Anomalous diffusion, nonlinear fractional Fokker–Planck equation and solutions

Abstract

We obtain new exact classes of solutions for the nonlinear fractional Fokker-Planck-like equation0) and a drift force F = −k1x + kγ x|x| γ−1 (k1, kγ , γ ∈ R). Connection with nonextensive statistical mechanics based on Tsallis entropy is also discussed.

Self-similar solutions to a density-dependent reaction-diffusion model

“…m+1 > 2, which the exact solution is known [19][20][21]. To this end, it can be interpreted that the nonlinear reactiondiffusion Eq.…”

“…(7). Similar to [19][20][21], the solution of Eq. (8) is assumed to be the normalized scaled function of the type…”

On non-extensive statistics, chaos and fractal strings

“…It seems that the ergodic hypothesis of ordinary Statistical Mechanics (that trajectories cover uniformly all of phase space ) is violated in many physical processes which nevertheless follow the principle of Tsallis non-extensive statistics. The latter non-extensive Statistics has applications in many areas of physics like in turbulence, nuclear high energy physics and cosmic rays astrophysics [17] ; anomalous diffusion of quarks in quark-gluon plasma, the solar neutrino problem [36], the identification of the edge of quantum chaos [3 ], nonlinear fractional Fokker-Planck equation [34], selfsimilar time series [33] , etc...…”

The precise time-dependent solution of the Fokker–Planck equation with anomalous diffusion

“…Anomalous diffusion is random motion having <(Δx) 2 > ~ t ν with ν≠1 and therefore there is no constant diffusion coefficient (D may be space/velocity dependent [3][4][5][6][7][8][9]) and the associated probability distribution is non-Gaussian or non-MB/power-law distributions [10][11][12][13][14][15][16][17]. Many nonlinear FP equations which appear to be some "fractal structure" are frequently constructed to describe the systems which behave anomalous diffusion [18][19][20][21][22][23]. It is interesting that they found the steady-state solution following a power-law q-distribution in nonextensive statistics [24].…”