  We study the thermalization of the classical Klein-Gordon equation under a
u^4 interaction. We numerically show that even in the presence of strong
nonlinearities, the local thermodynamic equilibrium state exhibits a weakly
nonlinear behavior in a renormalized wave basis. The renormalized basis is
defined locally in time by a linear transformation and the requirement of
vanishing wave-wave correlations. We show that the renormalized waves oscillate
around one frequency, and that the frequency dispersion relation undergoes a
nonlinear shift proportional to the mean square field. In addition, the
renormalized waves exhibit a Planck like spectrum. Namely, there is
equipartition of energy in the low frequency modes described by a Boltzmann
distribution, followed by a linear exponential decay in the high frequency
modes.
