where P i,j refers to the ͑i , j͒th component of the probability density found using the complete stochastic system of transformed equations ͓Fig. 4͑a͔͒ and Q i,j refers to the ͑i , j͒th component of the probability density found using the reduced system of equations ͓Fig. 4͑b͔͒. In our numerical computation of the relative entropy, we have added 10 −10 to each P ij and Q ij . This eliminates the possibility of having a Q ij =0 in the denominator of Eq. ͑23͒ and does not have much of an effect on the relative entropy. If the two histograms were identical, then the relative entropy given by Eq. ͑23͒ would be d KL = 0. The two histograms shown in Figs. 4͑a͒ and 4͑b͒ have a relative entropy of d KL = 0.0391, which means that the two histograms, while not identical, are quantitatively very similar. However, one cannot rely entirely on the histograms alone to say that the solutions of the complete system and the reduced system agree. As we have seen in Figs. 3͑a͒ and 3͑b͒, the two solutions have differing amplitudes and are out of phase with one another. It is important to note that these features are not picked up by the histograms of Fig. 4 .