Anomalous diffusion in the presence of external forces: exact time-dependent solutions and entropy

1995/11/02 by Constantino Tsallis, Dirk Jan Bukman · 2 citations
Physics and Astronomy · #cond-mat

paper · pdf · doi:10.1103/physreve.54.r2197

11 pages, RevTeX, 3 uuencoded postscript figures

arxiv created 1995/11/02 · arxiv updated 2009/11/30

Abstract

The optimization of the usual entropy S₁[p]=-∫ du p(u) ln p(u) under appropriate constraints is closely related to the Gaussian form of the exact time-dependent solution of the Fokker-Planck equation describing an important class of normal diffusions. We show here that the optimization of the generalized entropic form S_q[p]=\1- ∫ du [p(u)]^q\/(q-1) (with q=1+μ-ν∈ \bf \calR) is closely related to the calculation of the exact time-dependent solutions of a generalized, nonlinear, Fokker Planck equation, namely (∂)/(∂ t)p^μ= -(∂)/(∂ x)[F(x)p^μ]+D \frac∂² ∂ x²p^ν, associated with anomalous diffusion in the presence of the external force F(x)=k₁-k₂x. Consequently, paradigmatic types of normal (q=1) and anomalous (q ≠ 1) diffusions occurring in a great variety of physical situations become unified in a single picture.

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