Discrete random walk models for space–time fractional diffusion

Abstract

A physical-mathematical approach to anomalous diffusion may be based on generalized diffusion equations (containing derivatives of fractional order in space or/and time) and related random walk models. By space-time fractional diffusion equation we mean an evolution equation obtained from the standard linear diffusion equation by replacing the second-order space derivative with a Riesz-Feller derivative of order α ∈ (0, 2] and skewness θ (|θ| ≤ min {α, 2 − α}), and the first-order time derivative with a Caputo derivative of order β ∈ (0, 1] . Such evolution equation implies for the flux a fractional Fick's law which accounts for spatial and temporal non-locality. The fundamental solution (for the Cauchy problem) of the fractional diffusion equation can be interpreted as a probability density evolving in time of a peculiar self-similar stochastic process that we view as a generalized diffusion process. By adopting appropriate finite-difference schemes of solution, we generate models of random walk discrete in space and time suitable for simulating random variables whose spatial probability density evolves in time according to this fractional diffusion equation.1

Time–space fabric underlying anomalous diffusion

“…The fundamental solution of equation (2) is the time-dependent Lévy probability density distribution (fat tailed distribution α=1, β<1), in which 2β is the stability index of Lévy distribution [1][2][3] . Equation (2) also underlies the 3 fractional Brownian motion (long time range correlation, α<1, β=1), in which α is the memory strength index of process 4,13 , and the smaller α, the stronger memory.…”

“…These two anomalous statistics are often considered the statistical mechanism leading to anomalous diffusion (1) and accordingly β α η = can be derived. [1][2][3][4][5]15 When α=1, β<1, equation (1) leads to the diverging moment of higher than 2β order…”

“…For β <1, the mean square displacement diverges, [1][2][3][4]15,16 which implies the potential energy can not trap the particle. Following this view, the mean kinetic energy for a finite mass m also diverges.…”

“…PACS numbers: 61.43. Hv, 47.27.Qb, 65.40.Fb, 05.40.Ca, 43.20.Bi Anomalous diffusion is one of the most important concepts in modern physics [1][2][3][4] and is present in extremely diverse engineering fields such as charges transport in amorphous semiconductor, 5 vibration and acoustic dissipation in soft matter, 6 magnetic plasma, 7 polymer dynamics, 8 turbulence 9 and quantum processes 10 among many other problems. 4 However, it is noted that the anomaly is mostly introduced in a descriptive level of statistical representation or phenomenological modeling.…”

Stability and convergence of an implicit numerical method for the space and time fractional Bloch–Torrey equation

“…Equations (1.4) and (1.5) directly lead to the stretched exponential expression described by equation (1.3). At present, a growing number of works in science and engineering deal with dynamical systems described by fractional-order equations that involve derivatives and integrals of non-integer order [16][17][18][19][20]. These new models are more adequate than the previously used integer-order models, because fractional-order derivatives and integrals enable the description of the memory and hereditary properties of different substances [21].…”