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

Area of Science:

• Fluid dynamics
• Dynamical systems theory
• Statistical mechanics

Background:

• Low-dimensional dynamical systems model mixing in fluid and granular flows.
• The
• cutting and shuffling
• system is a one-dimensional discontinuous dynamical system used to study mixing.
• Finite-time mixing properties and the effect of diffusion are investigated.

Purpose of the Study:

• To conduct a comprehensive computational study of finite-time mixing properties in a cutting and shuffling dynamical system.
• To analyze the impact of diffusion on mixing efficiency.
• To identify universal behaviors and predict mixing times.

Main Methods:

• Introduction of fit functions for mixing metrics: number of cutting interfaces and a mixing norm.
• Computation of averages of mixing metrics across different permutations (shuffling protocols).
• Analysis of mixing norm decay curves and prediction of critical iterations using diffusion arguments.

Main Results:

• Averages of mixing metrics serve as robust descriptors of mixing, independent of permutation choice.
• Mixing norm decay curves collapse onto a universal stretched-exponential profile when rescaled by e-folding time.
• A predicted
• stopping time
• based on diffusion arguments aligns with the e-folding time.

Conclusions:

• The study demonstrates universality in the mixing dynamics of the cutting and shuffling system.
• Diffusion significantly sharpens the transition from unmixed to mixed states, exhibiting a
• cut-off
• phenomenon.
• The findings provide evidence for cut-off phenomena in interval exchange maps.