Optical Kinetic Theory of Nonlinear Multi-mode Photonic Networks

Recent experimental developments in multimode nonlinear photonic circuits (MMNPC), have motivated the development of an optical thermodynamic theory that describes the equilibrium properties of an initial beam excitation. However, a non-equilibrium transport theory for these systems, when they are in contact with thermal reservoirs, is still {\it terra incognita}. Here, by combining Landauer and kinematics formalisms we develop a one-parameter scaling theory that describes the transport in one-dimensional MMNPCs from a ballistic to a diffusive regime. We also derive a photonic version of the Wiedemann -Franz law that connects the thermal and power conductivities. Our work paves the way toward a fundamental understanding of the transport properties of MMNPC and may be useful for the design of all-optical cooling protocols.