| Summary: | The second-order Kuramoto equation describes the synchronization of coupled oscillators with inertia, which occur, for example, in power grids. On the contrary to the first-order Kuramoto equation, its synchronization transition behavior is significantly less known. In the case of Gaussian self-frequencies, it is discontinuous, in contrast to the continuous transition for the first-order Kuramoto equation. Herein, we investigate this transition on large 2D and 3D lattices and provide numerical evidence of hybrid phase transitions, whereby the oscillator phases <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>θ</mi><mi>i</mi></msub></semantics></math></inline-formula> exhibit a crossover, while the frequency is spread over a real phase transition in 3D. Thus, a lower critical dimension <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>d</mi><mi>l</mi><mi>O</mi></msubsup><mo>=</mo><mn>2</mn></mrow></semantics></math></inline-formula> is expected for the frequencies and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>d</mi><mi>l</mi><mi>R</mi></msubsup><mo>=</mo><mn>4</mn></mrow></semantics></math></inline-formula> for phases such as that in the massless case. We provide numerical estimates for the critical exponents, finding that the frequency spread decays as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>∼</mo><msup><mi>t</mi><mrow><mo>−</mo><mi>d</mi><mo>/</mo><mn>2</mn></mrow></msup></mrow></semantics></math></inline-formula> in the case of an aligned initial state of the phases in agreement with the linear approximation. In 3D, however, in the case of the initially random distribution of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>θ</mi><mi>i</mi></msub></semantics></math></inline-formula>, we find a faster decay, characterized by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>∼</mo><msup><mi>t</mi><mrow><mo>−</mo><mn>1.8</mn><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msup></mrow></semantics></math></inline-formula> as the consequence of enhanced nonlinearities which appear by the random phase fluctuations.
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