Predicting mixing via resonances: Application to spherical piecewise isometries

We present an analytic method to find the areas of nonmixing regions in orientation-preserving spherical piecewise isometries (PWIs), and apply it to determine the mixing efficacy of a class of spherical PWIs derived from granular flow in a biaxial tumbler. We show that mixing efficacy has a complex distribution across the protocol space, with local minima in mixing efficacy, termed resonances, that can be determined analytically. These resonances are caused by the interaction of two mode-locking-like phenomena.

Figure 1

The BST PWI for ${\theta }_z={\theta }_x=\pi /4$. The solid blue, black, and red curves show where cutting occurs, and the dashed lines show the rotation axes. Adapted with permission from Park et al. [18] (©2016 AIP Publishing).

Figure 2

Bottom view of the action of the BST PWI. The HS is cut along the curves ${\mathcal{D}}_{1-3}$ shown in (a) and recombined as shown in (b).

Figure 3

Exceptional sets (gray) for the BST PWI with the cutting lines ${\mathcal{D}}_{1-3}$ shown red, green, and blue, respectively. Cells with base periodic itinerary $P_4\to P_1\to P_1$ $\left(411={41}^2\right)$ are shown in dark red, and the conjugate pair with base periodic itinerary $P_3\to P_2\to P_1$ (321) is shown in light blue. (a) For ${\theta }_z={\theta }_x=4\pi /15$ cells are circular and $\overline{E}$ has positive area. (b) For ${\theta }_z={\theta }_x=arccos[(-1+\sqrt{5})/2]$ the union of the regular pentagonal cells (dark red, light blue), the irregular quadrilateral cells (orange with itinerary ${2131}^2)$, and the irregular triangular cells (white with itineraries ${31}^2{41}^2{21}^3$ and ${32131}^{4}21)$ perfectly tile the HS, which means $\overline{E}$ has zero area. (c) For $({\theta }_z,{\theta }_x)=(0.9960,0.5748)$, both circular and polygonal cells exist, and $\overline{E}$ has finite area. Irregular quadrilateral cells are shown orange, cells with base itinerary ${41}^2{21}^2$ are shown light red, and their conjugate with base itinerary ${321}^{2}21$ is shown in dark blue.

Figure 4

Characterization of the area of the exceptional set by domain discretization. (a) The exceptional set for $({\theta }_z,{\theta }_x)=(\pi /2,\pi /4)$. (b) Unfolded isocube half and isocube half mapped onto the HS, with $N=12\times 2^3\times 2^3$ boxes. (c) Isocube half with grid shown in gray and boxes containing a point in the exceptional set colored black. (d) The same as (c) except $N=12\times 2^6\times 2^6$ boxes are used, the isocube grid is not shown. (Adapted with permission from Park et al. [19], ©2017 American Physical Society.)

Figure 5

(a) Distribution of $\mathrm{\Phi }$ across the protocol space, sampled at increments of $\pi /1800$ in ${\theta }_z,{\theta }_x$. $\mathrm{\Phi }$ is normalized such that zero coverage (white) corresponds to $\mathrm{\Phi }=0$, and complete coverage (black) corresponds to $\mathrm{\Phi }=1$. Lines of constant ${\theta }_x/{\theta }_z=tan\beta$ are shown dashed white for $\beta =\pi /\left(2m\right),m=3,4,5$. (b)–(e) Example exceptional sets corresponding to protocols indicated by blue arrows. (b), (e) Resonant protocols, i.e., local minima of $\mathrm{\Phi }$, where ${\theta }_z,{\theta }_x$ are as in Figs. 3 and 3. (c), (d) Protocols with $\mathrm{\Phi }\sim 1$: (c) $({\theta }_z,{\theta }_x)=(1.25,0.93)$; (d) $({\theta }_z,{\theta }_x)=(0.8,0.64)$.

Figure 6

Exceptional sets (gray) in the limit as ${\theta }_x\to 0$. (a) ${\theta }_z=\pi /3$. (b) ${\theta }_z=\pi /\pi$.

Figure 7

(a) Radius of cells with base itinerary ${41}^2,r_{{41}^2}$, from Eq. (16). The cell does not exist in the gray region beyond the solid red, green, and white annihilation boundaries, which satisfy Eqs. (21, 22, 23) (denoting $d(\mathit{x},\mathcal{C})$ by $d_{\mathcal{C}}$ for short). The dashed red and green, red and white, and green and white curves represent protocols where the center of the cell is equidistant to two of the boundaries, given by Eqs. (17, 18, 19), e.g., cell centers for protocols along the red and green curve are equidistant to ${\mathcal{D}}_1$ and ${\mathcal{D}}_2$ $\left(d_{{\mathcal{D}}_1}=d_{{\mathcal{D}}_2}\right)$. The protocol where all three dashed curves intersect, ${\theta }_z={\theta }_x={\theta }^{*}=arccos[(-1+\sqrt{5})/2]\approx 0.9046\approx 51.{83}^{\circ }$, has maximal cell radius [see Fig. 3]. Exceptional sets for the white-outlined black points on the blue-sided square with side length $\pi /9$ centered on the maximal protocol are shown in Fig. 8. (b) The distribution of $\mathrm{\Phi }$ from Fig. 5 overlayed with the annihilation boundaries (solid) and equidistance curves (dashed) for the ${41}^2$ itinerary.

