On the semiclassical limit of the Schrödinger-Lohe model and concentration estimates

Abstract

We study the semiclassical limit of quantum synchronization model and concentration estimates for the resulting limit model. From the Schrödinger-Lohe model, we rigorously derive the Vlasov-Lohe model using Wigner transform and Wigner measure method. In semiclassical limit, generalized Wigner distributions to the Schrödinger-Lohe model converge to a set of Wigner measures which corresponds to a weak solution to the Vlasov-Lohe model, and then we show the asymptotic collective behaviors of the Vlasov-Lohe model. When one-body potentials are identical, we show that complete synchronization emerges for the Vlasov-Lohe model. In contrast, for non-identical potentials the lack of boundedness results in practical synchronization for the integrals of solutions. Moreover, we construct a global existence of classical solutions to the Vlasov-Lohe model using the standard method of characteristics. Analysis in this work can deal with possibly non-identical potentials in which their differences are constant. © 2024 Author(s).