| Prof. | Edouard BREZIN | President of the jury | Prof. | Marc-Olivier COPPENS | Reviewer | Prof. | Bertrand DUPLANTIER | Reviewer | Dr. | Marcel FILOCHE | Examiner | Prof. | Bernard SAPOVAL | Supervisor | Prof. | Wendelin WERNER | Examiner | Prof. | Michel LAPIDUS | Invited member |
The primary objective of the thesis is to develop a theoretical approach of the various Laplacian transfer phenomena towards irregular interfaces: stationary diffusion through semi-permeable membranes, electric transport towards non blocking electrodes (in an electrolyte), heterogeneous catalysis on catalytic surfaces. The influence of an irregular geometry, crucial for these phenomena, can be fully taken into account using a mathematical operator called Dirichlet-to-Neumann operator. Its spectral properties completely determine the linear response of the system in question.
An extensive numerical study of different aspects of the Laplacian transfer towards irregular interfaces, modelled by deterministic or stochastic Koch boundaries, has led us to numerous results. Most importantly, it has been established that the proportion of eigenmodes of the Dirichlet-to-Neumann operator contributing to the impedance of the interface is very low. It has been shown that its eigenvalues can be interpreted as the inverses of characteristic lengths of the boundary. An analytical model of the impedance of self-similar fractals was developed. In particular, this model is based on a hierarchy of characteristic scales of the boundary and allows to study finite generations (prefractals) of arbitrarily high order.
The fast random walk algorithm has been adapted to Koch boundaries in order to study the density of the harmonic measure which represents the distribution of hitting probabilities (analog of the primary current in electrochemistry). Its multifractal dimensions have been computed with a very high accuracy using a conjectured extension of the logarithmic development of local multifractal exponents for regular fractals.
At last, the experimental study of Koch electrodes showed that our theoretical approach allows to take into account the geometrical irregularity without specific knowledge of the microscopic transport mechanism. This result opens a new branch of possible applications, in electrochemistry or in other fields.