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

Driven anomalous diffusions (such as those occurring in some surface growths) are currently described through the nonlinear Fokker-Planck-like equation $({{∂}}{{∂}t}){p}^{{μ}}={-}({{∂}}{{∂}x})[F(x){p}^{{μ}}]+D({{{∂}}²}{{∂}{x}²}){p}^{{ν}}$ [$({μ}, {ν}){∈}{R}²$; $F(x)={k}₁{-}{k}₂x$ 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 ${S}q[p]={{1{-}{∫}du{[p(u)]}q}}{(q{-}1)}$ (with $q{∈}R$), in a completely analogous manner through which the usual entropy ${S}₁[p]={-}{∫}dup(u)lnp(u)$ is known to provide the correct ansatz for exactly solving the standard Fokker-Planck equation (${μ}={ν}=1$). This remarkably simple unification of normal diffusion ($q=1$), superdiffusion ($q>1$) and subdiffusion ($q<1$) occurs with $q=1+{μ}{-}{ν}$.