q-Gaussians in the porous-medium equation: stability and time evolution

TL;DR

This paper investigates the stability and time evolution of q-Gaussian solutions to the porous-medium equation, revealing how initial distributions relax to asymptotic states with a q-dependent exponential rule, with implications for long-range interacting systems.

Contribution

It provides a combined numerical and analytical study of q-Gaussian stability in the porous-medium equation, highlighting the q-exponential relaxation behavior and slow convergence for certain initial conditions.

Findings

Initial q-Gaussians evolve towards asymptotic solutions with a q-dependent relaxation rule.

Relaxation can be very slow when transforming infinite-variance to finite-variance distributions.

Results may explain slow relaxation phenomena in long-range interacting many-body systems.

Abstract

The stability of $q$-Gaussian distributions as particular solutions of the linear diffusion equation and its generalized nonlinear form, $\pderiv{P(x,t)}{t} = D \pderiv{^2 [P(x,t)]^{2-q}}{x^2}$, the \emph{porous-medium equation}, is investigated through both numerical and analytical approaches. It is shown that an \emph{initial} $q$-Gaussian, characterized by an index $q_i$, approaches the \emph{final}, asymptotic solution, characterized by an index $q$, in such a way that the relaxation rule for the kurtosis evolves in time according to a $q$-exponential, with a \emph{relaxation} index $q_{\rm rel} \equiv q_{\rm rel}(q)$. In some cases, particularly when one attempts to transform an infinite-variance distribution ($q_i \ge 5/3$) into a finite-variance one ($q<5/3$), the relaxation towards the asymptotic solution may occur very slowly in time. This fact might shed some light on the slow relaxation, for some long-range-interacting many-body Hamiltonian systems, from long-standing quasi-stationary states to the ultimate thermal equilibrium state.