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Nonlinear Collective Behaviour: Networks,
Swarming and Reaction Diffusion Dynamics
Continuum models of large-scale coherence in dense assemblies of self-propelled particles
Dense assemblies of active self-propelled particles, such as vigorously shaken anisotropic grains, microtubules interacting with molecular motor, and hydrodynamically entrained swimming bacteria, often exhibit large-scale patterns of collective motion whose correlation length greatly exceeds the size of individual particle.
Despite a vast difference in the physical mechanisms controlling onset of large-scale coherence, continuum description of such systems can be derived from an analogy with a dilute gas of inelastically colliding granular rods. Thus, starting from a generic stochastic microscopic model of inelastic polar rod-like particles with an anisotropic interaction kernel, we derive set of equations for the local rods concentration and orientation. Above certain critical density of rods the model exhibits spontaneous orientation instability and onset of large-scale coherence.
For the system of microtubules interacting with molecular motors we demonstrate that the orientation instability leads to the formation of vortices and asters seen in recent experiments.
Similar approach is applied to colonies of swimming bacteria Bacillus subtilis confined in a thin fluid film. The model is formulated in term of two-dimensional equations for local density and orientation of bacteria coupled to the low Reynolds number Navier-Stokes equation for the fluid flow velocity.
We demonstrate that this system exhibits formation of dynamic large-scale patterns with the typical scale determined by the density of bacteria.
Modelling nonequilibrium clustering and active nematic order in collective motion of self-propelled rods
Motivated by observations regarding collective motion of gliding bacteria on a substrate, we have derived two models describing rod-shaped self-propelled particles.
The first model takes into account explicitly the shape of the rods and assumes interaction due to volume exclusion. Simulations with up to 200 rods exhibit spontaneous formation of clusters indicated by a bimodal cluster size distribution already at low densities. Clustering is shown to require both the rod-shape and the active motion of the particle and is hence a genuine nonequilibrium phenomenon. Equations for the cluster size distribution are derived and reproduce the simulation findings well.
Our second model describes self-propelled particles with apolar liquid-crystal like interactions analogous to the Vicsek model for self-propelled particles with polar orientation. Such an interaction mimicks the volume exclusion interaction of rods. Simulations carried out with up to 16.384 particles reveal different ordering scenarios upon decrease of the noise. For high density, first nematic order appears with homogeneous density and is connected with a phase transition correctly described by a mean-field theory of the angular orientation of particles. A second transition leading to clustering and inhomogeneous density of particles is found in the ordered phase. At low transition, the homogeneous state without orientational becomes directly unstable to formation of clusters connected with orientational order similar to the finding in the first model with hard rods. Again the appearance of the ordered clustering phase is well described by a transition from a unimodal to a bimodal cluster size distribution.
Joint work with Fernando Peruani and Andreas Deutsch
Spatial Epidemics: Emergent Trade-offs and Evolutionary Cycling
Spread of diseases in human populations can exhibit large scale patterns. An often observed pattern is a so-called "wave of epidemic spread". We demonstrate that in a simple contact network model such epidemic waves can emerge spontaneously. Spatial patterns can have profound effects on selection of disease properties, such as infectiousness and duration of the infectious period. We have recently shown  that in this system spatial pattern formation and natural selection can generate an emergent trade-off between infectiousness and infection period, while the system is maximizing outbreak frequency. However, for larger differences in outbreak frequency, the system can display turbulent interface patterns, which can reverse the selection pressure towards maximizing secondary infections. In this way the spatial epidemic system can move into a so-called "evolutionary cycling" regime, where the switch in direction of selection is caused by a phase transition in the system's spatial pattern.
* Work in collaboration with Marijn van Ballegooijen
 van Ballegooijen W.M. & M.C. Boerlijst (2004) "Emergent trade-offs and selection for outbreak frequency in spatial epidemics", Proc. Nat. Acad. Sci. 101: 18246-18250.
