ACMS Abstracts: Fall 2013
Rob Sturman (Leeds)
Lecture 1: The ergodic hierarchy & uniform hyperbolicity
Ergodic theory provides a hierarchy of behaviours of increasing complexity, essentially covering dynamics from indecomposable, through mixing, to apparently random. Demonstrating that a (real) system possesses any of these properties is typically difficult, but one important class of system - uniformly hyperbolic systems - make the ergodic hierarchy immediately accessible. Uniform hyperbolicity is a strong condition, and so we also describe its weaker counterpart, non-uniform hyperbolicity. Our chief example in this lecture is the Arnold Cat Map, an example of a hyperbolic toral automorphism.
Lecture 2: Non-uniform hyperbolicity & Pesin theory
The connection between non-uniform hyperbolicity and the ergodic hierarchy is more difficult, but is made possible by the work of Yakov Pesin in 1977, and extensions due to Katok & Strelcyn. Here we illustrate the theory with a linked twist map, a paradigmatic example of non-uniformly hyperbolic system.
Seminar: Rates of mixing in models of fluid flow
The exponential complexity of chaotic advection might be reasonably assumed to produce exponential rates of mixing. However, experiments suggest that in practice, boundaries slow mixing rates down. We give rigorous results from smooth ergodic theory which establishes polynomial mixing rates for linked twist maps, a class of simple models of bounded flows.
Reed Ogrosky (UW)
Long-wave modeling of a viscous liquid film inside a vertical tube
Viscous film flows that coat the interior of a cylinder arise naturally in industrial and biological settings and have a free surface that is unstable to long-wave disturbances. I will discuss a series of recently derived long-wave asymptotic models of these flows, and an accompanying set of experiments, in the (i) absence and (ii) presence of a steady, upwards pressure-driven airflow. In the absence of airflow, the long-wave models show excellent agreement with experiments. In the presence of airflow, the agreement is primarily qualitative in nature, though modifications to the modeling of the free surface stress improve the agreement. In both cases, the long-wave models capture qualitative features of the flow seen in the experiments that are not captured by their popular thin-film model counterparts.
One of the most influential equations in the kinetic theory of gases is the so-called Boltzmann equation, describing the time evolution of the probability density of a particle in dilute gas. While widely used, and intuitive, the Boltzmann equation poses an interesting conceptual problem: as an equation, it is irreversible in time. However, one assumes that such equation arises from reversible Newtonian laws, which seems like a contradiction. This raises the following question: can one achieve an irreversible system from a reversible one, in macroscopic time scales? If so, how can it happen? In his 1956 paper, Marc Kac presented an attempt to solve this problem in a particular settings of the spatially homogeneous Boltzmann equation. Kac considered a model of N indistinguishable particles, with one dimensional velocities, that undergo a random binary collision process. Under the property of chaoticity, defined by Kac, he managed to show that when one takes the number of particles to infinity, the limit of the first marginal of the N-particle distribution function satisfies a caricature of the Boltzmann equation. Besides giving an intuitive explanation to how the Boltzmann equation can arise as a mean field limit, Kac hoped to use his model to study the properties of the Boltzmann equation - specifically, the rate of convergence to equilibrium.
We will start our talk by describing the Boltzmann equation and explaining the model Kac proposed. We will then discuss the so-called spectral gap problem, posed by Kac, which attempted, unsuccessfully, to find an exponential rate of conversion to equilibrium of the mean field limit using the natural normed structure. At this point we'll discuss a different, less linear, approach to the problem - the so-called entropy method, and show that this too is unsuccessful, but has more potential to work with more delicate investigation. In our talk we will also mention McKean model, which is an extension of Kac's model to the case where the velocities are not one dimensional. Time permitting we will discuss more relevant concepts such as entropic chaos, and the effects of moments of the mean field limit on chaoticity and entropic chaos.
Shamgar Gurevich (UW)
The incidence and cross methods for efficient radar detection
I will explain a model of radar detection and its digital form. The latter enables us to introduce techniques from Applied Algebra (construction of specific vectors using commutative groups of operators, and generalizations of Fast Fourier Transform techniques) to suggest new efficient algorithms for radar detection. I will explain these methods, and I will demonstrate an application to the Inhomogeneous Radar Scene Problem, formulated in our interaction with engineers from General Motors (GM), who want to develop sensitive radar devices for cars.
This is a joint work with Alexander Fish (Mathematics, Sydney) and is part from a joint project with Igal Bilik (GM), Akbar Sayeed (ECE, Madison), Oded Schwartz (EECS, Berkeley), and Kobi Scheim (GM).
Ronen Avni (UW)
Mathematical model for cell motility driven by active gel
Cell crawling is a highly complex integrated process involving three distinct activities: protrusion adhesion and contraction, and also three players: the plasma membrane (car body), the actin network engine) and the adhesion points (clutch). The actin network consists of actin polymers and many other types of molecules, e.g. molecular motors, which dynamically attach to and detach from the network, making it a biological gel. Furthermore, energy is consumed in the form of ATP due to both the activity of molecular motors and the polymerization at the filament tips; thus the system is far from thermodynamic equilibrium. These characteristics make the above system unique and responsible for a wide range of phenomena (different force-velocity relationships) and behaviors (contraction, elongation, rotation, formation of dynamic structures)
Like the story on the blind men and the elephant, previous works considered only parts of the complex process, neglecting other sub-processes, or using unrealistic assumptions. Our goal was to derive a mathematical model for the whole system that can predict the rich variety of behaviors. For this purpose we had to identify the major players and integrate previous works into a one coherent mathematical model with (almost) no arbitrary constraints, adding our own mathematical description where needed. The model we derived consists of several temporal and spatial scales, molecular processes e.g. capping / branching to macro processes; furthermore, we used a hydrodynamic approach, hence could relate local dynamic events on the boundary to the bulk inside the domain. We focused on the processes near the leading edge that drive the system, i.e. the complexity comes in the b.c., and termed this filaments-membrane dynamics “the polymerization machinery”.
In my talk I will describe the mathematical model we derived and its relation to previous works. I will also describe a numerical simulation we derived for a free-surface flow of complex fluid in arbitrary geometries. Finally I will discuss open questions and opportunities in this line of research.
Shilpa Khatri (UNC)