Statistics of an ensemble of nonlinear classical oscillators in the field of an external harmonic force when the oscillators have two stable states

An ensemble of nonlinear classical oscillators in a thermal bath and in the field of an harmonic external force is considered. Systematic application of the method of averaging with respect to the fast variables makes it possible to obtain approximate analytic solutions for the distribution function in a steady regime in different regions of the phase space separated by a separatrix. Matching of the solutions belonging to different regions of the phase space confirms the conjecture that there is an excess population in the region of oscillator energies in which the frequency of the eigenmodes of the oscillator is close to the frequency of the external force.