Collective dynamics of active particles
I will present an overview of recent results obtained about the collective motion of active or self-propelled particles, both polar and nematic. I will in particular discuss the nature of the transition to orientational order in such systems, the properties of the ordered phase specific to the non-equilibrium nature of the problem, and candidates for continuous mesoscopic descriptions.
Collective motion and decision-making in animal groups
Collective organization is everywhere, both around us and within us. Our brains are composed of billions of interconnected cells communicating with chemical and electrical signals. Our bodies are formed from clustering, communicating cells, and we ourselves are integrated in our own collective human society. Elsewhere in the natural world hundreds of thousands of blind army ants coordinate a massive raid across the rainforest floor, a flock of birds arcs and ripples while descending to roost and a fish school convulses, as if one entity, when attacked by a predator. How can animal groups move in unison? How does individual behaviour produce group dynamics? How do animal societies make informed unanimous decisions? From ant swarms to flocking birds, from consensus decision-making in fish schools to that among humans, I will discuss how, and why, coordinated collective patterns are generated in biological systems.
Epidemics on networks
From the point of view of an infective agent, contacts between two already infected hosts are wasted. And if the host population is structured as a network, repeated contacts between the same individuals are bound to occur. Accordingly the structure can have a large impact on the spread of the infective agent and one would like to determine this impact more precisely by way of an analysis of models. So far this is largely wishful thinking! In the lecture I'll review some known results and then try to formulate open questions.
Differential and Mutual (Cross-)Diffusion Effects on Pattern Formation in Reaction-Diffusion Systems
Turing was among the first to point out that substantial differences in diffusion rates between reactive species can lead to new types of pattern formation. Recent experimental work on chemically reactive microemulsions has revealed a variety of novel patterns, which can be simulated if one incorporates the differences in diffusion rates between aqueous and nonaqueous species. Models in which the diffusion of one species depends upon the spatial distribution of another (mutual or cross-diffusion) give rise to a similarly rich variety of patterns, even with equal diagonal diffusion coefficients, and we have obtained experimental evidence that cross-diffusion occurs in BZ-AOT microemulsions. Both mechanisms may be relevant to the nonlinear collective behavior of biological and social systems.
Spatiotemporal Patterns of Infectious Diseases - New Approaches to the Forecast of Epidemics
Many infectious diseases are transmitted from person to person and human travel is responsible for their geographical spread. In order to model, forecast, and control the spread of epidemics, one needs to know the statistical mechanics of human travel. How can we obtain reliable information on travelling statistics, if people can travel using very different means of transportation from bikes to planes? We have studied this problem empirically and theoretically using the dispersal of dollar bills as a proxy. The time dependent probability density obtained in this way exhibits pronounced spatiotemporal scaling and anomalous diffusion, which mathematically can be described very accurately in terms of a bifractional diffusion equation with few parameters.
*work in collaboration with D. Brockmann and L. Hufnagel
 D. Brockmann, L. Hufnagel, and T. Geisel, "The scaling laws of human travel", Nature 439, 462 (2006).
 L. Hufnagel, D. Brockmann, and T. Geisel, "Forecast and control of epidemics in a globalized world", PNAS, 101, 15124 (2004).
Physical Aspects of Evolutionary Transitions to Multicellularity
Raymond E. Goldstein
An important issue in evolutionary biology is the emergence of multicellular organisms from unicellular individuals. The accompanying differentiation from motile totipotent unicellular organisms to multicellular ones having cells specialized into reproductive (germ) and vegetative (soma) functions, such as motility, implies both costs and benefits, the analysis of which involves the physics of buoyancy, diffusion, and mixing. In this talk, I discuss recent results on this transition in a model lineage: the volvocine green algae. Particle Imaging Velocimetry of fluid flows generated by these organisms show that they exist in the regime of very large Peclet numbers, where the scaling of nutrient uptake rates with organism size is highly nontrivial. In concert with metabolic studies of deflagellated colonies, investigations of phenotypic plasticity under nutrient-deprived conditions, and theoretical studies of transport in the high-Peclet number regime, we find that flagella-generated fluid flows enhance the nutrient uptake rate per cell, and thereby provide a driving force for evolutionary transitions to multicellularity. Thus, there is a link between motility, mixing, and multicellularity.
