A mean-field limit of the Lohe matrix model and emergent dynamics

TL;DR

This paper rigorously derives a mean-field limit of the Lohe matrix model, a system describing collective unitary dynamics, leading to a Vlasov-type equation and analyzing conditions for emergent synchronization.

Contribution

It introduces a rigorous mean-field limit for the Lohe matrix model and explores emergent synchronization phenomena based on initial conditions and coupling strength.

Findings

Derivation of a Vlasov-type equation from the Lohe matrix model.

Identification of conditions for emergent synchronous dynamics.

Analysis of the impact of initial data and coupling strength on synchronization.

Abstract

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 Schr\"{o}dinger 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.