For comparison to the LS method approach, an alternative though ineffective mixing strategy is choosing cut locations randomly while using the same interval exchange. This random-cut mixing approach results in suboptimal mixing states with respect to homogeneity, although they may have higher entropy . Mixing randomly is similar to the type of mixing in a shuffled deck of cards where the most desirable outcome maximizes the uncertainty of the next card. Compare this with the most desireable outcome of homogeneity, which is a perfectly interleaved deck of cards-one that, after 26 perfect riffle shuffles, returns the deck to its original state. Homogeneity in the case studied here results in the most mixed state having a "perfect crystal" structure of alternating species . Hence, random cutting-andshuffling provides a comparison for mixing effectiveness that tends toward a high entropy mixed state different from the well-ordered optimally mixed state. The probability of creating new interfaces at each step (a measure of potential mixing) with random cuts is 1-