Eur. Phys. J. B 66, 537-546 (2008)
https://doi.org/10.1140/epjb/e2008-00451-y

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

¹ Centro Brasileiro de Pesquisas Físicas, Rua Xavier Sigaud 150, Rio de Janeiro, RJ, 22290-180, Brazil
² Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico, 87501, USA

Corresponding author: ^a fdnobre@cbpf.br

Received: 2 June 2008
Revised: 2 October 2008
Published online: 12 December 2008

Abstract

The stability of q-Gaussian distributions as particular solutions of the linear diffusion equation and its generalized nonlinear form, , the porous-medium equation, is investigated through both numerical and analytical approaches. An analysis of the kurtosis of the distributions strongly suggests that an initial q-Gaussian, characterized by an index q_i, approaches asymptotically the final, analytic solution of the porous-medium equation, 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 relaxation index q_rel ≡q_rel(q). In some cases, particularly when one attempts to transform an infinite-variance distribution (q_i ≥ 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.