectivity evolve in small systems only. Newcomers get extinct almost immediately in symmetric systems. The distribution in species lifetimes is determined for antisymmetric systems. The replicator systems investigated do not show any sign of self-organized criticality. The generalized Lotka-Volterra system is shown to be a tedious way of implementing the replicator system.

Effect of aspect ratio on the development of order in vibrated granular rods

We investigate ordering of granular rods in a container subject to vibrations in a gravitational field as a function of number density of the rods. We study rods with three different length to diameter aspect ratios A(r)= 5, 10, and 15. The measurements are performed in three dimensions using x-ray computer tomography to visualize the rods in the entire container. We first discuss a method to extract the position and orientation of the rods from the scans which enables us to obtain statistical measures of the degree of order in the packing. We find that the rods with A(r)=5 phase separate into domains with vertical and horizontal orientation as the number density of the rods is increased, whereas, for A(r)=10 and 15 the rods are predominately oriented vertically in layers. By calculating two-point spatial correlation functions, we further show that long range hexagonal order occurs within a layer when the rods are oriented along the vertical axis. Thus, our experiments find that long range order increases rapidly in granular rods with growing anisotropy.

Capturing self-propelled particles in a moving microwedge

Catching fish with a fishing net is typically done either by dragging a fishing net through quiescent water or by placing a stationary basket trap into a stream. We transfer these general concepts to micron-sized self-motile particles moving in a solvent at low Reynolds number and study their collective trapping behavior by means of computer simulations of a two-dimensional system of self-propelled rods. A chevron-shaped obstacle is dragged through the active suspension with a constant speed v and acts as a trapping "net." Three trapping states can be identified corresponding to no trapping, partial trapping, and complete trapping and their relative stability is studied as a function of the apex angle of the wedge, the swimmer density, and the drag speed v. When the net is dragged along the inner wedge, complete trapping is facilitated and a partially trapped state changes into a complete trapping state if the drag speed exceeds a certain value. Reversing the drag direction leads to a reentrant transition from no trapping to complete trapping and then back to no trapping upon increasing the drag speed along the outer wedge contour. The transition to complete trapping is marked by a templated self-assembly of rods forming polar smectic structures anchored onto the inner contour of the wedge. Our predictions can be verified in experiments of artificial or microbial swimmers confined in microfluidic trapping devices.

Order, intermittency, and pressure fluctuations in a system of proliferating rods

Nonmotile elongated bacteria confined in two-dimensional open microchannels can exhibit collective motion and form dense monolayers with nematic order if the cells proliferate, i.e., grow and divide. Using soft molecular dynamics simulations of a system of rods interacting through short range mechanical forces, we study the effects of the cell growth rate, the cell aspect ratio, and the sliding friction on nematic ordering and on pressure fluctuations in confined environments. Our results indicate that rods with aspect ratios >3.0 reach quasiperfect nematic states at low sliding friction. At higher frictions, the global nematic order parameter shows intermittent fluctuations due to sudden losses of order and the time intervals between these bursts are power-law distributed. The pressure transverse to the channel axis can vary abruptly in time and shows hysteresis due to lateral crowding effects. The longitudinal pressure field is on average correlated to nematic order, but it is locally very heterogeneous and its distribution follows an inverse power law, in sharp contrast with nonactive granular systems. We discuss some implications of these findings for tissue growth.

Emergent states in dense systems of active rods: from swarming to turbulence

Dense suspensions of self-propelled rod-like particles exhibit a fascinating variety of non-equilibrium phenomena. By means of computer simulations of a minimal model for rigid self-propelled colloidal rods with variable shape we explore the generic diagram of emerging states over a large range of rod densities and aspect ratios. The dynamics is studied using a simple numerical scheme for the overdamped noiseless frictional dynamics of a many-body system in which steric forces are dominant over hydrodynamic ones. The different emergent states are identified by various characteristic correlation functions and suitable order parameter fields. At low density and aspect ratio, a disordered phase with no coherent motion precedes a highly cooperative swarming state with giant number fluctuations at large aspect ratio. Conversely, at high densities weakly anisometric particles show a distinct jamming transition whereas slender particles form dynamic laning patterns. In between there is a large window corresponding to strongly vortical, turbulent flow. The different dynamical states should be verifiable in systems of swimming bacteria and artificial rod-like micro-swimmers.

Diffusion of granular rods on a rough vibrated substrate

We investigate the effect of shape anisotropy and number density on the dynamics of granular rods on a substrate with experiments using a mono-layer of bead chains in a vibrated container. Statistical measures of the translational and rotational degrees of freedom indicate a dramatic slowing down of dynamics because of steric interactions at a value well below the highest packing fraction of the chains. In particular, the in-plane orientation auto-correlation function decays exponentially with time at low densities, but increasingly slowly as density is increased with a form which is not described by a simple exponential function. While the mean square displacement of the chain center of mass is observed to grow linearly at low densities, it is observed to grow increasingly slowly and non-linearly as number density is increased. Decomposing the displacements parallel and perpendicular to the long axis of the chain, we find that the ratio of diffusion in the parallel and perpendicular direction to their long axis is less than one in the dilute regime. However, as the density of the chains is increased, the ratio rapidly increases above one with a greater value for higher aspect ratios. This anisotropic behavior can be explained by considering a higher effective drag on the rods by the substrate in the perpendicular direction compared with the parallel direction, and by tube-like dynamics at higher densities.

