Laplace operator eigenfunctions in complex geometries

by D. S. Grebenkov

Email: denis.grebenkov@polytechnique.edu

Diffusive transport in complex geometries is ubiquitous
in nature and industry: biochemical regulation in cells,
diffusion-limited reactions in catalysis, to name a few.
The interplay between geometrical complexity and stochastic
character of diffusion remains poorly understood, especially
in three dimensions. A spectral formulation of diffusion
problems is a fundamental way to reveal the role of the
geometry and to explain various striking phenomena observed 
in experiments. In this light, restricted diffusion in complex 
geometries can be investigated through the properties of the 
Laplace operator eigenfunctions.

The first part of this project consists in a numerical
computation of the eigenfunctions in several model
geometries: von Koch snowflakes, random packs of disks
and spheres, and 3d labyrinth structures. In the second
part, the properties of these eigenfunctions have to be
thoroughly investigated, in particular, an emergence of 
spatially localized eigenmodes and its relation to
geometrical complexity. In the third part, the role of
the localized eigenmodes for various diffusive phenomena
has to be revealed, with special focus on nuclear 
magnetic resonance applications (matrix formalism). 

The project necessarily relies on using theoretical
and numerical tools. A basic knowledge in mathematical
physics, spectral analysis, probability theory and/or
partial differential equations is recommended. 
Programming and computer skills are particularly 
welcome. The internship can eventually be continued
a PhD thesis.


References:

D. S. Grebenkov, NMR survey of reflected Brownian motion,
Rev. Mod. Phys. 79, 1077-1137 (2007).

D. S. Grebenkov, Laplacian Eigenfunctions in NMR I. A 
Numerical Tool, Conc. Magn. Reson. A 32, 277-301 (2008).

D. S. Grebenkov, Residence times and other functionals 
of reflected Brownian motion, Phys. Rev. E 76, 041139 
(2007).