On the occurrence of oscillatory modulations in the power-law behavior of dynamic and kinetic processes in fractals

TL;DR

This paper explores how fractals with discrete scale invariance exhibit oscillatory modulations in their dynamic and kinetic behaviors, linking spatial and temporal scaling properties through numerical analysis.

Contribution

It proposes a conjecture that the interplay between physical processes and fractal symmetry causes log-periodic modulations in observables, supported by numerical tests.

Findings

Log-periodic modulations observed in random walks on fractals

Fundamental time scaling ratio $ au = b_1^z$ confirmed numerically

Oscillatory behaviors linked to fractal symmetry properties

Abstract

The dynamic and kinetic behavior of processes occurring in fractals with spatial discrete scale invariance (DSI) is considered. Spatial DSI implies the existence of a fundamental scaling ratio (b_1). We address time-dependent physical processes, which as a consequence of the time evolution develop a characteristic length of the form $\xi \propto t^{1/z}$, where z is the dynamic exponent. So, we conjecture that the interplay between the physical process and the symmetry properties of the fractal leads to the occurrence of time DSI evidenced by soft log-periodic modulations of physical observables, with a fundamental time scaling ratio given by $\tau = b_1 ^z$. The conjecture is tested numerically for random walks, and representative systems of broad universality classes in the fields of irreversible and equilibrium critical phenomena.