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We study the thermalization of the classical Klein-Gordon equation under au^4 interaction. We numerically show that even in the presence of strongnonlinearities, the local thermodynamic equilibrium state exhibits a weaklynonlinear behavior in a renormalized wave basis. The renormalized basis isdefined locally in time by a linear transformation and the requirement ofvanishing wave-wave correlations. We show that the renormalized waves oscillatearound one frequency, and that the frequency dispersion relation undergoes anonlinear shift proportional to the mean square field. In addition, therenormalized waves exhibit a Planck like spectrum. Namely, there isequipartition of energy in the low frequency modes described by a Boltzmanndistribution, followed by a linear exponential decay in the high frequencymodes.

Renormalized waves and thermalization of the Klein-Gordon equation: What sound does a nonlinear string make?