Cut-and-Shuffle Mixing On a Hemispherical Shell and a Line Segment


Mixing by cutting-and-shuffling (like that for a deck of cards or a Rubik's cube) is a paradigm that has not been studied in detail even though it can be applied in a variety of situations including the mixing of granular materials. Mathematically, cutting- and-shuffling is described by piecewise isometries (PWIs), which isometrically (without distortion) rearrange solid pieces cut from an original object to re-form the same object. In such systems, mixing is created by the cut discontinuities in the PWI. The canonical PWI example of a bi-rotated hemispherical shell (rotated alternately about two axes by two different amounts), which is inspired by the mixing of granular material in a half- filled spherical tumbler rotated about two axes, is the main focus of study here. The region of the domain that is eligible for cut-and-shuffle mixing by a PWI is referred to as the exceptional set, and it has a fat fractal structure, shown quantitatively for many examples. By examining exceptional sets for a wide variety of parameter values (rotation axis orientations and rotation amounts) using a high performance computing algorithm, changes in the amount of potential mixing are examined. In particular, the relative placement of the two rotation axes can break symmetries that limit the potential mixing area found in previous studies using orthogonally oriented rotation axes. Additionally, a class of strictly non-mixing PWI parameter values, which create tiled patterns on the hemispherical shell, are identified. However, these potential mixing regions are not necessarily mixing, and, for some parameter combinations, barriers to mixing exist within the exceptional set. Using a type of return plot or a coloring method for rendering the exceptional set, the connectivity of the exceptional set is measured by examining orbits within the potential mixing region under the action of the PWI, and isolated non-mixing regions are located using a measure of how frequently an orbit returns to the discontinuities responsible for cut-and-shuffle mixing. Measurements of how often interactions with the discontinuities occur can identify mixing subsets within the exceptional set. The combined measures of the size of the exceptional set and its connectivity characterize the long-term mixing behavior of different hemispherical shell PWIs, and the methods developed here can be expanded to other discontinuous systems in general. The second topic examined in this dissertation is the application of machine learning to cut-and-shuffle mixing in one dimension. Because of the simplicity of the system and richness of its dynamics, cut-and-shuffle mixing is an instructive candidate system with which to evaluate the potentialities and limitations of machine learning (ML) as an approach to solve more difficult mixing problems. I focus on a specific subset of cut-and-shuffle systems, the one-dimensional interval exchange transform. This class of cut-and-shuffling mixing operations is well studied, and it has a simple associated mixing method, the longest segment (LS) method, that works well in many situations. I use supervised learning to train neural networks (NNs) to emulate the LS mixing algorithm to mix a one-dimensional domain of two species and find that a generic deep NN is able to emulate the LS method with good accuracy but is unable to generalize to conditions significantly outside its training repertoire. The challenges in defining the mixing problem and generalizing a ML mixing approach are indicative of those expected for more complex systems where optimal or near optimal mixing methods remain unknown.

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