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 ${K}_{n}\ensuremath{\subset}{E}^{n}$ is a compact metric space ${C}^{n}$ with the spectral dimension ${d}_{s}={d}_{\ensuremath{\ell}}=ngD$, where $D$ and ${d}_{\ensuremath{\ell}}=n$ 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 ${C}^{n}$ into ${K}_{n}\ensuremath{\subset}{E}^{n}$ defines the quenched L\'evy 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 L\'evy flight on ${K}_{n}\ensuremath{\subset}{E}^{n}$, 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.