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We show that, in general, any complex weakly nonlinear highly multimodesystem can reach thermodynamic equilibrium that is characterized by a uniquetemperature and chemical potential. The conditions leading to either positiveor negative temperatures are explicitly obtained in terms of the linearspectrum of the system, the input power, and the corresponding Hamiltonianinvariant. Pertinent examples illustrating these results are provided invarious scenarios.

Thermodynamic conditions governing the optical temperature and chemical potential in nonlinear highly multimoded photonic systems