Cutting and shuffling with diffusion: Evidence for cut-offs in interval exchange maps

TL;DR

This study investigates how diffusion influences mixing in a one-dimensional cutting and shuffling system, revealing universal decay patterns and evidence of a cut-off phenomenon similar to finite Markov chains.

Contribution

It provides a comprehensive computational analysis of mixing metrics, introduces fit functions for these metrics, and demonstrates the emergence of universality and cut-off phenomena in interval exchange maps with diffusion.

Findings

Mixing norm decay follows a stretched-exponential profile across protocols.

A predictive stopping time aligns well with the decay e-folding time.

Higher Péclet numbers lead to sharper transitions from unmixed to mixed states.

Abstract

Low-dimensional dynamical systems are fruitful models for mixing in fluid and granular flows. We study a one-dimensional discontinuous dynamical system (termed "cutting and shuffling" of a line segment), and we present a comprehensive computational study of its finite-time mixing properties including the effect of diffusion. To explore a large parameter space, we introduce fit functions for two mixing metrics of choice: the number of cutting interfaces (a standard quantity in dynamical systems theory of interval exchange transformations) and a mixing norm (a more physical measure of mixing). We compute averages of the mixing metrics across different permutations (shuffling protocols), showing that the latter averages are a robust descriptor of mixing for any permutation choice. If the decay of the normalized mixing norm is plotted against the number of map iterations rescaled by the characteristic e-folding time, then universality emerges: mixing norm decay curves across all cutting and shuffling protocols collapse onto a single stretched-exponential profile. Next, we predict this critical number of iterations using the average length of unmixed subsegments of continuous color during cutting and shuffling and a Batchelor-scale-type diffusion argument. This prediction, called a "stopping time" for finite Markov chains, compares well with the e-folding time of the stretched-exponential fit. Finally, we quantify the effect of diffusion on cutting and shuffling through a P\'eclet number (a dimensionless inverse diffusivity), showing that the system transitions more sharply from an unmixed initial state to a mixed final state as the P\'eclet number becomes large. Our numerical investigation of cutting and shuffling of a line segment in the presence of diffusion thus present evidence for the latter phenomenon, known as a "cut-off" for finite Markov chains, in interval exchange maps.