Uniform stability and emergent dynamics of particle and kinetic Lohe matrix models

Abstract

We study asymptotic behaviors of the second-order Lohe matrix model on the unitary group. For the emergent behaviors, we show that a solution to the Lohe matrix model with identical Hamiltonians always converges to equilibrium using a gradient flow approach. Moreover, we establish the uniform-in-time stability with respect to initial data using the complete aggregation estimate, which can be applied to obtain the corresponding mean-field kinetic equation for the one-oscillator distribution function on the phase space. In fact, we rigorously show that Wasserstein-2 distance between two solutions in different regimes converges to zero in any finite time interval, when the number of oscillators tends to infinity. For the asymptotic emergent dynamics of the kinetic equation, we use an energy estimate and temporal evolution for the diameter to provide sufficient frameworks leading to the complete and practical aggregations for identical and non-identical Hamiltonians, respectively.