Seung-Yeop Lee,
We formulate a reaction-diffusion model for bacterial branching growth, and use this model to study possible scenarios during the life of a colony: The use of chemotactic signaling, the appearance of chirality, the emergence of mutations and the stress of antibiotics.
Elucidation of the electrostatic properties of biomolecules has become a standard practice in molecular biophysics. Foremost among the models used to evaluate the electrostatic potential is the Poisson-Boltzmann equation, however, existing solution methods have limited the scope of accurate calculations to relatively small biomolecular systems. Two new numerical techniques will be presented which enable the parallel solution of the Poisson-Boltzmann equation for supramolecular structures orders of magnitude larger in size than those accessible with traditional methods. As a demonstration of this methodology, electrostatic potentials have been calculated for large microtubule and ribosome structures. The results point to the likely role of electrostatics in a variety of activities of these structures.
We consider two-dimensional turbulence for the case in which a drag force linear in the velocity is present. Such a drag force occurs in all cases where two dimensionality is justified. We find modifications from the dragless power law exponent of the wavenumber-energy-spectrum associated with the enstrophy cascade. The case with drag is also predicted to desplay intermittency (in contrast to the case without drag). Comparisons with numerical simulations will be given.
Expression of any single gene in a living cell is dependent upon presence of the products of other genes, known as transcription factors. This cross-regulation is essential for development and function of an organism. Mathematically, it means that the master equations for the protein concentrations are coupled in a highly non--linear manner. Interestingly, analysis of this nonlinear system can be considerably simplified in the limit when the maximum concentrations of the transcription factors are much larger than the thresholds of their catalytic (inhibiting) action. In particular, the search of the fixed points of this system is reduced to a "semi-linear" problem, somewhat analogous to electric circuit with ideal diodes. We have developed an intuitive diagrammatic representation of the problem, which allows one to find the fixed points, and to study their stability. In this framework, I will discuss the criterion for multistability of a genetic network. It will be argued that the multistability is crucial for understanding many important aspects of the problem, such as robustness, cell fate differentiation, and pattern formation. As an example, I will apply some of these ideas to early development of Drosophila embryo.
That the properties of materials depend on size is commonly expected and often observed. At sufficiently small sizes such dependencies may go beyond mere scaling with size, manifesting themselves in physical and chemical behavior that is new and different from that found at larger sizes. Such circumstances, when small is different in an essential way , may occur when one (or more) of the physical dimensions of the material aggregate approaches a length-scale characteristic to a physical phenomenon (with different phenomena being characterized by different length-scales), and similarly in the time domain. Associated with the above is the sensitivity of sufficiently small materials aggregates to shape. Not only can one affect the properties of a confined system by varying its shape, but, most importantly, is the spontaneous shape-selection occurring in such systems, originating from the ability of finite systems to adjust their shape (and structure) in order to minimize their (free) energy. Basic research of these and related issues underlies future technologies, from nano-scale machines, nanotribological systems, cellular injections, and nanocatalysis, to miniaturization of electronic circuitry and novel information storage and retrieval systems.In this talk we discuss and illustrate the above issues through large-scale classical and quantum mechanical simulations of several nano-scale systems. Topics include: (i) Formation mechanisms, mechanical, and quantized conductance properties of metal and semiconductor nanowires and their interconnections [1]; (ii) Atomic-scale friction, control of friction, and nanotribological processes in lubricated junctions [2]; (iii) Generation, stability and breakup of nanojets [3]; (iv) Catalysis by small gold and palladium clusters [4]; (v) Spontaneous symmetry breaking leading to formation of crystallized clusters (electron molecules) in individual two-dimensional quantum dots, and quantum-dot-molecules [5], (vi) Emergence of magnetism in free and surface-supported small palladium clusters [6], and (vii) Charge Transport in DNA (Science, August 19, 2001).
References
1. U. Landman et al, Microscopic Mechanisms and Dynamics of Adhesion, Microindentation and Fracture, Science 248, 454 (1990); U. Landman et al., Metal-Semiconductor Nanocontacts: Silicon Nanowires, Phys. Rev. Lett. 85, 1958 (2000).
2. B. Bhushan, J.N. Israelachvili and U. Landman, Nanotribology: Friction, Wear and Lubrication at the Atomic Scale, Nature 374, 607 (1995); J. Gao, W.D. Luedtke, and U. Landman, Friction Control in Thin-Film Lubrication, J. Phys. Chem. Chem. B 102, 5033 (1998).
3. M. Moseler and U. Landman, Formation, Stability and Breakup of Nanojets, Science 289, 1165 (2000).
4. A. Sanchez et al., When Gold is not Noble: Nanoscale Gold Catalysts, J. Phys. Chem. A 103, 9573 (1999); S. Abbet, U. Heiz, H. Hakkinen, and U. Landman, CO Oxdidation on a Single Pd Atom Model Catalyst, Phys. Rev. Lett, 86, 5950 (2001).
5. C. Yannouleas and U. Landman, Spontaneous Symmetry Breaking in Quantum Dots and Dot-Molecules, Phys. Rev. Lett. 82, 5325 (1999); ibid., Collective and Independent-Particle Motion in Two-Electron Artificial Atoms, Phys. Rev. Lett. 85, 1726 (2000); Coupling and Dissociation in Artificial Molecules, Euor.. Phys. J D 16, 373 (2001).
6. M. Moseler, H. Hakkinen, R.N. Barnett, and U. Landman, Structural and Spin Isomers of Neutral and Anionic Palladium clusters, Phys. Rev. Lett. 86, 2545 (2001).
For polymer systems, it has been conjectured that entropic effects lead to a spontaneous tightening of knots. However, this cannot be directly observed, only indirectly through radius of gyration. I will instead be discussing a system for which such an observation is possible. This system is a vibrated granular chain, and combines aspect of polymer and granular systems.I will discuss the entropic reasons for tightening in equilibrium. Experimental results will be given, which show a much sharper behavior than expected for an equilibrium system, as well as a clear breakdown of detailed balance. To interpret these results, I will give a dynamical explanation for the tightening, qualitatively valid for both equilibrium and nonequilibrium processes. This approach is hoped to elucidate the role of entropy in nonequilibrium systems.
Chemotaxis in many bacterial species is made possible by the remarkable and bizarre dynamics of their multiple, rotating, helical flagella. They bundle and de-bundle as their rotary motors episodically change rotational direction. When the flagella are bundled, the bacterium moves linearly, but the dissolution of the bundle leads to a tumbling event that effectively randomizes the cell's orientation. The motor reversal that initiates the tumbling not only torques the flagella oppositely, but also reverse the chirality of the filament, turning a left-handed helix into a right-handed helix. Hotani has performed careful experiments on helical flagella in external flows and he observed that regions within the filament periodically flip to the opposite chirality, and that those domains propagate stably downstream. I'll present a dynamical model for this phenomenon based on the existence of two competing locally stable states of opposite chirality whose interaction with the flow is through the torque they produce. The model displays a number of the key features seen in the experiments.
