일시 : 2024년 8월 26, 27일 오후 2시~4시, 8월 28일 오전 10시~12시

장소 : 전북대학교 자연과학대학 본관 417호, 308호

주관 : 전북대학교 순수및응용수학연구소

발표자 : 이진엽 박사(University of Basel)

제목 : Mean-field limit of large-particle interacting systems

초록 : This lecture series will delve into the intricate dynamics of large-particle interacting systems, focusing on the derivation of mean-field equations and the analysis of multi-body interactions. The first two lectures will concentrate on fermionic systems, exploring the derivation of the Vlasov equation from the fermionic many-body Schrödinger system using advanced mathematical tools like the Husimi measure. The final lecture will shift focus to hypergraphs, examining nonlinear averaging dynamics in systems characterized by three-body interactions.

세부강연내용 :

Lecture 1: Derivation of the Vlasov equation from the fermionic many-body Schrödinger system using the Husimi measure I

This lecture introduces the audience to the derivation of the Vlasov equation, a fundamental equation in statistical mechanics, from the underlying fermionic many-body Schrödinger system. The session will cover the essential concepts of the many-body Schrödinger equation, and the use of the Wigner and Husimi measures to link quantum mechanical descriptions with classical mean-field dynamics. The discussion will also include strategies for managing the residual terms that arise during the derivation process.

Lecture 2: Derivation of the Vlasov equation from the fermionic many-body Schrödinger system using the Husimi measure II

Building on the foundations laid in the first lecture, this session will continue to explore the application of the Husimi measure in deriving the Vlasov equation. We will delve deeper into the formalism, offering detailed insights into the challenges and solutions encountered in this approach. By the end of this lecture, attendees will have a comprehensive understanding of how quantum many-body dynamics can be rigorously connected to classical mean-field models.

Lecture 3: Convergence of nonlinear averaging dynamics with three-body interactions on hypergraphs

In the final lecture, the focus will shift from fermionic systems to the study of multi-body interactions within hypergraphs. Specifically, we will investigate discrete-time dynamics on a 3-uniform hypergraph where nodes update their states based on a nonlinearly-weighted average of neighboring pairs’ states, capturing complex group effects such as peer pressure. Unlike linear models, this system converges not to a simple average of initial states, but to a shifted value. The lecture will present a rigorous proof of convergence under certain conditions, offering insights into the behavior of complex networked systems.