Local Einstein relation for fractals

TL;DR

This paper investigates the local Einstein relation on fractals, demonstrating that power-law behaviors of random walks and electrical resistance are connected globally and locally, with analytical and numerical validation.

Contribution

It introduces a generalized local Einstein relation for fractals, extending the classical relation to scale-dependent regimes.

Findings

Power laws describe random walk displacement and resistance on fractals.

The Einstein relation holds globally and can be generalized locally.

Analytical derivations are confirmed by numerical simulations.

Abstract

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.