Abstract
We argue that a non-Markovian random walk on a fractal can be treated as a Markovian process in a fractional dimensional space with a suitable metric. This allows us to define the fractional dimensional space allied to the fractal as the -dimensional space equipped with the metric induced by the fractal topology. The relation between the number of effective spatial degrees of freedom of walkers on the fractal () and fractal dimensionalities is deduced. The intrinsic time of random walk in is inferred. The Laplacian operator in is constructed. This allows us to map physical problems on fractals into the corresponding problems in . In this way, essential features of physics on fractals are revealed. Particularly, subdiffusion on path-connected fractals is elucidated. The Coulomb potential of a point charge on a fractal embedded in the Euclidean space is derived. Intriguing attributes of some types of fractals are highlighted.
- Received 18 May 2015
- Revised 22 September 2015
DOI:https://doi.org/10.1103/PhysRevE.92.062146
©2015 American Physical Society