Studies in Statistical Mechanics

该补助金由材料研究，物理和数学科学部门支持。 该研究涵盖了统计力学的广泛计划，旨在更好地理解起源于其微观成分集体行为的宏观现象。 所使用的方法从严格的数学分析到计算机模拟。 将研究的主题包括：（一）导致空间（时间）模式形成的对称性破坏转变是集体现象的迷人而重要的例子。 它们是涌现行为的范例，在单个原子或分子的性质中没有对应物。 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图案的形成。 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：对应于宏观结构的微观状态（具有给定约束）的数量的对数。 这转化为在固定温度下自由能的最小化，对于非平衡系统，没有这样的一般原则。 然而，我们最近发现了一个严格的推广自由能模型非平衡系统。 将这项工作扩展到更现实的系统将提供一个框架rpattern形成在非平衡与一些现在享有的一般性平衡。 (ii)合金和流体混合物在从均匀高温相淬火到共存区之后的行为继续提供具有挑战性的问题。 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的情况，其中动力学决定许多重要的物理性质。 我们的方法来解决这些问题，包括分析推导的宏观方程描述相分离的微观模型，这些高度非线性方程的解决方案的调查，和计算机模拟。 当引起相分离的相互作用是长程Kac势时，发生重要的简化。 这些系统允许调查不同相位之间的接口（孤子）的详细属性。 (iii)非平衡现象的其他方面的工作包括动力系统方法载流系统中的时间演变是由恒温确定性非哈密顿动力学;研究非常大的，形式上无限的，哈密顿系统，可以分为一个子系统和水库;微观推导宏观（流体动力学类型）方程的现实系统。(iv)大量的宏观现象，如沸腾和冻结，可以被视为原子世界（有效地）经典。 这在低温或纳米尺寸下不再如此。 有一个是在量子的世界里，现象要丰富得多，计算要困难得多。 将继续研究本质上是量子力学的平衡和非平衡系统。 即使是最简单的模型系统的薛定谔时间演化是非常复杂和迷人的，一旦超越微扰理论。 (v)生物系统现在处于科学的前沿。 统计力学的思想在生物学中已经有许多应用。 从生态学和流行病学系统到免疫系统和大脑的神经网络模型。 虽然这些应用中只有少数真正达到了目标，但统计力学的方法论似乎确实提供了正确的框架，用于描述更高层次的模式或行为如何从大量相互作用的简单实体的活动中出现。 将对其中一些主题进行研究。该补助金由材料研究，物理和数学科学部门支持。 该研究涵盖了统计力学的广泛计划，旨在更好地理解起源于其微观成分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。

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.***