Previous Talks: 2000
Unusual ordering of hard-core particles sliding on fluctuating surfaces
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.
Construction of a Large Software System for a High Energy Particle Physics Experiment
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.
Fluid dynamics of bacterial suspensions: from interactions of individual organisms to collective order and quasi 2-d turbulence at Re<<1.
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 ! <--
The evolution of language
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?
Mathematics of Sea Ice
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.
Mathematics of Sea Ice
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.
Thermodynamics of Proteins: Experiments and Hierarchical Models
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 . 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 . 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 .
. A. Hansen, M.H. Jensen, K. Sneppen and G. Zocchi, Eur. Journ. Phys B 6, 157 (1998).
. A. Hansen, M.H. Jensen, K. Sneppen and G. Zocchi, Eur. Journ. Phys B, 10, 193 (1999); Europhys. Lett. 50, 120 (2000).
. 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).
Computational Fluid Dynamics: A High-LevelPerspective
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.
Noah's Flood; Historical Event Or Myth?
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?
Stratified Kolmogorov Flow
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.
The Coalescence-Cascade of a Drop
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.
Blow-up in parabolic problems
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.
Computational Design of Peptides that Activate Genomic DNA-Derived Brain Protein Receptors that are Without Known Natural Messengers
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.
Formation, Breakup and Stability of Nanojets
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.
On the Sound of Snapping Shrimp
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.
Towards Understanding Chaos in Higher Dimensional Systems
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).
Image Segmentation and Energy Dissipation
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.
Ionic Channels: Natural Nanotubes that Select between Ions
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 ≈. 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.
Glassy models for granular relaxation and history-dependent jamming
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.
Phase Transformations in One-Dimensional Ising Model of Finite Size
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.
Force distributions for Jammed and Unjammed Systems
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.
Things we can do with a viscous liquid: wrinkles and singularities
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.
Surface Singularities and Jet Eruption
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.
Snap, Jump, and Wiggle: Motion in Micellar Fluids
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.
Protein Evolution - How far back can we see?
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.
Contextual effects on orientation selectivity: beyond the ring
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.]
Generic Behavior of Reversible Cellular Automata
Renormalization Group Approach to Underresolved Computation
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.
Strongly coupled chaotic maps: collective behavior, universality, and models.
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.
Numerical Simulation of Axisymmetric Free-Surface Flows
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.
Some PDE Aspects of Thin Film Growth
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.
The Time-Dependent Ginzburg-Landau Equations as a Dynamical System
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.
Continued Fractions Hierarchy of Rotation Numbers in Planar Dynamics
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.
Vortex Matter as a Soft Condensed Matter System
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".
Swollen Onions: Dissolution of Multi-lamellar Vesicles
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.
Space-Time Adaptivity for Transport Applications
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.
Multistability in Delayed Feedback Control
Multistability readily arises in physiological delayed feedback control mechanisms. Here we show that conditions for multistability to occur in a recurrent loop comprised of a limit cycle oscillator subjected to pulsatile delayed feedback can be obtained from the measured phase resetting properties of the oscillator. Moreover, the basin of attraction can be determined for each attractor. Since the basins of attraction are known, it is possible, in principle, to use adaptive control techniques to regulate switches between attractors. The results are illustrated with experiments involving a time-delayed analog electronic circuit and with experiments involving a time-delayed recurrent loop involving an invertebrate neuron. Potential applications of these findings include the development of secure encoding-decoding devices and for the development of a `brain defibrillator' to treat human epilepsy.
We study the combined effects of chaotic advection and molecular diffusion on a region of pollutant in time periodic recirculating flows. We prove that the flux function and the width of the stochastic zone in the non-diffusive systems have a non-monotonic frequency dependence. Furthermore, these systems have an adiabatic transport mechanism which is inherently different from the moderate and fast frequency regimes (the relevant scale for the frequency will be defined). These different Lagrangian non-diffusive mechanisms of transport imply, as we demonstrate numerically, that diffusive, low frequency (high frequency) stirring leads to efficient transport on shorter (longer) time scales. This is a joint work with A.C. Poje.
Solitary Adventures in Discrete Worlds
The properties of solitary nonlinear waves in some continuum Hamiltonian nonlinear systems, such as the sine Gordon, \phi^4 or non-linear Schrodinger equations, are well-known. On the other hand, it is very interesting to study (spatially) discrete versions of these models. The motivation stems not only from the natural numerical discretizations in order to study these PDE's on a computer but also because many of the applications are inherently discrete. Such applications range from simple systems such as coupled torsion pendula to very exciting technological applications in arrays of coupled Josephson junctions and can go as far as the breathing oscillations of DNA and the local denaturation of the Crick-Watson double strand.
This talk will be concerned with the dramatic modifications that discreteness may entail when present in these systems. In particular, we will see how continuum solitons rather than propagating merrily will now get decelerated, trapped and eventually pinned between two sites of the lattice (for strong discreteness). We will trace this behavior numerically as a function of the lattice spacing and getting insights from the numerical experiments we will seek the theoretical origins of this behavior. Using analytical (Evans function, asymptotics beyond all orders, singular perturbation theory) and mixed analytical/numerical techniques (discrete Evans function, linear stability, bisection/continued fraction methods) we will study the spectrum of the kink-like structures. Hamiltonian dispersive normal form theory will then permit us to analyze the mechanism of internal dissipation of the energy (albeit in a Hamiltonian system!) from the coherent structure to the extended wave excitations (i.e. from the localized modes into the modes of the essential spectrum). In this way, we will try to present the complete picture and theoretical analysis of the coherent structure behavior and to link it to the relevant applications. Possible future extensions of this work will also be highlighted.
Continuum Field Description of Crack Propagation
We develop a continuum field model for crack propagation in brittle amorphous solids. The model is represented by equations for elastic displacements combined with the order parameter equation, which accounts for the dynamics of defects. This model captures all important phenomenology of crack propagation: crack initiation, propagation, dynamic fracture instability, sound emission, crack branching and fragmentation.
Is Quantum Mechanics Nonlocal?
Various reasons have been given for supposing that quantum mechanics - and the real world, insofar as quantum mechanics is an accurate description of it - is nonlocal. These include: instantaneous collapse of a wave function when a measurement is made; the peculiar properties of entangled states of spatially separated particles, including violations of Bell inequalities; the finite extent in space of Newton-Wigner states in relativistic quantum theory. The talk will introduce and then analyze these ideas, with particular emphasis on entangled states, to see whether they indicate that quantum theory is nonlocal, or simply non-classical.
Moving Mesh Methods
Meshes, both topologically regular and unstructured, which smoothly deform during a time-dependent simulation have been found useful in many situations. This talk presents some recent advances in the understanding of why such methods are effective. Since I don't want to mislead anyone about the content of the talk, I remark that most of the discussion will be theoretical, not computational. Much of the material presented is joint work with Yingjie Liu.