\(q\)-Gaussians in the porous-medium equation: stability and time evolution. (English) Zbl 1188.82064
Summary: The stability of \(q\)-Gaussian distributions as particular solutions of the linear diffusion equation and its generalized nonlinear form, \(\frac{\partial P(x,t)}{\partial t}=D\frac{\partial^2[P(x,t)]^{2-q}}{\partial x^2}\), 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 \(q_{\mathrm{rel}}\equiv q_{\mathrm{rel}}(q)\). In some cases, particularly when one attempts to transform an infinite-variance distribution \((q_i\geq 5/3)\) 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.
MSC:
| 82C70 | Transport processes in time-dependent statistical mechanics |
| 76S05 | Flows in porous media; filtration; seepage |
| 82C05 | Classical dynamic and nonequilibrium statistical mechanics (general) |
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