Time-fractional Caputo derivative versus other integro-differential operators in generalized Fokker-Planck and generalized Langevin equations

Abstract

Fractional diffusion and Fokker-Planck equations are widely used tools to describe anomalous diffusion in a large variety of complex systems. The equivalent formulations in terms of Caputo or Riemann-Liouville fractional derivatives can be derived as continuum limits of continuous time random walks and are associated with the Mittag-Leffler relaxation of Fourier modes, interpolating between a short-time stretched exponential and a long-time inverse power-law scaling. More recently, a number of other integro-differential operators have been proposed, including the Caputo-Fabrizio and Atangana-Baleanu forms. Moreover, the conformable derivative has been introduced. We here study the dynamics of the associated generalized Fokker-Planck equations from the perspective of the moments, the time averaged mean squared displacements, and the autocovariance functions. We also study generalized Langevin equations based on these generalized operators. The differences between the Fokker-Planck and Langevin equations with different integro-differential operators are discussed and compared with the dynamic behavior of established models of scaled Brownian motion and fractional Brownian motion. We demonstrate that the integro-differential operators with exponential and Mittag-Leffler kernels are not suitable to be introduced to Fokker-Planck and Langevin equations for the physically relevant diffusion scenarios discussed in our paper. The conformable and Caputo Langevin equations are unveiled to share similar properties with scaled and fractional Brownian motion, respectively.

arXiv comment26 pages, 7 figures, RevTeX

References in corpus (32)

• Anomalous transport in the crowded world of biological cells
• The fundamental solution of the space-time fractional diffusion equation
• Time-space fabric underlying anomalous diffusion
• Ergodic Properties of Fractional Brownian-Langevin Motion
• Characterizing microstructure of living tissues with time-dependent diffusion
• Anomalous diffusion of phospholipids and cholesterols in a lipid bilayer and its origins
• Physical Pictures of Transport in Heterogeneous Media: Advection-Dispersion, Random Walk and Fractional Derivative Formulations
• Analytical properties and applications of the Wright function
• Scaled Brownian motion: a paradoxical process with a time dependent diffusivity for the description of anomalous diffusion
• The dynamical foundation of fractal stream chemistry: The origin of extremely long retention times
• The role of fractional time-derivative operators on anomalous diffusion
• Why fractional derivatives with nonsingular kernels should not be used
• First passage time statistics of Brownian motion with purely time dependent drift and diffusion
• Inequivalence of time and ensemble averages in ergodic systems: exponential versus power-law relaxation in confinement
• Scaled Brownian motion as a mean field model for continuous time random walks
• Diffusion and Relaxation Controlled by Tempered α-stable Processes
• Finite-time effects and ultraweak ergodicity breaking in superdiffusive dynamics
• A comment on some new definitions of fractional derivative
• Nonstationary Increments, Scaling Distributions, and Variable Diffusion Processes in Financial Markets
• Nonergodic dynamics of force-free granular gases
• On the nature of the conformable derivative and its applications to physics
• Distributed order Hausdorff derivative diffusion model to characterize non-Fickian diffusion in porous media
• Variational Approach and Deformed Derivatives
• Different anomalous diffusion regimes measured in the dynamics of tracer particles in actin networks
• Operator solutions for fractional Fokker-Planck equations
• Dual Conformable Derivative: Definition, Simple Properties and Perspectives for Applications
• Rate equations, spatial moments, and concentration profiles for mobile-immobile models with power-law and mixed waiting time distributions
• Modeling Single-File Diffusion by Step Fractional Brownian Motion and Generalized Fractional Langevin Equation
• Anomalous dynamical scaling determines universal critical singularities
• Universal singularities of anomalous diffusion in the Richardson class
• Fluctuation relations for anomalous dynamics generated by time-fractional Fokker-Planck equations
• Optimal control of a nonlocal thermistor problem with ABC fractional time derivatives