Figure 8

Exceptional sets for the protocols marked by white-outlined black circles on the blue-sided square in Fig. 7. Cells with base itinerary ${41}^2$ are dark red; conjugate cells with base itinerary 321 are light blue. The cell with itinerary ${41}^2$ in $P_4$ touches one boundary in (a), (b), (f), (i); two boundaries in (c), (d), (h); and all three boundaries in (e), corresponding to the resonance ${\theta }_z={\theta }_x={\theta }^{*}=arccos[(-1+\sqrt{5})/2]$. Note that the cell with itinerary ${41}^2$ in $P_4$ is difficult to see when viewing the HS from below in (a) and (b), as it is close to the boundary $\partial S$ and has small radius. In (g) no cell with itinerary ${41}^2$ exists, period-4 cells with base itinerary ${41}^3$ are orange, and their conjugate cells (itinerary ${321}^2)$ are green; period-6 cells with base itineraries ${21}^2{31}^2$ and ${2131}^3$ are light red and dark blue, respectively. Protocols $({\theta }_z,{\theta }_x)$ are (a) $({\theta }^{*}-\pi /18,{\theta }^{*}+\pi /18)$, (b) $(0.8243,{\theta }^{*}+\pi /18)$ (approximate values correspond to intersections between equidistant curves and the blue-sided square in Fig. 7), (c) $({\theta }^{*}+\pi /18,{\theta }^{*}+\pi /18)$, (d) $({\theta }^{*}-\pi /18,0.9607)$, (e) $({\theta }^{*},{\theta }^{*})$, (f) $({\theta }^{*}+\pi /18,0.8243)$, (g) $({\theta }^{*}-\pi /18,{\theta }^{*}-\pi /18)$, (h) $(0.9607,{\theta }^{*}-\pi /18)$, (i) $({\theta }^{*}+\pi /18,{\theta }^{*}-\pi /18)$.

Figure 9

Distribution of $\mathrm{\Phi }$ from Fig. 5 shown with annihilation boundaries (solid) and equidistance curves (dashed) for 10 low-period itineraries (different colors). The resonances (local minima in $\mathrm{\Phi })$ occur at protocols where all three equidistance curves of an itinerary meet, and are labeled for each itinerary by dotted arrows.

Figure 10

Exceptional sets of the BST PWI for resonant protocols. For each resonance the corresponding itinerary that is tangent to three boundaries is indicated, its cells are colored dark red, and the cells of the conjugate itinerary are colored light blue. The ${\theta }_z^{*}$ axis represents the rational multiple of $\pi$ that the resonance is attached to via the tongues, and $m$ is the wrapping multiplicity of the resonant itinerary.

Figure 11

(a) Distribution of $\mathrm{\Phi }$ (Fig. 5) with regions containing cells corresponding to the itineraries in Fig. 9 colored white (and their reflection across the line ${\theta }_z={\theta }_x)$. Annihilation boundaries for the itineraries are shown in gray. (b), (c) Exceptional sets outside the annihilation boundaries that are predicted to produce a high degree of mixing. (b) $({\theta }_z,{\theta }_x)=(1.25,0.93)$. (c) $({\theta }_z,{\theta }_x)=(0.8,0.64)$.

Figure 12

The exceptional set for the BST PWI with ${\theta }_z={\theta }_x=4\pi /15$. Points in $P_1–P_4$ are colored gray, green, blue, and red, respectively. The image of $P_4$ under the rotation $R_{{\theta }_x}^{\mathit{u}}$, where $\mathit{u}=R_{{\theta }_z}^z(-1,0,0)$, is shown as the red points in $P_3$ (blue), illustrating the conjugacy between $P_4$ and $P_3$. The image of $P_4$ under $M_{{\theta }_z,{\theta }_x}{R}_{{\theta }_x}^{\mathit{u}}$ is shown as the red points contained in $P_2$ (green). The black dashed symmetry line $z=-x$ corresponding to the reflection-reversal symmetry (A5) yields symmetries in the $P_3$ and $P_4$ atoms also.

Figure 13

The BST PWI in the limit as ${\theta }_z,{\theta }_x\to 0$ with constant ratio ${\theta }_x/{\theta }_z=tan\beta$. The rotation axis $z=tan(\pi /2+\beta )x$ is indicated by the dashed black line. The segments of $\partial S$ where particles experience the different periodic boundary conditions described in Eq. (C4) are colored red, green, purple, and blue, respectively. Example particle trajectories that meet the red and green segments of $\partial S$ are shown in orange and light blue, starting from the points marked with circles. Note that when the light blue point meets the green segment of $\partial S$, it has two images due to the multivalued periodic boundary conditions.

Figure 14

Particle trajectories in the limit as ${\theta }_z,{\theta }_x\to 0$ and at small values of ${\theta }_z,{\theta }_x$, for constant ratios ${\theta }_x/{\theta }_z=tan\beta$. The first column shows the trajectories of four particles with streamlines that meet the boundary at $\theta =\beta ,3\beta /4,\beta /2,0$ colored blue, green, orange, and red, respectively. In the other columns, the same particles are tracked for ${\theta }_z$ as shown and ${\theta }_x=tan\left(\beta \right){\theta }_z$, combined with the exceptional set shown in gray.

Figure 15

The exceptional set (gray) for the BST PWI with $({\theta }_z,{\theta }_x)=(1.012,0.3796)$. The cells with base itineraries ${41}^2,{41}^2{21}^2,{41}^2{\left({21}^2\right)}^2$ and their conjugates are colored. The great circle ${\mathcal{C}}^{*}$ that passes through the centers of all the cells in $P_4$ and the point $\mathit{u}$ where ${\mathcal{D}}_{1-3}$ meet is shown in orange.