Anomalous diffusion in the presence of external forces: Exact time-dependent solutions and their thermostatistical basis

Abstract

Driven anomalous diffusions (such as those occurring in some surface growths) are currently described through the nonlinear Fokker-Planck-like equation $(\frac{\ensuremath{\partial}}{\ensuremath{\partial}t}){p}^{\ensuremath{\mu}}=\ensuremath{-}(\frac{\ensuremath{\partial}}{\ensuremath{\partial}x})[F(x){p}^{\ensuremath{\mu}}]+D(\frac{{\ensuremath{\partial}}^{2}}{\ensuremath{\partial}{x}^{2}}){p}^{\ensuremath{\nu}}$ [$(\ensuremath{\mu}, \ensuremath{\nu})\ensuremath{\in}{\mathcal{R}}^{2}$; $F(x)={k}_{1}\ensuremath{-}{k}_{2}x$ is the external force; ${k}_{2}>~0$]. We exhibit here the (physically relevant) exact solution for all $(x, t)$. This solution was found through an ansatz based on the generalized entropic form ${S}_{q}[p]=\frac{{1\ensuremath{-}\ensuremath{\int}\mathrm{du}{[p(u)]}^{q}}}{(q\ensuremath{-}1)}$ (with $q\ensuremath{\in}\mathcal{R}$), in a completely analogous manner through which the usual entropy ${S}_{1}[p]=\ensuremath{-}\ensuremath{\int}\mathrm{dup}(u)\mathrm{ln}p(u)$ is known to provide the correct ansatz for exactly solving the standard Fokker-Planck equation ($\ensuremath{\mu}=\ensuremath{\nu}=1$). This remarkably simple unification of normal diffusion ($q=1$), superdiffusion ($q>1$) and subdiffusion ($q<1$) occurs with $q=1+\ensuremath{\mu}\ensuremath{-}\ensuremath{\nu}$.