FOR 1548: Geometry and Physics of Spatial Random Systems

FOR 1548：空间随机系统的几何和物理

• 批准号: 173504944
• 金额: --
• 依托单位: Institut für Theoretische Physik
• 依托单位国家: 德国
• 项目类别: Research Units
• 财政年份: 2011
• 资助国家: 德国
• 起止时间: 2010-12-31 至 2018-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/173504944?language=en
• 关键词: 1548; Geometry; Physics; Spatial; Random

Spatially complex structured matter such as foams, gels and porous media is of increasing technological importance due to its material properties. These properties are highly dependent on the shape of the spatial structure. But the shape of disordered matter is a remarkably incoherent concept and the relationship between geometric and physical properties of spatially structured matter is far from being well understood. The interplay between liquid flow through porous rock and the shape of the pores, between growth laws for foams and their cell shape and geometry, and between the mechanical and geometric properties of synthetic or biological materials (e.g. wood) remain active topics of research in the physical sciences. The purpose of the interdisciplinary Research Unit is to develop the stochastic geometry methodology necessary to significantly enhance the current understanding of the relationships between geometric and physical properties of spatial condensed matter. Stochastic geometry provides and analyses mathematical models for random spatial geometric structures and is the only mathematical discipline that can cope with both the geometric and the statistical properties of complex disordered systems. Fundamental mathematical examples are Voronoi tessellations based on random point patterns, random systems of non-overlapping balls or other convex bodies (packings), union sets of randomly scattered (possibly overlapping) particles (Boolean models) and excursion or level sets of (Gaussian) random fields. Stochastic geometry is using and developing a wide range of mathematical techniques, for instance, from probability theory, convex and integral geometry, geometric measure theory and differential and discrete geometry. A successful treatment of the proposed topics requires the synthesis and improvement of existing mathematical tools, the creation of new concepts and techniques at the borderline between geometry and probability theory and a state-of-the-art knowledge in the physics of complex materials. The main topics are tensor valuations, mean value and distributional analysis of tessellations and hard-core models, the exploration of geometric descriptors for Boolean models and geometric properties of random fields and continuum percolation models. One of the six projects is concerned with image analysis and statistical problems raised by the other projects.

空间复杂结构物质，如泡沫、凝胶和多孔介质，由于其材料特性而具有越来越重要的技术意义。这些特性高度依赖于空间结构的形状。但是无序物质的形状是一个非常不连贯的概念，空间结构物质的几何和物理性质之间的关系还远没有得到很好的理解。通过多孔岩石的液体流动和孔隙形状之间的相互作用，泡沫的生长规律和它们的细胞形状和几何形状之间的相互作用，以及合成或生物材料（例如木材）的机械和几何性质之间的相互作用，仍然是物理科学研究的活跃课题。该跨学科研究单位的目的是开发必要的随机几何方法，以显着提高目前的理解空间凝聚态物质的几何和物理性质之间的关系。随机几何提供和分析随机空间几何结构的数学模型，是唯一可以科普复杂无序系统的几何和统计特性的数学学科。基本的数学例子是基于随机点模式的Voronoi镶嵌，非重叠球或其他凸体（填充）的随机系统，随机分散（可能重叠）粒子的并集（布尔模型）和（高斯）随机场的偏移或水平集。随机几何正在使用和发展广泛的数学技术，例如，从概率论，凸和积分几何，几何测度理论和微分和离散几何。一个成功的治疗所提出的主题需要综合和改进现有的数学工具，创造新的概念和技术之间的边界几何和概率论和国家的最先进的知识在复杂材料的物理学。主要议题是张量估值，平均值和分布分析的镶嵌和硬核模型，几何描述符的布尔模型和几何性质的随机场和连续渗流模型的探索。六个项目之一涉及其他项目提出的图像分析和统计问题。