Many-Body Theory of Synchronization by Long-Range Interactions

Abstract

Synchronization of coupled oscillators on a $d$-dimensional lattice with the power-law coupling $G(r)=g_0/r^{\alpha }$ and randomly distributed intrinsic frequency is analyzed. A systematic perturbation theory is developed to calculate the order parameter profile and correlation functions in powers of $\epsilon =\alpha /d-1$. For $\alpha \le d$, the system exhibits a sharp synchronization transition as described by the conventional mean-field theory. For $\alpha >d$, the transition is smeared by the quenched disorder, and the macroscopic order parameter $\Avψ$ decays slowly with $g_0$ as $|\Avψ| \propto g_0^2$.

4 pages, 2 figures

References in corpus (2)

• Entrainment transition in populations of random frequency oscillators
• Synchronization of oscillators with long range power law interactions

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