Driven anomalous diffusions (such as those occurring in some surface growths) are currently described through the nonlinear Fokker-Planck-like equation $\left(\frac{\partial }{\partial t}\right)p^{\mu }=-\left(\frac{\partial }{\partial x}\right)\left[F\right(x\left)\mathrm{}{p}^{\mu }\right]+D\left(\frac{{\partial }^2}{\partial x^2}\right)p^{\nu }$ [$(\mu , \nu )\in {\mathcal{R}}^2$; $F\left(x\right)=k_1-k_{2}x$ 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 $S_q\mathrm{}\left[p\right]=\frac{\{1-\int \mathrm{du}{\mathrm{}\left[p\right(u\left)\right]}^q\}}{\mathrm{}(q-1\mathrm{})\mathrm{}}$ (with $q\in \mathcal{R}$), in a completely analogous manner through which the usual entropy $S_1\mathrm{}\left[p\right]=-\int \mathrm{dup}\left(u\right)\mathrm{}\ln p\left(u\right)\mathrm{}$ is known to provide the correct ansatz for exactly solving the standard Fokker-Planck equation ($\mu =\nu =1\mathrm{}$). This remarkably simple unification of normal diffusion ($q=1\mathrm{}$), superdiffusion ($q>\mathrm{}1\mathrm{}$) and subdiffusion ($q<\mathrm{}1\mathrm{}$) occurs with $q=1+\mu -\nu$.

• Received 20 May 1996

DOI:https://doi.org/10.1103/PhysRevE.54.R2197