Diffusion on fractal phase spaces and entropy production

Complex systems may show strongly non-exponential relaxation, the origins of these behavior seem to be related to the fractal structure of the phase space. In this article, we consider the diffusion on a dilute hypercube, this stochastic process is a coarse-grained model for the time evolution of a glassy system, where both the dynamics and the structure of the phase space was considered as simple as possible. We characterize this process using two quantities: a memory function and the Tsallis entropy. The first one is a measure of mixing while the second is related to non-extensivity. We characterize quantitatively their relationship. Finally, we discuss the implications for other glassy and complex systems. (C) 2004 Elsevier B.V. All rights reserved.