Chemically Powered Nanodimers
Molecular motors play important roles in transport in biological systems. These molecular machines are powered by chemical energy and operate in the regime of low Reynolds number hydrodynamics. Recently a class of simple inorganic molecular motors has been constructed and studied experimentally [1,2]. These motors are bimetallic rods, one end of which is chemically active. The talk will describe simple mesoscopic models for the motion of such nanomotors. The motor consists of two linked spheres, one of which catalyzes the conversion between two chemical species. The chemical species interact differently with the two spheres in the dimer. The nano-dimer motor is solvated by a molecules treated at a mesoscopic level whose evolution is governed by multi-particle collision dynamics. The dynamics conserves mass, momentum and energy so that coupling between the nanomotor and the hydrodynamic modes of the solvent is treated correctly. The simulations allow one to explore the mechanisms of the chemically powered motion and the effects of fluctuations on the motor dynamics.
* Work in collaboration with Gunnar Rueckner.
 W. F. Paxton, et al., "Catalytic Nanomotors: Autonomous Movement of Striped Nanorods", J. Am. Chem. Soc. (JACS), 126 (41), 13424 (2004).
 S. Fournier-Bidoz, et al. "Synthetic Self-Propelled Nanorotors", Chem. Commun., (4), 441 (2005).
A new hypothesis towards the onset of cardiac fibrillation
Kyoung J. Lee
Ventricular fibrillation (VF) is one of the most deadly cardiac arrhythmia during which different parts of the ventricle beat asynchronously. A growing body of evidence shows that rotating spiral waves (or reentries) are a precursor to VF: Self-organized spiral waves in heart tissue are life-threatening because their fast rotation frequency (5-10 Hz) drives an abnormally rapid contraction known as ventricular tachycardia (VT), which subsequently becomes unstable and decays to VF. Another, equally important, precursor of VF is “alternans,” a beat-to-beat temporal alternation in the sequence of heart beats. Our recent in vitro studies have shown that these two phenomena can coexist as a form of period-2 oscillatory reentry having line defects. Thus the underlying mechanism for VF very likely stems from the instability of this unusual spiral reentry. Here, I will discuss some important recent findings and future directions.
Nonlinear relaxation dynamics in elastic networks and design principles of molecular machines
Alexander S. Mikhailov
Analyzing nonlinear conformational relaxation dynamics in elastic networks corresponding to two classical motor proteins, we find that they respond by well-defined internal mechanical motions to various initial deformations and that these motions are robust against external perturbations. We show that this behavior is not characteristic for random elastic networks. However, special network architectures with such properties can be designed by evolutionary optimization methods. Using them, an example of an artificial elastic network, operating as a cyclic machine powered by ligand binding, is constructed.
Pinning and depinning for localized waves in heterogeneous media
Particle-like (spatially localized) dissipative patterns arise in many fields such as chemical reaction, gas-discharge system, liquid crystal, binary convection, and morphogenesis. A typical model system treated here is a three-component reaction diffusion system of one-activator-two-inhibitor type. We discuss about the dynamics of moving particles in heterogeneous media, especially we are interested in the case in which traveling pulses or spots encounter heterogeneities of bump type. A variety of dynamics are produced such as penetration, annihilation, rebound, splitting, and relaxing to an ordered pattern, i.e., pinning. It turns out that global bifurcation such as heteroclinic one and a basin switching controlled by the unstable manifolds of saddless called scattors play a crucial role for the transition from pinning to depinning. We also discuss a wave-generator created near the jump of heterogeneity. A dynamical system view point obtained here is basic for the understanding of the collective behavior of many particles in heterogeneous media.
