Mixing and the fractal geometry of piecewise isometries

Paul P. Park, Thomas F. Lynn, Paul B. Umbanhowar, Julio M. Ottino, Richard M. Lueptow

Research output: Contribution to journalArticlepeer-review

8 Scopus citations


Mathematical concepts often have applicability in areas that may have surprised their original developers. This is the case with piecewise isometries (PWIs), which transform an object by cutting it into pieces that are then rearranged to reconstruct the original object, and which also provide a paradigm to study mixing via cutting and shuffling in physical sciences and engineering. Every PWI is characterized by a geometric structure called the exceptional set, E, whose complement comprises nonmixing regions in the domain. Varying the parameters that define the PWI changes both the structure of E as well as the degree of mixing the PWI produces, which begs the question of how to determine which parameters produce the best mixing. Motivated by mixing of yield stress materials, for example granular media, in physical systems, we use numerical simulations of PWIs on a hemispherical shell and examine how the fat fractal properties of E relate to the degree of mixing for any particular PWI. We present numerical evidence that the fractional coverage of E negatively correlates with the intensity of segregation, a standard measure for the degree of mixing, which suggests that fundamental properties of E such as fractional coverage can be used to predict the effectiveness of a particular PWI as a mixing mechanism.

Original languageEnglish (US)
Article number042208
JournalPhysical Review E
Issue number4
StatePublished - Apr 14 2017

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Statistics and Probability
  • Condensed Matter Physics


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