The propagation of a soliton pattern through one-dimensional medium with weakly disordered dispersion is considered. Solitons, perturbed by this disorder, radiate. The emergence of a long-range interaction between the solitons, mediated by the radiation, is reported. Basic soliton patterns are analyzed. The interaction is triple and is extremely sensitive to the phase mismatch and relative spatial separations within the pattern. The phenomenon is a generic feature of any problem explaining adiabatic evolution of solitons through a medium with frozen disorder. This is a joint work with I. Gabitov, I. Kolokolov, and V. Lebedev to appear in October issue of JETP Lett.
I am interested in the evolution of spatial representation. I present here a radical revision of how the mammalian brain sees, understands and maps its external world. This process, mediated by the hippocampus, has implications not only for the evolution of spatial navigation but for the evolution of a complex cognitive trait from simple antecedents. Based on a new model of hippocampal evolution, I propose that the hippocampus maps space with two independent representations, that are mediated by the 'old' (dentate gyrus) and the 'new' (Ammon's horn) hippocampal structures. The two maps are based on different stimuli: distributed and discrete stimuli, respectively. The coactivation of these parallel maps leads to the cognitive map, which I argue is a property that can only emerge from the collaboration of these two maps. The parallel map theory of hippocampal function has important implications: it explains paradoxes of spatial learning in rats with hippocampal lesions, patterns of sex differences in learning in rats, and why the hippocampus continues to generate new cells, even in the adult mammal. It also predicts sex differences in humans, not only in how men and women navigate the world, but also in abstract thought processes that are spatially encoded. Thus the understanding of a primitive universal, the need to navigate in space, may have important consequences for the development of higher cognitive processes.
Standing waves, parametrically excited on the free surface of a fluid byperiodic acceleration, are found in a wide variety of patterns: squares,hexagons, quasipatterns, superlattices, etc. The more exotic of these patterns are readily obtained in laboratory experiments by employing a periodic forcing function that has (at least) two frequency components. This introduces a number of control parameters to the problem: the amplitudes, frequencies and relative phases of the forcing components. Our aim is to understand the role of each in the pattern formation problem in a systematic fashion. We use methods of equivariant bifurcation theory to make some headway with this, focusing on the weakly inviscid situation. Our analysis identifies a subtle stabilization mechanism for superlattice patterns that involves resonant interactions of waves driven by the two frequency components. We also use general symmetry arguments to derive the form of the resonant coefficients in the relevant bifurcation problem in several cases, showing how these depend strongly on the forcing frequency ratio and on the relative phase of the two driving terms. Additional scaling laws follow from the weakly broken time reversal symmetry and an underlying Hamiltonian structure. The predicted scaling laws, etc. are confirmed by numerically calculating coefficients for the resonant triad amplitude equations from the quasipotential formulation of the gravity-capillary wave problem due to Zhang and Vinals. This talk highlights collaborative work with Jeff Porter and Chad Topaz.
Turbulence is one of those notorious problems in classical and nonlinear physics, which so far has not surrendered to the cohorts of physicists. Over almost a century many tactics have been invented to derive the statistical properties of fully developed turbulence from the fundamental hydrodynamical equations, but with little success. Contrary to these theoretical top-down approaches, a bottom-up approach is presented. The emphasis is put on understanding the data and the relationship between various, apparently different data-motivated models first. From there, more sophisticated phenomenological models are developed, which are expected to bridge the gap between data and fundamental hydrodynamical equations.
Granular gases spontaneously separate into dense and dilute regions. Here we experimentally and theoretically demonstrate that the cluster formation and its breakage are fundamentally different due to the lack of time reversability: For a vibro-fluidized granular gas in N connected compartments the cluster formation process is gradual, via several metastable states, whereas the collapse of the cluster is very abrupt. The observed cluster lifetime (as a function of the driving intensity) is analytically calculated within a flux model, making use of the self-similarity of the process. After collapse, the cluster diffuses out into the uniform distribution in a self-similar way, with an anomalous diffusion exponent 1/3.
It is often the case that successful simulations of experiments result from a collaboration of the experimentalists and the modelers, because the relation between the two groups allows transfer of details about the weaknesses of the tools that each group must use. The aim of the work discussed in this talk is move part of this relationship to the experiment and the simulation by allowing the simulation to use partial and flawed experimental information, such as isolated measurements, shadow-like projections and/or qualitative information to confine simulation results. We will give TOY examples in which we have only qualitative information about initial conditions and a few isolated measurements which illustrate how we can guess initial conditions and experimental parameters that give agreement between the experiments and the simulations. Techniques that incorporate uncertainty in the measured values will be examined.
BE WARNED, this is a report on work that is quite preliminary. If you are expecting an encyclopedic view, you will be disappointed. On the other hand, since we don't know what we are doing, your insights will be much appreciated.
There has been much recent activity on the problem of crumpling of elastic membranes, both here at the Univ. of Chicago, and at many other places. I will give an (elementary) overview of some of the mathematical questions that come up through the study of crumpled sheets. I will then discuss some of the tools and techniques that are used to study this problem. Finally, I will give my (very biased) perspective on some of the recent results that have been obtained, and the many questions that spring from these results.
A continuum theory of partially fluidized granular flows is developed. The theory is based on a combination of the equations for the flow velocity and shear stresses coupled with the order parameter equation which describes the transition between flowing and static components of the granular system. We apply this theory to several important granular problems: avalanche flow in deep and shallow inclined layers, and shear granular flows between two plates. We carry out quantitative comparisons between the theory and experiment.
Every day, whenever water flows out from an overturned cap, we observe the Rayleigh-Taylor instability. Turbulent mixing caused by this instability is a long-standing problem in many physical and technological applications (inertial confinement fusion, supernova, flames, etc.). The cascades of energy and the dynamics of large-scale coherent structure are fundamental issues. This large-scale structure is a periodic array of regular bubbles and singular spikes. To study its dynamics we propose new approach based on symmetry theory. First we analyze the local properties, and derive from the conservation laws a dynamical system governing the bubble dynamics. Due to formation of the singularities, the regular asymptotic solution to the dynamical system for a family, and we choose the fastest stable solution in the family as the physically dominant one. The dependence of the bubble motion on the acceleration history as well as on the flow symmetry is analyzed. It is shown that 3D bubbles in RTI conserve near-circular contour, and the 3D-2D dimensional crossover is discontinuous. Then we consider global properties, and study the structural stability and the transitions associated with the growth of the flow length scale. Both local and global analysis lead us to a conclusion that in RT turbulent mixing a balance between the inverse and direct cascades is required to keep isotropy of the flow. The theory eliminates discrepancies between previous approaches, explains existing experiments, predicts new ways of the bubble front evolution, and establishes control parameters to be monitored in experiments.
A central problem in computer vision is the detectionof objects of a particular class in complex images containing multiple objects. Detection refers not only to computing the location, but other parameters describing the particular instantiation of the object in the image. This can have the form of a non-linear deformation defined on a `prototypical' example of the object. I will describe two continuum based variational problems for finding the deformation of a prototype to a simple image containing only one object, at more or less the scale of the prototype. These are solved using a coarse to fine gradient descent procedure. In complex images location and scale need to be identified prior to the computation of the deformation. This is impossible to do with the original continuum based cost functions. Instead a coarse and discrete approximation to these functions is formulated, allowing for very efficient identification of candidate locations and scales.
