The Role of Fractional Time-Derivative Operators on Anomalous Diffusion

Frontiers in Physics

Tateishi, Angel A.²  Lenzi, Ervin K.³  Ribeiro, Haroldo V.⁶ 

[1] , Brazil;Departamento de Fígica Federal do Paranásica, Universidade Estadual de Maringásica, Universidade Estadual de Ponta Grossa, Brazil;sica, Universidade Tecnoló

关键词: fractional-time operators;    anomalous diffusion;    Continuous time random walk;    Fractional order derivatives;    fractional calculus;   

DOI  :  10.3389/fphy.2017.00052

学科分类：物理（综合）

来源: Frontiers

【 摘 要 】

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 suggest that these new operators may be a simple and efficient way for incorporating different structural aspects into the system, opening new possibilities for modeling and investigating anomalous diffusive processes.

【 授权许可】

CC BY