Particles, Stochastic Partial Differential Equations, Random Fields

粒子、随机偏微分方程、随机场

• 批准号: 9703648
• 负责人: Peter Kotelenez
• 金额: $ 9.4万
• 依托单位: Case Western Reserve University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-07-15 至 2002-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9703648&HistoricalAwards=false
• 关键词: Particles; Stochastic; Partial; Differential; Equations

9703648 Kotelenez The investigator will work on the derivation of physically meaningful stochastic partial differential equations (SPDEs) from systems of diffusing and interacting particles. The main difference from the traditional approach for N diffusing particles is that the fluctuations will be represented by "correlated" Brownian motions (representing the fluctuation forces), where the strength of the correlations depends on the distance of two given particles and a parameter, the so-called correlation length. Consequently, the SPDEs will depend on the correlation length as a parameter. Macroscopic and central limit theorems are expected, as the correlation length tends to zero. Further, the investigator will study the qualitative properties of the SPDEs. Homogeneous, isotropic and stationary random fields shall be derived as solutions of the SPDEs, and the spectral analysis of the random fields in terms of the coefficients of the SPDEs shall be conducted. Mathematically, the proposed work will build a bridge between 4 different areas of mathematics, namely particle systems, SPDEs, random fields and partial differential equations. Many physical, biological and economic phenomena consist of a large number of components. The large number and the rapid changes of the components make an exact description of the time evolution of all components very difficult , if not impossible. Examples are turbulence, chemical reactions, diffusion, spreading of diseases (viruses, bacteria), population growth, environmental pollution, climate, weather, financial markets etc. To build a mathematical model of those phenomena, the real world systems of atoms, molecules, viruses etc. are replaced by random particle systems whose time evolution can serve as a good predictor for the real world systems. A more traditional model for the same phenomena are macroscopic (deterministic) partial differential equations, many of which, more than a century ago, became the ma in theoretical tools in different areas of physics and engineering and also, more recently, in some areas of the biosciences and economics. Such an equation typically describes "how much mass (of some matter) is at a given place at a given time." Mathematically, most of those macroscopic equations can be derived from particle systems by a limit procedure, assuming that the number of particles becomes infinite and their respective masses become very small, and that the motion of particles is always statistically uncorrelated. This derivation also explains why the macroscopic equations often do not correctly predict the time evolution of the real world systems, but that in the long run they appear to be an average of many observations. Macroscopic equations, however, are easier to compute than particle systems. The investigator will extend the model of macroscopic partial differential equations to a model of mezoscopic (stochastic) partial differential equations. The mezoscopic equations shall be derived from particle systems in the same way as the macroscopic equations, but under more realistic assumptions on the particle systems, namely that the motion of particles is statistically correlated, when they are close to one another. As a result the mezoscopic equations should be in better agreement with observations than the macroscopic equations, while preserving the computational simplicity of the latter ones. Moreover, the macroscopic equations will appear as the limiting case of the mezoscopic equations. Mezoscopic equations will have diverse applications in physical chemistry, the biosciences, fluid mechanics and financial markets.

9703648 Kotelenez 研究员将致力于从扩散和相互作用的粒子系统中推导具有物理意义的随机偏微分方程（SPDE）。 与 N 个扩散粒子的传统方法的主要区别在于，波动将由“相关”布朗运动（代表波动力）来表示，其中相关性的强度取决于两个给定粒子的距离和一个参数，即所谓的相关长度。因此，SPDE 将依赖于相关长度作为参数。随着相关长度趋于零，宏观和中心极限定理是可预期的。此外，研究人员将研究 SPDE 的定性特性。应导出均匀、各向同性和平稳随机场作为 SPDE 的解，并根据 SPDE 系数对随机场进行谱分析。 在数学上，拟议的工作将在 4 个不同的数学领域之间架起一座桥梁，即粒子系统、SPDE、随机场和偏微分方程。 许多物理、生物和经济现象由大量组成部分组成。组件的数量庞大且变化迅速，使得准确描述所有组件的时间演变即使不是不可能，也是非常困难的。 例如湍流、化学反应、扩散、疾病（病毒、细菌）传播、人口增长、环境污染、气候、天气、金融市场等。为了建立这些现象的数学模型，现实世界的原子、分子、病毒等系统被随机粒子系统取代，随机粒子系统的时间演化可以作为现实世界系统的良好预测器。对于相同现象，更传统的模型是宏观（确定性）偏微分方程，其中许多模型在一个多世纪前成为物理和工程不同领域以及最近生物科学和经济学某些领域的主要理论工具。这样的方程通常描述“在给定时间给定地点有多少（某种物质）的质量”。从数学上讲，大多数宏观方程都可以通过极限程序从粒子系统导出，假设粒子的数量变得无限，它们各自的质量变得非常小，并且粒子的运动总是在统计上不相关。这一推导还解释了为什么宏观方程通常不能正确预测现实世界系统的时间演化，但从长远来看，它们似乎是许多观测值的平均值。 然而，宏观方程比粒子系统更容易计算。研究者将宏观偏微分方程模型扩展到细观（随机）偏微分方程模型。介观方程应以与宏观方程相同的方式从粒子系统导出，但对粒子系统做出更现实的假设，即当粒子彼此接近时，粒子的运动在统计上相关。 因此，介观方程应该比宏观方程更符合观测结果，同时保持后者的计算简单性。此外，宏观方程将表现为细观方程的极限情况。 介观方程将在物理化学、生物科学、流体力学和金融市场中有多种应用。