RS Fellow - EPSRC grant (2016): Spatial fragmentations

RS 研究员 - EPSRC 拨款 (2016)：空间碎片

• 批准号: EP/R005249/1
• 负责人: Matthew Roberts
• 金额: $ 25.94万
• 依托单位: University of Bath
• 依托单位国家: 英国
• 项目类别: Fellowship
• 财政年份: 2017
• 资助国家: 英国
• 起止时间: 2017 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FR005249%2F1
• 关键词: RS; Fellow; EPSRC; grant; 2016; RS; Fellow; EPSRC; grant; 2016

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.

当大一些东西被分解成较小的碎片（例如地震中的岩石）时，这些碎片的破裂可能如何取决于大量参数，例如它们的大小，形状和结构。例如，长而薄的岩石可能比紧凑的块更快地破裂。一个称为碎片过程的数学概念还描述了随着时间的流逝，这些对象的集合将随机分解成越来越小的碎片。但是，分手如何发生的规则可能非常限制：一件零件的破裂以及它的速度如何破裂，只能依赖其大小。这使我们可以就这些过程给出不错的数学结果，但对于理解真实对象不是很有用。在这个项目中，我们建议研究一个简单的碎片模型，在该模型中，阻断破裂的速度确实取决于它们的形状和大小。我们从一个正方形开始，一段时间后将其分解为两个随机尺寸的矩形。然后，这两个矩形也这样做：每个矩形都等待一段时间，然后分成两个。但是他们等待之前的时间取决于其高度与宽度的比率。实际上，与岩石示例完全一样，长薄碎片可能比大约方形的碎片更快地破裂。目的是，这项工作将是理解更复杂的破碎系统的第一步，在这种情况下，分手规则可能非常复杂。但是，即使我们计划研究的相对简单的对象也被用作称为马氏体雪崩的现实生活过程的模型。这些描述了晶体原子结构中杂质形成的方式 - 例如金属熔化然后迅速冷却以提高其强度。