The role of fractional time-derivative operators on anomalous diffusion

Abstract

The generalized diffusion equations with fractional order derivatives have shown be quite efficient to describe the diffusion in complex systems, with the advantage of producing exact expressions for the underlying diffusive properties. Recently, researchers have proposed different fractional-time operators (namely: the Caputo-Fabrizio and Atangana-Baleanu) which, differently from the well-known Riemann-Liouville operator, are defined by non-singular memory kernels. Here we proposed to use these new operators to generalize the usual diffusion equation. By analyzing the corresponding fractional diffusion equations within the continuous time random walk framework, we obtained waiting time distributions characterized by exponential, stretched exponential, and power-law functions, as well as a crossover between two behaviors. For the mean square displacement, we found crossovers between usual and confined diffusion, and between usual and sub-diffusion. We obtained the exact expressions for the probability distributions, where non-Gaussian and stationary distributions emerged. This former feature is remarkable because the fractional diffusion equation is solved without external forces and subjected to the free diffusion boundary conditions. We have further shown that these new fractional diffusion equations are related to diffusive processes with stochastic resetting, and to fractional diffusion equations with derivatives of distributed order. Thus, our results show that these new operators are a simple and efficient way for incorporating different structural aspects into the system, opening new possibilities for modeling and investigating anomalous diffusive processes.

References in corpus (9)

• Anomalous transport in the crowded world of biological cells
• Diffusion with Stochastic Resetting
• Retarding Sub- and Accelerating Super-Diffusion Governed by Distributed Order Fractional Diffusion Equations
• CTRW Pathways to the Fractional Diffusion Equation
• The Continuous Time Random Walk, still trendy: Fifty-year history, state of art, and outlook
• Subdiffusion on a Fractal Comb
• Transient superdiffusion and long-range correlations in the motility patterns of trypanosomatid flagellate protozoa
• Investigating the interplay between mechanisms of anomalous diffusion via fractional Brownian walks on a comb-like structure
• On anomalous diffusion and the out of equilibrium response function in one-dimensional models

Cited by in corpus (10)

• Stochastic resetting on comb-like structures
• Solutions of systems with the Caputo-Fabrizio fractional delta derivative on time scales
• Traveling wave solutions of some important Wick-type fractional stochastic nonlinear partial differential equations
• A complex analysis approach to Atangana-Baleanu fractional calculus
• Fractional Fokker-Planck equation from non-singular kernel operators
• Quenched and Annealed Disorder Mechanisms in Comb-Models with Fractional Operators
• Time-fractional Caputo derivative versus other integro-differential operators in generalized Fokker-Planck and generalized Langevin equations
• Riemann-Liouville derivative over the space of integrable distributions
• Emergent time crystal from a fractional Langevin equation with white and colored noise
• A numerical method for a class of nonlinear fractional advection-diffusion equations