A Mean-Field Limit of the Lohe Matrix Model and Emergent Dynamics

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 for the non-abelian generalization of the Kuramoto 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 Lohe kinetic equation in terms of the initial data and the coupling strength.