Motivated by experiments performed both with a surface forces apparatus and by neutron scattering, we discuss theoretically the behavior of grafted polymer layers under a hydrodynamic shear.
In the existing models, chain configurations are stationnary; we take explicitly into account the diffusion of the chain end points in the direction perpendicular to the grafting surface; this diffusive motion allows an exchange of the chain configurations. Each chain is subjected to the shear only when its end point stands in the thin region at the edge of the grafted layer where the flow penetrates.
We calculate both the extension of a chain in the direction of the flow and the tension on the grafting surface. As long as the relevant Deborah number (the dimensionless shear rate) remains small, these quantities remain small of the order of hte values that they reach by thermal fluctuations. This is in agreement with the recent neutron scattering experiments but in strong disagreement with the old surface force experiment where a strong deformation of the chains is observed. Our mean field approach also predicts a thinning of the grafted layer due to the shear.
The following free surface problems will be presented, two forpotential flow and two for Stokes flow. 1. The design of the pressure pulse for a drop-on-demand ink-jet-printer, and the associated pinch-off. 2. How cusps are rounded by surface tension in the Hele-Shaw flow of an initially circular blob of fluid withdrawn through a non-concentric sink. 3. The collision of two deformable drops in a viscous suspension undergoing shear. 4. The rheology of an emulsion at moderate concentration.
The presentation will be partly about mechanics, including some unresolved problems, and partly about numerical methods, including some recent ideas such as the use of B-splines to represent the unknown surface distributions in 2D calculations and the use of radial basis functions to calculate the curvature of a 3D surface.
Most talks in this series describe how one can use computation to help answer scientific questions. This talk will instead be about using science to improve the computational infrastructure. I will discuss two design questions relating to the Internet architecture. These questions involve extending the Internet architecture to improve Quality-of-Service (QoS); one extension is to allow bandwidth reservation and the other is to use multiple priority levels for streaming media. The emphasis will be on using extremely simple models to explore qualitative aspects of these design issues. This talk will be self-contained, and no knowledge of networking or of Internet architecture is required.
My research aims to elucidate how proteins attain their folded conformation within biologically relevant time scales. Proteins sample a vast number of conformations on their way to the folded state and the study of folding is best approached from a statistical standpoint. Concepts borrowed from the well-established field of statistical mechanics have provided considerable insight into the folding problem. The energy landscape of the protein may be described as a minimally rough surface, in which a strong energy bias towards the native state "funnels" the protein towards its biologically active conformation. Roughness of the surface, which hinders this funneling, can be due to both energetic factors (associated with the formation of incorrect but stabilizing interactions) and to topological factors (geometric constraints leading to the premature formation of native interactions). I will discuss the origins of topological roughness (or "frustration") and how the native state "shape" of the protein affects the folding process. My approach involves a combination of off-lattice minimalist models in which the protein is described in a coarse-grained manner and fully atomic models, which provide a detailed representation of both the protein and the solvent.
The digital revolution has given us a clear distinction between hardware and software, between channels and the content they carry, between physical science and computer science, but it is right at these boundaries between the bits of the digital world and the atoms of our physical world that the most compelling opportunities and problems in information technology lay. I will discuss the science underlying the integration of information with its physical properties over length scales from atomic nuclei to planetary networks, and discuss its implications for the life of people, and their machines. Examples will be drawn from projects addressing global development, creative expression, and appropriate interfaces.
I will try to describe quantum computation to computer scientists unfamiliar with quantum mechanics, without boring quantum physicists unfamiliar with its recent application to computation. People familiar with both subjects will have to settle for the pleasure of understanding everything they hear, though they may find my point of view amusing. Or irritating.
Many interesting problems involve a combination of Hamiltonian mechanics with some dissipative dynamics. In this work, a large space contains a friction-free lower dimensional region embedded within it. A particle located in that embedded region will undergo a Hamiltonian dynamics. Particles with trajectories in some nearby regions are drawn into the space. Other nearby orbits make the particle escape, 'bail out', of this neighborbood of the Hamiltonian region and go someplace far away. Escape and reinjection into the neighborhhood permit all kinds of interesting orbital behaviors.
We can design this space so that orbits of our choosing bail out, and others stay. Specifically, we build our system so the KAM trajectories stay near the embedded region. (KAM trajectories are particularly interesting motion of the Hamiltonian system, which show marginal stability and an integrable behavior.) A particle initially on the embedded region is subjected to small amounts of noise, which then serves to move it slightly out of the embedding region. Such a particle typically bails out of the neighborhood of the embedded region after some time, except if it lies on a KAM orbit. This procedure thus allows us to "surgically" separate KAM orbits from the other Hamiltonian orbits.
We analyze this bailout in terms of the fluctuation amplitude around the stable embedding for infinitesimally small noise. We show that the bailout process consists of two distinct phases, an avoidance phase, where the fluctuations are small but whose amplitude acquire arbitrarily large prefactors, and a detachment phase, where the fluctuations become finite in size even for infinitesimally small noise.
The floor of the earth's ocean has been created over the past 150 million years by plate tectonics. Continental plates are moving apart and new ocean floor is perpetually solidified at the mid-ocean ridges. From satellite data and ship soundings we have an excellent knowledge of the bathymetry of the ocean floors and find three distinct morphological features: transform faults and fracture zones, a spreading velocity dependent mid ocean ridge profile, and microplates. Midocean ridge dynamics combines the full complexity of fluid flow, phase transformation, and fracture. I will show in my talk that a simple table top experiment can capture the complex dynamics of earth like processes. Our experiments pose a challenge to theorists: Is it possible to develop a consistent theoretical model that captures the complexity of our table top experiments and possibly that of the earth.
[Joint work with Boualem Khouider (UdM) and Andy Majda (Courant).]
Turbulence enhances the speed of propagation of premixed flames via mixing in the preheat zone. Capturing this phenomenon is a huge computational challenge because, to do so accurately, one would need to resolve the wide range of length scales induced by turbulence, chemical reactions, and their interaction.
On the other hand, a rigorous homogenization approach to describe the turbulent flame propagation in the asymptotic limit of a very thin reaction zone has been developed by Majda and Souganidis (1994) in an idealized context. In this talk, I will describe our attempts at bridging the gap between the mathematical understanding in the asymptotic limit and practical simulations requirements.
This involves purely numerical issues (designing a novel, robust numerical solver for the effective Hamiltonian of the flame for a variety of flows); fundamental modeling issues (using the numerical database and formal asymptotics to parameterize the burning speed turbulent enhancement as a function of a flame residence time - this lead us to a new quantitative explanation for the so-called "bending" effect); large scale computational issues (possible interpretation of the homogenized limit not just as a mathematical concept by also in the practical context of the validation of large scale simulations by comparison with resolved computations at finite values of the flame thickness.)
