Asymptotic solutions of a nonlinear diffusive equation in the framework of $\kappa$-generalized statistical mechanics

TL;DR

This paper investigates the long-term behavior of a nonlinear diffusive equation within $$-generalized statistical mechanics, revealing that solutions tend to approach the $$-Gaussian despite not being scale invariant.

Contribution

The study demonstrates through analysis and simulations that solutions of the $$-generalized diffusive equation asymptotically approach the $$-Gaussian, challenging previous assumptions about scale invariance.

Findings

Solutions tend to the $$-Gaussian over time.

The $$-Gaussian is not a scale invariant solution.

Numerical simulations support the asymptotic approach.

Abstract

The asymptotic behavior of a nonlinear diffusive equation obtained in the framework of the $\kappa$-generalized statistical mechanics is studied. The analysis based on the classical Lie symmetry shows that the $\kappa$-Gaussian function is not a scale invariant solution of the generalized diffusive equation. Notwithstanding, several numerical simulations, with different initial conditions, show that the solutions asymptotically approach to the $\kappa$-Gaussian function. Simple argument based on a time-dependent transformation performed on the related $\kappa$-generalized Fokker-Planck equation, supports this conclusion.