Local Einstein relation for fractals

We study single random walks and the electrical resistance for fractals obtained as the limit of a sequence of periodic structures. In the long-scale regime, power laws describe both the mean-square displacement of a random walk as a function of time and the electrical resistance as a function of length. We show that the corresponding power-law exponents satisfy the Einstein relation. For shorter scales, where these exponents depend on length, we find how the Einstein relation can be generalized to hold locally. All these findings were analytically derived and confirmed by numerical simulations.

Automated summary

We study the properties of random walks and electrical resistance in fractals obtained as the limit of a sequence of periodic structures. In the long-scale regime, power laws describe the mean-square displacement of a random walk with time and the electrical resistance with length. We provide analytical derivations and numerical simulations to show that the power-law exponents satisfy the Einstein relation, and we also find a local generalization of the Einstein relation at shorter scales.