Collective Dynamics of Lohe type aggregation models

TL;DR

This paper reviews recent advances in understanding collective behaviors in Lohe type aggregation models, focusing on classical and quantum systems, with theoretical conditions and numerical simulations.

Contribution

It provides a unified framework of sufficient conditions for collective dynamics in Lohe models, including new results for the Lohe tensor and Schrödinger-Lohe models.

Findings

Sufficient conditions for collective behavior derived via Lyapunov functionals.

Numerical simulations illustrating dynamics of Schrödinger-Lohe model.

Unified theoretical approach applicable to classical and quantum models.

Abstract

In this paper, we review state-of-the-art results on the collective behaviors for Lohe type first-order aggregation models. Collective behaviors of classical and quantum many-body systems have received lots of attention from diverse scientific disciplines such as applied mathematics, control theory in engineering, nonlinear dynamics of statistical physics, etc. To model such collective dynamics, several phenomenological models were proposed in literature and their emergent dynamics were extensively studied in recent years. Among them, we present two Lohe type models: the Lohe tensor (LT) model and the Schrodinger-Lohe mode}, and present several sufficient conditions in unified frameworks via the Lyapunov functional approach for state diameters and dynamical systems theory approach for two-point correlation functions. We also present several numerical simulation results for the SL model.