PDE and Applied Mathematics Seminar[ Edit ]
Moderator: Yehuda Pinchover
Dispersion of particles in chaotic, turbulent or random flows has beenstudied for a long time. It is known that the action of advection on largespatial and temporal scales typically can be described as an (anisotropic)normal diffusion process. In random but strongly correlated velocityfields, an anomalous diffusion is possible. Anomalous diffusion ispossible also in spatially regular velocity fields in the presence ofLagrangian chaos.It is less known that an anomalous transport can take place in steadytwo-dimensional flows, in the absence of any kind of chaos. In the presenttalk, we discuss two examples of such behavior.The first example is the deterministic advection in spatially periodic,steady two-dimensional velocity fields, which include stagnation pointsor solid obstacles, so that the passage time is infinite along somestreamlines. The large-time asymptotics of the dispersion law is analyzedusing the special flow construction (a flow built over the mapping).Depending on the type of the passage time singularity, the asymptoticdispersion law can correspond to either subdiffusion or superdiffusion.The analytical predictions match the results of numerical simulations.The second example is the diffusion-advection problem in spatiallyperiodic, steady two-dimensional flows that contain closed cells, possiblyseparated by jets. The anomalous dispersion is predicted and foundnumerically on an intermediate time interval. On the large time scale, anormal diffusion (enhanced by the flow) takes place. The dispersiondisplays peculiar aging properties.
Joint work with M.A. Zaks, P. Poeschke and I.M. Sokolov, Humboldt University of Berlin, Germany
After summarizing 1D periodic Jacobi matrices, I will define periodic Jacobi matrices on infinite trees. I'll discuss the few known results and some interesting examples and then discuss lots and lots of interesting conjectures. This is joint work mainly with Nir Avni and Jonathan Breuer but also with Jacob Christensen, Gil Kilai and Maxim Zinchenko.
It is on the spectral theory of a class of operators on trees, for which there has been literature on the random case even in the theoretical physics literature but I am not aware of any application to anything close to real physics so this is probably better as a math talk but I leave it to you to sort it out if you are interested. I don’t care at all which it is called or even if it is jointly sponsored.