Local and average behaviour in inhomogeneous superdiffusive media

Abstract

We consider a random walk on one-dimensional inhomogeneous graphs built from Cantor fractals. Our study is motivated by recent experiments that demonstrated superdiffusion of light in complex disordered materials, thereby termed Lévy glasses. We introduce a geometric parameter α which plays a role analogous to the exponent characterising the step length distribution in random systems. We study the large-time behaviour of both local and average observables; for the latter case, we distinguish two different types of averages, respectively over the set of all initial sites and over the scattering sites only. The 'single long-jump approximation” is applied to analytically determine the different asymptotic behaviour as a function of α and to understand their origin. We also discuss the possibility that the root of the mean square displacement and the characteristic length of the walker distribution may grow according to different power laws; this anomalous behaviour is typical of processes characterised by Lévy statistics and here, in particular, it is shown to influence average quantities.

Keywords:

• Lévy walks
• anomalous transport and diffusion
• fractals
• superdiffusive media
• scaling
• inhomogeneous disorder

Acknowledgements

We acknowledge useful discussions with P. Barthelemy, J. Bertolotti, R. Livi, D.S. Wiersma, and K. Vynck. This work is partially supported by the MIUR project PRIN 2008 Non-Linearity and Disorder in Classical and Quantum Processes.