A Spectral Approach to Survival Probabilities in Porous Media

We consider a diffusive process in a bounded domain with heterogeneously distributed traps, reactive regions or relaxing sinks. This is a mathematical model for chemical reactors with heterogeneous spatial distributions of catalytic germs, for biological cells with specific arrangements of organelles, and for mineral porous media with relaxing agents in NMR experiments. We propose a spectral approach for computing survival probabilities which are represented in the form of a spectral decomposition over the Laplace operator eigenfunctions. We illustrate the performances of the approach by considering diffusion inside the unit disk filled with reactive regions of various shapes and reactivities. The role of the spatial arrangement of these regions and its influence on the overall reaction rate are investigated in the long-time regime. When the reactivity is finite, a uniform filling of the disk is shown to provide the highest reaction rate. Although the heterogeneity tends to reduce the reaction rate, reactive regions can still be heterogeneously arranged to get nearly optimal performances.