Lohe矩阵模型的平均场极限与突现动力学
A mean-field limit of the Lohe matrix model and emergent dynamics
摘要：
Lohe矩阵模型是描述酉群流形中群元集体动力学的连续时间动力系统，它是Kuramoto相位模型的非阿贝尔推广的玩具模型。在没有耦合的情况下，它归结为具有常数哈密顿量的有限维解耦自由薛定谔方程。本文研究了Lohe矩阵模型的严格平均场极限，它导致相应相空间上密度函数的Vlasov型方程。我们还提供了两个不同的设置的紧急同步动力学的动力学Lohe方程的初始数据和耦合强度。
英文摘要：
The Lohe matrix model is a continuous-time dynamical system describing the collective dynamics of group elements in the unitary group manifold, and it has been introduced as a toy model of a non abelian generalization of the Kuramoto phase model. In the absence of couplings, it reduces to the finite-dimensional decoupled free Schrödinger equations with constant Hamiltonians. In this paper, we study a rigorous mean-field limit of the Lohe matrix model which results in a Vlasov type equation for the probability density function on the corresponding phase space. We also provide two different settings for the emergent synchronous dynamics of the kinetic Lohe equation in terms of the initial data and the coupling strength.
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