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

The stability of q-Gaussian distributions as particular solutions of the linear diffusion equation and its generalized nonlinear form, partial derivative P(x,t)/partial derivative t = D partial derivative(2)[P(x,t)](2-)q/partial derivative x(2), 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) equivalent to 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.