APPLICATIONS OF ANDERSON LOCALIZATION TO DYNAMICAL SYSTEMS AND EVOLUTIONARY PDE

该提案由两部分组成。第一部分讨论光滑映射的正熵问题。其概念如下。拿一块材料，比方说一块球状的面团。可以改变它的形状，比如把它弄成一个圆盘。然后把它收集起来，卷起来，重新塑造成一个球。将该过程重复多次。自然，这种材料的成分，比如盐和其他物质，均匀地分布在每立方英寸的材料上。这就是所谓的“混合”。在物理学中，如果混合率很高，就可以用一个叫做熵的量来衡量。这其中就涉及到了相当复杂的熵方程。准确地估计熵是一个很难的数学问题。一个好消息是，熵可以与所讨论的变换的一些几何特征有关。例如，坚固的一块材料在一个方向上“层叠”，它在互补方向上变得“薄”，这是有道理的。因此，在收集回材料后，“混合”进行得更快。仅对特殊变换的熵进行了评估。对于大多数自然的例子来说，这个问题是完全开放的。1969年，苏联物理学家B·奇里科夫发现了一张简单的正方形地图，它是粒子力学、加速器、等离子体物理学中的重要模型。这张地图后来被命名为奇里科夫地图，也是标准地图。这张地图依赖于衡量其总强度的主要参数。奇里科夫制作的数值模拟令人信服地表明，如果强度不低，那么正方形的主要部分表现出快速混合，或者正如数学家所说的那样“熵为正”。用数学上的确定性来评价标准地图的熵是一个非常困难的问题。在目前的提案中，我们计划开发一种全新的标准地图方法。这张图自然会产生一个完全不同的物理系统--量子力学。有一种猜想认为，在这个量子力学系统的某些特殊性质中隐藏着熵的计算。1958年，P.Anderson发现了量子力学中的全新现象。这种现象后来被称为安德森本地化。这一发现是1977年授予安德森、莫特和范·弗莱克的诺贝尔奖的主要引文之一。我们计划将这一现象用于解决熵问题。著名的KdV方程是以D.Korteweg和G.de Vries的名字命名的，他们提出了它作为某些特殊波的传播模型。人们需要知道初始输入才能解这个方程。空间周期性输入的求解方法是近40年前发展起来的。几年前，这一领域的主要专家之一P.Deift提出了一个问题，即开发一种不是周期性的而是不同周期的周期性集合的混合的初始输入方法。在提案的第二部分，我们计划进一步发展我们在过去8年中在这个问题上取得的进展。

The proposal has two parts. The first part addresses positive entropy problem for smooth maps. The concept is as follows. Take a piece of material, say a ball--shaped piece of dough.  Transform its shape, say by flattering into a disk. Then gather it back roll and shape back into a ball. Repeat the process a number of times. Naturally, the components of the material, say salt and others, get evenly distributed per each cubic inch of the material. That is called "mixing". In physics if the rate of the mixing is high it can be measured by a quantity called entropy. The entropy equation is rather involved. An accurate evaluation of the entropy is a hard mathematical problem. A good news is that the entropy can be related to some geometrical characteristics of the transformation in question. For instance, it makes sense that the stronger one "stratches" the piece of the material in one direction, the "thinner" it gets in the complimentary direction. So, after gathering back the material, the "mixing" goes faster. Only for special transformations the entropy has been evaluated. For most natural examples the problem is wide open. In 1969 Soviet physicist B.Chirikov discovered a simple map of the square which serves as an important model in mechanics of particles, accelerators, plasma physics. The map was later named Chirikov map and also standard map. The map depends on its major parameter which measures its total intensity. Chirikov produced numerical simulations which show convincingly that if the intensity is not low, then the main portion of the square exhibits fast mixing, or as mathematicians put it "the entropy is positive". To evaluate the standard map entropy with mathematical certainty proved to be a very hard problem. In the current proposal we plan to develop a completely new method for standard map. The map naturally produces a system which belongs to completely different physics,-quantum mechanics. There is a conjecture that the entropy evaluation is hidden in certain special propertie of this quantum mechanical system.  In 1958 P.Anderson discovered absolutely new phenomenon in quantum mechanics. The phenomenon was later called Anderson localization. The discovery was one of the major citations for 1977 Nobel prize awarded to Anderson jointly with Mott and Van Vleck. We plan to use this phenomenon for entropy problem. The famous KdV equation was named after D.Korteweg and G. de Vries, who proposed it as a model for propagation of certain special waves. One needs to know the initial input to solve the equation. The solution method was developed for input periodic in space almost 40 years ago. Several years ago P.Deift , who is one of the leading experts in this field, stated a problem to develop a method for initial input which is not periodic but a mixture of a collection of periodic with different periods. In the second part of the proposal we plan to develop further the progress we made in this problem in last 8 years.