Deceleration of one-dimensional mixing by discontinuous mappings

We present a computational study of a simple one-dimensional map with dynamics composed of stretching, permutations of equally sized cells, and diffusion. We observe that the combination of the aforementioned dynamics results in eigenmodes with long-time exponential decay rates. The decay rate of the eigenmodes is shown to be dependent on the choice of permutation and changes nonmonotonically with the diffusion coefficient for many of the permutations. The global mixing rate of the map $M$ in the limit of vanishing diffusivity approximates well the decay rates of the eigenmodes for small diffusivity, however this global mixing rate does not bound the rates for all values of the diffusion coefficient. This counterintuitively predicts a deceleration in the asymptotic mixing rate with an increasing diffusivity rate. The implications of the results on finite time mixing are discussed.

Figure 1

The action of the baker's map (5) is to contract the $x$ direction, stretch the $y$ direction, and reassemble onto the unit torus. Taking a $y$ independent initial condition, here $c(x,y)=\mathrm{gray}$ for $x<1/2$ and $c(x,y)=\mathrm{white}$ for $x\ge 1/2$, the baker's map is reduced to a one-dimensional map on the unit interval.

Figure 2

The initial stages of variance decay for representative permutations. Permutations which introduce new interface $S_5\setminus S_5^R$ deplete the variance quicker than the identity of rotational permutations $S_5^R$ in the initial three iterations. The diffusion coefficient is $\kappa ={10}^{-5}$ with the initial condition $c^{\left(0\right)}\left(x\right)=cos\left(2\pi x\right)$.

Figure 3

Variance decay profiles for representative permutations on a linear-log plot. The dashed lines show the variance decay predicted by the second leading eigenvalues ${\lambda }_2$. The diffusion coefficient is $\kappa ={10}^{-5}$ with the initial condition $c^{\left(0\right)}\left(x\right)=cos\left(2\pi x\right)$.

Figure 4

The image shows the first few iterations for the composed map with $\sigma \circ M_B$ with $\sigma =\left(34\right),\kappa ={10}^{-4}$ and the initial condition $c^{\left(0\right)}\left(x\right)=cos\left(2\pi x\right)$. The color scale is adjusted at each iteration to clearly show the persistent pattern emerging.

Figure 5

$|{\lambda }_2|$ is plotted against $\kappa$ for a number of permutations. Convergence in the limit $\kappa \to 0$ occurs for some of the permutations, whereas $|{\lambda }_2|$ changed nonmonotonically for others. The dashed lines represent theoretical upper and lower bounds in the limit $\kappa =0$. The symbols are used to distinguish the profiles and do not represent data points.

Figure 6

Graphical representations of (a) a one-dimensional approximation of the baker's map, the halving map, (b) the halving map composed with the permutation $\sigma =\left(15243\right)$, denoted $(\sigma \circ M_B)$, and (c) the preimage map ${(\sigma \circ M_B)}^{-1}$ is shown with an appropriate Markov partition.

Figure 7

The variance $\psi \left(j\right)/\psi \left(0\right)$ is plotted for $\sigma \circ M_B$ with $\sigma =\left(2354\right)$, initial condition $c^{\left(0\right)}\left(x\right)=cos\left(2\pi x\right)$ for three diffusivity values. The linear-log plot is used to better distinguish the iterates in which $\psi \left(j\right)/\psi \left(0\right)$ crosses the 95% threshold, represented by the the dashed line. $\psi \left(j\right)/\psi \left(0\right)$ crosses the threshold at the earliest time when $\kappa ={10}^{-5}$, the smallest value.

Figure 8

The $L_t^{\infty }/L_0^{\infty }$ norm is plotted for $\sigma \circ M_B$ with $\sigma =\left(2354\right)$, initial condition $c^{\left(0\right)}\left(x\right)=cos\left(2\pi x\right)$ for three diffusivity values. The linear-log plot is used to better distinguish the iterates in which $L_t^{\infty }/L_0^{\infty }$ crosses the 95% threshold, represented by the dashed line. Similar to the variance, $L_t^{\infty }/L_0^{\infty }$ crosses the threshold at the earliest time when $\kappa ={10}^{-5}$, the smallest value.