AF: Small: Markov Chains and Mass Action Kinetics

AF：小：马尔可夫链和质量作用动力学

• 批准号: 2231095
• 负责人: Alistair Sinclair
• 金额: $ 60万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-01-01 至 2025-12-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2231095&HistoricalAwards=false
• 关键词: AF; Small; Markov; Chains; Mass

This project studies various types of dynamical processes that model computational and physical phenomena, including equilibrium and non-equilibrium statistical physics, genetic evolution via reproduction, and chemical reaction networks. The overarching goal is to leverage mathematical techniques and intuition developed for the analysis of algorithms in theoretical Computer Science to improve our understanding of these complex processes. Thus the project is an example of the application of a so-called “computational lens” to questions in other scientific fields, such as physics, chemistry and biology, which has proven very powerful in many cases. In addition to its scientific diversity, the project also affords ample opportunity for the training of graduate students, as well as curricular innovations at graduate and undergraduate levels.The project focuses on three major types of dynamics. The first type is classical reversible Markov chains, which are widely used to understand the evolution to equilibrium of systems in statistical physics, as well as algorithms for random sampling and approximate counting, and optimization methods such as simulated annealing. The second type is less classical “non-reversible” Markov chains, which describe the behavior of physical systems that are held out of equilibrium by an interaction with some external entity such as a heat-bath or particle reservoir. And the third type, known as "mass action kinetics”, capture the behavior of systems of species that repeatedly interact to produce new species, as in, for example, chemical reaction networks, genetic evolution via reproduction, genetic algorithms and Boltzmann’s model of an ideal gas. The project will add to our existing knowledge of reversible Markov chains, which are already well understood using sophisticated mathematical techniques, and will also develop the emerging theory of non-reversible Markov chains and mass action kinetics, both of which remain mathematically challenging and relatively far less well understood.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目研究了各种类型的动态过程，这些过程对计算和物理现象进行了建模，包括平衡和非平衡统计物理，通过繁殖通过遗传进化以及化学反应网络。总体目标是利用用于分析理论计算机科学算法的数学技术和直觉，以提高我们对这些复杂过程的理解。该项目是将所谓的“计算镜头”应用于其他科学领域（例如物理学，化学和生物学）中的问题的一个示例，在许多情况下，该问题在许多情况下都非常强大。除科​​学多样性外，该项目还为研究生培训以及研究生和本科级别的现代创新提供了充足的机会。该项目侧重于三种主要的动态类型。第一种类型是经典可逆的马尔可夫链，它被广泛用于了解统计物理中系统平衡的演变，以及用于随机采样和近似计数的算法，以及优化方法，例如模拟退火。第二种类型是不太经典的“非可逆”马尔可夫链，它描述了通过与某些外部实体（例如热浴或粒子储层）的相互作用来避免平衡的物理系统的行为。第三种被称为“质量作用动力学”的类型，捕获了反复相互作用以产生新物种的物种系统的行为，例如，化学反应网络，通过繁殖，遗传算法和鲍尔茨曼的理想气体模型，遗传进化。该项目将增加我们对可逆的马尔可夫链的了解，这些知识已经使用复杂的数学技术已经很好地理解了，还将发展出非可逆的马尔可夫链和大众行动动力学的新兴理论，它们在数学上保持挑战，这在数学上保持挑战，相对较少地理解，这反映了NSF的构建范围，其诚实的构建范围是诚实的传统，其法定性的传统是构建的，其法定性的构建范围是构建的，这是该挑战的范围。审查标准。