Cutting and Shuffling a Line Segment: Mixing by Interval Exchange Transformations

TL;DR

This paper computationally investigates finite-time mixing of a line segment through cutting and shuffling, demonstrating effective mixing with simple interval exchange transformations and practical measures, despite the lack of chaos.

Contribution

It introduces practical mixing metrics for finite iterations and shows that good mixing is achievable with simple interval exchange transformations.

Findings

Good mixing achieved after few iterations with six or seven intervals.

Irreducible permutation order leads to effective mixing.

Mixing effectiveness persists despite non-chaotic nature of the map.

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.