Gradient flows with random energies
We introduce some random energies and associated gradient flows motivated by the evolution of phase boundaries in heterogeneous media. The talk will focus on new effects due to the random nature of the energies and on the interplay of techniques from statistical mechanics and analysis.
Density functional theory and optimal transportation with Coulomb cost
Density functional theory is a widely used variational model of the electronic structure of molecules, clusters, and bulk matter (it underlies most current numerical electronic structure predictions in physics, chemistry, materials science, molecular biology). There exists an 'exact' density functional whose minimizer co-incides with the correct electron density predicted by full many-body quantum mechanics (the so-called Hohenberg-Kohn functional). But its construction is too indirect to be computationally usable. Explicit functionals are obtained via 'closure relations' expressing the electron pair density or the N-point density in terms of its one-point marginal (alias single-particle density).
In the semi-classical limit, there emerges an exact closure relation: the N-point density is the solution of an N-body optimal transport problem with Coulomb cost. An interesting and unusual feature of this problem is that the cost decreases (rather than increases) with distance. The limit problem is known in the physics literature since 1999, but the fact that it has the structure of an OT problem and can be usefully analyzed via OT theory was first noticed by us in 2011 (C.Cotar, G.F., C.Klueppelberg, arxiv.org/abs/1104.0603, 2011 and CPAM 66, 548-599, 2013).
In the talk I will begin by informally explaining the connection quantum mechanics -- density functional theory -- optimal transport (not assuming expertise in any of these fields). I will then discuss
- qualitative theory of OT with Coulomb cost, including the question
whether ''Kantorovich minimizers'' must be ''Monge minimizers'' (yes for 2 particles, open for N particles)
- exactly soluble examples, including N particles, 2 sites (arxiv.org/abs/1304.0679, 2013, with C.B.Mendl, B.Pass, C.Cotar, C.Klueppelberg)
- the limit of infinitely many particles (Preprint, 2013, with C.Cotar and B.Pass).
Mathematical Aspects of Materials Engineering
Materials Science and Engineering is a scientific discipline which is concerned with complex physical systems, vaguely defined boundary conditions and typically chemically ‚dirty‘ substances. Nevertheless, at the end of the day, a materials engineer has to come up with a prediction of material properties for given processing conditions with high reliability, for instance for the toughness of a turbine blade in a jet engine which has to perform reliably and reproducibly in extreme environments.
Unfortunately, contrary to common belief of materials engineers, the processing conditions do not constitute the state parameters of the properties of a material; rather the engineering properties of a processed product are determined by both the overall chemical composition and the internal distribution of elements and crystal defects, also referred to as microstructure. Since the overall chemical composition remains unchanged, the microstructure serves as state parameter of material properties. For predicting material properties with given processing history, the evolution of microstructure has to formulated, usually over many processing steps.
For select examples it will be demonstrated, how to set up constitutive equations, what microstructural variable to define, which mathematical tools to use and eventually, which quantitative predictions can be made.
Landau damping and one-dimensional waves in plasmas
This talk will begin with a brief review of Landau's theory for the linear stability of plasmas in which the electrons satisfy a Maxwellian distribution. The nonlinear expansion of such a plasma in a tube will then be considered analytically and numerically, with emphasis on the effect of Landau damping on the shock waves that would occur if the ions were cold.
Imaging in complex media
In the emerging interdisciplinary science of imaging, in all its forms, sensor imaging in complex media has a special place. This is because of the mathematical challenges it poses as well as because of the many applications that depend on its success, from high-resolution medical imaging to seismic imaging, satellite imaging, etc. I will give a brief overview of the mathematical issues that come up and then introduce correlation based, or interferometric methods that are well suited to deal with complex media. I will give examples from seismic imaging where correlation based methods have had a huge impact recently.
Particle models for the Stefan problem
I will report on some recent works on stochastic particle systems whose evolution mimics a free boundary problem. Prototype is the d=1 simple symmetric exclusion process (SSEP) restricted to configurations which have a rightmost particle and a leftmost hole.
The process is defined by adding to the SSEP a birth-death generator which at some small rate kills the rightmost particle and, independently, the leftmost hole. The model thus describes a microscopic diffusion (SSEP) in the free boundary interval between the leftmost hole and rightmost particle. Hydrodynamic limit (with proper scaling of space and time) is studied and established.
N1∕3 scaling in the 2D Ising model and the Airy diffusion
I will consider the behavior of the phase separation interface of the 2D Ising model in the vicinity of the wall. Properly scaled, it converges to the diffusion process, with a drift expressed via Airy function. Joint work with D. Ioffe and Y. Velenik.
Panagiotis E. Souganidis
Periodic approximations of the effective nonlinearities in stochastic homogenization
In this lecture I discuss how to approximate the effective nonlinearities in the stochastic homogenization of first- and second- order nonlinear partial differential equations by the effective nonlinearities of appropriate chosen periodic problems. Under mixing hypotheses I will also present some error estimates.
June 24 - 26, 2013
University of Leipzig
Max-Planck-Institut für Mathematik in den Naturwissenschaften
Max-Planck-Institut für Mathematik in den Naturwissenschaften
Max Planck Institute for Mathematics in the Sciences
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