Interacting Particle Systems and Nonlinear Partial Differential Equations

相互作用的粒子系统和非线性偏微分方程

基本信息

批准号：
2009549
负责人：
Natasa Pavlovic
金额：
$ 30.94万
依托单位：
University of Texas at Austin
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-07-15 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2009549&HistoricalAwards=false
关键词：
Interacting Particle Systems Nonlinear Partial

项目摘要

Analysis of large systems of interacting particles is key for predicting and understanding various phenomena arising in different contexts, from physics (in understanding e.g. boson stars) to social studies (when modeling social networks). Since the number of particles is usually very large one would like to understand qualitative and quantitative properties of such systems of particles through some macroscopic, averaged characteristics. In order to identify macroscopic behavior of multi-particle systems, it is helpful to study the asymptotic behavior when the number of particles approaches infinity, with the assumption that the limit will approximate properties observed in the systems with a large finite number of particles. An example of an important phenomenon that describes such macroscopic behavior of a large system of particles is the Bose-Einstein condensation (BEC), which is a state of the matter of a dilute Bose gas at very low temperatures when the gas moves as a single particle. Although the BEC was predicted in early days of quantum mechanics by Bose and Einstein, the first experimental realization came in 1995 (subsequently recognized by a Nobel Prize in physics in 2001). Mathematical models have been developed to understand such phenomena. Those models connect large quantum systems of interacting particles and nonlinear partial differential equations (PDE) that are derived from such systems in the limit of the number of particles going to infinity. However there are still many challenging problems on both ends, that could benefit from an interdisciplinary perspective, and the Principal Investigator will work on these. The PI will continue to explore diverse ways to disseminate the knowledge obtained from the proposed projects via designing and teaching new courses (e.g. the PI designed and taught multiple courses for graduate students at summer schools), training and mentoring graduate students and postdocs, and via organizing as well as attending seminars and research meetings.problems. With fundamental works on derivation of effective equations from quantum many body systems (e.g. nonlinear Schrodinger equation) and effective equations from classical many particle systems (e.g. Boltzmann equation) a new channel of communication between mathematical physics and nonlinear PDE communities has opened, contributing to advances in both areas. In particular, recently remarkable progress has been achieved in the rigorous derivation of nonlinear Schroedinger (NLS) equations from quantum systems of interacting bosons. Motivated by that progress, about a decade ago, the PI and her colleague Chen started studying connections between quantum many particle systems and NLS equation, and consequently together with their collaborators (including 11 PhD students and 3 postdocs) they developed the program of studying quantum many particle systems via ideas and techniques that originated in the context of 1 particle nonlinear PDE, namely the NLS. In the current project, the PI and collaborators will significantly expand the span of the above program to include: derivation of qualitative aspects of nonlinear PDE, such as being Hamiltonian or integrable, and Analysis of classical systems of particles that lead to new kinetic equations.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.

对大型相互作用粒子系统的分析是预测和理解在不同情况下（从物理学（例如，例如玻色子星）到社会研究（建模社交网络）的不同情况下产生的各种现象。由于颗粒的数量通常很大，因此希望通过某些宏观的平均特征来理解此类粒子系统的定性和定量特性。为了识别多粒子系统的宏观行为，当颗粒数量接近无穷大时，研究渐近行为是有帮助的，假设极限将在具有较大有限粒子数量的系统中观察到的限制。一个重要现象的一个例子，描述了大型颗粒系统的这种宏观行为的是Bose-Einstein凝结（BEC），它是当气体作为单个气体移动时在非常低的温度下稀释的Bose气体问题的状态粒子。尽管Bose和Einstein在量子力学的早期预测了BEC，但第一个实验性实现是在1995年（随后在2001年获得诺贝尔物理学奖的）。已经开发了数学模型来理解这种现象。这些模型将相互作用颗粒的大量子系统和非线性偏微分方程（PDE）连接起来，这些系统是从此类系统中得出的，以无限段的颗粒数量的极限。但是，两端仍然存在许多具有挑战性的问题，这可能从跨学科的角度受益，主要研究人员将致力于这些问题。 PI将继续探索通过设计和教授新课程从拟议项目中获得的知识传播的多种方式（例如，PI设计和教授暑期学校的研究生多个课程），培训和指导研究生和博士后，以及通过组织和参加研讨会和研究会议。问题。借助从量子推导有效方程的基本工作，许多身体系统（例如非线性schrodinger方程）和经典的许多粒子系统（例如Boltzmann方程）的有效方程式（例如，Boltzmann方程）是数学物理学和非线性PDE社区之间的新通信渠道在这两个地区。特别是，最近在相互作用玻色子的量子系统的非线性Schroedinger（NLS）方程的严格推导中取得了显着的进步。在大约十年前，PI和她的同事Chen开始研究量子许多粒子系统和NLS方程之间的连接，因此，在大约十年前，他们的合作者（包括11名Phd学生和3名PostDocs），他们开发了研究量子的计划，从而开发了研究量子的计划许多粒子系统通过源自1个粒子非线性PDE的上下文的思想和技术，即NLS。在当前项目中，PI和合作者将大大扩展上述程序的跨度，包括：非线性PDE定性方面的推导，例如是哈密顿量或可集成的，以及对导致新动力学方程的经典粒子系统的分析。该奖项反映了NSF的法定使命，并通过使用基金会的知识分子和更广泛的影响审查标准进行评估而被认为值得支持。

项目成果

期刊论文数量（7）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Small Data Global Well-Posedness for a Boltzmann Equation via Bilinear Spacetime Estimates

通过双线性时空估计的玻尔兹曼方程的小数据全局适定性

DOI：
10.1007/s00205-021-01613-y
发表时间：
2021
期刊：
Archive for Rational Mechanics and Analysis
影响因子：
2.5
作者：
Chen, Thomas;Denlinger, Ryan;Pavlović, Nataša
通讯作者：
Pavlović, Nataša

Global well-posedness of a binary–ternary Boltzmann equation

DOI：
10.4171/aihpc/9
发表时间：
2019-10
期刊：
Annales de l'Institut Henri Poincaré C, Analyse non linéaire
影响因子：
0
作者：
Ioakeim Ampatzoglou;I. Gamba;N. Pavlović;M. Taskovic
通讯作者：
Ioakeim Ampatzoglou;I. Gamba;N. Pavlović;M. Taskovic

Susan Friedlander's Contributions in Mathematical Fluid Dynamics

苏珊·弗里德兰德 (Susan Friedlander) 在数学流体动力学方面的贡献

DOI：
10.1090/noti2237
发表时间：
2021
期刊：
Notices of the American Mathematical Society
影响因子：
0
作者：
Cheskidov, Alexey;Glatt-Holtz, Nathan;Pavlovic, Natasa;Shvydkoy, Roman;Vicol, Vlad
通讯作者：
Vicol, Vlad

A Rigorous Derivation of a Boltzmann System for a Mixture of Hard-Sphere Gases

硬球气体混合物玻尔兹曼系统的严格推导