Mechano-electric feedback is a complex concept that has been studied in the clinical community for over a century, and may have arrhythmogenic consequences. Mathematical modelling is a useful tool to study cardiac arrhythmogenesis, however most electrophysiological simulation studies have not accounted for the mechanical function of the heart. Here we introduce the concept of a contracting excitable medium in order to study the basic effects of mechanical deformations on re-entrant cardiac arrhythmias. We illustrate the electromechanical model formulation using results from a specific implementation that combines a modified Aliev-Panfilov 1996 ventricular excitation model with the nonlinear stress equilibrium equations that govern tissue mechanics. The Aliev-Panfilov model has been used previously to study various types of propagation using 2D and 3D representations of cardiac tissue, and anatomically accurate models of the heart. The model reproduces most of the regimes important for understanding re-entrant electrical activity, such as stationary rotation of spiral waves, meandering of spiral waves and spiral wave break up. We have extended the Aliev-Panfilov model to include an approximation of the actively developed stress during contraction to investigate the effects of mechanical deformation on cardiac excitation and the various types of spiral wave activity. Passive mechanical properties are defined using an isotropic, homogeneous Mooney-Rivlin constitutive law. Propagating waves of excitation initiate tissue contraction by generating active stresses that add to the passive axial stress components. The coupled electromechanical model is solved using a hybrid approach that incorporates finite difference and nonhomogeneous finite element techniques for the electrical activity and tissue mechanics, respectively.
We study the effects of mechanical activity on different regimes of wave propagation in cardiac tissue. We show that for some parameter values, mechano-electric feedback can induce automaticity in an array of otherwise non-oscillatory cardiac cells, and illustrate the spatio-temporal dynamics of these pacemakers. We also demonstrate the significant effects of mechano-electric feedback on re-entrant wave dynamics. We found that contraction can induce drift of an otherwise stationary rotating spiral wave. This drift settles to an attractor that can be located either at the center or periphery of the tissue, depending on its excitability. Finally, we observed that in some cases mechano-electric feedback can induce breakup of a single re-entrant wave into complex fibrillatory patterns.
We conclude that tissue mechanics significantly contributes to the dynamics of cardiac electrical activity, and that a coupled electromechanical-mechanoelectric feedback framework should be adopted for future electrophysiological modelling studies.
Stochastic Multi-Agents and Motion of Simple Objects
L. Schimansky-Geier & W. Ebeling
In the lecture we introduce equations of motion of particles which convert energy into motion by negative friction. If including furtheron random forces which act on the particles these active Brownian particles may describe the motion of simple organisms which results in a diffusional motion of the particles. External forces as well as chemotactical interactions may be included into the description for the purpose to decribe more complex situtations. We present several typical situations and corresponding attractors of particles in external fields and if interacting.
A few animals have prefered turning angles along their random motion. This attribute is modeled by forces acting perpendicularly to the motion which yield a drastic reduction of the diffusion coefficient compared to the situation of free motion. A biological reason for this restricted motion is seen in an enhanced food-consumption of these animals compared to the case of free diffusion.
Collective Behavior in Excitable Media: Interacting Particle-Like Waves
We describe studies of interacting particle-like waves in the photosensitive Belousov-Zhabotinsky reaction. Unstable waves are stabilized by global feedback that affects the overall excitability of the medium, and the motion of these waves is controlled by imposing excitability gradients that are regulated by a secondary feedback loop. Waves interact via a Lennard-Jones potential in which there are attractive forces at long distances and repulsive forces at short distances. Processional motion is the most common behavior, where waves align with one another to varying degrees depending on the strength of the potential. Rotational motion is also observed, which may occur for the same parameters as processional motion depending on initial conditions. We also discuss other modes of behavior and an analysis of the wave interaction in terms of the gradient of the potential.
* In collaboration with Mark Tinsley and Aaron Steele.
Continuous phase transition in collective motion: experiments and models
Detailed investigations on the nature of the phase transition in the scalar noise model (SNM) of collective motion will be presented. Our results confirm the original findings, i.e., that the disorder-order transition in the SNM is a continuous, second order phase transition for small particle velocities (v < 0.1). However, for large velocities (v > 0.3) we find a strong anisotropy in the particle diffusion in contrast to the isotropic diffusion for small velocities. The interplay between the anisotropic diffusion and the periodic boundary conditions leads to an artificial symmetry breaking of the solutions (directionally quantized density waves) and, a consequence, to a first order transition like behavior.
