What’s wrong with the diffusion equation? Nonlocal models of anomalous transport - M2P2 Seminar

Abstract : Transport modeling is a key element of applied sciences and engineering, and a fertile area of applied mathematics. In the past, a significant amount of work has been devoted to models formulated in terms of advection-diffusion type equations. However, relatively recently there has been growing experimental and theoretical evidence that these models fail to describe anomalous non-diffusive transport (e.g., super-diffusive and sub-diffusive transport). To describe these phenomena, nonlocal models introduce nonlocal flux-gradient relations and formulate transport in terms of partial integro-differential equations. The goal of this lecture is to present a tutorial introduction to nonlocal transport modeling with emphasis on applications. We will start with an overview of nonlocal (in space and time) transport, and the statistical foundations of nonlocal models based on the theory of continuous time random walks driven by general Levy processes. Following this, we will review several applications including: (i) Nonlocal particle and heat transport in fluids and plasmas; (ii) Fluctuation-driven transport in the nonlocal Fokker-Planck equation (e.g., Levy ratchets); and (iii) Nonlocal reaction diffusion systems (e.g., front acceleration in the nonlocal Fisher-Kolmogorov equation).