Connections between tagged particles and hydrodynamics in some interacting systems

拟议的项目考虑的问题方面的各种标度限制的标记，或区别的粒子，粒子的质量密度的时空演化，或流体力学，以及它们的潜在的连接在某些系统的相互作用的随机游动，如简单的排斥和零程过程。 该提案将集中在相关的，但较少研究的情况下，这些系统是不平衡的，当粒子的相互作用不是有限范围，当系统是无序的。 重点研究的具体问题包括：（1）对称简单排斥过程中标记粒子的大偏差，（2）长程排斥模型中标记粒子的流体动力学和稳定过程极限，（3）随机速率零程系统中的流体动力学和标记粒子极限，（4）有界速率为零时标记粒子的扩散极限。理解大规模系统中与其他标记交互的标记的运动在许多应用环境中是重要的，例如交通，队列，流体，颗粒和分子流，也是统计物理学长期关注的问题，涉及制定布朗运动的物理基础。 在分析中的主要困难之一是，从其他粒子的体积贡献是不容易纳入渐近的标记粒子。 在这种情况下，该提案的主旨是开发数学技术来处理复杂的微观粒子相互作用，以便可以捕获不同类型的宏观标记粒子行为。

The proposed project considers problems with respect to various scaling limits of a tagged, or distinguished particle, the space-time evolution of the mass density of particles, or hydrodynamics, and their underlying connections in certain systems of interacting random walks such as simple exclusion and zero- range processes. The proposal will concentrate on relevant but less studied situations when these systems are out of equilibrium, when particles interactions are not finite-range, and when the systems are disordered. Specific questions focused upon include (1) large deviations of a tagged particle in symmetric simple exclusion processes, (2) hydrodynamics, and stable process limits of a tagged particle in long range exclusion models, (3) hydrodynamics and tagged particle limits in zero-range systems with random rates, and (4) diffusive limits of a tagged particle in bounded rate zero- range processes.Understanding the motion of a tagged interacting with others in large scale systems is important in many applied contexts such as traffic, queues, fluids, and granular and molecular flows, and is also a long standing concern in statistical physics with regard to formulating the physical basis of Brownian motion. One of the main difficulties in the analysis is that the bulk contribution from the other particles is not easily incorporated into the asymptotics of the tagged particle. In this context, a main thrust of the proposal is to develop mathematical techniques to handle complex microscopic particle interactions so that different types of macroscopic tagged particle behaviors may be captured.