From a nonlinear string to a weakly interacting Bose gas

The main goal of this paper is to provide a deeper physical understanding of the classical field method - a method that is widely and successfully used for description of the atomic Bose - Einstein condensates at finite temperatures. To this end, we start with a genuine nonlinear classical system - interacting oscillators. We investigate the dynamics and show that the system approaches thermal equilibrium. We show that extra assumptions are necessary to interpret the results in quantum terms. The quantum interpretation is inevitably related to the cut- off of high momenta modes. A proper choice of the cut- off momentum is equivalent to the introduction of the Planck constant. The presented approach sheds more light on the foundations of the classical field method.