Random walk in chemical space of Cantor dust as a paradigm of superdiffusion

We point out that the chemical space of a totally disconnected Cantor dust K_nEn is a compact metric space Cn with the spectral dimension d _s=d=n>D, where D and d=n are the fractal and chemical dimensions of K_n, respectively. Hence, we can define a random walk in the chemical space as a Markovian Gaussian process. The mapping of a random walk in Cn into K_nEn defines the quenched Lévy flight on the Cantor dust with a single step duration independent of the step length. The equations, describing the superdiffusion and diffusion-reaction front propagation ruled by the local quenched Lévy flight on K_nEn, are derived. The use of these equations to model superdiffusive phenomena, observed in some physical systems in which propagators decay faster than algebraically, is discussed.