Molecular theory of anomalous diffusion

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 ($<r^2(t)>\sim t^γ$ with $0 <γ\leq 1.5$) 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 $γ$. 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.

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