Large Deviation, Kinetic Limit and Gelation

大偏差、动力学极限和凝胶化

• 批准号: 1106526
• 负责人: Fraydoun Rezakhanlou
• 金额: $ 30万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-08-15 至 2015-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1106526&HistoricalAwards=false
• 关键词: Large; Deviation; Kinetic; Limit; Gelation

An outstanding and long studied problem in statistical mechanics is to establish the connection between the microscopic world and its macroscopic behavior. The investigator's research concerns stochastic and deterministic models associated with the evolution of gases, the formation of gels and crystals, and Hamiltonian systems with random potentials. As the first step, one derives a partial differential equation for the macroscopic evolution of such stochastic models. Roughly, after a suitable scaling, the density of particles in a gas with grazing collisions converges to a solution of Landau Equation, a rough interface modeled by Hamilton-Jacobi PDE with impurity homogenizes to a homogeneous Hamilton-Jacobi equation, and a chain of particles with random potential are governed by an effective potential. Many issues related to these models are not fully understood. Large-deviations type questions for the homogenization and a central limit theorem type result for gases would provide us with valuable information about the corresponding microscopic models. Also, simplified models for coagulation phenomenon should help us to discover the rules governing the mysterious interaction between gels and sols.Our world appears differently at different scales! For example a fluid or a gas is a collection of an enormous number of molecules that collide incessantly and move erratically without any particular aim. How do these molecules then manage to organize themselves in such a way as to form a flow pattern on a large scale? Roughly the reason is that the local conservation laws impose constraints not immediately visible on the microscopic scale. As an another example, consider a solid to which further material sticks from the ambient atmosphere. The process of the attachment is a function of a huge variety of growth mechanisms depending on the materials involved, their temperature, composition, etc. Following the tradition of statistical mechanics, one studies simplified models which nevertheless captures some of the essential physics.The investigator's research concerns the relationship between the microscopic structure and the macroscopic behavior of fluids, gases and crystals. The analysis of the mathematical models consisting of a large number of components is proved to be useful in understanding the intricate behavior of our microscopic world such as the formation of crystals, the emergence of clots in blood, the roughness of the surface boundary of solids and occurrence of shocks in fluids.

建立微观世界与宏观行为之间的联系是统计力学中一个突出而长期研究的问题。研究人员的研究涉及与气体的演化，凝胶和晶体的形成以及具有随机势的哈密顿系统相关的随机和确定性模型。作为第一步，推导出这样的随机模型的宏观演化的偏微分方程。粗略地说，经过适当的尺度变换，气体中的粒子密度在掠射碰撞下收敛于朗道方程的解，含杂质的粗糙界面由Hamilton-Jacobi偏微分方程均匀化为齐次Hamilton-Jacobi方程，具有随机势的粒子链由有效势控制.与这些模型相关的许多问题尚未完全理解。大偏差类型的问题的均匀化和中心极限定理类型的气体的结果将为我们提供有价值的信息，相应的微观模型。此外，凝结现象的简化模型应该有助于我们发现控制凝胶和溶胶之间神秘相互作用的规则。例如，流体或气体是大量分子的集合，这些分子不停地碰撞，不规则地运动，没有任何特定的目标。那么这些分子又是如何组织起来，形成大规模的流动模式的呢？ 大致的原因是，局部守恒定律施加的约束在微观尺度上并不直接可见。作为另一个例子，考虑固体，其他材料从环境大气粘到该固体上。 附着的过程是一个巨大的各种各样的生长机制的函数，这取决于所涉及的材料，它们的温度，成分等。遵循统计力学的传统，人们研究简化的模型，但捕获一些基本的物理。研究人员的研究关注流体，气体和晶体的微观结构和宏观行为之间的关系。由大量组件组成的数学模型的分析被证明是有用的，在理解我们的微观世界的复杂行为，如晶体的形成，血液中凝块的出现，固体表面边界的粗糙度和流体中的冲击的发生。