Analysis of the effective long time-behaviour of molecular systems

打高尔夫球意味着：将高尔夫球从发球台上给定的初始位置，穿过山丘和山谷的景观，移动到作为最终位置的球洞。这个项目的目的在某种程度上简化了，目的是了解如何在我们习惯的三维空间中打球，而是在数百维的复杂景观中打球。虽然这听起来像是数学上的愚蠢，但这实际上发生在许多复杂的系统中，比如分子或DNA。例如，分子可以以几种截然不同的状态存在，有时称为构象。这些州可以被描绘成复杂能源格局的井。假设分子由N个原子组成，每个原子由3个空间坐标和3个动量坐标描述。那么这些井就在6N维的空间里。从分子的一种构象(其中一个井，相当于化学上的T形)开始，原子将花费大部分时间在那个井中相互推挤，然后发生罕见的自发涨落，将反应物的原子抬起越过势垒进入下一个山谷，即另一个构象的井(对应于高尔夫类比中的洞)。这是一个所谓罕见事件的例子。虽然这些事件确实很少见，但它们通常会携带有关相关系统的关键信息。因此，人们想要理解和预测这些转变和罕见的事件。然而，直接分子模拟需要解析原子时间尺度，而罕见事件发生在时间尺度上，可能会大许多个数量级；这使得直接模拟即使在最大的超级计算机上也是不可行的。相反，这个项目的目标是得到严格简化的模型，以捕捉高维复杂系统的有效长时间行为。特别关注的将是罕见事件和过渡状态。将调查各种模型问题，选择这些问题来捕捉各种应用领域中一些更复杂的问题中存在的关键挑战。

Playing golf means: moving the golf ball from the given initial position on the tee through a landscape of hills and valleys to the hole as the final location. Somewhat simplified, the aim of this project is to understand how to play golf not in the three-dimensional space we are accustomed to, but in a complicated landscape with hundreds of dimensions.While this sounds like a mathematical folly, this is what in fact happens in many complex systems, such as molecules or DNA. For example, molecules can exist in several distinctly different states, sometimes called conformations. These states can be pictured as wells of a complex energy landscape. Let us say that the molecule consists of N atoms, each described by 3 space coordinates and 3 momentum coordinates. Then the wells are in a 6N dimensional space. Starting with one conformation of the molecule (one of the wells, as the chemical equivalent of the tee), the atoms will then spend most of the time jostling around in that well before a rare spontaneous fluctuation occurs that lifts the atoms of the reactant over the barrier into the next valley, the well of the other conformation (corresponding to the hole in the golf analogy). This is an example of a so-called rare event. And while these events are indeed rare, they normally carry crucial information on the system in question. So one would like to understand and predict these transitions and rare events. However, a direct molecular simulation would need to resolve the atomistic timescale, while rare events take place on timescales which can be larger by many orders of magnitude; this renders direct simulations unfeasible even on the largest supercomputers.Instead, the aim of this project is to derive rigorously reduced models that capture the effective long-time behaviour ofhigh-dimensional complex systems. A particular focus will be on rare events and transition states. A variety of model problems will be investigated, chosen to capture key challenges present in a number of more complicated problems in various application areas.