What is the alternative to the Alexander–Orbach relation?. (25th January 2016) - Sokolov, Igor M
Abstract: The Alexander–Orbach (AO) relation d w = 2 d f / d s connecting the fractal dimension of a random walk's (RW) trajectory d w or the exponent of anomalous diffusion α = 2/ d w on a fractal structure with the fractal and spectral dimension of the structure itself plays a key role in discussion of dynamical properties of complex systems including living cells and single biomolecules. This relation however does not hold universally and breaks down for some structures like diffusion limited aggregates and Eden trees. We show that the alternative to the AO relation is the explicit dependence of the coefficient of the anomalous diffusion on the system's size, i.e. the absence of its thermodynamical limit. The prerequisite for its breakdown is the dependence of the local structure of possible steps of the RW on the system's size. The discussion is illustrated by the examples of diffusion on a Koch curve (AO-conform) and on a Cantor dust (violating AO relation).