Our preliminary results suggest that the situation is the same if flocking takes place in three dimensions.
We also investigated the transition experimentally, by recording the swarming-like collective migration of a large number of keratocytes (tissue cells obtained from the scales of goldfish) using long-term videomicroscopy. By increasing the overall density of the migrating cells, we were able to demonstrate a kinetic phase transition from a disordered into an ordered state. Near the critical density a complex picture emerges with interacting clusters of cells moving in groups.
Motivated by these experiments we have constructed a flocking model that exhibits a continuous transition to the ordered phase, while assuming only short-range interactions and no explicit information about the knowledge of the direction of motion of the neighbours. Placing the cells into microfabricated arenas we obsereved a spectacular whirling pattern of motion which we could reproduce in the simulations as well.
* In collaboration with I. Daruka, B. Gonci, Zs. Juranyi, M. Nagy
The ticking of a circadian clock
Pieter Rein ten Wolde
In a recent series of ground-breaking experiments, Nakajima et al. [Science 308, 414-415 (2005)] showed that the three cyanobacterial clock proteins KaiA, KaiB, and KaiC are sufficient in vitro to generate circadian phosphorylation of KaiC. Here, we present a mathematical model of the Kai system. At its heart is the assumption that KaiC can exist in two conformational states, one favoring phosphorylation and the other dephosphorylation. Eachindividual KaiC hexamer then has a propensity to be phosphorylated in a cyclic manner. To generate macroscopic oscillations, however, the phosphorylation cycles of the different hexamers must be synchronized. We propose a novel synchronisation mechanism based on differential affinity: KaiA stimulates KaiC phosphorylation, but the limited supply of KaiA dimers binds preferentially to those KaiC hexamers that are falling behind in the oscillation. KaiB sequesters KaiA and stabilizes the dephosphorylating KaiC state. We show that our model can reproduce a wide range of published data, including the observed insensitivity of the oscillation period to variations in temperature, and that it makes nontrivial predictions about the effects of varying the concentrations of the Kai proteins.
Pulse dynamics in a three-component system
Peter van Heijster, Arjen Doelman & Tasso J. Kaper
We investigate various types of pulse solutions in a certain singularly perturbed, bi-stable, three component, reaction-diffusion system, that has a natural two-component limit. Using geometric singular perturbation theory and the Evans function approach, we study the existence, stability and bifurcations of standing pulses, travelling pulses and double-pulse solutions.
Clusters and Loops in a Reaction-Advection-Diffusion System
We construct a phenomenological model describing aggregated spots and a loop structure. Our model is based on the Gray-Scott model which is supplemented by a global coupling term and advection terms. One of the species makes a field proportional to its concentration, which induces the advection. By numerically investigating the model, we show that the system has a transition from aggregated spots to a loop which wanders around chaotically or reaches a stationary state. We also show the reduced equation of motion of spots in which the advection term can be interpreted as a long-range attractive force. Relation to a similar transition observed in a recent gas discharge experiment is discussed.
Transport in a periodic velocity field without attractors
Felix Müller & Lutz Schimansky-Geier
We consider the transport of over-damped particles in a two-dimensional periodic velocity field. This field possesses extended lines of fixed points where the deterministic motion stops. Additive noise makes the lines penetrable and results in an oscillatory motion along tori. We characterize the stochastic motion by the probability distribution density, the stationary mean velocity and mean times of escape from bounded domains. For intermediate noise intensities the fluctuations enhance the transport of the particles compared to the deterministic case. A fast dichotomic modulation of asymmetry enhances fluxes.
