(A) The log of the KL divergence between the free head spatial probability densities and corresponding to the cosine potential used in the main text and respectively. The minimum KL divergence occurs at , which is the quartic expansion of the cosine potential. The stars indicate the potentials used to compute diffusion contours in panels B–D. The diffusion contours are shown for (B) the harmonic potential , as well as potentials with (C) , and (D) . The harmonic potential produces diffusion that is nearly identical to the cosine potential (Figure 3C), while the contour shown in C has only subtle differences. The diffusion shown in D is qualitatively different because the potential has an additional energy minimum giving rise to a new peak in the distribution. In all cases we set , as in the main text.