Advertisement

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

  • François GolseEmail author
  • Seung-Yeal Ha
Article

Abstract

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.

Notes

Acknowledgements

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.

References

  1. 1.
    Acebron, J.A., Bonilla, L.L., Pérez Vicente, C.J.P., Ritort, F., Spigler, R.: The Kuramoto model: a simple paradigm for synchronization phenomena. Rev. Mod. Phys. 77, 137–185, 2005ADSCrossRefGoogle Scholar
  2. 2.
    Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and the Space of Probability Measures, 2nd edn. Birkhäuser Verlag AG, Basel 2008zbMATHGoogle Scholar
  3. 3.
    Ameri, V., Eghbali-arani, M., Mari, A., Farace, A., Kheirandish, F., Giovannetti, V., Fazio, R.: Mutual information as an order parameter for quantum sychronization. Phys. Rev. A 91, 012301, 2015ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Buck, J., Buck, E.: Biology of synchronous flashing of fireflies. Nature 211, 562, 1966ADSCrossRefGoogle Scholar
  5. 5.
    Chandra, S., Girvan, M., Ott, E.: Continuous versus discontinuous transitions in the D-dimensional generalized kuramoto model: odd D is different. Phys. Rev. X 9, 011002, 2019Google Scholar
  6. 6.
    Cho, J., Choi, S.-H., Ha, S.-Y.: Practical quantum synchronization for the Schrodinger–Lohe system. J. Phys. A 49, 205203, 2016ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Choi, S.-H., Ha, S.-Y.: Complete entrainment of Lohe oscillators under attractive and repulsive couplings. SIAM. J. App. Dyn. 13, 1417–1441, 2013MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Choi, Y., Ha, S.-Y., Jung, S., Kim, Y.: Asymptotic formation and orbital stability of phase-locked states for the Kuramoto model. Physica D 241, 735–754, 2012ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chopra, N., Spong, M.W.: On exponential synchronization of Kuramoto oscillators. IEEE Trans. Autom. Control 54, 353–357, 2009MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    DeVille, L.: Synchronization and stability for quantum Kuramoto. J. Stat. Phys. 174, 160–187, 2019ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Dong, J.-G., Xue, X.: Synchronization analysis of Kuramoto oscillators. Commun. Math. Sci. 11, 465–480, 2013MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Dörfler, F., Bullo, F.: Synchronization in complex networks of phase oscillators: a survey. Automatica 50, 1539–1564, 2014MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Dörfler, F., Bullo, F.: Exploring synchronization in complex oscillator networks. IIEEE 51st Annual Conference on Decision and Control (CDC), 7157–7170 (2012)Google Scholar
  14. 14.
    Dörfler, F., Bullo, F.: On the critical coupling for Kuramoto oscillators. SIAM. J. Appl. Dyn. Syst. 10, 1070–1099, 2011MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Golse, F., Paul, T.: The Schrödinger equation in the mean-field and semiclassical regime. Arch. Ration. Mech. Anal. 223, 57–94, 2017MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Golse, F., Mouhot, C., Paul, T.: On the mean-field and classical limits of quantum mechanics. Commun. Math. Phys. 343, 165–205, 2016ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Ha, S.-Y., Kim, H.W., Ryoo, S.W.: Emergence of phase-locked states for the Kuramoto model in a large coupling regime. Commun. Math. Sci. 14, 1073–1091, 2016MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Ha, S.-Y., Ko, D., Ryoo, S.W.: On the relaxation dynamics of Lohe oscillators on some Riemannian manifolds. J. Stat. Phys. 172, 1427–1478, 2018ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Ha, S.-Y., Ko, D., Ryoo, S.W.: Emergent dynamcis of a generalized Lohe model on some class of Lie groups. J. Stat. Phys. 168, 171–207, 2017ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Ha, S.-Y., Ko, D., Park, J., Zhang, X.: Collective synchronization of classical and quantum oscillators. EMS Surv. Math. Sci. 3, 209–267, 2016ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Ha, S.-Y., Ryoo, S.W.: On the emergence and orbital stability of phase-locked states for the Lohe model. J. Stat. Phys. 163, 411–439, 2016ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Hush, M.R., Li, W., Genway, S., Lesanovsky, I., Armour, A.: Spin correlations as a probe of quantum synchronization in trapped-ion phonon lasers. Phys. Rev. A 91, 061401, 2015ADSCrossRefGoogle Scholar
