Deceleration of one-dimensional mixing by discontinuous mappings

We present a computational study of a simple one-dimensional map with dynamics composed of stretching, permutations of equal sized cells, and diffusion. We observe that the combination of the aforementioned dynamics results in eigenmodes with long-time exponential decay rates. The decay rate of the eigenmodes is shown to be dependent on the choice of permutation and changes non-monotonically with the diffusion coefficient for many of the permutations. The global mixing rate of the map M in the limit of vanishing diffusivity approximates well the decay rates of the eigenmodes for small diffusivity, however this global mixing rate does not bound the rates for all values of the diffusion coefficient. This counter-intuitively predicts a deceleration in the asymptotic mixing rate with increasing diffusivity rate. The implication of the results on finite time mixing are discussed.