RS Fellow - EPSRC grant (2016): Spatial fragmentations

当一些大的东西被分解成更小的碎片时--例如地震中的岩石--这些碎片究竟如何破碎可能取决于一系列参数，比如它们的大小、形状和结构。例如，长而薄的岩石块可能比致密的岩石块破碎得更快。一个被称为碎片化过程的数学概念也描述了一个对象的集合，随着时间的推移，这些对象随机地分解成越来越小的碎片。然而，关于如何分解的规则可能是相当有限制性的：一个棋子如何分解，以及它分解的速度，只允许取决于它的大小。这使我们能够给出关于这些过程的很好的数学结果，但对于理解真实的物体不是很有用。在这个项目中，我们建议研究一个简单的破碎模型，其中块破碎的速度取决于它们的形状和大小。我们开始从一个正方形开始，过了一段时间，它分裂成两个随机大小的矩形。这两个长方形做同样的事情：每个长方形等待一段时间，然后分成两个。但是它们在这样做之前等待的时间取决于它们的高度与宽度的比例。事实上，就像在岩石的例子中一样，细长的碎片可能比大致方形的碎片更快地破碎。其目的是，这项工作将是朝着了解更复杂的碎裂系统迈出的第一步，在这种系统中，碎裂规则可能非常复杂。但是，即使是我们计划研究的相对简单的物体，也已被用作现实生活中称为马氏体雪崩的过程的模型。这些描述了杂质在晶体原子结构中形成的方式--例如当金属熔化然后快速冷却以增加其强度时。

When something big gets broken up into smaller pieces - for example rocks in an earthquake - how exactly the pieces break might depend on a whole host of parameters, such as their size, shape and structure. For example, long, thin pieces of rock are likely to break more quickly than compact blocks. A mathematical concept called a fragmentation process also describes a collection of objects that break up randomly into smaller and smaller pieces as time passes. However, the rules of how the breaking up happens can be quite restrictive: how a piece breaks, and how quickly it breaks, is only allowed to depend on its size. This allows us to give nice mathematical results about these processes but is not very useful for understanding real objects. In this project we propose to study a simple model of fragmentation where how quickly blocks break does depend on their shape as well as their size. We begin with a square, which after some time breaks up into two randomly-sized rectangles. These two rectangles then do the same: each of them waits some time and then splits into two. But the time they wait before doing so depends on the ratio of their height to their width. In fact, exactly as in the rocks example, long thin pieces are likely to break more quickly than roughly square pieces. The aim is that this work will be a first step towards understanding more complex fragmentation systems, where the break-up rules might be very complicated. But even the relatively simple object that we plan to study has been used as a model for real-life processes called martensitic avalanches. These describe the way in which impurities form in crystalline atomic structures - such as when metals are melted and then cooled quickly in order to increase their strength.