Patterns in Nature and In the Laboratory

自然界和实验室中的模式

• 批准号: 0501243
• 负责人: Alan Newell
• 金额: $ 14万
• 依托单位: University of Arizona
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-01 至 2009-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0501243&HistoricalAwards=false
• 关键词: Patterns; Nature; Laboratory

The striking similarities of pattern textures arising fromdiverse microscopic systems suggest that patterns are macroscopicobjects whose behaviors depend mainly on certain overall large scalecommon symmetries rather than on microscopic details. The proposer hasbeen associated with efforts to exploit the macroscopic character ofpatterns for over thirty years. The main ideas involve averaging overmany pattern wavelengths and introducing macroscopic order parameters,such as the pattern wavevector which, unlike the microscopic variablessuch as velocity and temperature, vary slowly over distances of thepattern wavelength. Near onset, that is for values of a stressparameter (e.g., Rayleigh number, buckling load) close to the thresholdvalue at which the uniform state is unstable, the order parameters arethe slowly varying pattern amplitude and phase gradient and certain soft(Goldstone) modes which ay be present (e.g., mean drift in low tomoderate Prandte number convecting fluids). They satisfy universalpde's (Newell-Whitehead-Segel, complex Gingburg-Landau). Far fromonset, the amplitudes are slaved to the phase gradients and then thephase gradients and the soft modes also satisfy universal pde's(Cross-Newell). In certain cases, these pde's are gradient flows andthe phase gradient fields to which the pattern relaxes are minimizers of functionals which belong to the harmonic map family. The first aim of the proposed research is to continue work on plant patterns and plant phyllotaxis. The second aim is to continue work on epidermal ridge formation. The third aim is to develop further insight into the role of soft (Goldstone) modes in elastic sheet buckling. The fourth aim is to continue our investigations into patterns far from onset and in particular those described by the regularized Cross-Newell (RCN)equation.Patterns are ubiquitous. One sees them in nature as sandripples, as cloud structures, as fingerprints, as tiger stripes andleopard spots, in the morphology of plants, and in geologicalformations. In the laboratory they arise in experiments on heattransport by convection, or ferrimagnetic garnet films, on wide aperture ("fat") laser beams, and on flame fronts. They are all over literature. Open any journal these days in physics, chemistry, biology, ecology, engineering, the mathematical sciences and you will find several articles devoted to pattern formation. One of the reasons for this interest is a curiosity as to why complicated microscopic systemsorganize themselves into ordered structures. A second reason is thatmore powerful methods (experimental, numerical, theoretical) arebecoming available to help us understand patterns. A third reason isthat, in addition to scientific understanding, there may also betechnological payoffs such as control of powerful laser arrays orinformation storage. But perhaps the purest reason is the simplest.Like all beautiful things, they are mysterious and intriguing and fullof open challenges for the fertile imagination to enjoy and explore.

来自不同微观系统的图案纹理的惊人相似性表明，图案是宏观对象，其行为主要取决于某些整体的大尺度常见对称性，而不是微观细节。这位提出者三十多年来一直致力于探索图案的宏观特征。其主要思想包括平均过多的图案波长，并引入宏观有序参数，例如图案波矢，与速度和温度等微观变量不同，它随图案波长的距离变化缓慢。对于接近临界值的应力参数(如瑞利数、屈曲载荷)，在均匀状态不稳定的阈值附近，序参数是缓慢变化的模式振幅和相位梯度以及可能存在的某些软(Goldstone)模式(如低中普朗特数对流流体中的平均漂移)。它们满足普尼阿尔德的(Newell-Whitehead-Segel，Complex Gingburg-Landau)。远离子集，振幅受制于相位梯度，然后相位梯度和软模也满足普适的PDE(Cross-Newell)。在某些情况下，这些PDE是梯度流，图案松弛到的相位梯度场是属于调和映射族的泛函的最小化。拟议研究的第一个目标是继续研究植物模式和植物叶分类。第二个目标是继续研究表皮脊的形成。第三个目标是进一步深入了解软(Goldstone)模式在弹性薄板屈曲中的作用。第四个目标是继续我们对远离起病的模式的研究，特别是那些由正则化的Cross-Newell(RCN)方程描述的模式。人们在自然界中将它们视为沙波、云结构、指纹、虎纹和豹斑，在植物的形态中，在地质构造中。在实验室中，它们出现在对流或亚铁磁性石榴石薄膜、大口径(“FAT”)激光光束和火焰前沿的热传输实验中。它们在文学中随处可见。打开当今任何一本有关物理、化学、生物、生态学、工程学、数学科学的期刊，你都会找到几篇关于图案形成的文章。引起这种兴趣的原因之一是好奇为什么复杂的微观系统会将自己组织成有序的结构。第二个原因是，更强大的方法(实验的、数值的、理论的)正在变得可以帮助我们理解模式。第三个原因是，除了科学上的理解，还可能有技术上的回报，比如控制强大的激光阵列或信息存储。但也许最纯粹的理由是最简单的。就像所有美丽的事物一样，它们是神秘的、耐人寻味的，充满了开放的挑战，供丰富的想象力欣赏和探索。