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

Abstract

The optimization of the usual entropy $S_1[p]=-\int dup(u)lnp(u)$ 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 $S_q[p]=\{1-\int du[p(u){]}^q\}/(q-1)$ (with $q=1+\mu -\nu \in \mathcal{R}$) is closely related to the calculation of the exact time-dependent solutions of a generalized, nonlinear, Fokker Planck equation, namely $\frac{\partial }{\partial t}{p}^{\mu }=-\frac{\partial }{\partial x}[F(x)p^{\mu }]+D\frac{{\partial }^2}{\partial x^2}{p}^{\nu }$, associated with anomalous diffusion in the presence of the external force $F(x)=k_1-k_{2}x$. Consequently, paradigmatic types of normal ($q=1$) and anomalous ($q\ne 1$) diffusions occurring in a great variety of physical situations become unified in a single picture.

11 pages, RevTeX, 3 uuencoded postscript figures

References in corpus (2)

• Thermodynamic Description of the Relaxation of Two-Dimensional Euler Turbulence Using Tsallis Statistics
• Generalized Statistics and Solar Neutrinos

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