Stochastic Models for Anomalous Diffusion

反常扩散的随机模型

• 批准号: 0803360
• 负责人: Mark Meerschaert
• 金额: $ 29.99万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-08-01 至 2012-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0803360&HistoricalAwards=false
• 关键词: Stochastic; Models; Anomalous; Diffusion

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.

经典扩散是一种分子碰撞引起的物质扩散的数学模型。同样的模型也描述了各种分散现象，其中扩散是由于其他机制，如速度对比，跳跃和捕获。当扩散速度比经典模型预测的快（超扩散）或慢（亚扩散）时，就会发生异常扩散。这种现象可以在空气和水中污染物的扩散、生物物种的分散、分子通过细胞膜的运动以及股票价格的波动中看到。随机方法识别确定性扩散方程背后的随机运动。它们描述了反常扩散方程的物理原理，并通过粒子跟踪方法促进数值求解，其中大量的代理人通过时间和空间来模仿随机物理模型。幂律静止时间导致反常扩散方程的分数阶时间导数，而幂律运动导致空间的分数阶导数。但是，幂律的适用范围在理论上受到技术条件的限制。拟议的研究将细化和扩展的异常扩散的随机模型，放宽这些技术条件，允许任意的幂律指数。它还将包括其他随机模型，通过使用三角形阵列限制和马尔可夫过程方法，导致广泛的时空伪微分方程模型的运动和传播。反常扩散方程背后的随机过程模型是连续时间随机游动的标度极限，其中随机等待时间介于随机运动之间。标度极限是服从于非马尔可夫击中时间过程的马尔可夫过程。由此产生的模型将为物理学、生物学和金融学的重要应用提供坚实的基础。生物或化学制剂一旦释放到空气或水中，就会移动和扩散。 拟议的研究将允许对这些污染物的传播进行更准确的建模。 以前的研究已经记录了空气和水中的异常传播。 快速扩散可能导致污染物比预期更早到达下游。 准确预测这一风险因素对于保护饮用水源、正确评估通过空气传播的生物或化学攻击的风险以及设计安全的核废料处置库都很重要。 缓慢释放是一个相关的问题，在清理水污染的努力中已经观察到。 改进的模式将增强国家？的能力，预算的超级基金补救工作。 在药物跨越细胞边界的运动、股票市场价格变化以及先进复合材料形成中的分子迁移中也观察到异常扩散。 拟议的研究将为更准确的医疗药物输送设计系统，更好地管理退休投资组合以及使用复合材料改进制造奠定基础。