Anomalous diffusion associated with nonlinear fractional derivative Fokker-Planck-like equation: Exact time-dependent solutions

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dc.creatorBologna, Mauro
dc.creatorTsallis, Constantino
dc.creatorGrigolini, Paolo
dc.date2000-03-30
dc.date.accessioned2026-07-07T12:38:17Z
dc.date.available2026-07-07T12:38:17Z
dc.descriptionWe consider the $d=1$ nonlinear Fokker-Planck-like equation with fractional derivatives $\frac{\partial}{\partial t}P(x,t)=D \frac{\partial^γ}{\partial x^γ}[P(x,t) ]^ν$. Exact time-dependent solutions are found for $ ν= \frac{2-γ}{1+ γ}$ ($-\infty<γ\leq 2$). By considering the long-distance {\it asymptotic} behavior of these solutions, a connection is established, namely $q=\frac{γ+3}{γ+1}$ ($0<γ\le 2$), with the solutions optimizing the nonextensive entropy characterized by index $q$ . Interestingly enough, this relation coincides with the one already known for Lévy-like superdiffusion (i.e., $ν=1$ and $0<γ\le 2$). Finally, for $(γ,ν)=(2, 0)$ we obtain $q=5/3$ which differs from the value $q=2$ corresponding to the $γ=2$ solutions available in the literature ($ν<1$ porous medium equation), thus exhibiting nonuniform convergence.
dc.description3 figures
dc.identifierhttps://arxiv.org/abs/cond-mat/0003482
dc.identifierhttp://arxiv.org/abs/cond-mat/0003482
dc.identifierPhys.Rev.E62:2213-2218,2000
dc.identifierdoi:10.1103/PhysRevE.62.2213
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/218709
dc.subjectStatistical Mechanics
dc.titleAnomalous diffusion associated with nonlinear fractional derivative Fokker-Planck-like equation: Exact time-dependent solutions
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