Wednesday, May 4, 2011 - 16:30 , Location: Skiles 006 , Maria Westdickenberg , School of Math, Georgia Tech , Organizer: Michael Loss
The logarithmic Sobolev inequality (LSI) is a powerful tool for studying convergence to equilibrium in spin systems. The Bakry-Emery criterion implies LSI in the case of a convex Hamiltonian. What can be said in the nonconvex case? We present two recent sufficient conditions for LSI. The first is a Bakry-Emery-type criterion that requires only LSI (not convexity) for the single-site conditional measures. The second is a two-scale condition: An LSI on the microscopic scale (conditional measures) and an LSI on the macroscopic scale (marginal measure) are combined to prove a global LSI. We extend the two-scale method to derive an abstract theorem for convergence to the hydrodynamic limit which we then apply to the example of Guo-Papanicolaou-Varadhan. We also survey some new results.This work is joint with Grunewald, Otto, and Villani.
Wednesday, April 27, 2011 - 16:30 , Location: Skiles 006 , Günter Stolz , Dept. of Math. University of Alabama at Birmingham , Organizer: Michael Loss
We will show that a quantum xy-spin chain which is exposed to a randomexterior magnetic field satisfies a zero-velocity Lieb-Robinson bound. Thiscan be interpreted as dynamical localization for the spin chain or asabsence of information transport. We will also discuss a general result,which says that zero velocity LR-bounds in a quantum spin system implyexponential decay of ground state correlations. This is joint work withRobert Sims and Eman Hamza and motivated by recent works of Burrell-Osborneas well as Hastings.
Wednesday, April 20, 2011 - 15:00 , Location: Physics Howey L5 , John Bush , Department of Mathematics, MIT , Organizer:
Hosted by Predrag Cvitanović, School of Physics, Georgia Tech.
Yves Couder and coworkers have recently reported the results of a startling series of experiments in which droplets bouncing on a fluid surface exhibit wave-particle duality and, as a consequence, several dynamical features previously thought to be peculiar to the microscopic realm, including single-particle diffraction, interference, tunneling and quantized orbits. We explore this fluid system in light of the Madelung transformation, whereby Schrodinger's equation is recast in a hydrodynamic form. Doing so reveals a remarkable correspondence between bouncing droplets and subatomic particles, and provides rationale for the observed macroscopic quantum behaviour. New experiments are presented, and indicate the potential value of this hydrodynamic approach to both visualizing and understanding quantum mechanics.
Wednesday, April 13, 2011 - 16:30 , Location: Skiles 006 , Yuri Bakhtin , Georgia Tech , Organizer: Michael Loss
I will talk about sequential decision making models based ondiffusion along heteroclinic networks of dynamical systems, i.e.,multiple saddle-type equilibrium points connected by heteroclinicorbits. The goal is to give a precise description of the asymptoticbehavior in the limit of vanishing noise.In particular, I will interpret exit times for stochastic dynamics asdecision making times and give a result on their asymptotic behavior.I will report on extensive data on decision making in no a priori biassetting obtained in a psychology experiment that I ran with JoshuaCorrell (University of Chicago),and compare the data with my theoretical results. I will also showthat the same kind of limiting distribution for exit times appears innonequilibrium models of statistical mechanics.
Monday, April 11, 2011 - 15:00 , Location: Howey W505 , Divarkar Viswanath , Department of Mathematics, University of Michigan , Organizer:
Host: Predrag Cvitanovic, School of Physics
Suppose that x(t) is a signal generated by a chaotic system and that the signal has been recorded in the interval [0,T]. We ask: What is the largest value t_f such that the signal can be predicted in the interval (T,T+t_f] using the history of the signal and nothing more? We show that the answer to this question is contained in a major result of modern information theory proved by Wyner, Ziv, Ornstein, and Weiss. All current algorithms for predicting chaotic series assume that if a pattern of events in some interval in the past is similar to the pattern of events leading up to the present moment, the pattern from the past can be used to predict the chaotic signal. Unfortunately, this intuitively reasonable idea is fundamentally deficient and all current predictors fall well short of the Wyner-Ziv bound. We explain why the current methods are deficient and develop some ideas for deriving an optimal predictor. [This talk is based on joint work with X. Liang and K. Serkh]. To view and/or participate in the webinar from wherever you are, click on:EVO.caltech.edu/evoNext/koala.jnlp?meeting=MvM2Ml2M2tDvDn9n9nDe9v
Wednesday, April 6, 2011 - 16:30 , Location: Skiles 006 , Leander Geisinger , University of Stuttgart , Organizer: Michael Loss
We study the sum of the negative eigenvalues of the Dirichlet Laplace operatoron a bounded domain in the semiclassical limit. We give a new proof thatyields not only the Weyl term but also the second asymptotic term involvingthe surface area of the boundary of the domain.The proof is valid under weak smoothness assumptions on the boundary and theresult can be extended to non-local, non-smooth operators like fractionalpowers of the Dirichlet Laplacian.(This is joint work with Rupert L. Frank.)
Wednesday, March 30, 2011 - 16:30 , Location: Skiles 006 , Yulia Karpeshina , Dept. of Mathematics, University of Alabama, Birmingham , Organizer: Michael Loss
We study a two dimensional Schrödinger operator for a limit-periodic potential. We prove that the spectrum contains a semiaxis and there is a family of generalized eigenfunctions at every point of this semiaxis with the following properties. First, the eigenfunctions are close to plane waves in the high energy region. Second, the isoenergetic curves in the space of momenta corresponding to these eigenfunctions have a form of slightly distorted circles with holes (Cantor type structure). Third, the spectrum corresponding to these eigenfunctions (the semiaxis) is absolutely continuous.
Wednesday, March 16, 2011 - 16:30 , Location: Skiles 006 , Jean Bellissard , Georgia Tech , Organizer: Michael Loss
The motivation is to compute the spectral properties of the Schrodinger operator describing an electron in a quasicrystal. The talk will focus on the case of the Fibonacci sequence (one dimension), to illustrate the method. Then the Wannier transform will be defined. It will be shown that the Hamiltonian can be seen as a direct integral over operators with discrete spectra, in a way similar to the construction of band spectra for crystal. A discussion of the differences with crystal will conclude this talk.This is joint work with Giuseppe De Nittis and Vida Milani
Wednesday, March 9, 2011 - 16:30 , Location: Skiles 006 , Rafael D. Benguria , Physics Department, Catholic University of Chile , Organizer: Michael Loss
An explicit asymptotic expression for the ground-state energy of the Pekar-Tomasevich functional for the N-polaron is found, when the positive repulsion parameter U of the electrons is less than twice the coupling constant of the polaron. This is joint workwith Gonzalo Bley.
Wednesday, February 23, 2011 - 16:30 , Location: Skiles 006 , Professor Almut Burchard , Department of Mathematics, University of Toronto , Organizer: Michael Loss
I will describe recent work with Marina Chugunovaand Ben Stephens on the evolution of a thin-filmequation that models a "coating flow" on a horizontalcylinder. Formally, the equation defines a gradientflow with respect to an energy that controls theH^1-norm.We show that for each given mass there exists aunique steady state, given by a droplet hanging from thebottom of the cylinder that meets the dry region withzero contact angle. The droplet minimizes the energy andattracts all strong solutions that satisfy certain energyand entropy inequalities. (Such solutions exist for arbitraryinitial values of finite energy and entropy, but it is notknown if they are unique.) The distance of any solutionfrom the steady state decays no faster than a power law.