Exact correlations in the nonequilibrium stationary state of the noisy Kuramoto model. (English) Zbl 1480.82013
Summary: We obtain exact results on autocorrelation of the order parameter in the nonequilibrium stationary state of a paradigmatic model of spontaneous collective synchronization, the Kuramoto model of coupled oscillators, evolving in presence of Gaussian, white noise. The method relies on an exact mapping of the stationary-state dynamics of the model in the thermodynamic limit to the noisy dynamics of a single, non-uniform oscillator, and allows to obtain besides the Kuramoto model the autocorrelation in the equilibrium stationary state of a related model of long-range interactions, the Brownian mean-field model. Both the models show a phase transition between a synchronized and an incoherent phase at a critical value of the noise strength. Our results indicate that in the two phases as well as at the critical point, the autocorrelation for both the model decays as an exponential with a rate that increases continuously with the noise strength.
MSC:
| 82C22 | Interacting particle systems in time-dependent statistical mechanics |
| 60H40 | White noise theory |
| 82C31 | Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics |
| 82C26 | Dynamic and nonequilibrium phase transitions (general) in statistical mechanics |
| 82C27 | Dynamic critical phenomena in statistical mechanics |
References:
| [1] | Huang, K., Statistical Mechanics, (1987), New York: Wiley, New York · Zbl 1041.82500 |
| [2] | Livi, R.; Politi, P., Nonequilibrium Statistical Physics: a Modern Perspective, (2017), Cambridge: Cambridge University Press, Cambridge · Zbl 1451.82001 |
| [3] | Kubo, R., The fluctuation-dissipation theorem, Rep. Prog. Phys., 29, 255, (1966) · Zbl 0163.23102 · doi:10.1088/0034-4885/29/1/306 |
| [4] | Pikovsky, A.; Rosenblum, M.; Kurths, J., Synchronization: a Universal Concept in Nonlinear Sciences, (2001), Cambridge: Cambridge University Press, Cambridge · Zbl 0993.37002 |
| [5] | Bier, M.; Bakker, B. M.; Westerhoff, H. V., How yeast cells synchronize their glycolytic oscillations: a perturbation analytic treatment, Biophys. J., 78, 1087, (2000) · doi:10.1016/S0006-3495(00)76667-7 |
| [6] | Buck, J., Synchronous rhythmic flashing of fireflies. II, Q. Rev. Biol., 63, 265, (1988) · doi:10.1086/415929 |
| [7] | Wiesenfeld, K.; Colet, P.; Strogatz, S. H., Frequency locking in Josephson arrays: connection with the Kuramoto model, Phys. Rev. E, 57, 1563, (1998) · doi:10.1103/PhysRevE.57.1563 |
| [8] | Hirosawa, K.; Kittaka, S.; Oishi, Y.; Kannari, F.; Yanagisawa, T., Phase locking in a Nd:YVO_4 waveguide laser array using Talbot cavity, Opt. Express, 21, 24952, (2013) · doi:10.1364/OE.21.024952 |
| [9] | Rohden, M.; Sorge, A.; Timme, M.; Witthaut, D., Self-organized synchronization in decentralized power grids, Phys. Rev. Lett., 109, (2012) · doi:10.1103/PhysRevLett.109.064101 |
| [10] | Kuramoto, Y., Chemical Oscillations, Waves and Turbulence, (1984), Berlin: Springer, Berlin · Zbl 0558.76051 |
| [11] | Strogatz, S. H., From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators, Physica D, 143, 1, (2000) · Zbl 0956.00057 · doi:10.1016/S0167-2789(00)00094-4 |
| [12] | Acebron, J. A.; Bonilla, L. L.; Vicente, C. J P.; Ritort, F.; Spigler, R., The Kuramoto model: a simple paradigm for synchronization phenomena, Rev. Mod. Phys., 77, 137, (2005) · doi:10.1103/RevModPhys.77.137 |
| [13] | Gupta, S.; Campa, A.; Ruffo, S., Kuramoto model of synchronization: equilibrium and nonequilibrium aspects, J. Stat. Mech., (2014) · Zbl 1456.34060 · doi:10.1088/1742-5468/14/08/R08001 |
| [14] | Gupta, S.; Campa, A.; Ruffo, S., Statistical Physics of Synchronization, (2018), Berlin: Springer, Berlin · Zbl 1407.82003 |
| [15] | Sakaguchi, H., Cooperative phenomena in coupled oscillator systems under external fields, Prog. Theor. Phys., 79, 39, (1988) · doi:10.1143/PTP.79.39 |
| [16] | Chavanis, P. H., The Brownian mean field model, Eur. Phys. J. B, 87, 120, (2014) · doi:10.1140/epjb/e2014-40586-6 |
| [17] | Antoni, M.; Ruffo, S., Clustering and relaxation in Hamiltonian long-range dynamics, Phys. Rev. E, 52, 2361, (1995) · doi:10.1103/PhysRevE.52.2361 |
| [18] | Gupta, S., Spontaneous collective synchronization in the Kuramoto model with additional non-local interactions, J. Phys. A: Math. Theor., 50, (2017) · Zbl 1456.82734 · doi:10.1088/1751-8121/aa88d7 |
| [19] | Campa, A.; Dauxois, T.; Fanelli, D.; Ruffo, S., Physics of Long-range Interacting Systems, (2014), Oxford: Oxford University Press, Oxford · Zbl 1303.82003 |
| [20] | Papoulis, A., Probability, Random Variables and Stochastic Processes, (1965), New York: McGraw-Hill, New York · Zbl 0191.46704 |
| [21] | Strogatz, S. H., Nonlinear Dynamics and Chaos: with Applications to Physics, Biology, Chemistry, and Engineering, (2014), Boulder, CO: Westview Press, Boulder, CO |
| [22] | Risken, H., The Fokker-Planck Equation: Methods of Solution and Applications, (1996), Berlin: Springer, Berlin · Zbl 0866.60071 |
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