Effective degrees of freedom of a random walk on a fractal

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 $\nu$-dimensional space $F^{\nu }$ 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 ($\nu$) and fractal dimensionalities is deduced. The intrinsic time of random walk in $F^{\nu }$ is inferred. The Laplacian operator in $F^{\nu }$ is constructed. This allows us to map physical problems on fractals into the corresponding problems in $F^{\nu }$. 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.