Variational Problems, Stability and Dynamics

变分问题、稳定性和动力学

• 批准号: 1764254
• 负责人: Eric Carlen
• 金额: $ 27万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-06-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1764254&HistoricalAwards=false
• 关键词: Variational; Problems; Stability; Dynamics

This research project is aimed at solving mathematical problems that are not only significant as mathematics, but that have been suggested by problems arising in physics, quantum information theory and biology. The analysis of the equations governing physical processes often requires a precise quantitative understanding of the relative sizes of various quantities involved in these processes, and this is provided by mathematical inequalities. An example familiar to the general public can be paraphrased as saying that ``entropy never decreases'' in physical processes; the inequality in question says that the entropy at a later time is at least as large as the entropy at an earlier time. For more specific physical systems one can say much more, and the search for a better understanding of physical processes, and the evolution equations that govern them, is in part a quest for new and more precise mathematical inequalities governing these processes. The connection between mathematical inequalities and physical processes runs both ways, so that in trying to prove such an inequality, one may try to relate it to a simple and well understood evolution equation. This area of research has been fruitful not only in producing results that are of interest to a wider scientific community, but also in engaging the interest of Ph.D. students. The intellectual merit of the research is that it will produce not only significant new mathematics, but results that are relevant to the physical sciences and engineering as well. This applicability in other fields guarantee a broad impact of the work, which is further enhanced by the involvement of students, contributing to training of the next generation of researchers.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.

该研究项目的目的是解决数学问题，这些问题不仅与数学一样重要，而且是由于物理学，量子信息理论和生物学出现的问题所表明的。对处理物理过程的方程式的分析通常需要对这些过程所涉及的各种数量的相对大小的相对大小进行精确的定量理解，这是由数学不平等所提供的。可以说一个公众熟悉的例子说：``熵永远不会减少''在物理过程中。 问题不平等说，以后的熵至少与较早的熵一样大。对于更具体的物理系统，人们可以说更多，并且寻求更好地理解物理过程，以及控制它们的演化方程，部分是寻求管理这些过程的新的，更精确的数学不平等现象。数学不平等和物理过程之间的联系都会两种方式运行，因此，在试图证明这种不平等时，人们可能会尝试将其与简单且知名度广泛的进化方程联系起来。这一研究领域不仅在产生更广泛的科学界感兴趣的结果方面富有成果，而且还吸引了博士学位的兴趣。学生。这项研究的智力优点在于，它不仅会产生重要的新数学，而且还会产生与物理科学和工程相关的结果。在其他领域的这种适用性保证了这项工作的广泛影响，这是通过学生参与进一步增强的，这是有助于培训下一代研究人员的培训。该奖项反映了NSF的法定任务，并被认为是值得通过使用该评估的支持基金会的智力优点和更广泛的影响评论标准。

项目成果

Quantitative bounds on the rate of approach to equilibrium for some one-dimensional stochastic nonlinear Schrödinger equations

一些一维随机非线性薛定谔方程接近平衡速率的定量界限

• DOI: 10.1088/1361-6544/aae69c
• 发表时间: 2019
• 期刊: Nonlinearity
• 影响因子: 1.7
• 作者: Carlen, Eric A;Fröhlich, Jürg;Lebowitz, Joel;Wang, Wei-Min
• 通讯作者: Wang, Wei-Min

A Kac model with exclusion

具有排除的 Kac 模型

• DOI: 10.1214/22-aihp1276
• 发表时间: 2023
• 期刊: Probabilités et Statistiques
• 作者: Carlen, Eric;Wennberg, Bernt
• 通讯作者: Wennberg, Bernt

On the Convolution Inequality f ≥ f ⋆ f

关于卷积不等式 f → f → f

• DOI: 10.1093/imrn/rnaa350
• 发表时间: 2021
• 期刊: International Mathematics Research Notices
• 影响因子: 1
• 作者: Carlen, Eric A;Jauslin, Ian;Lieb, Elliott H;Loss, Michael P
• 通讯作者: Loss, Michael P

Spectral gaps for reversible Markov processes with chaotic invariant measures: The Kac process with hard sphere collisions in three dimensions

具有混沌不变测度的可逆马尔可夫过程的谱间隙：具有三维硬球碰撞的 Kac 过程

• DOI: 10.1214/20-aop1437
• 发表时间: 2020
• 期刊: The Annals of Probability
• 作者: Carlen, Eric;Carvalho, Maria;Loss, Michael
• 通讯作者: Loss, Michael

Complete positivity and self-adjointness

• DOI: 10.1016/j.laa.2020.10.038
• 发表时间: 2021-02-15
• 期刊: LINEAR ALGEBRA AND ITS APPLICATIONS
• 影响因子: 1.1
• 作者: Amorim, Erik;Carlen, Eric A.
• 通讯作者: Carlen, Eric A.