Dynamics of the Statistical Mechanics Models

建议：DMS-9800860首席研究员：Senya Shlosman摘要：Shlosman将研究受随机动力学约束的系统在系统初始状态远离平衡的情况下的行为。典型的例子是统计力学模型的Glauber动力学，该模型的参数与相变区的参数接近(但不同)。在很长一段时间里，这个系统的行为就好像它处于“错误的”均衡，直到它最终经历了向“真实的”(和唯一的)平衡的转变。(非均衡！)模型在跃迁前的状态应该被认为是系统的亚稳态。Shlosman指出，这样的状态族是平衡相曲线通过临界点的光滑延续。这种转变的发生是因为在非平衡相中产生和生长了平衡相的“液滴”。PI将研究这些亚稳态的寿命。可以编写一个显式的表达式来描述它。这个公式非常有趣，特别是包含了一个与伍尔夫液滴的表面能相对应的项。解释是，临界液滴的形状是由Wulff结构给出的。这解决了关于文学中已经存在的主题的争论。本研究项目的主题是对亚稳态现象的严格研究。物质亚稳态的一个很好的例子是过冷水(即冷却到冰点以下的水，但仍是液体)。这种水的状态确实可以在自然界中观察到。过冷水的特点是，它不能长时间保持液态：一段时间后，它突然冻结。亚稳态的概念在物理学中是众所周知的，它在物理学中扮演着重要的角色。到目前为止，缺乏的是在微观统计机制层面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“认真”理解亚稳态的工具。为此，提出了“临界水滴”的概念(在水的情况下，它是一块微小的冰)。Shlosman将研究这种液滴的形成过程，并研究这种液滴的生长过程，直到它最终吸收整个样本(通过与其他生长的液滴结合)。整个理论看起来相当完整和令人满意。

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.