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

Abstract

The optimization of the usual entropy $S_1[p]=-\int du p(u) ln p(u)$ 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 $S_q[p]=\{1- \int du [p(u)]^q\}/(q-1)$ (with $q=1+\mu-\nu \in {\bf \cal{R}}$) 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 $F(x)=k_1-k_2x$. Consequently, paradigmatic types of normal ($q=1$) and anomalous ($q \neq 1$) diffusions occurring in a great variety of physical situations become unified in a single picture.