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The mean-field optical phase transition in multimode equal-coupling photonicnetworks is studied by temporal evolution of the nonlinear equations of motionof the coupled modes. Analogies to statistical mechanics models of interactingclassical spins, built upon the correspondence between complex-valued modes andtwo-component spins, are employed to define two-component and single-componentorder parameters. A comprehensive finite-size scaling analysis is performed toestimate critical points and exponents of a second-order phase transition,driven by the optical energy per mode. Equilibrium properties of the system arecompared to exact results whenever applicable. Considering various parametersettings, our results confirm the mean-field nature of the transition andestablish the critical line in the nonlinearity--energy-density phase diagram.Critical scaling leads to infer the upper critical dimension $d_c=4$.Connection to thermodynamic quantities is established by means of a kinetictemperature with appropriate zero-temperature limit. In the low temperaturephase (low energy per mode), spins align. In the high temperature phase (largeenergy per mode), spins rotate independent of one another.

Critical-point phenomena and finite-size scaling in mean-field equal-coupling photonic networks