Flows, Polymers and Random Media

项目拟开展随机流动、抛物型安德森模型、聚合物相变和随机薛定谔算子等问题的研究。在随机流领域，PI将继续致力于随机流运动下被动示踪剂的分布。在这里，目标将是研究两个不相交的被动示踪剂体在流动下移动的联合渐近分布。这也将研究湍流的情况下，示踪剂携带的柯尔莫哥洛夫速度场。其他的研究将涉及被称为Kraichnan流的奇异流的行为。在大致标记为抛物线安德森模型的区域，PI将研究所谓的发电机问题。这涉及到在紊流介质（如恒星表面）中产生磁场的模型。发电机猜想是恒星的磁场会呈指数增长。PI将尝试建立这种指数增长，并检查当反雷诺数趋于零时指数常数的渐近性。这个建议的总体精神是对随机和混沌介质存在下的物理现象进行数学研究。在对随机流动的调查中，浮游生物微粒被洋流携带时的分布或最近在墨西哥湾发生的灾难中石油的扩散就提供了这方面的例子。这项研究的目的是给出由随机电流携带的粒子体的分布和形状的信息。另一个混沌介质现象的例子是年轻恒星中磁场的产生。推测磁场强度呈快速增长趋势，数学模型还应在太阳黑子的小区域内具有高度聚焦的磁场强度。这将使我们对这些对地球事件有影响的黑点的发展有更多的了解。另一个项目涉及聚合物链的行为。这项工作的一个方面将是详细研究聚合物的相变以及长度与温度的关系对相变性质的影响。聚合物研究的另一个方面是测量随机环境对聚合物形状的影响。到目前为止，非常嘈杂的环境已经被证明会迫使聚合物形成特定的形状，也就是说，强烈的随机性降低了模型中聚合物的自由度。我们的目标是更深入地了解这种特殊形状的本质。

The PI proposes to carry on research on problems in stochastic flows, the parabolic Anderson model, polymer phase transitions and random Schrodinger operators. In the area of stochastic flows, the PI will continue efforts on the distribution of passive tracers under the motion of stochastic flows. Here, the goal will be to investigate the joint asymptotic distribution of two disjoint bodies of passive tracers moving under the flow. This will also be investigated for the case of turbulence where the tracers are carried by Kolmogorov velocity fields. Other research will be carried out involving the behavior of the singular flows called Kraichnan flows. In the area roughly labeled the parabolic Anderson model, the PI will investigate the so-called dynamo problem. This involves a model for the generation of magnetic fields in turbulent media such as on the surface of a star. The dynamo conjecture is that the magnetic field of a star will exhibit exponential growth. The PI will attempt to establish this exponential growth and examine the asymptotics of the exponential constant as the inverse Reynolds number goes to zero.The general spirit of this proposal is to pursue a mathematical investigation ofphysical phenomena in the presence of random and chaotic media. Examples of thisin the investigations in stochastic flows are provided by the distribution of plankton particles when carried by ocean currents or the spread of oil as in the recent disaster in the Gulf of Mexico. The goal of this study is to give information on the distribution and shape of a body of particles being carried by a random current. Another example of phenomena in chaotic media is the creation of magnetic fields in young stars. The field strength is conjectured to exhibit rapid growth and the mathematical model should also have the high focusing of the magnetic field strength in small regions which are sun spots. This will lead to more understanding of the development of these spots which have an effect on events on earth. Another project relates to behavior of polymer chains. One aspect of this work will be to make a detailed study of the phase transitions of polymers and the effect the relation of length to temperature has on the nature of the transition. Another aspect of the polymer study is to gauge the effect that a random environment has on the shape of a polymer. So far, very noisy environments have been shown to force the polymer into a particular shape, that is intense randomness reduces the degrees of freedom of the polymer in the model. We aim to gain more insight into the nature of this particular shape.