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Glassy states and superrelaxation in populations of coupled phase oscillators

Large networks of coupled oscillators appear in many branches of science, so that the kinds of phenomena they exhibit are not only of intrinsic interest but also of very wide importance. In 1975, Kuramoto proposed an analytically tractable model to describe such systems, which has since been successfully applied in many contexts and remains a subject of intensive research. Some related problems, however, remain unclarified for decades, such as the properties of the oscillator glass state discovered by Daido in 1992. Here we present a detailed analysis of a very general form of the Kuramoto model. In particular, we find the conditions when it can exhibit glassy behavior, which represents a kind of synchronous disorder in the present case. Furthermore, we discover a new and intriguing phenomenon that we refer to as {\it superrelaxation} where, for a class of parameter distributions, the oscillators feel no interaction at all during relaxation to incoherence, a phenomenon reminiscent of superfluidity or superconductivity. Our findings offer the possibility of creating glassy states and observing superrelaxation in real systems, thus paving the way to a cascade of applications and further research in the field.