Stochastic Models for Anomalous Diffusion

Michigan State University, East Lansing MI

Investigators

Abstract

Classical diffusion is a mathematical model for the spread of agents due to molecular collisions. The same model also describes various dispersion phenomena, where the spreading is due to other mechanisms such as velocity contrasts, hopping, and trapping. Anomalous diffusion occurs when the rate of spreading is either faster (super-diffusion) or slower (subdiffusion) than the classical model predicts. This phenomenon is seen in the spread of contaminants in air and water, the dispersion of biological species, the movement of molecules through cell membranes, and the fluctuations of stock prices. Stochastic methods identify the random motions behind the deterministic diffusion equations. They describe the physical principles that underlie the anomalous diffusion equations, and facilitate numerical solution by the method of particle tracking, where a large number of agents are followed through time and space to mimic the stochastic physical model. Power law resting times lead to fractional time derivatives in the anomalous diffusion equations, while power law movements lead to fractional derivatives in space. However, the range of power laws is restricted by technical conditions in the theory. The proposed research will refine and extend the stochastic models of anomalous diffusion, relaxing these technical conditions to allow an arbitrary power law index. It will also encompass alternative stochastic models that lead to a wide range of space-time pseudo-differential equation models for movement and spreading, by using triangular array limits and Markov process methods. The stochastic process models behind the anomalous diffusion equations are scaling limits of continuous time random walks, where random waiting times intervene between random motions. The scaling limits are Markov processes subordinated to non-Markovian hitting time processes. The resulting models should provide a sound basis for important applications in geophysics, biology, and finance. A biological or chemical agent, once released into the air or water, will move and spread. The proposed research will allow a more accurate modeling of the spreading of these contaminants. Previous research has documented anomalous spreading in both air and water. Fast spreading can cause pollutants to arrive downstream earlier than expected. Accurate prediction of this risk factor is important for protecting sources of drinking water, for properly assessing the risk from a biological or chemical attack via airborne release, and for designing a safe repository for nuclear waste. Slow release is a related issue that has been observed in efforts to clean up water pollution. Improved models will enhance the nation?s ability to budget for superfund remediation efforts. Anomalous spreading is also observed in the movement of drugs across cell boundaries, stock market price changes, and the migration of molecules in the formation of advanced composite materials. The proposed research will build a foundation for more accurate medical drug delivery design systems, better management of retirement portfolios, and improved manufacturing using composite materials.