Optimized mixing by cutting-and-shuffling

Lachlan D. Smith*, Paul B. Umbanhowar, Julio M. Ottino, Richard M. Lueptow

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


Mixing by cutting-and-shuffling can be understood and predicted using dynamical systems based tools and techniques. In existing studies, mixing is generated by maps that repeat the same cutand- shuffle process at every iteration in a "fixed" manner. However, mixing can be greatly improved by varying the cut-and-shuffle parameters at each step using a "variable" approach. To demonstrate this approach, we show how to optimize mixing by cutting-and-shuffling on the one-dimensional line interval, known as an interval exchange transformation (IET). Mixing can be significantly improved by optimizing variable protocols, especially for initial conditions more complex than just a simple twocolor line interval. While we show that optimal variable IETs can be found analytically for arbitrary numbers of iterations, for more complex cutting-and-shuffling systems, computationally expensive numerical optimization methods are required. Furthermore, the number of control parameters grows linearly with the number of iterations in variable systems. Therefore, optimizing over large numbers of iterations is generally computationally prohibitive. We demonstrate an ad hoc approach to cuttingand- shuffling that is computationally inexpensive and guarantees that the mixing metric is within a constant factor of the optimum. This ad hoc approach yields significantly better mixing than fixed IETs, which are known to produce weak-mixing, because cut pieces never reconnect. The heuristic principles of this method can be applied to more general cutting-and-shuffling systems.

Original languageEnglish (US)
Pages (from-to)2544-2573
Number of pages30
JournalSIAM Journal on Applied Dynamical Systems
Issue number4
StatePublished - 2018


  • cutting-and-shuffling
  • interval exchange transformation
  • mixing optimization

ASJC Scopus subject areas

  • Analysis
  • Modeling and Simulation


Dive into the research topics of 'Optimized mixing by cutting-and-shuffling'. Together they form a unique fingerprint.

Cite this