Nucleation, Drift, and Decay of Phase Bubbles in Period-2 Oscillatory Wave Trains
Jin-Sung Park, Sung-Jae Woo, Okyu Kwon, Tae Yun Kim, and Kyoung J. Lee
Abstract: We report experimental observations of phase bubbles, simply closed boundaries between domains oscillating $2\pi$ out of phase, associated with period-2 oscillatory traveling waves in a Belousov-Zhabotinsky reaction-diffusion system. These phase bubbles nucleate unpredictably and spontaneously through a fast localized phase slip in the traveling waves, drift away from the spiral core in an oscillatory fashion, and gradually shrink to disappear. Their oscillatory drift in the radial direction is an attribute of Doppler effect, while their lateral shrinkage is an attribute of the local curvature. Similar phenomenon is also captured in a simple, three-species reaction-diffusion model that support period-2 oscillatory wave trains.
Intercellular Calcium Communications in Cultured Networks of Glia
Jin-Sung Park, Byeong Ha Jeong, Joon Hoan Kim, Cheol Hong Min, and Kyoung J. Lee
Abstract: A series of recent studies augments the role of glia as regulatory agents of neuronal activity, undertaking active roles in brain computation. Intercellular and extracellular calcium kinetics are believed to be an important mediator for the interplay among neurons and glia. In mature neuron-glia co-culture systems, populations of neurons often exhibit globally synchronized fast 'calcium spikes' that are accompanied by electrical bursts of action potentials. The glia in the same culture system do often show slower calcium wave activities that possibly modulate the neuronal activity. We examine the property of these slowly propagating calcium waves in pure glia networks under various pharmacological conditions and with different extrinsic stimulations. We find that there are at least two distinct modes of propagation: (1) calcium puffs - circular wave fronts (which are initiated by localized sources) that propagate for some distance and extinguish by themselves and (2) small irregular wavelets that often emerge in the wake of some dominant pacemaker waves to last for quite some time circulating along local recurrent networks. We investigate on the spatio-temporal dynamics of cytosolic calcium levels for concentration changes of extracellular calcium and magnesium. As the concentration of extracellular calcium ion becomes lower, calcium waves begin to generate at a number of puff sites and their propagating ranges are also more increased. As well, a long range correlation of these glial calcium dynamics is obviously increased by periodic stimulus through local electrical pulse.
Self-propelled particles: from individual to collective behavior
Fernando Peruani, Luis G. Morelli, Andreas Deutsch, and Markus Baer
We study general aspects of non-interacting self-propelled particles (SPPs) with fluctuations in the speed and the direction of motion in two dimensions. We consider the case in which fluctuations in the speed are not correlated to fluctuations in the direction of motion, and assume that both processes can be described by independent characteristic time-scales. We investigate two different angular dynamics which correspond to persistent and directed random walks and show the occurrence of a complex transient that exhibits a series of alternating regimes of motion. We also show additive corrections to the diffusion coefficient. The characteristic time-scales are also exposed in the velocity autocorrelation, which is a sum of exponential forms .
Inspired by the motion of anisotropic self-propelled objects as myxobacteria , we apply the obtained results for non-interacting SPPs to understand orientational ordering and clustering in a simple model of SPPs that interact locally by a liquid-crystal alignment mechanism. It is observed through extensive simulations that the behavior of the system at high and low densities is remarkably different. In particular, we show that at low density orientational order emerges from the interplay between local orientation and cluster formation, whose dynamics is explained in terms of a reaction equation approach .
 F. Peruani and L. G. Morelli, to appear in Phys. Rev. Lett.
 F. Peruani, A. Deutsch, and M. Baer, Phys. Rev. E, 74, 030904(R) (2006).
 F. Peruani, A. Deutsch, and M. Baer (in preparation).
Beyond the Keller-Segel Model – Microscopic Description of Chemotactic Bacteria
Pawel Romanczuk, Udo Erdmann & Lutz Schimansky-Geier
Complex spatiotemporal patterns of cell clusters were observed in colonies of chemotactic bacteria such as Escherichia coli or Sallmonella typhimurium. The production of a potent chemoattractor by the bacteria themselves as a reaction to certain nutrients is the essential factor for this pattern formation. Additional collective dynamics, such as collective translocation and rotation of bacterial clusters were reported from experiments on bacterial colonies.
