Study of diffusion-reaction processes in branched structures

by D. S. Grebenkov

Email: denis.grebenkov@polytechnique.edu

Numerous transport processes in nature and industry are governed by
molecular diffusion (called also Brownian motion, random walks, Wiener
process, etc.). Moreover, certain biological systems exhibit branched
structures as, for instance, lungs or kidney. It is therefore
important to study transport processes in these systems and to
understand the role of their branching structure. This problem has
been largely studied in the group "Physics of Irregularity" of
LPMC. In particular, we have developed a technique for analytical
computation of stationary diffusion in a tree (a graph without loops)
taking into account reaction/absorption [1].

The goal of the internship is 

1) to generalize this approach to non-stationary diffusion;

2) to calculate different temporal and spatial characteristics of the
distribution of diffusing particles (e.g., the distribution of
diffusional fluxes, mean time to cross the tree, etc.);

3) to compare the results for symmetric and non-symmetric trees, to
develop the concept of symmetrization proposed in [1];

4) to deduce potential physiological consequences for oxygen diffusion
in the lung acinus.

[1] D. S. Grebenkov, M. Filoche, B. Sapoval, M. Felici,
Phys. Rev. Lett 94, 050602 (2005).