How to capture active particles

In many applications, it is important to catch collections of autonomously navigating microbes and man-made microswimmers in a controlled way. Using computer simulation of a two-dimensional system of self-propelled rods we show that a static chevron-shaped wall represents an excellent trapping device for self-motile particles. Its catching efficiency can be controlled by varying the apex angle of the trap which defines the sharpness of the cusp. Upon decreasing the angle we find a sequence of three emergent states: no trapping at wide angles followed by a sharp transition towards complete trapping at medium angles and a crossover to partial trapping at small cusp angles. A generic trapping "phase diagram" maps out the conditions at which the capture of active particles at a given density is rendered optimal.

Computer simulations of the collapse of columns formed by elongated grains

A numerical investigation of the collapse of granular columns has been conducted. In particular, we address the effect of the grain shape on the properties of the collapse. We show that the final runout and height of the deposits scale as a power law of the initial aspect ratio of the column, a, independently of the elongation of the grains used. We describe this process in terms of an energy balance, and construct an "inertial number" that can be used to describe the flow in terms of a recently proposed granular rheology. We argue that an effective friction that results from this dimensionless quantity explains why the shape of the grains is irrelevant for the final properties of the collapse.

Concentration dependent diffusion of self-propelled rods

We examine the persistent random motion of self-propelled rods (SPR) as a function of the area fraction phi and study the effect of steric interactions on their diffusion properties. SPR of length l and width w are fabricated with a spherocylindrical head attached to a beaded chain tail, and show directed motion on a vibrated substrate. The mean square displacement (MSD) on the substrate grows linearly with time t for phi<w/l, before displaying caging as phi is increased, and stops well below the close packing limit. Direction autocorrelations decay progressively slower with phi. We describe the observed decrease of SPR propagation speed c(phi) with a tube model. Further, MSD parallel to the SPR collapse with tau=l/c(phi) and scales as (lt/tau);{2}, and MSD in the perpendicular direction grows progressively slower than l{2}t/tau with phi, consistent with dynamics inside a thinning tube.

Biomechanical ordering of dense cell populations

The structure of bacterial populations is governed by the interplay of many physical and biological factors, ranging from properties of surrounding aqueous media and substrates to cell-cell communication and gene expression in individual cells. The biomechanical interactions arising from the growth and division of individual cells in confined environments are ubiquitous, yet little work has focused on this fundamental aspect of colony formation. We analyze the spatial organization of Escherichia coli growing in a microfluidic chemostat. We find that growth and expansion of a dense colony of cells leads to a dynamical transition from an isotropic disordered phase to a nematic phase characterized by orientational alignment of rod-like cells. We develop a continuum model of collective cell dynamics based on equations for local cell density, velocity, and the tensor order parameter. We use this model and discrete element simulations to elucidate the mechanism of cell ordering and quantify the relationship between the dynamics of cell proliferation and the spatial structure of the population.

Coefficient of tangential restitution for viscoelastic spheres

The collision of frictional granular particles may be described by an interaction force whose normal component is that of viscoelastic spheres while the tangential part is described by the model by Cundall and Strack (Géotechnique 29, 47 (1979)) being the most popular tangential collision model in Molecular Dynamics simulations. Albeit being a rather complicated model, governed by 5 phenomenological parameters and 2 independent initial conditions, we find that it is described by 3 independent parameters only. Surprisingly, in a wide range of parameters the corresponding coefficient of tangential restitution, epsilont, is well described by the simple Coulomb law with a cut-off at epsilont = 0. A more complex behavior of the coefficient of restitution as a function on the normal and tangential components of the impact velocity, gn and gt, including negative values of epsilont, is found only for very small ratio gt/gn. For the analysis presented here we neglect dissipation of the interaction in normal direction.

Energy dissipation and dispersion effects in granular media

The strong interactions between particles will make the energy within the granular materials propagate through the network of contacts and be partly dissipated. Establishing a model that can clearly classify the dissipation and dispersion effects is crucial for the understanding of the global behaviors in the granular materials. For particles with rate-independent material, the dissipation effects come from the local plastic deformation and can be constrained at the energy level by using energetic restitution coefficients. On the other hand, the dispersion effects should depend on the intrinsic nature of the interaction law between two particles. In terms of a bistiffness compliant contact model that obeys the energetical constraint defined by the energetic coefficients, our recent work related to the issue of multiple impacts indicates that the propagation of energy during collisions can be represented by a distributing law. In particular, this law shows that the dispersion effects are dominated by the relative contact stiffness and the relative potential energy stored at the contact points. In this paper, we will apply our theory to the investigation of the wave behavior in granular chain systems. The comparisons between our numerical results and the experimental ones by Falcon, [Eur. Phys. J. B 5, 111 (1998)] for a column of beads colliding against a wall show very good agreement and confirm some conclusions proposed by Falcon Other numerical results associated with the case of several particles impacting a chain, and the collisions between two so-called solitary waves in a Hertzian type chain are also presented.

