CMI Silver Jubilee Lecture
V Balakrishnan, Indian Institute of Technology Madras
Random Walks on Sierpinski Fractals
Wednesday, September 9, 2015
Random walks on fractal structures are useful in modelling a variety of phenomena such as diffusive transport in porous media, reaction kinetics on substrates, heterogeneous catalysis, anomalous diffusion in amorphous media, and so on. While a number of scaling relations and effective dimensionalities are known with regard to diffusion on fractals, exact solutions to random walk problems on such structures are relatively rare. After an overview of the salient features of random walks on fractals, I consider random walks on the class of Sierpinski graphs of various fractal dimensions, as these are deterministic fractals that are relatively simple, but yet have nontrivial characteristics such as lacunarity, ramification, etc. A real-space-renormalization-like recursive method enables us to find analytic expressions for mean first-passage times for Markovian random walks on arbitrary finite-generation Sierpinski fractals. The effect of including non-nearest neighbour jumps is also elucidated. More generally, the exact scaling function of waiting-time densities for arbitrary non-Markovian continuous-time random walks on these hierarchical structures is also determined.