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Phase transitions, chaos and dynamical anomalies in long-range models of coupled oscillators

Andrea Rapisarda, Alessandro Pluchino
Long-range interacting systems have been intensively studied in the last years and new methodologies have been developed in the attempt to understand their intriguing features. One of the most promising directions is the combination of statistical mechanics tools and methods adopted in dynamical systems. In particular, phase transitions have been extensively explored in both conservative and dissipative long-range systems. The Hamiltonian mean-field (HMF) model [1] and the Kuramoto model [2] represent two paradigmatic toy models, the former conservative and the latter dissipative, for many real systems with long-range forces and have several interesting applications. Both models share the same order parameter and display a spontaneous phase transition from a homogeneous/incoherent phase to a magnetized/synchronized one.
In this paper we address the chaotic behavior of the synchronization phase transition in the Kuramoto model and we discuss the relationship with analogous features found in the Hamiltonian mean-field (HMF) model. Our results support the connection between the two models, which can be considered as limiting cases (dissipative and conservative, respectively) of a more general dynamical system of damped/driven coupled pendula [3]. We also discuss the similarities of the dynamical anomalies found in the out-of-equilibrium regime of the two models and address the possible application of generalized Tsallis statistics [4] to explain this anomalous behavior [5].


[1] Dauxois T., Latora V., Rapisarda A., Ruffo S. And Torcini A., Dynamics and
Thermodynamics of Systems with Long Range Interactions, Lect. Notes
Phys.,Vol. 602 (Springer) 2002, p. 458.

[2] Kuramoto Y., Chemical Oscillations, Waves, and Turbulence (Springer, Berlin)
Acebron J. A., Bonilla L. L., Perez Vicente C. J., Ritort F. and Spigler R., Rev.
Mod. Phys., 77 (2005) 137.

[3] Miritello G., Pluchino and A., Rapisarda A. Europhys. Lett. 85 (2009)

[4] Tsallis C. , J. Stat. Phys. 52 (1988) 479;
Gell-Mann M., Tsallis C. (Eds.), Nonextensive Entropy - Interdisciplinary
Applications, Oxford University Press, New York, 2004;
Tsallis C., Introduction to nonextensive statistical mechanics: approaching a
complex world, (Springer) 2009.

[5] Pluchino A., Rapisarda A.,Tsallis C., Europhys. Lett. 80 (2007) 26002;
Pluchino A., Rapisarda A. and Tsallis C., Physica A, 387 (2008) 3121;
Pluchino A., Rapisarda and A.,Tsallis C., Europhys. Lett. (2009) in press;
Miritello G., Pluchino A. and Rapisarda A., (2009) to be submitted.
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