Swarming and swirling in self-propelled polar granular rods

Using experiments with anisotropic vibrated rods and quasi-2D numerical simulations, we show that shape plays an important role in the collective dynamics of self-propelled (SP) particles. We demonstrate that SP rods exhibit local ordering, aggregation at the side walls, and clustering absent in round SP particles. Furthermore, we find that at sufficiently strong excitation SP rods engage in a persistent swirling motion in which the velocity is strongly correlated with particle orientation.

Vertical ordering of rods under vertical vibration

Granular media composed of elongated particles rearrange and order vertically upon vertical vibration. We perform pseudo-two-dimensional discrete element model simulations and show that this phenomenon also takes place with no help from vertical walls. We quantitatively analyze the sizes of voids forming during vibrations and consider a void-filling mechanism to explain the observed vertical ordering. Void filling can explain why short rods are less prone to align vertically than long ones. We cannot, however, explain, invoking just void filling, the existence of an optimum acceleration to promote vertical ordering and its dependence on particle length. We finally introduce an interpretation of the phenomenon, by considering the energetic barriers that particles have to overcome to exit a horizontal or a vertical lattice. By comparing these energetic thresholds with the peak mean particle fluctuant kinetic energy, we identify three different regimes. In the intermediate regime a vertical lattice is stable, while a horizontal is not. This interpretation succeeds in reconciling both dependencies on vibration acceleration and on particle length.

Angular velocity distribution of a granular planar rotator in a thermalized bath

The kinetics of a granular planar rotator with a fixed center undergoing inelastic collisions with bath particles is analyzed both numerically and analytically by means of the Boltzmann equation. The angular velocity distribution evolves from quasi-Gaussian in the Brownian limit to an algebraic decay in the limit of an infinitely light particle. In addition, we compare this model to that of a planar rotator with a free center and discuss the prospects for experimental confirmation of these results.

Swirling motion in a system of vibrated elongated particles

Large-scale collective motion emerging in a monolayer of vertically vibrated elongated particles is studied. The motion is characterized by recurring swirls, with the characteristic scale exceeding several times the size of an individual particle. Our experiments identified a small horizontal component of the oscillatory acceleration of the vibrating plate in combination with orientation-dependent bottom friction (with respect to horizontal acceleration) as a source for the swirl formation. We developed a continuum model operating with the velocity field and local alignment tensor, which is in qualitative agreement with the experiment.

Stochastic dynamics of a rod bouncing upon a vibrating surface

We describe the behavior of a rod bouncing upon a horizontal surface which is undergoing sinusoidal vertical vibration. The predictions of computer simulations are compared with experiments in which a stainless-steel rod bounces upon a metal-coated glass surface. We find that, as the dimensionless acceleration parameter Gamma is increased appreciably above unity, the motion of a long rod passes from periodic or near-periodic motion into stochastic dynamics. Within this stochastic regime the statistics of the times between impacts follow distributions with tails of approximately Gaussian form while the probability distributions of the angles at impact have tails that are close to exponential. We determine the dependence of each distribution upon the length of the rod, upon frequency, and on Gamma. The statistics of the total energy and of the translational and rotational components each approximately follow a Boltzmann distribution in their tails, the translational and rotational energy components being strongly correlated. The time-averaged mean vertical translational energy is significantly larger than the mean rotational energy, and both are considerably larger than the energy associated with horizontal motion.

Theory of self-assembly of microtubules and motors

We derive a model describing spatiotemporal organization of an array of microtubules interacting via molecular motors. Starting from a stochastic model of inelastic polar rods with a generic anisotropic interaction kernel, we obtain a set of equations for the local rods concentration and orientation. At large enough mean density of rods and concentration of motors, the model describes an orientational instability. We demonstrate that the orientational instability leads to the formation of vortices and (for large density and/or kernel anisotropy) asters seen in recent experiments. We derive the specific form of the interaction kernel from the detailed analysis of microscopic interaction of two filaments mediated by a moving molecular motor and extend our results to include variable motor density and motor attachment to the substrate.

Dynamics of a bouncing dimer

We investigate the dynamics of a dimer bouncing on a vertically oscillated plate. The dimer, composed of two spheres rigidly connected by a light rod, exhibits several modes depending on initial and driving conditions. The first excited mode has a novel horizontal drift in which one end of the dimer stays on the plate during most of the cycle, while the other end bounces in phase with the plate. The speed and direction of the drift depend on the aspect ratio of the dimer. We employ event-driven simulations based on a detailed treatment of frictional interactions between the dimer and the plate in order to elucidate the nature of the transport mechanism in the drift mode.