Dynamics of the Statistical Mechanics Models

统计力学模型的动力学

• 批准号: 9800860
• 负责人: Abel Klein
• 金额: $ 6.88万
• 依托单位: University of California-Irvine
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-07-01 至 2001-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9800860&HistoricalAwards=false
• 关键词: Dynamics; Statistical; Mechanics; Models

Proposal: DMS-9800860 Principal Investigator: Senya Shlosman Abstract: Shlosman will study the behavior of a system subject to stochastic dynamics in the situation where the initial state of the system is far from equilibrium. The typical example is the Glauber dynamics of the model of statistical mechanics, which model is taken with parameters close to (but different from) those of the phase transition region. For a long time the system behaves as if it were in the "wrong" equilibrium, until it finally undergoes the transition to the "true" (and unique) one. The (non-equilibrium!) states of the model before transition are what should be considered the metastable states of the system. Shlosman shows that such family of states is a smooth continuation of the curve of equilibrium phases through the critical point. The transition happens because of the creation and growth of "droplets" of the equilibrium phase inside the non-equilibrium phase. The PI will study the lifetime of these metastable states. It is possible to write an explicit expression describing it. The formula is quite interesting and contains, in particular, a term corresponding to the surface energy of the Wulff droplet. The explanation is that the shape of the critical droplet is given by the Wulff construction. This settles a controversy on the subject that has existed in the literature. The subject of this research project is the rigorous study of the phenomenon of metastability. A good example of the metastable state of matter is supercooled water (i.e., water cooled below its freezing temperature, yet remaining liquid). This state of water can indeed be observed in nature. The characteristic feature of supercooled water is that it cannot stay liquid for a very long time: after a while, it suddenly freezes. The concept of metastable state is well known in physics, where it plays an important role. What has hitherto been lacking is the rigorous understanding of this phenomenon on the level of microscopic statistical mechan ics. As one famous scientist put it, "The concepts of the metastable state and of fractional dimension are useful, provided they are not taken too seriously." Shlosman's research is aimed precisely at providing the tools to understand metastability one hundred percent "seriously." To this end, the concept of a "critical droplet" (which is a microscopic piece of ice in the case of water) is developed. Shlosman will study the process through which such a droplet is created and examine the process of growth of this droplet, until it eventually absorbs the whole sample (by coalescing with other growing droplets). The overall theory looks quite complete and satisfactory.

提案：DMS-9800860主要研究者：Senya Shlosman 摘要：Shlosman将研究一个系统的行为受到随机动力学的情况下，系统的初始状态是远离平衡。典型的例子是统计力学模型的Glauber动力学，该模型的参数接近（但不同于）相变区的参数。在很长一段时间内，系统的行为就好像它处于“错误的”平衡状态，直到它最终过渡到“真正的”（和唯一的）平衡状态。非平衡（Non-equilibrium！转换之前的模型状态是应该被认为是系统的亚稳态。Shlosman指出，这类态族是平衡相曲线通过临界点的光滑延续。转变的发生是因为在非平衡相内部平衡相的“液滴”的产生和生长。PI将研究这些亚稳态的寿命。这个公式很有趣，特别是包含了一个对应于武尔夫液滴表面能的项。其解释是临界液滴的形状由Wulff结构给出。这解决了文献中存在的关于这个问题的争议。 本研究项目的主题是对亚稳态现象的严格研究。物质的亚稳态的一个很好的例子是过冷水（即，水冷却到其冰点以下，但仍保持液体）。水的这种状态在自然界中确实可以观察到。过冷水的特点是它不能长时间保持液态：过一段时间后，它会突然结冰。亚稳态的概念在物理学中是众所周知的，它在其中起着重要的作用。迄今为止所缺乏的是在微观统计力学水平上对这一现象的严格理解。正如一位著名的科学家所说：“亚稳态和分数维的概念是有用的，只要它们不被太认真地对待。“Shlosman的研究正是为了提供百分之百理解亚稳态的工具”。为此，发展了“临界液滴”（在水的情况下是一块微小的冰）的概念。Shlosman将研究这种液滴产生的过程，并检查这种液滴的生长过程，直到它最终吸收整个样品（通过与其他生长的液滴合并）。整个理论看起来相当完整和令人满意。