Abstract : In recent years, kinetic-type models emerged to be a powerful mathematical framework for the description of emerging patterns in systems composed by a large number of agents. Furthermore, the natural multiscale nature of these equations, linking microscopic unobservable social forces to macroscopic measurable patterns, permits an efficient investigation of collective phenomena in a heterogeneity of disciplines, like biology, social sciences and robotics. In this minisymposium we collect novel perspectives from experts actively working on these research problems.
[04948] Weak couplings of Lohe type aggregation models
Format : Online Talk on Zoom
Author(s) :
Seung Yeal Ha (Seoul National University)
Dohyun Kim (Sungkyunkwan University)
Hansol Park (Simon Fraser University)
Abstract : In this talk, we present a systematic algebraic approach for the weak coupling of Cauchy problems to multiple Lohe tensor models. For this, we identify an admissible Cauchy problem to the Lohe tensor (LT) model with a characteristic symbol consisting of four tuples in terms of a size vector, a natural frequency tensor, a coupling strength tensor and admissible initial configuration. In this way, the collection of all admissible Cauchy problems to the LT models is equivalent to the space of characteristic symbols. On the other hand, we introduce a binary operation, namely fusion operation" as a binary operation between characteristic symbols. It turns out that the fusion operation satisfies an associativity and admits the identity element in the space of characteristic symbols which naturally forms a monoid. By virtue of the fusion operation, the weakly coupled system of multi tensor models can be obtained by applying the fusion operation of multiple characteristic symbols corresponding to the Lohe tensor models. As a concrete example, we consider a weak coupling of the swarm sphere model and the Lohe matrix model, and provide sufficient framework leading to emergent dynamics to the proposed weakly coupled model. This is a joint work with Dohyun Kim (Sungshin Women’s Univ.) and Hansol Park (Simon Fraser Univ.)
[03897] Quantified overdamped limit for Kinetic Vlasov-Fokker-Planck equations
Format : Talk at Waseda University
Author(s) :
Oliver Tse (Eindhoven University of Technology)
Young-Pil Choi (Yonsei University)
Abstract : The study of the overdamped limit for the kinetic Fokker-Planck equation has been of interest since the seminal work of Kramers in 1940, where he formally discussed the convergence by introducing a coarse-graining map. This talk is based on work with Young-Pil Choi, where we provide a framework to establish quantitative estimates for the overdamped limit of the kinetic Vlasov-Fokker-Planck with singular interaction in terms of the 2-Wasserstein distance.
[04724] Phase-coupled models for synchronization with nonlocal temporal interactions
Format : Talk at Waseda University
Author(s) :
Myeongju Kang (Korea Institute for Advanced Study)
Abstract : We study the emergent dynamics and global well-posedness of the Kuramoto model with memory effect, which is a system of Volterra-type integro-differential equations on unit circle. We adopt nonlocal temporal interactions to design synchronized behavior of oscillators affected by memories of the past. We first establish the global well-posedness of the Kuramoto model with memory effect, and provide sufficient frameworks for uniform boundedness of the phase diameter. Then, we define an energy functional whose boundedness is guaranteed by the boundedness of the phase diameter. We show that energy functional is monotonically decreasing, which implies complete frequency synchronization. Moreover, when natural frequencies are all identical we show the emergence of complete phase synchronization.
[04208] Rigorous derivation of the Euler-Alignment model with singular communication weights from a kinetic Fokker-Planck-Alignment model
Format : Online Talk on Zoom
Author(s) :
Young-Pil Choi (Yonsei University)
Jeongho Kim (Kyung Hee University)
Abstract : We present a rigorous derivation of the isothermal Euler-alignment model with singular communication weights. We consider a hydrodynamic limit of a kinetic Fokker-Planck-alignment model, which is the nonlinear Fokker-Planck equation with the Cucker-Smale alignment force. Our analysis is based on the estimate of relative entropy between macroscopic quantities, together with careful analysis on the singular communication weights.
[04945] Sticky-particle Cucker-Smale dynamics and the entropic selection principle for the Euler-alignment system
Format : Talk at Waseda University
Author(s) :
Trevor Leslie (University of Southern California)
Changhui Tan (University of South Carolina)
Abstract : In this talk, I will discuss weak solutions to the Euler-alignment system for collective behaviors. I will introduce an entropic selection principle that serves to isolate a unique weak solution. Notably, the solution can be constructed and approximated using Cucker-Smale dynamics, with sticky particle collision rules. I will present an analytical convergence result, as well as the formation of finite and infinite time clusters.
[04190] Structure-preserving particle method for the Vlasov-Landau-Maxwell system
Format : Online Talk on Zoom
Author(s) :
Rafa Bailo (University of Oxford)
Jose Carrillo (University of Oxford)
Jingwei Hu (University of Washington)
Abstract : Vlasov-Landau-Maxwell equation is often considered as the first-principle physics model for plasmas. We introduce a novel particle method for this equation which preserves the basic physical properties such as conservation of mass, momentum, and energy, and even decay of entropy. The method is based on a proper regularization of the Landau collision operator so that it can be naturally coupled with the classical particle-in-cell (PIC) method to preserve the structure. Various plasma benchmark tests such as collisional Landau damping and two-stream instability will be presented.
[03372] On solutions to the kinetic Cucker-Smale model with singular communication weights
Format : Talk at Waseda University
Author(s) :
Jinwook Jung (Jeonbuk National University)
Young-Pil Choi (Yonsei University)
Abstract : In this talk, we investigate the existence of solutions to the kinetic Cucker-Smale model with singular communication weights $\phi(r) = r^{-\gamma}$. First, we establish the local-in-time well-posedness of strong solutions to the equation in a weighted Sobolev space for $\gamma \in [d-1, d+1/4)\setminus \{d\}$. Secondly, we present the existence of weak solutions for $\gamma \in [d-1, d)$ and also the uniqueness result when $\gamma =d-1$. This talk is based on the collaboration with Y.-P. Choi.
[03940] Interaction energy minimizers on bounded domains
Format : Online Talk on Zoom
Author(s) :
Ruiwen Shu (University of Georgia)
José Carrillo (University of Oxford)
Abstract : I will discuss the behavior of interaction energy minimizers on bounded domains. When the interaction potential is more singular than Newtonian, then mass does not tend to concentrate on the boundary; when it is Newtonian or less singular, then mass necessarily concentrate on the boundary for purely repulsive potentials. We also draw a connection between bounded-domain minimizers and whole-space minimizers.
[03933] Analytical approaches to the problem of emergence arising in systems of collective behavior
Format : Online Talk on Zoom
Author(s) :
Roman Shvydkoy (University of Illinois at Chicago)
Abstract : Emergence is a phenomenon of formation of collective outcomes in systems where communications between agents has local range. For a wide range of applications, such as swarming behavior of animals or exchange of opinions between individuals, such outcomes result in a globally aligned state or congregation of aligned clusters. The classical result of Cucker and Smale states that alignment is unconditional in flocks that have global communication with non-integrable radial tails. Proving a similar statement for purely local interactions is a challenging mathematical problem. In this talk we will overview three programs of research directed on understanding the emergent phenomena: statistical approach to generic alignment for agent-based systems, kinetic approach based on relaxation and hypocoercivity, and hydrodynamic models incorporating a novel way of interaction based on topological communication.