Numerical and physical experiments on two-dimensional (2d) turbulenceshow that the differences of transverse components of velocity field are well described by Gaussian statistics and Kolmogorov scaling exponents. In this case the dissipation fluctuations are irrelevant in the limit of small viscosity. In general, one can assume the existence of a critical space-dimensionality d=dc, at which the energy flux and all odd-order moments of velocity difference change sign and the dissipation fluctuations become dynamically unimportant. At dThe expressions for pressure and dissipation contributions to the exact equation for the generating function of transverse velocity differences are derived in the vicinity of d=dc. The resulting equation describes experimental data on two-dimensional turbulence and demonstrate onset of intermittency as d-dc>0 and r/L -> 0 in three-dimensional flows in close agreement with experimental data. In addition, some new exact relations between correlation functions of velocity differences are derived. It is also predicted that the single-point pdf of transverse velocity components in developing as well as in the large-scale stabilized two-dimensional turbulence is a gaussian.
Robots have motors and people have muscles. What for? To guide motions and to make up for lost mechanical energy. How much guidance is fundamentally needed for repetitive tasks? How much energy needs to be supplied for what losses? One approach to understanding the need for motors and controllers is what can be done without them.
Tad McGeer demonstrated (1988-1993) with simple computational models and with physical devices that uncontrolled human-like walking motions can be achieved with, to put it simply, sticks and hinges that walk downhill. The motions of these toy-like devices are energetically efficient (low specific transport cost) and stable (limit-cycles with linearized stability). We have found that, in principle, some of these devices can walk on arbitrarily small slopes and thus approach perfect efficiency and that Robot configurations that have this efficiency are reminiscent of the human design. These models can also limp (period 2), waltz (period 3) , and stumble (chaos). One of our devices has the unintuitive feature that it has no stable standing posture, yet can walk stably.
The basic theory is not novel: numerical search for limit cycles and numerical evaluation of their stability. However, two morals seem to be exposed: locomotion efficiency is based on avoidance of impacts, and stability comes from utilizing non-holonomic constraints.
At Argonne, Chicago and elsewhere work has begun to explore the concept of integrated whole room scale visual environments. These environments consist of group work rooms that have been augmented with multiple displays including: large-format whole wall displays (e.g. ActiveMural our high-resolution rear projected tiled display), driven by PC clusters, or multi-processor visualization engines, semi-immersive or immersive displays (Workbenches, ImmersaDesks, CAVEs), multiple desktop devices, and multiple front projection systems. These rooms may also have active or passive tracking systems, multiple channels of audio support, and support for multiple wireless hand-held controllers and navigation devices.
These room-sized environments can be linked via the national "Grid" to form compelling collaborative visualization environments (e.g. "The Access Grid"). We believe these systems represent a new type of visual application development target and delivery mechanism. We call these ensembles ActiveSpaces. In this talk I will explore with the audience some of the ideas we are working on to facilitate the delivery of high-end scientific visualization to groups of users and to create new types of electronically augmented spaces explicitly designed to support rapid collaborative exploration and visual analysis of complex data.
In the selective withdrawal experiment we lower a straw so that its tip hangs above a water-oil interface. We then withdraw the oil through the straw. When the withdrawal rate is low the interface below the straw deforms into a hump with a flow stagnation point located at the hump peak. As the withdrawal rate is increased, this hump grows in height and the curvature at the hump peak becomes very large. At the transition flow rate the interface makes a topological transition from being bounded to being unbounded in the vertical direction. The water is then entrained in a thin spout along with the oil and the stagnation point moves from the interfacial boundary to the interior of the lower fluid. For two-fluid systems with different straw diameters, viscosity ratios, density differences and surface tensions, I will present data showing interesting scaling and hysteretic behavior in the transition-flow-rate dependence on the straw heights. I will then discuss the scaling behavior in the steady state hump shapes as the system approaches the transition point. Finally, I will point out some experimental problems related to selective withdrawal and will try to draw the audience into a discussion of these experimental issues.
A very simple system that supports stress like a granular material is a pack of frictionless spheres of random sizes, deposited one at a time in a periodic box. David Head (University of Edinburgh) Alexei Tkachenko (Bell Labs) and I have been studying jamming and force chains in simulations of this system in two dimensions. First I'll describe how the simulation achieves a mechanically stable state by a simplified relaxation process that requires no translational motion. Next I'll show how forces propagate in this granular medium, corroborating previous theoretical postulates. The support for an applied small point force is concentrated along diagonal lines leading to the bottom: so-called light cones. The distribution of contact forces resembles those reported in physical jammed systems of simulated emulsions or real glass beads. We report how the force distribution evolves under various conditions of loading. We also report our search for organization of the contact forces into force chains, in which strong forces tend to occur in opposing pairs on a given particle.
The evolution of a foam is determined by the drainage flow of the continuous (liquid) phase and the coarsening (aging) of the gas bubbles. Free drainage experiments with slow and fast-coarsening gases show markedly different dynamics and elucidate the importance of the coupling of these two effects. Strong coarsening leads to accelerated drainage; however, the liquid flow also becomes self-limiting and cannot exceed a maximum drainage rate. A simple physical model incorporating foam drainage and diffusive coarsening shows quantitative agreement with experiment.
The statistical correlations between defects in the two dimensional complex Ginsburg-Landau model are discussed in the defect coarsening regime. In particular the defect-velocity probability distribution is determined. The spiral arms of the defects lead to a very different behavior for the order parameter correlation function in the scaling regime compared to the results for the related dissipative model.
This is joint work with Ed Ott. We propose that an useful approach to the modeling of periodically forced extended systems is through continuum coupled maps (CCMs). CCMs are discrete time, continuous space models, mapping a continuous spatially varying field Xi_n(x) from time n to time n+1.
The efficacy of CCM models is illustrated by application to experiments of Umbanhowar et al. on vertically vibrated granular layers. We first derive an appropriate CCM model for this system, using simple physical considerations (essentially dimensional analysis). We then present a framework for the analysis of pattern selection in CCM models using a truncated modal expansion. Through the analysis, we show how the model reproduces the observed experimental behavior. We also obtain some other results (scaling laws) that are experimentally testable predictions from our model. We conclude with a discussion of the limitations of our model for the vibrated granular layer systems, and extensions of this approach to other (non-granular) periodically forced, strongly dissipative systems.
We study a system of hard-core particles sliding locally downwards on a fluctuating surface. For certain surfaces, the system exhibits a novel steady state in which most strikingly, phase ordering coexists with large-scale fluctuations. The distribution of the particle cluster sizes varies as a power law, and gives rise to many of the unusual spatial properties of this ordered state. Insight into the origin of this phenomenon is obtained by studying coarse-grained depth models of the hill-valley profile of the underlying surfaces.
The BaBar experiment at the Stanford Linear Accelerator Center has produced over 100 Terabytes of data and is expected to produce 300 Terabytes per year, soon. These data require extensive processing prior to and after storage. The 4-million lines-of-code system that performs this task was written, from scratch, in C++ by a group of people distributed all over the world. This talk will discuss the process of building this system and will discuss some aspects of the system architecture. The talk will not spend substantial time on database design nor hardware architecture.
