Journal of Statistical Physics

, Volume 174, Issue 3, pp 622–655 | Cite as

Particle and Kinetic Models for Swarming Particles on a Sphere and Stability Properties

  • Seung-Yeal Ha
  • Dohyun Kim
  • Jaeseung Lee
  • Se Eun NohEmail author


We present particle and kinetic models for the description of swarming particles on a sphere in the presence of random noises, and study their stability properties. In the absence of noises, the proposed particle model can be reduced from the Lohe matrix model for quantum synchronization, and the kinetic model can be formally derived from the particle swarming model using the BBGKY hierarchy. For the particle model without noises, we show that it is uniformly stable with respect to initial data in a Lebesgue norm. This uniform stability and particle-in-cell method yield a global existence of a measure-valued solution to the corresponding inviscid kinetic model. We also show that the incoherent state for the kinetic model is nonlinearly stable, as long as the ratio between noise strength and coupling strength is sufficiently large.


Emergence Kuramoto model Lohe model Nonlinear stability Swarm model 

Mathematics Subject Classification

92D25 74A25 76N10 


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Authors and Affiliations

  1. 1.Department of Mathematical Sciences and Research Institute of MathematicsSeoul National UniversitySeoulRepublic of Korea
  2. 2.Korea Institute for Advanced StudySeoulRepublic of Korea
  3. 3.Department of Mathematical SciencesSeoul National UniversitySeoulRepublic of Korea
  4. 4.The Research Institute of Basic SciencesSeoul National UniversitySeoulRepublic of Korea
  5. 5.Department of MathematicsMyongji UniversityYong-InRepublic of Korea

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