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.

该研究项目旨在解决不仅具有数学意义的数学问题，而且还涉及物理学、量子信息论和生物学中出现的问题。对控制物理过程的方程的分析通常需要对这些过程中涉及的各种量的相对大小进行精确的定量理解，而这是由数学不等式提供的。一个公众熟悉的例子可以解释为物理过程中“熵永远不会减少”； 所讨论的不等式表明，稍后时间的熵至少与早期时间的熵一样大。对于更具体的物理系统，我们可以说得更多，而寻求更好地理解物理过程以及控制它们的演化方程，部分是为了寻求新的、更精确的控制这些过程的数学不等式。数学不等式和物理过程之间的联系是双向的，因此在试图证明这种不等式时，人们可能会尝试将其与一个简单且易于理解的进化方程联系起来。这一研究领域取得了丰硕的成果，不仅产生了更广泛的科学界感兴趣的结果，而且还引起了博士生的兴趣。学生。这项研究的智力价值在于，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学相关的结果。这种在其他领域的适用性保证了这项工作的广泛影响，学生的参与进一步增强了这种影响，有助于培养下一代研究人员。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

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

Recovery and the Data Processing Inequality for Quasi-Entropies

准熵的恢复和数据处理不等式

• DOI: 10.1109/tit.2018.2812038
• 发表时间: 2018
• 期刊: IEEE Transactions on Information Theory
• 影响因子: 2.5
• 作者: Carlen, Eric A.;Vershynina, Anna
• 通讯作者: Vershynina, Anna