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. 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.

References in corpus (17)

• Lévy walks
• The fundamental solution of the space-time fractional diffusion equation
• Anomalous diffusion: A basic mechanism for the evolution of inhomogeneous systems
• Nonlinear mean field Fokker-Planck equations. Application to the chemotaxis of biological populations
• Anomalous Diffusion of Inertial, Weakly Damped Particles
• Consequences of the H-Theorem from Nonlinear Fokker-Planck Equations
• A nonextensive approach to the dynamics of financial observables
• Stochastic approach to equilibrium and nonequilibrium thermodynamics
• Weakly non-ergodic Statistical Physics
• On the diffusive anomalies in a long-range Hamiltonian system
• Superstatistical generalised Langevin equation: non-Gaussian viscoelastic anomalous diffusion
• q-Gaussians in the porous-medium equation: stability and time evolution
• Thermodynamic Framework for Compact q-Gaussian Distributions
• Approach of complexity in nature: Entropic nonuniqueness
• Dynamic Scaling of Polymer Gels Comprising Nanoparticles
• Self-similar Turing Patterns: An Anomalous diffusion consequence
• Nonextensive Kinetic Theory and H-Theorem in General Relativity

Cited by in corpus (4)

• Anomalous diffusion originated by two Markovian hopping-trap mechanisms
• Mittag-Leffler functions in superstatistics
• Rayleigh-Bénard instability in a horizontal porous layer with anomalous diffusion
• Random diffusivity scenarios behind anomalous non-Gaussian diffusion