Cutting and shuffling a line segment: Mixing by interval exchange transformations

Marissa K. Krotter, Ivan C. Christov, Julio M. Ottino, Richard M. Lueptow*

*Corresponding author for this work

Research output: Contribution to journalArticle

15 Scopus citations

Abstract

We present a computational study of finite-time mixing of a line segment by cutting and shuffling. A family of one-dimensional interval exchange transformations is constructed as a model system in which to study these types of mixing processes. Illustrative examples of the mixing behaviors, including pathological cases that violate the assumptions of the known governing theorems and lead to poor mixing, are shown. Since the mathematical theory applies as the number of iterations of the map goes to infinity, we introduce practical measures of mixing (the percent unmixed and the number of intermaterial interfaces) that can be computed over given (finite) numbers of iterations. We find that good mixing can be achieved after a finite number of iterations of a one-dimensional cutting and shuffling map, even though such a map cannot be considered chaotic in the usual sense and/or it may not fulfill the conditions of the ergodic theorems for interval exchange transformations. Specifically, good shuffling can occur with only six or seven intervals of roughly the same length, as long as the rearrangement order is an irreducible permutation. This study has implications for a number of mixing processes in which discontinuities arise either by construction or due to the underlying physics.

Original languageEnglish (US)
Article number1230041
JournalInternational Journal of Bifurcation and Chaos
Volume22
Issue number12
DOIs
StatePublished - Jan 1 2012

Keywords

  • Cutting and shuffling
  • granular mixing
  • lamellar structures
  • piecewise isometries

ASJC Scopus subject areas

  • Modeling and Simulation
  • Engineering (miscellaneous)
  • General
  • Applied Mathematics

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