Quenched and Annealed Disorder Mechanisms in Comb-Models with Fractional Operators

• Main Authors: Tateishi, A. A, Ribeiro, H. V, Sandev, T, Petreska, I, Lenzi, E. K
• Format: Journal Article
• Language: English
• Published: 13-02-2020
• Subjects: Physics - Statistical Mechanics
• DOI: 10.48550/arxiv.2002.05433

Phys. Rev. E 101, 022135 (2020) Recent experimental findings on anomalous diffusion have demanded novel
models that combine annealed (temporal) and quenched (spatial or static)
disorder mechanisms. The comb-model is a simplified description of diffusion on
percolation clusters, where the comb-like structure mimics quenched disorder
mechanisms and yields a subdiffusive regime. Here we extend the comb-model to
simultaneously account for quenched and annealed disorder mechanisms. To do so,
we replace usual derivatives in the comb diffusion equation by different
fractional time-derivative operators and the conventional comb-like structure
by a generalized fractal structure. Our hybrid comb-models thus represent a
diffusion where different comb-like structures describe different quenched
disorder mechanisms, and the fractional operators account for various annealed
disorders mechanisms. We find exact solutions for the diffusion propagator and
mean square displacement in terms of different memory kernels used for defining
the fractional operators. Among other findings, we show that these models
describe crossovers from subdiffusion to Brownian or confined diffusions,
situations emerging in empirical results. These results reveal the critical
role of interactions between geometrical restrictions and memory effects on
modeling anomalous diffusion.