  We present a Master Equation formulation based on a Markovian random walk
model that exhibits sub-diffusion, classical diffusion and super-diffusion as a
function of a single parameter. The non-classical diffusive behavior is
generated by allowing for interactions between a population of walkers. At the
macroscopic level, this gives rise to a nonlinear Fokker-Planck equation. The
diffusive behavior is reflected not only in the mean-squared displacement
($<r^2(t)>\sim t^{\gamma}$ with $0 <\gamma \leq 1.5$) but also in the existence
of self-similar scaling solutions of the Fokker-Planck equation. We give a
physical interpretation of sub- and super-diffusion in terms of the attractive
and repulsive interactions between the diffusing particles and we discuss
analytically the limiting values of the exponent $\gamma$. Simulations based on
the Master Equation are shown to be in agreement with the analytical solutions
of the nonlinear Fokker-Planck equation in all three diffusion regimes.