  23. 23.
    Goychuk, I., Casado-Pascual, J., Morillo, M., Lehmann, J., Hänggi, P.: Quantum stochastic synchronization. Phys. Rev. Lett. 97, 210601, 2006ADSCrossRefGoogle Scholar
  24. 24.
    Giorgi, G.L., Galve, F., Manzano, G., Colet, P., Zambrini, R.: Quantum correlations and mutual synchronization. Phys. Rev. A 85, 052101, 2012ADSCrossRefGoogle Scholar
  25. 25.
    Kimble, H.J.: The quantum internet. Nature 453, 1023–1030, 2008ADSCrossRefGoogle Scholar
  26. 26.
    Kuramoto, Y.: Chemical Oscillations, Waves and Turbulence. Springer, Berlin 1984CrossRefzbMATHGoogle Scholar
  27. 27.
    Kuramoto, Y.: International symposium on mathematical problems in mathematical physics. Lect. Notes Theor. Phys. 30, 420, 1975ADSCrossRefGoogle Scholar
  28. 28.
    Lohe, M.A.: Quantum synchronization over quantum networks. J. Phys. A Math. Theor. 43, 465301, 2010ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Lohe, M.A.: Non-abelian Kuramoto model and synchronization. J. Phys. A Math. Theor. 42, 395101, 2009ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Markdahl, J.: A topological obstruction to almost global synchronization on Riemannian manifolds. arXiv:1808.00862v3
  31. 31.
    Markdahl, J. Thunberg, J., Goncalves, J.: High-dimensional Kuramoto models on Stiefel manifolds synchronize complex networks almost globally. arXiv:1807.10233v2
  32. 32.
    Mirollo, R., Strogatz, S.H.: The spectrum of the partially locked state for the Kuramoto model. J. Nonlinear Sci. 17, 309–347, 2007ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Mirollo, R., Strogatz, S.H.: The spectrum of the locked state for the Kuramoto model of coupled oscillators. Physica D 205, 249–266, 2005ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Mirollo, R., Strogatz, S.H.: Stability of incoherence in a population of coupled oscillators. J. Stat. Phys. 63, 613–635, 1991ADSMathSciNetCrossRefGoogle Scholar
  35. 35.
    Olfati-Saber, R.: Swarms on sphere: a programmable swarm with synchronous behaviors like oscillator networks. IEEE 45th Conference on Decision and Control (CDC), 5060–5066 (2006)Google Scholar
  36. 36.
    Peskin, C.S.: Mathematical Aspects of Heart Physiology. Courant Institute of Mathematical Sciences, New York 1975zbMATHGoogle Scholar
  37. 37.
    Pikovsky, A., Rosenblum, M., Kurths, J.: Synchronization: A Universal Concept in Nonlinear Sciences. Cambridge University Press, Cambridge 2001CrossRefzbMATHGoogle Scholar
  38. 38.
    Ritchie, L.M., Lohe, M.A., Williams, A.G.: Synchronization of relativistic particles in the hyperbolic Kuramoto model. Chaos 28, 053116, 2018ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Strogatz, S.H.: From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators. Physica D 143, 1–20, 2000ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Verwoerd, M., Mason, O.: On computing the critical coupling coefficient for the Kuramoto model on a complete bipartite graph. SIAM J. Appl. Dyn. Syst. 8, 417–453, 2009ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Verwoerd, M., Mason, O.: Global phase-locking in finite populations of phase-coupled oscillators. SIAM J. Appl. Dyn. Syst. 7, 134–160, 2008ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Villani, C.: Topics on Optimal Transportation. American Mathematical Society, Providence, RI 2003CrossRefzbMATHGoogle Scholar
  43. 43.
    Winfree, A.T.: Biological rhythms and the behavior of populations of coupled oscillators. J. Theor. Biol. 16, 15–42, 1967CrossRefGoogle Scholar
  44. 44.
    Winfree, A.T.: The Geometry of Biological Time. Springer, New York 1980CrossRefzbMATHGoogle Scholar
  45. 45.
    Xu, M., Tieri, D.A., Fine, E.C., Thompson, J.K., Holland, M.J.: Quantum synchronization of two ensembles of atoms. Phys. Rev. Lett. 113, 154101, 2014ADSCrossRefGoogle Scholar
  46. 46.
    Zhu, B., Schachenmayer, J., Xu, M., Herrera, F., Restrepo, J.G., Holland, M.J., Rey, A.M.: Synchronization of interacting dipoles. New J. Phys. 17, 083063, 2015ADSCrossRefGoogle Scholar

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

Personalised recommendations