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

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 Péclet number (a dimensionless inverse diffusivity), showing that the system transitions more sharply from an unmixed initial state to a mixed final state as the Péclet number becomes large. Our numerical investigation of cutting and shuffling of a line segment in the presence of diffusion thus presents evidence for the latter phenomenon, known as a "cut-off" for finite Markov chains, in interval exchange maps.

Optimal linear response for Markov Hilbert-Schmidt integral operators and stochastic dynamical systems

“…Before we present the explicit formula for the optimal solution, we will reformulate the optimization problem (45) to simplify the analysis. We first note that since the objective function in (45) is linear in Ṫ , the maximum will occur on S T 0 , ∩ ∂B 1 . Combining this with the fact that we only need c ∈ span{f 0 } ⊥ , we consider the following reformulation of ( 45):…”

“…The combined effect of the "cutting and shuffling" of interval exchanges with diffusion on mixing rates has been widely studied, e.g. [2,43,16,34,45], including investigations of the impact of changing the diffusion or the interval exchange on mixing. The very general type of formal map optimisation we consider here has not been attempted before, and we hope that our novel techniques will stimulate interesting new research questions and motivate more sophisticated experiments in the field of mixing optimisation.…”

Potentialities and Limitations of Machine Learning to Solve Mixing Problems: A Case Study

“…In one dimension, cutting-and-shuffling is described mathematically by interval exchange transforms (IETs) [42][43][44][45][46][47][48][49][50], a subset of broader cut-and-shuffle mixing actions called piecewise isometries (PWIs) [41,[51][52][53]. An IET cuts a one dimensional domain into intervals that are then reordered according to a specific permutation.…”

“…An IET cuts a one dimensional domain into intervals that are then reordered according to a specific permutation. The mixing dynamics of IET systems have been studied extensively due to the relative simplicity of IETs (compared to higher dimensional PWIs and mixing in fluid systems) and the surprising richness of their resulting dynamics [42][43][44][45][46][47][48][49][50][54][55][56][57][58]. Similar to the reproduction of a dynamical system [36] or a portion of it [37,38], here we emulate an established algorithm for mixing a one-dimensional domain of two species using supervised learning to train a NN.…”

Cutting and shuffling a hemisphere: Nonorthogonal axes

“…Cutting-andshuffling is a mixing mechanism that is far less understood, but is particularly relevant to systems with flow discontinuities, such as granular materials [3][4][5][6][7][8], valved fluid flow [9,10], thrust faults in geology [11][12][13], and, of course, the typical example of mixing a deck of cards [14][15][16]. In one dimension, cutting-and-shuffling is described mathematically by interval exchange transforms (IETs) [17][18][19][20][21][22][23][24][25][26][27][28] which naturally extend to higher dimensions under the framework of piecewise isometries (PWIs). PWIs, which cut an object into pieces and spatially rearrange them to form the original shape, can produce complex dynamics despite their relative simplicity [29][30][31][32][33][34][35][36][37].…”