This presentation concerns the astonishing diversity of individual and collective dynamic phenomena exhibited by swimming bacteria ( Bacillus subtilis ), at concentrations ranging from dilute to close-packed. Topics covered will include 1) the distribution of swimming velocities, 2) binary interactions, 3) influence of bounding geometry on the velocity probability densities for speed and direction of swimming, 4) consumption/supply - driven bioconvection patterns, and 5) chaotic dynamics of populations at high volume fraction, where the trajectories of inert tracers include intermittent "trapping", long flights, and transport exponents reminiscent of the superdiffusion found in "2-d turbulence". Approaches to modelling some of these phenomena will be presented, e.g. bioconvection and some possible mechanisms for energy balance and long range coherence required for "turbulence" at low Reynolds Number. --> Videos ! <--
I will give an overview of the recent work that has been done in an attempt to create a mathematical formulation of the evolution of language. I will speak about the two major components of the language: the lexicon and the grammar. In a sense, languages evolve like individuals in a population: the fittest ones survive and get passed down generations, the less fit get eliminated. The two driving forces of evolution, selection and mutation (i.e. the mistakes when learning a language), can be incorporated into a simple system of ODE's called the evolutionary equations. Within this framework, it is possible to get some analitical insights into the dynamics of the language. One of the questions we ask is how accurate children have to learn the language of their parents in order for the population to be able to maintain a coherent language? Another one is what are the evolutionary forces that shape the Chomskian Universal Grammar?
Sea ice is a composite of pure ice with brine and air inclusions. It is distinguished from many other porous media, such as sandstones or bone, in that its microstructure and bulk material properties depend strongly on temperature. Above a critical value of around -5 degrees C, sea ice is permeable, allowing transport of brine, nutrients, and heat through the ice. These processes play an important role in air-sea-ice interactions, in the life cycles of sea ice algae, and in remote sensing of the pack. Recently we have used percolation theory to model the transition in the transport properties of sea ice. We give an overview of these results, and how they explain data we took in Antarctica. We also describe recent work on inverse scattering algorithms for recovering the physical properties of sea ice via electromagnetic remote sensing, and how percolation processes come into play. At the conclusion, we will show a short video on a recent winter expedition into the Antarctic sea ice pack.
In the chaotic case (time-periodic velocity field), the scalar evolves to a complex recurrent pattern that subsequently decays without change of form, as first noted in a numerical simulation by Pierrehumbert. The typical path length per cycle of the forcing and the Reynolds number are shown to govern the decay rate, but the dependence is strikingly non-monotonic. The time evolution of various statistical measures of the scalar field provides a quantitative description of the interplay between stretching and molecular diffusion. It is surprising to note that diffusion does not broaden the striations of the scalar field, We have explored the effects of many flow variables including periodic and nonperiodic forcing in both space and time. Particle tracking over long perios of time is also used to study the transient mixing process. Weakly turbulent flows (obtained by reducing the viscosity) are shown to mix much more efficiently than chaotic flows in the same geometry.
The thermodynamical properties of protein are very well documented experimentally. Two first order phase transitions are found: the well-known ``warm'' unfolding around 60 C and the less known ``cold'' unfolding around 0 C. To explain these data, we propose a protein model based on a hierarchy of constraints that force the protein to follow certain pathways when changing conformation [1]. The model exhibits a first order phase transition, cooperativity and is exactly solvable. The model is extended to explicitly take into account the coupling between the protein and water degrees of freedom. In a statistical mechanics treatment we obtain both the cold and the warm unfolding transitions and reproduce qualitatively the known experimental results. We argue that the two transitions ends in a critical point at a given temperature and chemical potential of the surrounding water [2]. In order to characterize the sharpness of the transition we weight multiple pathways for the folding and show that most transitions generically are two-state like in accordance with experiments on single domain proteins [3].
[1]. A. Hansen, M.H. Jensen, K. Sneppen and G. Zocchi, Eur. Journ. Phys B 6, 157 (1998).
[2]. A. Hansen, M.H. Jensen, K. Sneppen and G. Zocchi, Eur. Journ. Phys B, 10, 193 (1999); Europhys. Lett. 50, 120 (2000).
[3]. P.G. Dommersnes, A. Hansen, M.H. Jensen and K. Sneppen, ``Parametrization of Multiple Pathways in Proteins: Fast Folding versus Tight Transitions'', cond-mat/0006304 (2000).
The spectacular developments in computer hardware and software over the past half-century have revolutionized what can be done and what can be expected to be done via simulations of fluid dynamics. In this talk, we will review progress, try to make forecasts of future advances, and point out various pitfalls that can be encountered. A discussion will be given of the status of diverse methods, including direct simulation, large-eddy simulation, lattice methods, and the like.
Geologic data have been interpreted to show that a catastrophic flood occurred 7600 years in the Black Sea. Was anyone there?
Last week, the discovery of remnants of human habitation under the Black Sea was announced. This is believed to be the first proof that people thrived along an ancient shoreline before it was inundated by a great flood thousands of years ago.
Was this event the source of the Noah's Flood story and other flood Myths?
In this study we investigate stratified Kolmogorov shear flow. We derive the amplitude equations for this system and solve them numerically to explore the effect of a weak stabilizing stratification. We then explore the non-diffusive limit of this system, and solve amplitude equations for this system to study the weakly nonlinear evolution of the internal boundary layer in the stratification. We further solve the full 2-dimensional system and investigate the different dynamics as we vary the Peclet number.
When a drop is released from a nozzle very close to a liquid surface, it will sit momentarily before coalescing into the bottom layer. High-speed video imaging reveals that the coalescence process is not instantaneous, but rather takes place in a cascade where each step generates a smaller daughter drop.
This cascade is self-similar, with each step generating a new drop about one half the original diameter. We have observed up to 6 steps in this cascade, generating drops as small as 180 $\mu m$ in diameter. Using ultra-high-speed video, with frame rates as high as 40500 f/s, we have measured the time associated with each partial coalescence. This time scales very well with the surface tension time-scale.
The coalescence cascade will however not proceed ad infinitum due to viscous effects, as the Reynolds number of the process is proportional to square root of drop diameter. Viscous forces will thereby become increasingly important as the drops become smaller.
We will furthermore present some recent results from impacts using granular materials, thus eliminating the effect of surface tension. The results could be very useful in separating inertial and surface tension effects, as well as building constitutive laws for rapidly moving granular media.
Reference: Thoroddsen, S. T. and Takehara, K. ``The coalescence-cascade of a drop'', to appear in Physics of Fluids.
In this seminar, I will present an overview of several types of singularities that can occur in parabolic equations. Several of the examples that I will exhibit, will concern examples of singularities that appear in the so called Keller Segel system that has been extensively used in the study of chemotactic aggregation of biological organisms. The analogies between the type of singularities that occur in this system with the ones that take place in another systems, like the Kompaneets equation used in plasma physics and the classical Stefan problem in solidification will be also discussed during the seminar.
Many of the amino acid polymeric protein products of human genome sequences have homologies with familiar transmembrane receptors, but are without either known natural messengers, "ligands," or physiological functions. The current approach to drug discovery for these "orphan receptors" is called "high throughput screening" and involves multimillion dollar factories that robotically screen up to a hundred thousand chemical candidates per day for biochemical signs of receptor activation. When an active compound is found (with successes in the 1-2 per 100,000 range) it is characterized by its 3D spatial geometry and charge distribution, generating a physical model called a "pharmacophore" which drives drug companies' programs of combinatorial substitution and biological testing in their search for more potent and specific ligands.
