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In this work we discuss a few ways to create chaotic families that are not entropically chaotic on Kac's Sphere. We present two types of examples: limiting convex combination of an entropically chaotic family with a particularly 'bad' non-entropic family, and two explicitly computable families that vary rapidly with $N$, causing loss of support on the sphere or high entropic tails.

A few ways to destroy entropic chaoticity on Kac’s Sphere