An inverse scaling approach to a multi-state random activation energy model

A multi-state random activation energy model is introduced. The time evolution of the internal degrees of freedom is described in terms of a succession of activated jump processes. The heights of the energy barriers are independent random variables selected from a constant probability law. The underlying evolution equations are reduced to a continuous time random walk equation with the first jump treated in a different way. The probability distribution of internal degrees of freedom evolves towards a time independent form. Special attention is paid to the occurrence of fractal time. An inverse scaling approach shows that the occurrence of fractal time is related to the homogeneity of the probability distribution of the jump rates. If the distribution of energy barriers is a superposition of frozen canonical distributions this condition is automatically fulfilled. In this case explicit analytical solutions for the waiting time distributions are available. The analysis is extended to real space random walks. The connections with the diffusion in disordered media and the validity range of the model are also discussed.