Alejandro Perez Riascos

arXiv2019

Generalized space-time fractional dynamics in networks and lattices

We analyze generalized space-time fractional motions on undirected networks and lattices. The continuous-time random walk (CTRW) approach of Montroll and Weiss is employed to subordinate a space fractional walk to a generalization of the time-fractional Poisson renewal process. This process introduces a non-Markovian walk with long-time memory effects and fat-tailed characteristics in the waiting time density. We analyze `generalized space-time fractional diffusion' in the infinite $\it d$-dimensional integer lattice $\it \mathbb{Z}^d$. We obtain in the diffusion limit a `macroscopic' space-time fractional diffusion equation. Classical CTRW models such as with Laskin's fractional Poisson process and standard Poisson process which occur as special cases are also analyzed. The developed generalized space-time fractional CTRW model contains a four-dimensional parameter space and offers therefore a great flexibility to describe real-world situations in complex systems.

cond-mat.stat-mech↗

arXiv2016

Fractional random walk lattice dynamics

We analyze time-discrete and continuous `fractional' random walks on undirected regular networks with special focus on cubic periodic lattices in $n=1,2,3,..$ dimensions. The fractional random walk dynamics is governed by a master equation involving {\it fractional} powers of Laplacian matrices $L^{\fracα{2}}$}where $α=2$ recovers the normal walk. First we demonstrate that the interval $0<α\leq 2$ is admissible for the fractional random walk. We derive analytical expressions for fractional transition matrix and closely related the average return probabilities. We further obtain the fundamental matrix $Z^{(α)}$, and the mean relaxation time (Kemeny constant) for the fractional random walk. The representation for the fundamental matrix $Z^{(α)}$ relates fractional random walks with normal random walks. We show that the fractional transition matrix elements exhibit for large cubic $n$-dimensional lattices a power law decay of an $n$-dimensional infinite space Riesz fractional derivative type indicating emergence of Lévy flights. As a further footprint of Lévy flights in the $n$-dimensional space, the fractional transition matrix and fractional return probabilities are dominated for large times $t$ by slowly relaxing long-wave modes leading to a characteristic $t^{-\frac{n}α}$-decay. It can be concluded that, due to long range moves of fractional random walk, a small world property is emerging increasing the efficiency to explore the lattice when instead of a normal random walk a fractional random walk is chosen.

cond-mat.stat-mech↗