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

Ergodicity bounds for stable Ornstein-Uhlenbeck systems in Wasserstein distance with applications to cutoff stability

This article establishes cutoff stability also known as abrupt thermalization for generic multidimensional Hurwitz stable Ornstein-Uhlenbeck systems with (possibly degenerate) Lévy noise at fixed noise intensity. The results are based on several ergodicity quantitative lower and upper bounds some of which make use of the recently established shift linearity property of the Wasserstein-Kantorovich-Rubinstein distance by the authors. It covers such irregular systems like Jacobi chains and more general networks of coupled harmonic oscillators with a heat bath (including Lévy excitations) at constant temperature on the outer edges and the so-called Brownian gyrator.

Identifying invariant ergodic subsets and barriers to mixing by cutting and shuffling: Study in a birotated hemisphere

Mixing by cutting and shuffling can be mathematically described by the dynamics of piecewise isometries (PWIs), higher dimensional analogs of one-dimensional interval exchange transformations. In a two-dimensional domain under a PWI, the exceptional set, E[over ¯], which is created by the accumulation of cutting lines (the union of all iterates of cutting lines and all points that pass arbitrarily close to a cutting line), defines where mixing is possible but not guaranteed. There is structure within E[over ¯] that directly influences the mixing potential of the PWI. Here we provide computational and analytical formalisms for examining this structure by way of measuring the density and connectivity of ɛ-fattened cutting lines that form an approximation of E[over ¯]. For the example of a PWI on a hemispherical shell studied here, this approach reveals the subtle mixing behaviors and barriers to mixing formed by invariant ergodic subsets (confined orbits) within the fractal structure of the exceptional set. Some PWIs on the shell have provably nonergodic exceptional sets, which prevent mixing, while others have potentially ergodic exceptional sets where mixing is possible since ergodic exceptional sets have uniform cutting line density. For these latter exceptional sets, we show the connectivity of orbits in the PWI map through direct examination of orbit position and shape and through a two-dimensional return plot to explain the necessity of orbit connectivity for mixing.

Cutting and shuffling a hemisphere: Nonorthogonal axes

We examine the dynamics of cutting-and-shuffling a hemispherical shell driven by alternate rotation about two horizontal axes using the framework of piecewise isometry (PWI) theory. Previous restrictions on how the domain is cut-and-shuffled are relaxed to allow for nonorthogonal rotation axes, adding a new degree of freedom to the PWI. A new computational method for efficiently executing the cutting-and-shuffling using parallel processing allows for extensive parameter sweeps and investigations of mixing protocols that produce a low degree of mixing. Nonorthogonal rotation axes break some of the symmetries that produce poor mixing with orthogonal axes and increase the overall degree of mixing on average. Arnold tongues arising from rational ratios of rotation angles and their intersections, as in the orthogonal axes case, are responsible for many protocols with low degrees of mixing in the nonorthogonal-axes parameter space. Arnold tongue intersections along a fundamental symmetry plane of the system reveal a new and unexpected class of protocols whose dynamics are periodic, with exceptional sets forming polygonal tilings of the hemispherical shell.