Asympotic Problems in Random Transport

随机传输中的渐近问题

• 批准号: 0706974
• 负责人: Leonid Koralov
• 金额: $ 11万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-09-01 至 2011-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0706974&HistoricalAwards=false
• 关键词: Asympotic; Problems; Random; Transport

The main goal of this proposal is to investigate various aspects of transport phenomena. One type of problems concerns the motion of a particle in a random force field. We shall prove that the position, velocity, and energy of the particle, when scaled appropriately, converge, as time tends to infinity, to certain limiting processes. The limiting processes will be described explicitly in terms of the distribution of the force field. We shall also study measures carried by random velocity fields and describe their limiting behavior as time tends to infinity. Another set of problems concerns random perturbations of Hamiltonian flows. We shall strive to obtain a description of the effective behavior of randomly perturbed Hamiltonian flows on compact symplectic manifolds. A separate part of the proposal is devoted to an inverse problem for Gibbs fields. We shall prove that there always exist Gibbs fields with prescribed correlation functions, provided that the correlation functions satisfy certain consistency conditions. Most of the proposed problems naturally arise in the study of various phenomena in physics, oceanography, theory of turbulence, and statistical mechanics. Consider, for example, a particle moving in a force field, such as a charged particle in an electro-magnetic field. We intend to describe the behavior of the particle (such as its position, velocity and energy) based on the probabilistic properties of the field. The main assumption is that the time at which the particle is observed is much larger than the typical size of the force field. We shall also study the transport properties of random flows. One could think of a pollutant carried by atmospheric currents as a physical model for this type of problems. Based on the probabilistic properties of the flow, one wishes to describe the long time behavior of the substance carried by it. Many results in this direction have been obtained be several groups of researchers under various simplifying assumptions, such as the periodicity of the flow. We shall study the problem for a large class of physically relevant flows.

该提案的主要目标是调查交通现象的各个方面。 一类问题涉及随机力场中粒子的运动。我们将证明，当时间趋于无穷大时，粒子的位置、速度和能量在适当缩放时会收敛到某些极限过程。将根据力场的分布来明确地描述限制过程。我们还将研究随机速度场所携带的测量值，并描述它们在时间趋于无穷大时的极限行为。另一组问题涉及哈密顿流的随机扰动。我们将努力获得紧辛流形上随机扰动哈密顿流的有效行为的描述。该提案的一个单独部分致力于解决吉布斯场的反问题。我们将证明，只要相关函数满足一定的一致性条件，总是存在具有规定相关函数的吉布斯场。 大多数提出的问题自然出现在物理学、海洋学、湍流理论和统计力学的各种现象的研究中。例如，考虑在力场中移动的粒子，例如电磁场中的带电粒子。我们打算根据场的概率属性来描述粒子的行为（例如其位置、速度和能量）。主要假设是观察粒子的时间远大于力场的典型大小。我们还将研究随机流的传输特性。人们可以将大气流携带的污染物视为此类问题的物理模型。基于流动的概率特性，人们希望描述其所携带的物质的长期行为。几组研究人员在各种简化假设（例如流动的周期性）下已经获得了这个方向的许多结果。我们将研究一大类物理相关流的问题。