Description of diffusion on discrete arrays through system of piecewise linear maps

It is shown that diffusionʹs dynamic on any discrete array, including fractals, can be described by piecewise linear maps. The proposed method, which is geometric in its nature, allows to construct a system of piecewise linear maps based on the nearest neighbor coupling matrix of Cosenza and Kapral. We propose that the dynamics of the map system describes the diffusion dynamics on the array, which in our case is the Sierpinski gasket, due to the existence of an intrinsic homeomorphism between them. It is found that there is a one-to-one correspondence between the partitions of the map and the cells of the fractal. This fact allows to generate specific diffusion trajectories on the fractal in order to study the particle dwell time distribution. Since the method proposed here is completely deterministic, the diffusion on the map is anomalous: the mean square displacement scales with time to a power very near to 2. It is also shown that, as a consequence of the lattice autosimilarity and the finite precision of the calculations, this parameter remains fixed for several levels of construction of the Sierpinski gasket.