Extended finite-size scaling of synchronized coupled oscillators

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We present a systematic analysis of dynamic scaling in the time evolution of the phase order parameter for coupled oscillators with non-identical natural frequencies in terms of the Kuramoto model. This provides a comprehensive view of phase synchronization. In particular, we extend finite-size scaling (FSS) in the steady state to dynamics, determine critical exponents, and find the critical coupling strength. The dynamic scaling approach enables us to measure not only the FSS exponent associated with the correlation volume in finite systems but also thermodynamic critical exponents. Based on the extended FSS theory, we also discuss how the sampling of natural frequencies and thermal noise affect dynamic scaling, which is numerically confirmed.