  We generalize Einstein's master equation for random walk processes by
considering that the probability for a particle at position $r$ to make a jump
of length $j$ lattice sites, $P_j(r)$ is a functional of the particle
distribution function $f(r,t)$. By multiscale expansion, we obtain a
generalized advection-diffusion equation. We show that the power law $P_j(r)
\propto f(r)^{\alpha - 1}$ (with $\alpha > 1$) follows from the requirement
that the generalized equation admits of scaling solutions ($ f(r;t) =
t^{-\gamma}\phi (r/t^{\gamma}) $). The solutions have a $q$-exponential form
and are found to be in agreement with the results of Monte-Carlo simulations,
so providing a microscopic basis validating the nonlinear diffusion equation.
Although its hydrodynamic limit is equivalent to the phenomenological porous
media equation, there are extra terms which, in general, cannot be neglected as
evidenced by the Monte-Carlo computations.}
