# Diffusion in flows of granular materials

Flowing granular materials are an example of a heterogeneous complex system away from equilibrium. As a result, their dynamics are still poorly understood. One canonical example is granular flow in a slowly-rotating container. Granular materials do not perform Brownian motion, nevertheless diffusion is observed in such systems because agitation (flow) causes inelastic collisions between particles. It has been suggested that axial diffusion of granular matter in a rotating drum might be "anomalous" in the sense that the mean squared displacement of particles follows a power law in time with exponent less than unity. Further numerical and experimental studies have been unable to definitively confirm or disprove this observation. In this talk, we will first review the theory of fractional parabolic PDEs governing anomalous diffusion and discuss their physical origin. Next, we will show that such a "paradox" can be resolved using the theory of self-similar intermediate asymptotics of (nonlinear) parabolic PDEs, without the need to appeal to nonlocal effects such as those represented by fractional derivatives. Specifically, we will derive an analytical expression for the instantaneous scaling exponent of a macroscopic concentration profile. Then, by incorporating concentration-dependent diffusivity in the model, we will show the existence of a crossover from an anomalous scaling (consistent with experimental observations) to a normal diffusive scaling at very long times.