We asked the question, given only the DNA derived, receptor's amino acid sequence, could we computationally design new, short (15-20 mer) amino acid polymers, peptides, which could activate orphan receptors and thus shorten (and significantly cheapen) the process of new drug discovery. Our successes in this pursuit have involved the conversion of the receptors' amino acid sequences into a unified system of meaningful physical, quasi-thermodynamic quantities followed by the application of several signal processing and symbolic dynamical techniques to find one dimensional patterns which are then used as templates for peptide design. The sequences of receptors and other proteins that were transformed in these ways:
(1) Revealed diagnostic global familial patterns; for examples, Morlet wavelet transformations of protein sequences discriminated between helical, strand, mixed, poly and receptor proteins and in the latter located likely segments for ligand targeting.
(2) Led to sliding window computations of the local sequential Markovian metric entropy, which located segments of higher order and successfully marked the physiologically distinct sections of "polyproteins," those are post-translationally split up into multiple distinctive peptide messengers.
(3) Involved Karhuenen-Loeve-like orthogonal mode decomposition of receptor sequence, lagged autocovariance matrices and the construction of Broomhead-King-like eigenfunctions, which, when characterized by all poles, "maximum entropy" power spectra, demonstrated systematic matches between the modes of known peptide receptors and their ligands.
Inverting (3) for brain-related orphan receptor function by using the ligand-relevant, eigenfunction associated eigenvectors as templates, we designed 15 mer peptides. When 22 of these were synthesized and tested, 15 (68%) were statistically signficantly active in vitro and in vivo (in brain). This suggests , counterintuitively, that a one-dimensional approach to this apparently three dimensional protein folding-like problem can be useful. We think that Israelachvili's aqueous "hydrophobic long range attraction" (500 angstroms, 10-100 fold van der Waals forces) between matching segments of sequentially patterned hydrophobic amino acids lead to their hydrophobic aggregation, membrane receptor destabilization and physiological alteration.
*Major participants in this work include Karen A. Selz, Michael J. Owen and Michael F. Shlesinger.
Atomistic molecular dynamics simulations reveal formation of nanojets with velocities up to 400 m/s, created via pressurized injection of fluid propane through nanoscale convergent gold nozzles with heating or coating of the nozzle exterior surface to prevent formation of thick blocking films. The atomistic description is related to continuum hydrodynamic modeling through derivation of a stochastic lubrication equation which includes thermally triggered fluctuations whose influence on the dynamical evolution increases as the jet dimensions become smaller. Emergence of double-cone neck shapes is predicted when the jet approaches nanoscale molecular dimensions, deviating from the long-thread universal similarity solution obtained in the absence of such fluctuations.
Alpheus heterochaelis (``the snaping shrimp'') generates noise so loud that it disturbes submarine communication. It was believed that the noise is generated when the claw rapidly closes and its two sides hit each other. However, in this work we show with the help of high speed video (40000 frames/second) and parallel sound detection with a hydrophone that the origin of the noise in fact is a collapsing cavitation bubble: When rapidly closing the pair of sissors, the shrimp emits a thin water jet so fast that a cavitation bubble develops. This collapses and on collapse, it emits the sound. Our optical and acoustical measurements are supplemented through a simple theoretical model of the process.
In free surface flows, cusps can form under a variety of circumstances. Examples are drop coalescence, or rising bubbles in a viscous fluid. A particularly simple two-dimensional model system consists of two counter-rotating cylinders, submerged below the surface of a viscous fluid. In the absence of an outer fluid a two-dimensional cusp forms, which is stable at any value of the capillary number. However an outer fluid, typically air, will be drawn into the narrow cusp pushing its walls apart. We show that as a result stationary solutions no longer exist above a critical capillary number. Instead, a sheet forms, that is unstable to three-dimensional peturbations at its lower rim.
Recent advances in lasers and molecular beams make it possible to observe details of chemical reactions in even a femtosecond time scale. In these experiments, dynamical aspects of reactions are of interest such as the following. (1) How does the reaction path depend on initial conditions? (2) How does the energy distribution occur among degrees of freedom on the system? (3) To what extent is the process statistical? These questions are of fundamental importance in understanding molecular details of reactions such as intramolecular vibrational-energy redistribution (IVR), rates of reactions on a state-to-state basis, and dynamics of transition-state species.
On the other hand, dynamics of vibrationally excited molecules in gas phase is a typical example of Hamiltonian dynamics of many degrees of freedom. It is well known that generic Hamiltonian systems of many degrees of freedom exhibit chaos. Therefore, IVR is supposed to be closely related to chaotic motion of the molecules.
However, most of the studies on chaos so far have focused their attention to one-dimensional maps. In order to fill the gap between the study of chemical reactions and that of chaos, we need to investigare chaos of many degrees of freedom.
Our study is a step towards this direction. Our main results are two-fold. First, we will show that there exists a transition between lower dimensional chaos and a higher dimensional one. This transition is signalled by homoclinic (or heteroclinic) tangency between stable and unstable manifolds. Second, the symmetry of molecules plays an important role. Since the molecular systems are quantum, interference effects tend to suppress chaos. This phenomenon is revealed in the network of nonlinear resonances (Arnold web).
In this talk I will present some work in progress in vision research. We consider the problem of recognizing what parts of an image are perceived as being in the foreground. We use a variant of the Pao-Geiger-Rubin model, which uses an energy dissipation approach to this problem. The model is surface-based, rather than contour-based. Specifically, the edges in the image are not viewed as isolated contours, but are viewed as bounding a surface. Each local edge has a local hypothesis; for example, a north-south edge might think "the region immediately to the left of me is part of the figure". The model then uses energy dissipation methods to seek assignments of local hypotheses that are mutually agreeable, yielding a segmentation of the image that might be perceived. We test the model on various images to address questions like: Does the model "perceive" smaller objects to be in the foreground (the way we do)? Convex objects to be in the foreground (the way we do)? How does it perform on optical illusions that viewers report to have two different segmentations?
This is joint work with Nava Rubin and Anita Disney of the Center for Neural Science, NYU. I thank Davi Geiger (Courant, NYU), Bob Shapley (CNS, NYU), and Dave McLaughlin (Courant, NYU) for useful discussions.
