Electronic Journal of Probability

Synchronization for discrete mean-field rotators

Benedikt Jahnel and Christof Külske

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We analyze a non-reversible mean-field jump dynamics for discrete q-valued rotators and show in particular that it exhibits synchronization. The dynamics is the mean-field analogue of the lattice dynamics investigated by the same authors which provides an example of a non-ergodic interacting particle system on the basis of a mechanism suggested by Maes and Shlosman.

Based on the correspondence to an underlying model of continuous rotators via a discretization transformation we show the existence of a locally attractive periodic orbit of rotating measures. We also discuss global attractivity, using a free energy as a Lyapunov function and the linearization of the ODE which describes typical behavior of the empirical distribution vector.

Article information

Electron. J. Probab., Volume 19 (2014), paper no. 14, 26 pp.

Accepted: 20 January 2014
First available in Project Euclid: 4 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82B26: Phase transitions (general) 82C22: Interacting particle systems [See also 60K35]

Interacting particle systems non-equilibrium synchronization mean-field sytems discretization XY model clock model rotation dynamics attractive limit cycle

This work is licensed under a Creative Commons Attribution 3.0 License.


Jahnel, Benedikt; Külske, Christof. Synchronization for discrete mean-field rotators. Electron. J. Probab. 19 (2014), paper no. 14, 26 pp. doi:10.1214/EJP.v19-2948. https://projecteuclid.org/euclid.ejp/1465065656