Motivated by this observations we suggest a simple model for the description of bacterial colonies based on the concept of Active Brownian motion. Our model represents an interesting alternative to the usually employed “pure” reaction-diffusion equations as it allows us to study the macroscopic pattern formation of the colony, the mesoscopic dynamics of bacterial ensembles (swarming), as well as the microscopic dynamics of single cells.
We are able to reproduce the macroscopic behaviour, as well as the collective types of motion using Active Brownian particles including chemotaxis and velocity alignment. We compare analytical results for macroscopic pattern formation obtained from the overdamped limit approximation with numerical simulations and discuss the microscopic and collective dynamics of our model.
Self-moving droplet powered by chemical energy
In oil-water-surfactant systems, there appears the difference of the chemical potential of a surfactant in oil and aqueous phases by introducing appropriate initial distribution of the surfactant. In systems far from equilibrium it is possible to generate macroscopically ordered self motion from the microscopic flow of a surfactant through the interface. This conversion is realized by such as wetting, Marangoni effect, and generation of elastic phase. These self motions are particularly interesting because they can be considered as the energy transduction of chemical energy into mechanical energy in the isothermal condition, whose mechanism is different from conventional thermal engines. In this presentation, we will present the various self-motions of droplets.
1) We report the emergence of regular spontaneous motion of an oil droplet in the oil-water-surfactant system, composed of an organic phase with potassium iodide and iodine and an aqueous phase containing stearyl trimethyl ammonium chloride (STAC). This self motion is using the change in the wetting condition of the oil droplet and a substrate [1, 2].
2) We will also show the spontaneous motion in a two-phase system of an alcohol (pentanol) and water. Here the self motion is induced by the Marangoni flow generated on air-water interface. It is found that the spontaneous motion of an alcohol droplet changes markedly depending on the size of the droplet, and that regular motion is generated for suitable rage of the volume. Such a trend is discussed in relation to the instability of certain wave number, which is crucially dependent on the droplet size .
3) Finally, we will briefly talk about the spontaneous deformation of an oil droplet floating on an aqueous phase. It is found that the system is generating the elastic phase as it approaches its equilibrium. Interestingly, the thickness of the generated elastic phase becomes spatially inhomogeneous and as a result the droplet starts to deform like living cells .
 Y. Sumino, N. Magome, T. Hamada and K. Yoshikawa, Phys. Rev. Lett., 94, 068301 (2005)
 Y. Sumino, et. al. Phys. Rev. E, 72, 041603 (2005)
 K. Nagai, Y. Sumino, H. Kitahata and K. Yoshikawa, Phys. Rev. E, 71, 065301 (2005)
 Y. Sumino, H. Kitahata, H. Seto and K. Yoshikawa, submitting
Velocity dependence of the phase transition in the Scalar Noise Model of Self-Propelled Particles in 2D and 3D
Balázs Gönci, Máté Nagy & Tamás Vicsek
We study the nature of phase transition of flocking using the scalar noise model (SNM). We find that the behaviour of the model is similar in two- and three-dimensions in many aspects. For small particle velocities (v <= 0.1, which is an assumption of the original model), a second order, continuous disorder-order phase transition occurs. Increasing the particle velocities the phase transition changes to first order. In addition, for these velocities we found that an artificial symmetry breaking occurs in the average movement direction of the particles accompanied by the emergence of density waves, which are caused by the anisotropic diffusion of particles and the periodic boundary conditions.
Rheological and structural properties of active filament solutions
Falko Ziebert & Igor S. Aronson
The rheology and the structure of a semiflexible biofilament solution, like F-actin, interacting via molecular motors is probed by molecular dynamics simulations. Oscillatory external shear is used to measure the storage and loss moduli as a function of motor activity in a range of frequencies and for low shear rates. The overall effect of the motor activity on the rheological properties is interpreted as an increase of the temperature, with the effective temperature proportional to the density of motors. However, the effect of motors on the structural properties of the solution, such as the orientation correlation function, is opposite: the motors drastically increase the orientation correlation length whereas thermal fluctuations decrease it.