Protein channels conduct ions (Na+, K+, Ca++, and Cl-) through a narrow tunnel of fixed charge ('doping') thereby acting as gatekeepers or cells and cell compartments. Hundreds of types of channels are studied everyday in thousands of laboratories because of their biological and medical importance: a substantial fraction of all drugs used by physicians act directly or indirectly on channels. Channels are studied with the powerful techniques of molecular biology. The atoms of channels can be manipulated one at a time and the location of every atom can be determined within 0.3 Å. Ionic channels are 'holes in the wall' that use the simple physics of electrodiffusion to perform these important tasks. They have simple structure which is known in atomic detail in a few cases; more to come. They are ideal objects for mathematical and computational investigations. Computing the movement of spheres through a 'hole in the wall' should be easier than computing most other biological functions, yet it is nearly as important as any from a medical and technological point of view. The function of open channels can be described if the electric field and current flow are computed by the Poisson-Drift-Diffusion (called PNP, for Poisson Nernst Planck, in biology) equations and the channel protein is described as an invariant arrangement of fixed charges, not as an invariant potential of mean force or set of rate constants, as is done in the chemical and biological tradition. ThePoisson-Drift-Diffusion equations describe the flux of individual ions (each moving randomly in the Langevin trajectories of Brownian motion) in the mean electric field. They are nearly identical to the drift diffusion equations of semiconductor physics used there to describe the diffusion and migration of quasi-particles, holes and electrons. They are closely related to the Vlasov equations of plasma physics. Ionic channels form a biological system of great clinical significance and potential technological importance that can be immediately studied by the techniques of computational physics. Many of those techniques have not yet been used to analyze other biological systems. Perhaps they should be: the application of the even the lowest resolution techniques involving the Poisson-Drift-Diffusion equation has revolutionized the study of channels. An opportunity exists to apply the well established methods of computational physics to the central problems of computational biology. In my opinion, the plasmas of biology need to be analyzed like the plasmas of physics. The mathematics of semiconductors and ionized gases should be the starting point for the mathematics of ions and proteins, for the analysis of protein structure, protein folding, nucleic acids (i.e., DNA), and the binding of drugs to proteins and nucleic acids.
In this talk I shall discuss two schematic models for slow or `glassy' relaxation in driven systems. The first is an attempt to reproduce the results of the granular compaction experiments performed here at Chicago from a minimal set of assumptions, principally that the relaxation is similar to thermal activation, with an effective `temperature' that is coupled to the external driving. The second model, which is not specific to any particular material, has a similar mathematical basis but includes strain degrees of freedom, and appears to allow a strain-dependent jamming/unjamming transition, perhaps in the spirit of `jamming phase diagram' recently proposed by Andrea Liu and Sid Nagel [Nature vol. 396, p. 21 (1998)]. It is hoped that simple mathematical models such as these may aid our understanding of complex physical systems.
The Ising lattice of interacting spins is the simplest> possible microscopic model in which second order phase transition is expected to occur.Whether such transition does occur also in one dimensional lattice and what is the Curie temperature of the system was a point of contention since the time when Lenz and Ising proposed the model.We recently reexamined this model concentrating on question how the size of the system affects its properties and how one should define characteristic temperatures of various transitions that occur spontaneously in a finite-size system.Conclusions may have an impact on how we look on the Ising lattices in higher dimensions.
We measure the distributions of interparticle normal forces $P(F)$near the glass transition in supercooled liquids and compare them to those obtained in recent experiments on static granular packings. We find that the distributions $P(F)$ for glasses and static granular packings are very similar, showing a plateau or small peak at small forces. We propose that the formation of this peak signals the development of a yield stress in glasses and jammed systems.
When a bubble of air rises to the top of a highly viscous liquid, it forms a dome-shaped protuberance on the free surface. Unlike a soap bubble it bursts so slowly as to collapse under its own weight simultaneously, and folds into a wavy structure. This rippling effect occurs for both elastic and viscous sheets, and a theory for its onset is formulated. The growth of the corrugation is governed by the competition between gravitational and bending forces (shearing).
When the very same viscous liquid is drained out of the container a dimple is formed at the free surface and develop into a cusp. The interplay between the surface tension that tends to to keep the surface flat and the viscous forces "pinching" down the free surface, gives rise to a universal exponent of the height of the cusped interface versus the draining time elapsed before the dimple becomes a cusp.
The formation of self-focusing singularities and jets due to the collapse of standing waves on a fluid surface is studied using experiments, theory, and numerical calculations. A qualitative characterization of the singularity development from experimental observations is presented along with a detailed theoretical and numerical analysis of the process. The singularities focus inertial energy in the system and produce very high-speed jets which rise vertically from the surface. A similarity solution to the equations of motion which leads to the focusing is presented and compared with observation.
Non-Newtonian or viscoelastic fluids do many things which Newtonian fluids cannot do. Examples include rod climbing, the tubeless siphon, and cusp-like tails on rising bubbles. In this talk I will describe the even more peculiar behavior of aqueous micellar solutions in which the micelles take the form of long tubes (often called wormlike micelles). Our approach is both experimental and mathematical. By way of introduction to non-Newtonian fluid dynamics, I will present results on the spin down of a micellar fluid. I will then discuss new observations of the oscillations of bubbles (and spheres) rising (and falling) in a wormlike micellar fluid. We model these phenomena with various constitutive relations, and in particular focus on an ordinary differential equation model for the falling sphere in an infinite medium. For a Newtonian sphere this model is exact, for which we have proven that the sphere cannot oscillate. The work presented is in collaboration with Anand Jayaraman, Jon Jacobsen, and Andrea Young.
Today, sequence similarity searching is the most effective methodavailable for characterizing newly determined protein sequences. Similarity searching the bases of more than 80% of the gene assignments for the recently determined yeast, Haemophilus, and Methanococcus genomes. Similarity searching is popular because it is surprisingly effective. For the yeast genome, similarity searching found homologues to more than 75% of the yeast genes, and for the much more distantly related Methanococcus, homologues could be found for more than 50% of the genes.
However, finding homologues for 75% or 50% of the genes means 25% 50% of the genes were unidentified. Genes may be missed because they are novel - not present in other organisms. However, in most cases, these "non-homologous" genes may share a common ancestor with sequences in the databases, but the sequences have diverged so much that the homology cannot be detected by sequence comparison. Our goal is to develop more effective methods for protein sequence comparison, so that distant relationships that cannot be reliably inferred today can be detected.
The seminar will discuss the logical and statistical basis for the inference of homology from sequence similarity, demonstrating that inferences of homology based on sequence similarity are reliable. By comparing human proteins to the proteins in completely determined genomes (C. elegans, yeast, E. coli, M. jannaschii) we can estimate how far back in time we can look, and possibly discover "young" protein sequences. If many "young" proteins have emerged in the past 800 My, one might infer that discovering (or rediscovering) protein folds is easier than expected.
About 40 years ago Hubel and Wiesel discovered that neurons in the Visual Cortex (V1) of cats and primates respond selectively to oriented contrast edges and bars. They conjectured that converging axons from neurons of the Lateral Geniculate Nucleus (LGN), which themselves respond to spots of light on a contrasting background, could provide the anatomical substrate for edge detection. It has recently been shown in computational modelling studies that this mechanism cannot fully account for the selectivity of cortical neurons to more complex stimuli. Intracortical mechanisms are also necessary, in particular recurrent cortical excitation and lateral inhibition. Thus neighboring cortical neurons signalling similar orientation preferences cooperate, those signalling different preferences compete. This "Turing mechanism" was originally suggested as a cortical property by Wilson and Cowan in 1973. In the context of orientation tuning it is known as the "Ring Model".
