A physically based connection between fractional calculus and fractal geometry

TL;DR

This paper establishes a physical and geometrical connection between fractional calculus and fractal geometry, demonstrating how power-laws and fractional differential equations naturally emerge in systems with fractal structures.

Contribution

It introduces a novel relation linking fractional calculus to fractal geometry based on physical principles, especially in modeling phenomena on fractal structures.

Findings

Power-laws in natural phenomena relate to fractal dimensions.

Fractional differential equations describe dynamics on fractal geometries.

The order of fractional derivatives correlates with fractal dimensions.

Abstract

We show a relation between fractional calculus and fractals, based only on physical and geometrical considerations. The link has been found in the physical origins of the power-laws, ruling the evolution of many natural phenomena, whose long memory and hereditary properties are mathematically modelled by differential operators of non integer order. Dealing with the relevant example of a viscous fluid seeping through a fractal shaped porous medium, we show that, once a physical phenomenon or process takes place on an underlying fractal geometry, then a power-law naturally comes up in ruling its evolution, whose order is related to the anomalous dimension of such geometry, as well as to the model used to describe the physics involved. By linearizing the non linear dependence of the response of the system at hand to a proper forcing action then, exploiting the Boltzmann superposition principle, a fractional differential equation is found, describing the dynamics of the system itself. The order of such equation is again related to the anomalous dimension of the underlying geometry.