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  • Aizenman, Michael; Wehr, Jan. Rounding effects of quenched randomness on first-order phase transitions. Comm. Math. Phys. 130 (1990), no. 3, 489–528.
  • Arnołd, V. I. Ordinary differential equations. Translated from the Russian and edited by Richard A. Silverman. MIT Press, Cambridge, Mass.-London, 1978. ix+280 pp. ISBN: 0-262-51018-9
  • Balaban, Tadeusz; O'Carroll, Michael. Low temperature properties for correlation functions in classical $N$-vector spin models. Comm. Math. Phys. 199 (1999), no. 3, 493–520.
  • Bertini, Lorenzo; Giacomin, Giambattista; Pakdaman, Khashayar. Dynamical aspects of mean field plane rotators and the Kuramoto model. J. Stat. Phys. 138 (2010), no. 1-3, 270–290.
  • L. Bertini, G. Giacomin and C. Poquet: Synchronization and random long time dynamics for mean-field plane rotators, Probab. Theory Related Fields, 0178-8051, (2013).
  • Bovier, Anton. Statistical mechanics of disordered systems. A mathematical perspective. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 2006. xiv+312 pp. ISBN: 978-0-521-84991-3; 0-521-84991-8
  • Chassaing, Philippe; Mairesse, Jean. A non-ergodic probabilistic cellular automaton with a unique invariant measure. Stochastic Process. Appl. 121 (2011), no. 11, 2474–2487.
  • Collet, Francesca; Dai Pra, Paolo. The role of disorder in the dynamics of critical fluctuations of mean field models. Electron. J. Probab. 17 (2012), no. 26, 40 pp.
  • Cotar, Codina; Külske, Christof. Existence of random gradient states. Ann. Appl. Probab. 22 (2012), no. 4, 1650–1692.
  • N. Crawford: Random Field Induced Order in Low Dimension, EPL, 102, 36003 (2013).
  • Dai Pra, Paolo; den Hollander, Frank. McKean-Vlasov limit for interacting random processes in random media. J. Statist. Phys. 84 (1996), no. 3-4, 735–772.
  • Dembo, Amir; Zeitouni, Ofer. Large deviations techniques and applications. Corrected reprint of the second (1998) edition. Stochastic Modelling and Applied Probability, 38. Springer-Verlag, Berlin, 2010. xvi+396 pp. ISBN: 978-3-642-03310-0
  • Ellis, Richard S. Entropy, large deviations, and statistical mechanics. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 271. Springer-Verlag, New York, 1985. xiv+364 pp. ISBN: 0-387-96052-X
  • van Enter, A. C. D.; Fernández, R.; den Hollander, F.; Redig, F. A large-deviation view on dynamical Gibbs-non-Gibbs transitions. Mosc. Math. J. 10 (2010), no. 4, 687–711, 838.
  • van Enter, Aernout C. D.; Fernández, Roberto; Sokal, Alan D. Regularity properties and pathologies of position-space renormalization-group transformations: scope and limitations of Gibbsian theory. J. Statist. Phys. 72 (1993), no. 5-6, 879–1167.
  • van Enter, Aernout C. D.; Külske, Christof. Nonexistence of random gradient Gibbs measures in continuous interface models in $d=2$. Ann. Appl. Probab. 18 (2008), no. 1, 109–119.
  • van Enter, Aernout C. D.; Külske, Christof; Opoku, Alex A. Discrete approximations to vector spin models. J. Phys. A 44 (2011), no. 47, 475002, 11 pp.
  • van Enter, Aernout C. D.; Külske, Christof; Opoku, Alex A.; Ruszel, Wioletta M. Gibbs–non-Gibbs properties for $n$-vector lattice and mean-field models. Braz. J. Probab. Stat. 24 (2010), no. 2, 226–255.
  • Feng, Jin; Kurtz, Thomas G. Large deviations for stochastic processes. Mathematical Surveys and Monographs, 131. American Mathematical Society, Providence, RI, 2006. xii+410 pp. ISBN: 978-0-8218-4145-7; 0-8218-4145-9
  • Fröhlich, Jürg; Pfister, Charles. On the absence of spontaneous symmetry breaking and of crystalline ordering in two-dimensional systems. Comm. Math. Phys. 81 (1981), no. 2, 277–298.
  • Funaki, Tadahisa. Stochastic interface models. Lectures on probability theory and statistics, 103–274, Lecture Notes in Math., 1869, Springer, Berlin, 2005.
  • Fröhlich, Jürg; Spencer, Thomas. Massless phases and symmetry restoration in abelian gauge theories and spin systems. Comm. Math. Phys. 83 (1982), no. 3, 411–454.
  • Fröhlich, Jürg; Spencer, Thomas. The BerežinskiÄ­-Kosterlitz-Thouless transition (energy-entropy arguments and renormalization in defect gases). Scaling and self-similarity in physics (Bures-sur-Yvette, 1981/1982), 29–138, Progr. Phys., 7, Birkhäuser Boston, Boston, MA, 1983.
  • Fröhlich, J.; Simon, B.; Spencer, Thomas. Infrared bounds, phase transitions and continuous symmetry breaking. Comm. Math. Phys. 50 (1976), no. 1, 79–95.
  • Georgii, Hans-Otto. Gibbs measures and phase transitions. Second edition. de Gruyter Studies in Mathematics, 9. Walter de Gruyter & Co., Berlin, 2011. xiv+545 pp. ISBN: 978-3-11-025029-9
  • Giacomin, Giambattista; Pakdaman, Khashayar; Pellegrin, Xavier. Global attractor and asymptotic dynamics in the Kuramoto model for coupled noisy phase oscillators. Nonlinearity 25 (2012), no. 5, 1247–1273.
  • Giacomin, Giambattista; Pakdaman, Khashayar; Pellegrin, Xavier; Poquet, Christophe. Transitions in active rotator systems: invariant hyperbolic manifold approach. SIAM J. Math. Anal. 44 (2012), no. 6, 4165–4194.
  • den Hollander, Frank. Large deviations. Fields Institute Monographs, 14. American Mathematical Society, Providence, RI, 2000. x+143 pp. ISBN: 0-8218-1989-5
  • F. den Hollander, F. Redig and W. van Zuijlen: Gibbs-non-Gibbs dynamical transitions for mean-field interacting Brownian motions, arXiv:1312.3438 (2013).
  • B. Jahnel and C. Külske: A class of nonergodic interacting particle systems with unique invariant measure, accepted for publication in Ann. Appl. Probab., arXiv:1208.5433v2 (2012).
  • Külske, Christof; Le Ny, Arnaud; Redig, Frank. Relative entropy and variational properties of generalized Gibbsian measures. Ann. Probab. 32 (2004), no. 2, 1691–1726.
  • Külske, Christof; Opoku, Alex A. The posterior metric and the goodness of Gibbsianness for transforms of Gibbs measures. Electron. J. Probab. 13 (2008), no. 47, 1307–1344.
  • Külske, Christof; Opoku, Alex A. Continuous spin mean-field models: limiting kernels and Gibbs properties of local transforms. J. Math. Phys. 49 (2008), no. 12, 125215, 31 pp.
  • Külske, Christof; Le Ny, Arnaud. Spin-flip dynamics of the Curie-Weiss model: loss of Gibbsianness with possibly broken symmetry. Comm. Math. Phys. 271 (2007), no. 2, 431–454.
  • D.H. Lee, R.G. Caflisch, J.D. Joannopoulos and F.Y. Wu: Antiferromagnetic classical XY model: A mean-field analysis, Phys. Rev. B 29, 5 (1984).
  • Liggett, Thomas M. Interacting particle systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 276. Springer-Verlag, New York, 1985. xv+488 pp. ISBN: 0-387-96069-4
  • Maes, Christian; Shlosman, Senya. Rotating states in driven clock- and XY-models. J. Stat. Phys. 144 (2011), no. 6, 1238–1246.
  • Newman, C. M.; Schulman, L. S. Asymptotic symmetry: enhancement and stability. Phys. Rev. B (3) 26 (1982), no. 7, 3910–3914.
  • Oelschláger, Karl. A martingale approach to the law of large numbers for weakly interacting stochastic processes. Ann. Probab. 12 (1984), no. 2, 458–479.
  • Ortiz, G.; Cobanera, E.; Nussinov, Z. Dualities and the phase diagram of the $p$-clock model. Nuclear Phys. B 854 (2012), no. 3, 780–814.
  • Redig, Frank; Wang, Feijia. Gibbs-non-Gibbs transitions via large deviations: computable examples. J. Stat. Phys. 147 (2012), no. 6, 1094–1112.
  • H. Silver, N.E. Frankel and B.W. Ninham: A class of mean field models, J. Math. Phys., 468-474 (1972).