# Analytic approaches of the anomalous diffusion: a review

## TL;DR

This review summarizes analytic methods for describing anomalous diffusion, including fractional, nonlinear, and Langevin equations, highlighting their historical development, diversity, and recent research directions in physics.

## Contribution

It consolidates various analytic approaches to anomalous diffusion and discusses their evolution and current trends in the field.

## Key findings

- Multiple analytic formalisms effectively describe anomalous diffusion.
- Historical context links classical theories to modern approaches.
- Recent research explores new directions in anomalous diffusion modeling.

## 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.