Studies in Statistical Mechanics

统计力学研究

• 批准号: 0127926
• 负责人: Joel Lebowitz
• 金额: $ 45万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-01-01 至 2005-03-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0127926&HistoricalAwards=false
• 关键词: Studies; Statistical; Mechanics

This grant is supported by the Divisions of Materials Research, Physics, and Mathematical Sciences. The research covers a broad program in statistical mechanics with an aim to better understand macroscopic phenomena originating in the collective behavior of its microscopic constituents. The methods used range from rigorous mathematical analysis to computer simulations. Topics to be studied include: (i) Symmetry breaking transitions leading to the formation of spatial (temporal) patterns are fascinating and important examples of collective phenomena. They are paradigms of emergent behavior, with no counterpart in the properties of individual atoms or molecules. They occur in very diverse situations, ranging from crystallization in equilibrium systems, the development of rolls and cells in fluids heated from below to the formation of patterns in morphogenesis. For equilibrium systems the state observed with overwhelming probability is the one which maximizes the entropy: the logarithm of the number of microscopic states (with given constraints) corresponding to the macroscopic structure. This translates into minimization of the free energy at a fixed temperature, etc. For nonequilibrium systems there is no such general principle. We have however recently found a rigorous generalization of free energy to a model nonequilibrium system. Extending this work to more realistic systems will provide a framework fo rpattern formation in nonequilibrium with some of the generality now enjoyed by equilibrium. (ii) The behavior of alloys and fluid mixtures following a quench from a uniform high temperature phase into the coexistence region continues to offer challenging problems. Of particular theoretical interest and practical importance is the case of alloys with elastic interactions where the kinetics determine many of the important physical properties. Our approach to these problems includes analytic derivations of macroscopic equations describing phase segregation from microscopic models; investigations of the solution of these highly nonlinear equations; and, computer simulations. An important simplification occurs when the interactions giving rise to the phase segregation are long range Kac potentials. These systems permit the investigation of detailed properties of the interface (soliton) between different phases. (iii) Work on other aspects of nonequilibrium phenomena include dynamical systems approach to current carrying systems in which the time evolution is described by thermostated deterministic non-Hamiltonian dynamics; the study of very large, formally infinite, Hamiltonian systems which can be divided into a subsystem and reservoirs; microscopic derivation of macroscopic (hydrodynamics type) equations for realistic systems. (iv) A surprisingly large number of macroscopic phenomena, such as boiling and freezing, can be treated as if the atomic world was (effectively) classical. This is no longer so at low temperatures or ano sizes. There one is in the world of the quantum where the phenomena is much richer and calculations much harder. Work will continue on both equilibrium and nonequilibrium systems which are intrinsically quantum mechanical. The Schroedinger time evolution of even the simplest model system is exceedingly complex and fascinating once one goes beyond perturbation theory. (v) Biological systems are now at the frontier of science. There have already been many applications of statistical mechanical ideas to biology. These range from ecological and epidemiological systems to neural network models of the immune system and the brain. While only a few of these applications have really been on target, the methodology of statistical mechanics does seem to provide the right framework for describing how higher level patterns or behavior emerge from the activity of a multitude of interacting simpler entities. Research will be carried out on a number of these topics. %%%This grant is supported by the Divisions of Materials Research, Physics, and Mathematical Sciences. The research covers a broad program in statistical mechanics with an aim to better understand macroscopic phenomena originating in the collective behavior of its microscopic constituents. The methods used range from rigorous mathematical analysis to computer simulations.***

该基金由材料研究部、物理部和数学科学部提供支持。该研究涵盖了统计力学中的一个广泛项目，旨在更好地理解起源于其微观组成部分的集体行为的宏观现象。使用的方法从严格的数学分析到计算机模拟。要研究的主题包括：(i)导致空间（时间）模式形成的对称破缺转变是集体现象的迷人和重要的例子。它们是突现行为的范例，与单个原子或分子的特性没有对应关系。它们发生在非常不同的情况下，从平衡系统中的结晶，从下面加热的流体中的卷和细胞的发展到形态发生中的模式形成。对于平衡系统，以压倒性的概率观察到的状态是使熵最大化的状态：与宏观结构相对应的微观状态（具有给定约束）的对数。这转化为在固定温度下自由能的最小化，等等。对于非平衡系统，没有这样的一般原理。然而，我们最近发现了自由能对模型非平衡系统的严格推广。将这项工作扩展到更现实的系统，将为非平衡状态下的模式形成提供一个框架，该框架具有平衡状态现在所享有的一些普遍性。（二）合金和流体混合物从均匀高温相淬火进入共存区后的行为仍然是具有挑战性的问题。具有特殊理论意义和实际重要性的是具有弹性相互作用的合金，其中动力学决定了许多重要的物理性质。我们解决这些问题的方法包括从微观模型中解析推导描述相偏析的宏观方程；这类高度非线性方程解的研究还有，计算机模拟。当引起相偏析的相互作用是长距离Kac势时，发生了一个重要的简化。这些系统允许研究不同相之间界面（孤子）的详细性质。（iii）非平衡现象的其他方面的工作包括对载流系统的动力系统方法，其中时间演化由恒温确定性非哈密顿动力学描述；对非常大的、形式上无限的哈密顿系统的研究，这种系统可以分为子系统和储层；现实系统宏观（流体力学型）方程的微观推导。（iv）大量的宏观现象，如沸腾和冻结，可以被当作原子世界（实际上）是经典的来处理。这在低温或ano尺寸下不再如此。一个是在量子世界中，那里的现象更加丰富，计算也更加困难。工作将继续在本质上属于量子力学的平衡和非平衡系统上进行。即使是最简单的模型系统的薛定谔时间演化，一旦超越摄动理论，也是极其复杂和迷人的。生物系统现在处于科学的前沿。统计力学的思想在生物学上已经有了很多应用。这些范围从生态和流行病学系统到免疫系统和大脑的神经网络模型。虽然这些应用程序中只有少数真正达到了目标，但统计力学的方法似乎确实提供了正确的框架，用于描述如何从大量相互作用的简单实体的活动中产生更高层次的模式或行为。将对其中一些题目进行研究。该基金由材料研究部、物理部和数学科学部提供支持。该研究涵盖了统计力学中的一个广泛项目，旨在更好地理解起源于其微观组成部分的集体行为的宏观现象。使用的方法从严格的数学分析到计算机模拟都有