In this talk I will show how the ring model can be analyzed mathematically using the techniques of nonlinear dynamics. I will do this both with continuous neuron models and also with spiking neurons. In so doing I will describe novel methods for analyzing networks of spiking neurons recently introduced by Bressloff and Coombes in the UK. Such methods lead to the prediction that visual cortex cells can exhibit clustered spiking patterns when responding to stimuli, in a manner consistent with recent experiment findings of Gray and Singer that there exists a 40 Hz modulation of neural spiking patterns.
I will then describe recent discoveries concerning the longer ranged architecture of the visual cortex which suggests how to extend the ring model to cover, not just one local patch, but the entire visual cortex. The mathematical problems of dealing with the visual cortex as a whole are both more difficult and more interesting than those concerning a single patch. I will describe some of these problems, and some experimental predictions of the analysis relevant to normal context dependent visual perception, and to abnormal phenomena such as visual illusions of angle and geometric visual hallucinations. In so doing I will suggest how top-down influences from extra-striate cortex, V2 and beyond may also play a role.
[This talk is based on joint work with my former graduate students G.B. Ermentrout (1976-1980), M. Wiener (1992-1994) and T. Mundel (1993-1996), and recently with P. Bressloff (1998-) and M. Golubitsky (1999-), and my current student P. Thomas.]
Often one is interested in the dynamics of a spatially extended system only down to some appropriate level of detail known in advance. In such a case, it is wasteful or perhaps impossible to compute the dynamics at scales smaller than this limit, even though the problem may be complex and nonlinear. This "under-resolved computation" is considered here from a renormalization group (RG) perspective. Assumptions about the behaviour of the ignored degrees of freedom typically mean that even deterministic problems must be modeled as stochastic differential equations. The RG provides a natural framework for coarse-graining such problems up to the scale of interest.
In this talk, I will discuss both the successes and current limitations of this method.
Work performed in collaboration with Qing Hou and Alan McKane, and supported by NSF-DMR-93-14938.
Adapting statistical physics to deterministic dynamical systems with a large number of degrees of freedom is an ubiquitous question in nowadays physics. Coupled map lattices (CMLs) constitute perhaps one of the simplest models of spatio-temporal chaos, hence appear as a model of choice to test our ideas. Strongly-coupled chaotic maps generically display collective behavior emerging out of extensive chaos. The rich phenomenology exhibited by these systems, although much more complex than that of single maps, is nevertheless reminiscent of the self-similar structure of asymptotic trajectories observed in low-dimensional (temporal) chaos. After presenting general properties of collective behavior, I will show how an extension of the well-known renormalization group (RG) of unimodal maps holds for coupled systems. I will then present an approximation scheme that, taking into account the dynamics of spatial correlations, reproduces strikingly well the collective behavior of strongly-interacting maps.
The dynamics of free surface flows, and in particular the mechanisms for singularity formation at the interface of fluids with different physical properties, constitute a problem of high theoretical and practical interest. The applications include such commonplace devices as ink-jet printers and fuel injectors, oil extraction, and fiber spinning. While considerable theoretical and computational advances have been achieved in our understanding of the problem in certain limiting cases (such as the drop pinch-off in lubrication approximation), theoretical understanding of the general case is still lacking. We present a general numerical algorithm aimed at describing the dynamics of singularity formation in axially symmetric free surface flows for arbitrary Reynolds numbers. In order to improve the spatial resolution in the vicinity of the singular point the interface is treated as a mathematical discontinuity corresponding to the abrupt change in the fluid properties, rather than being artificially smeared over a finite region, as is usually done. As a particular application, we discuss the results of the direct numerical simulation of selective fluid withdrawal and compare them with recent experiments by Sid Nagel and Itai Cohen.
The microscopic mechanisms of epitaxial growth have been known for 50 years, but we are still far from mastering their mesoscopic consequences. I will discuss two topics of this type: (a) The analysis of coarsening during spiral growth (joint work with Tim Schulze). The starting point is a simple, geometric model of spiral growth, which gives a Hamilton-Jacobi equation for the height of the growing film. The coarsening behavior is obtained by examining the Hopf-Lax solution formula. (b) The analysis of coarsening during step-flow growth, associated with step-bunching (PhD thesis work of Cameron Connell). The starting point is a reaction-diffusion model proposed by J. Kandel and D. Weeks. The coarsening in this setting is due to collision of traveling waves.
In the first part of my talk, I will show that the TDGL equations of superconductivity define a dynamical system in a suitably chosen gauge. Then I will discuss the "frozen-field approximation" and its relation to the TDGL equations. I will illustrate with the results of some numerical simulations.
Global bifurcations such as crises of attractors, explosions of chaotic saddles, and metamorphoses of basin boundaries play a crucial role in understanding the dynamical evolution of physical systems. Global bifurcations in dissipative planar maps are typically caused by collisions of invariant manifolds of periodic orbits, whose dynamical behaviors are described by rotation numbers. We show that the rotation numbers of the periodic orbits created at certain important tangencies are determined by the continued fraction expansion of the rotation number of the orbit involved in the collision.
A variety of experimental techniques are now available for creating spatially resolved images of single vortices passing through Type-II superconductors and even for tracking their motions. This talk focuses on some of the recent progress in mining this rich vein of data. Images of vortex distributions created by Lorentz microscopy or Bitter decoration offer new qualitative and quantitative insights into the topology of the pinscape, or random pinning potential, on which the vortices are arrayed. Vortex correlations, similarly, make possible the first direct measurements of the vortex interaction potential using the characteristic energy scale for pinning as a reference. These measurements reveal a surprising analogy between vortices array on a quenched random pinscape and classical particles buffeted by random thermal forces. This analogy carries over to phase transitions in vortex ensembles revealed by recent torque magnetometry measurements. Understanding the kinetics of these phase transitions likely will require new insights into the mechanism of heat evolution and transportation through the superconducting "substrate".
When a lamellar phase of amphiphilic molecules is subjected to shear, it may transform into an array of close-packed multi-lamellar vesicles, called the `onion phase'. A theory will be presented for the behavior of the onion phase upon dilution. A unique feature of this system is the possibility to sustain a non-uniform pressure by tension in the lamellae. Tension enables the onions to remain stable beyond the unbinding point of a flat lamellar stack. The model accounts for various concentration profiles and interfaces, which develop in the onion as it dissolves. In particular, densely packed `onion cores' are shown to appear, as observed in experiments. The formation of interfaces and onion cores is an unusual example for interface stability in confined geometry.
Hyperbolic conservation laws and advection-dominated parabolic equations model a great number of physically interesting phenomena such as shallow water and contaminant transport. Solutions to such equations often have sharp, moving fronts and other local, fine-scale features. Locally conservative methods such as upwind-mixed methods are of interest because of their ability to approximate these fine-scale features without excessive smearing or spurious oscillations. However, the standard explicit time-stepping procedures for these methods can incur a strong time step restriction in the presence of spatially varying velocity fields or local mesh refinement. In order to reduce this drawback, upwind methods which allow the time step to vary spatially yet retain a maximum principle and strict local conservation are developed. First and (formally) second order in time schemes which allow for high resolution in space will be developed, and one-dimensional numerical results demonstrating the accuracy and stability of the methods will be given. In addition, continuous time a posteriori estimates for a model convection-diffusion equation will be discussed.
For questions contact
Seung-Yeop Lee,