Analytic approaches of the anomalous diffusion: A review

Abstract

This review article aims to stress and reunite some of the analytic formalism of the anomalous diffusive processes that have succeeded in their description. Also, it has the objective to discuss which of the new directions they have taken nowadays. The discussion is started by a brief historical report that starts with the studies of thermal machines and combines in theories such as the statistical mechanics of Boltzmann-Gibbs and the Brownian Movement. In this scenario, in the twentieth century, a series of experiments were reported that were not described by the usual model of diffusion. Such experiments paved the way for deeper investigation into anomalous diffusion, i.e. (∆x) 2 ∼ t α to α = 1. These processes are very abundant in physics, and the mechanisms for them to occur are diverse. For this reason, there are many possible ways of modelling the diffusive processes. This article discusses three analytic approaches to investigate anomalous diffusion: fractional diffusion equation, nonlinear diffusion equation and Langevin equation in the presence of fractional, coloured or multiplicative noises. All these formalisms presented different degrees of complexity and for this reason, they have succeeded in describing anomalous diffusion phenomena.

Mittag–Leffler Memory Kernel in Lévy Flights

“…In this way, the prediction of the walker in the next position may not only want local knowledge of walker but also of positions in earlier times. Thereby, the system has a dependence of past history, and reveals that the non-Markovian process can be described by use of CTRW theory [13,16]. Now, using the CTRW theory, we want to build a fractional diffusion equation that is associated with generalized Lévy process.…”

“…Among them, in super-statistical [1][2][3], diffusion with memory kernels [4][5][6], stochastic resetting process [7,8], controlled-diffusion [9][10][11], complex fluids [12], etc. In this scenario, a huge quantity of systems present a relation between a non-Gaussian distribution and anomalous diffusion process by nonlinear growth of the mean square displacement (MSD) in time [13,14], i.e., (∆x) 2 = 2K α t α , in which K α is a general diffusion coefficient with fractional dimension. The MSD relation is associated with different diffusive behaviors, classified as follows: 0 < α < 1, the system is sub-diffusive; α = 1 usual diffusion; and 1 < α < 2 occurs the super-diffusion.…”

“…Hence, investigate the mechanisms which lead to anomalous process and non-Gaussian distributions are central themes in mathematical of diffusion. In particular, the Lévy process (in Riesz-Feller sense [13]) can be determined through analytical calculus of probability distribution in CTRW approach, and applications for it is present in many fields of physics [17]. In this direction, the present work investigates how memory kernel modifies the Lévy process.…”

Spectral Analysis of Fractional Hyperbolic Diffusion Equations with Random Data

“…However, because of the initial conditions (43) and (44), the cases 0 < α ≤ 1 2 and 1 2 < α ≤ 1 can be written in the same form, as in eq. (46).…”

“…On the other hand, deviations from the standard diffusive behaviour are known to occur in many situations [3,44,47]. Among the different models of anomalous behaviour, an interesting one is provided by the use of fractional differential equations (FDE) [11].…”