  The mean-field optical phase transition in multimode equal-coupling photonic
networks is studied by temporal evolution of the nonlinear equations of motion
of the coupled modes. Analogies to statistical mechanics models of interacting
classical spins, built upon the correspondence between complex-valued modes and
two-component spins, are employed to define two-component and single-component
order parameters. A comprehensive finite-size scaling analysis is performed to
estimate critical points and exponents of a second-order phase transition,
driven by the optical energy per mode. Equilibrium properties of the system are
compared to exact results whenever applicable. Considering various parameter
settings, our results confirm the mean-field nature of the transition and
establish 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 kinetic
temperature with appropriate zero-temperature limit. In the low temperature
phase (low energy per mode), spins align. In the high temperature phase (large
energy per mode), spins rotate independent of one another.
