Logarithmic diffusion and porous media equations: a unified description

TL;DR

This paper introduces a unified framework for logarithmic diffusion and porous media equations, revealing their connection through a generalized diffusion equation within nonextensive thermostatistics, enhancing the modeling of anomalous diffusion.

Contribution

It presents a new unified equation linking porous media and logarithmic diffusion, expanding the theoretical understanding of anomalous diffusion processes.

Findings

Logarithmic diffusion as a limit case of nonlinear Fokker-Planck equations.

Solution characterized by a Lorentzian form indicating super diffusion.

Unified equation applicable in fractal dimensions within nonextensive thermostatistics.

Abstract

In this work we present the logarithmic diffusion equation as a limit case when the index that characterizes a nonlinear Fokker-Planck equation, in its diffusive term, goes to zero. A linear drift and a source term are considered in this equation. Its solution has a lorentzian form, consequently this equation characterizes a super diffusion like a L\'evy kind. In addition is obtained an equation that unifies the porous media and the logarithmic diffusion equations, including a generalized diffusion equation in fractal dimension. This unification is performed in the nonextensive thermostatistics context and increases the possibilities about the description of anomalous diffusive processes.