Dynamics and Variational Problems

摘要：研究人员将研究各种物理问题，从许多粒子系统平衡速率的概率研究，到物质在量子化辐射场存在下的稳定性。更具体地说，在提到的第一个问题中，他们将通过熵相关的李雅普诺夫函数来研究与多粒子系统平衡的距离。我们的目标是，在远离平衡状态的初始条件下，得到接近平衡的实际速率的不等式。在最近对分子碰撞的Kac模型的研究中，研究人员开发了一种方法来控制耗散率，使其成为系统中分子数量的函数。新的工作将把它扩展到其他距离平衡距离的测量，这些测量更适合于远离平衡的初始数据。第二个问题是如何理解考虑了对产生的相对论模型的物质稳定性问题。此外，还将研究与辐射相互作用的系统的自能和基态的存在性等问题。所有这些问题都将从非摄动的角度来处理。这是一个基本的观察，许多自然规律可以表述为最大化或最小化问题。因此，重要的是描述产生这些最小值或最大值的物理系统的构型，以及理解系统如何达到这些平衡构型。本建议在具体但基本的模型中研究这些问题。这些模型在物理上是多种多样的：既有经典模型，也有量子模型，既有确定性模型，也有概率模型。将这些问题和提案中讨论的许多其他问题联系在一起的共同数学线索是，它们都导致了变分学中具有挑战性的问题。因此，它们是联系在一起的，因为它们的解需要新的先验估计，以不等式的形式。这可以是场论中对基态能量的估计，也可以是运动论中对耗散速率的估计。理想情况下，这些不等式应该足够尖锐，可以作为精确计算的基础，并且可以应用于其他问题，而不是激发它们的问题，就像之前在Kac模型上的工作一样。在这个领域工作之所以特别吸引人，是因为它跨越了数学和其他科学学科的界限。

PI/Co-PI:Michael Loss/Eric Carlen, Georgia, TechDMS-0300349ABSTRACT:The investigators will study a variety of physical problems ranging from a probabilistic investigation of the rates of equilibration in many particle systems, to the stability of matter in the presence of a quantized radiation field. More specifically, in the first problem mentioned, they will be studying the distance from equilibrium of a many particle system as measured by entropyor related Lyapunov functions. The goal will be inequalities that yield realistic rates of approach to equilibrium for initial conditions that are far away from the equilibrium state. In recent work on the Kac model of molecular collisions, the investigators developed a method for controlling the rate of dissipation as a function of the number of molecules in the system. New work will extend this to other measures of the distance from equilibrium that is better suited to initial data far from equilibrium. The second problem is concerned with understanding the stability of matter problem for relativistic models with pair creation taken into account. Moreover, various questions concerning the self-energy of systems interacting with radiation and the existence of ground states will be investigated. All these problems will be treated from a non-perturbative perspective.It is a fundamental observation that many laws of nature can be formulated as maximization or minimization problems. It is therefore important to describe the configurations of the physical systems that yield these minima or maxima, as well as understanding how these equilibrium configurations are reached by the system. This proposal investigates such problems in specific but fundamental models. These models are physically diverse: both classical and quantum, deterministic and probabilistic. The common mathematical thread that binds these problems, and many others discussed in the proposal, is that they all lead to challenging problems in the calculus of variations. Thus, they are tied together in that their solutions require new a--priori estimates, in the form of inequalities. This can be an estimate on the ground state energy in field theory or an estimate on the rate of dissipation in kinetic theory. Ideally, these inequalities should be sharp enough to serve as the basis of exact calculations, and have applicability to problems other than the one that motivated them, as in the previous work on the Kac model. What makes it especially attractive to work in this area is that it cuts across the boundaries of mathematics and other scientific disciplines.