Resonance Broadening in Mode Coupling

The nonlinear interaction of $N$ oscillators, with $N$ not too small ($N\sim 10$), is treated statistically. A diffusion equation for the probability density in action space describes its evolution to a microcanonical distribution over the portion of the energy hyperplane for which the resonance widths (or relative-phase diffusivities) are nonvanishing. These widths are determined in terms of the resonance frequency mismatches and the nonlinear interaction coefficients by nonlinear coupled equations. For diffusion to occur, the interaction strengths must exceed the mismatches.