Flows, Polymers and Random Media

University Of California-Irvine, Irvine CA

Investigators

Abstract

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 of physical phenomena in the presence of random and chaotic media. Examples of this in 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.