On the equivalence between the Boltzmann equation and classical field theory at large occupation numbers

We consider a system made up of exictations of a neutral scalar field, \phi, having a \lambda\phi^4 interaction term. Starting from an ensemble where the occupation number f is large, but \lambda f is small, we develop a classical field theory description of the evolution of the system toward equilibrium. A Boltzmann equation naturally emerges in this description and we show by explicit calculation that the collision term is the same as that coming from elastic scattering. This shows the equivalence of a Boltzmann equation description and a classical field theory description of the same system.