A Pair of Oscillators Subject to Stochastic Forces

Suppose that a pair of oscillators, which are identical in nature, are weakly coupled to each other and under the influence of some external stochastic forces. The following model would then be appropriate  

The lowest-order perturbation theory applied to (5.5.7) yields d0a 

Rapidly oscillating quantities in J are now time averaged. After putting e equal to 1, we obtain 

Let us look into what occurs in further detail. Let y/+ denote the mean phase and y/ the relative phase (i.e., y/ -y/ y/2) and Qiy/-,t) the reduced distribution function with respect to the relative phase, or 

The implication of this equation is better understood by writing down the nonlinear Langevin equation equivalent to it  

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