Interacting Particle Systems

相互作用的粒子系统

• 批准号: 0301795
• 负责人: Thomas Liggett
• 金额: $ 57.85万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-07-01 至 2009-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0301795&HistoricalAwards=false
• 关键词: Interacting; Particle; Systems

0301795Liggett The project addresses several problems in interacting particle systems and related areas of probability theory. One collection of problems involves the stationary distributions for exclusion processes with drift in both one and higher dimensions. Recent work of the investigator showed the existence of stationary blocking measures in one dimension under weak assumptions. The next set of problems concerns the existence of nonblocking stationary profile measures in one dimension and of nonreversible stationary blocking measures in higher dimensions -- in neither case are there any known examples. Other problems involve (a) connections between the symmetric exclusion process and negative dependence, (b) the extent to which the known theory of the symmetric exclusion process extends to processes with a few asymmetries, (c) the analysis of some infinite systems motivated by the investigator's recent work on a class of models in sociology, and (d) the computation of the critical values for a family of reversible growth models on homogeneous trees. Many problems in the natural and social sciences involve the behavior of large numbers of individuals (people, molecules, cars, viruses, etc.) that interact in unpredictable ways. This lack of predictability is modeled by randomness. One of the fundamental contributions of probability theory in the past century is the realization that large populations display a significant amount of predictability and order in spite of the lack of predictability of the actions of the individuals in the population. This project is aimed at gaining a better understanding of several models of this general type. Among them are: (a) the exclusion process, which models particle motion, traffic flow, and the behavior of ribosomes in biology, (b) certain growth models that are partly motivated by tumor growth and conflicts between competing populations, and (c) models from sociology, in which relationships among individuals are altered by the exchange of gifts or rewards. In each case, the objective is to determine the long time behavior of the system.

0301795 Liggett该项目解决了相互作用粒子系统和概率论相关领域的几个问题。一个问题的集合涉及在一维和更高的维度漂移的排斥过程的平稳分布。最近的研究工作表明，在弱假设下，在一维中存在固定的阻塞措施。下一组问题涉及存在的nonblocking固定配置文件措施在一维和不可逆的固定阻塞措施在更高的维度-在这两种情况下都没有任何已知的例子。其他问题包括：（a）对称排斥过程和负相关之间的联系，（B）对称排斥过程的已知理论扩展到具有少数不对称性的过程的程度，（c）由研究者最近对社会学中一类模型的工作所激发的一些无限系统的分析，（d）齐次树上一类可逆增长模型临界值的计算。 自然科学和社会科学中的许多问题涉及大量个体的行为（人、分子、汽车、病毒等）。以不可预知的方式相互作用。这种可预测性的缺乏是由随机性模拟的。概率论在过去世纪的基本贡献之一是认识到，尽管群体中个体的行为缺乏可预测性，但大群体显示出显著的可预测性和有序性。该项目旨在更好地了解这种一般类型的几种模型。其中包括：（a）排斥过程，它模拟了生物学中的粒子运动、交通流和核糖体行为;（B）某些生长模型，部分是由肿瘤生长和竞争种群之间的冲突所激发的;（c）社会学模型，其中个体之间的关系通过交换礼物或奖励而改变。在每种情况下，目标是确定系统的长期行为。