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 (53)

• Superstatistics
• Lévy walks
• The fundamental solution of the space-time fractional diffusion equation
• In vivo anomalous diffusion and weak ergodicity breaking of lipid granules
• Black hole thermodynamical entropy
• Anomalous diffusion in the presence of external forces: exact time-dependent solutions and entropy
• Fractional Schrödinger equation in optics
• Fractional Langevin equation
• Brownian yet non-Gaussian diffusion: from superstatistics to subordination of diffusing diffusivities
• Ergodic Properties of Fractional Brownian-Langevin Motion
• Anomalous diffusion and ergodicity breaking in heterogeneous diffusion processes
• Active Walker Model for the Formation of Human and Animal Trail Systems
• A new fractional operator of variable order: application in the description of anomalous diffusion
• Hilfer-Prabhakar Derivatives and Some Applications
• Anomalous diffusion: A basic mechanism for the evolution of inhomogeneous systems
• Observation of anomalous diffusion and fractional self-similarity in one dimension
• Random diffusivity from stochastic equations: comparison of two models for Brownian yet non-Gaussian diffusion
• Nonlinear mean field Fokker-Planck equations. Application to the chemotaxis of biological populations
• Superstatistics: Theory and Applications
• The standard map: From Boltzmann-Gibbs statistics to Tsallis statistics
• Experimental validation of nonextensive scaling law in confined granular media
• Anomalous Diffusion of Inertial, Weakly Damped Particles
• Boundary problems for the fractional and tempered fractional operators
• Consequences of the H-Theorem from Nonlinear Fokker-Planck Equations
• A Double-Transition Scenario for Anomalous Diffusion in Glass-Forming Mixtures
• Theory of fractional-Lévy kinetics for cold atoms diffusing in optical lattices
• Nonlinear equation for anomalous diffusion: unified power-law and stretched exponential exact solution
• A nonextensive approach to the dynamics of financial observables
• Comb-like models for transport along spiny dendrites
• Stochastic approach to equilibrium and nonequilibrium thermodynamics
• q-exponential, Weibull, and q-Weibull distributions: an empirical analysis
• Weakly non-ergodic Statistical Physics
• On the diffusive anomalies in a long-range Hamiltonian system
• Superstatistical generalised Langevin equation: non-Gaussian viscoelastic anomalous diffusion
• Solving PDEs of fractional order using the unified transform method
• A Riesz basis Galerkin method for the tempered fractional Laplacian
• Transient superdiffusion and long-range correlations in the motility patterns of trypanosomatid flagellate protozoa
• q-Gaussians in the porous-medium equation: stability and time evolution
• Analytical results for long time behavior in anomalous diffusion
• Fractional Fokker-Planck equation from non-singular kernel operators
• Time independent fractional Schrodinger equation for generalized Mie-type potential in higher dimension framed with Jumarie type fractional derivative
• Renormalization group approach to nonextensive statistical mechanics
• Logarithmic diffusion and porous media equations: a unified description
• Thermodynamic Framework for Compact q-Gaussian Distributions
• Approach of complexity in nature: Entropic nonuniqueness
• Langevin equation in systems with also negative temperatures
• Role of dimensionality in preferential attachment growth in the Bianconi-Barabási model
• Dynamic Scaling of Polymer Gels Comprising Nanoparticles
• Generalised Langevin Equation Formulation for Anomalous Diffusion in the Ising Model at the Critical Temperature
• Self-similar Turing Patterns: An Anomalous diffusion consequence
• Nonextensive Kinetic Theory and H-Theorem in General Relativity
• Dissipative Effects in Nonlinear Klein-Gordon Dynamics
• Moderate deviations for the Langevin equation with strong damping

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• Rayleigh-Bénard instability in a horizontal porous layer with anomalous diffusion
• The Fluctuation-Dissipation Relations: Growth, Diffusion, and Beyond
• Conditional entropic approach to nonequilibrium complex systems with weak fluctuation correlation
• Relaxation of saturated random sequential adsorption packings of discorectangles aligned on a line
• Random diffusivity scenarios behind anomalous non-Gaussian diffusion