Monday, February 20, 2012 - 11:05am
1 hour (actually 50 minutes)
The study of transport is an active area of applied mathematics of interest to fluid mechanics, plasma physics, geophysics, engineering, and biology among other areas. A considerable amount of work has been done in the context of diffusion models in which, according to the Fourier-‐Fick’s prescription, the flux is assumed to depend on the instantaneous, local spatial gradient of the transported field. However, despiteits relative success, experimental, numerical, and theoretical results indicate that the diffusion paradigm fails to apply in the case of anomalous transport. Following an overview of anomalous transport we present an alternative(non-‐diffusive) class of models in which the flux and the gradient are related non-‐locally through integro-differential operators, of which fractional Laplacians are a particularly important special case. We discuss the statistical foundations of these models in the context of generalized random walks with memory (modeling non-‐locality in time) and jump statistics corresponding to general Levy processes (modeling non-‐locality in space). We discuss several applications including: (i) Turbulent transport in the presence of coherent structures; (ii) chaotic transport in rapidly rotating fluids; (iii) non-‐local fast heat transport in high temperature plasmas; (iv) front acceleration in the non-‐local Fisher-‐Kolmogorov equation, and (v) non-‐Gaussian fluctuation-‐driven transport in the non-‐local Fokker-‐Planck equation.