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

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 $q_{\mathrm{rel}}\equiv q_{\mathrm{rel}}(q)$. 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.

20 pages, 6 figures

References in corpus (7)

• Superdiffusion and non-Gaussian statistics in a driven-dissipative 2D dusty plasma
• Nonlinear mean field Fokker-Planck equations. Application to the chemotaxis of biological populations
• Defect turbulence and generalized statistical mechanics
• Out-of-equilibrium tricritical point in a system with long-range interactions
• Consequences of the H-Theorem from Nonlinear Fokker-Planck Equations
• A General Nonlinear Fokker-Planck Equation and its Associated Entropy
• A closer look at the indications of q-generalized Central Limit Theorem behavior in quasi-stationary states of the HMF model