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

  • François GolseEmail author
  • Seung-Yeal Ha


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.



The work of Seung-Yeal Ha is partially supported by National Research Foundation of Korea Grant (NRF-2017R1A2B2001864) funded by the Korea Government. This work was started, when the second author stayed National Center for Theoretical Sciences-Mathematics Division in National Taiwan University in the fall of 2015.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Ecole Polytechnique, CMLSPalaiseau CedexFrance
  2. 2.Department of Mathematical Sciences and Research Institute of MathematicsSeoul National UniversitySeoulRepublic of Korea
  3. 3.Korea Institute for Advanced StudySeoulRepublic of Korea

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