期刊名称：Proceedings of the National Academy of Sciences
印刷版ISSN：0027-8424
电子版ISSN：1091-6490
出版年度：2012
卷号：109
期号：40
页码：16012-16017
DOI：10.1073/pnas.1211110109
语种：English
出版社：The National Academy of Sciences of the United States of America
摘要：Granular materials do not perform Brownian motion, yet diffusion can be observed in such systems when agitation 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. We show two possible resolutions to this apparent paradox without the need to appeal to anomalous diffusion. First, we consider the evolution of arbitrary (non-point-source) initial data towards the self-similar intermediate asymptotics of diffusion by deriving an analytical expression for the instantaneous collapse exponent of the macroscopic concentration profiles. Second, we account for the concentration-dependent diffusivity in bidisperse mixtures, and we give an asymptotic argument for the self-similar behavior of such a diffusion process, for which an exact self-similar analytical solution does not exist. The theoretical arguments are verified through numerical simulations of the governing partial differential equations, showing that concentration-dependent diffusivity leads to two intermediate asymptotic regimes: one with an anomalous scaling that matches the experimental observations for naturally polydisperse granular materials, and another with a "normal" diffusive scaling (consistent with a "normal" random walk) at even longer times.
