Cutting and Shuffling: A New Dynamical Systems Paradigm for Mixing

工程师和科学家通常认为有三种方法来混合流体，这可以很容易地解释为在咖啡中混合奶油。扩散通过分子碰撞使一杯咖啡中的奶油分散；混乱的平流或搅拌产生了薄薄的一层咖啡和奶油，它们融合在一起；湍流通过强烈的搅拌或剧烈的搅拌使咖啡和奶油混合在一起。然而，简单的切割和洗牌技巧，就像洗牌或混合魔方的颜色一样，也是一种常见的混合方法。然而，在工程系统的背景下，如混合粉末和颗粒的药品，塑料树脂和化学品，它不是很好地理解。该奖项支持正式数学框架的发展，以预测通过切割和洗牌的混合，并得到计算和实验结果的验证。该理论受到抽象数学概念的启发，但其动机是混合和混合大块颗粒或颗粒的系统，这是化学和制药工业的常见过程。这项研究将基础数学与实际应用联系起来，同时培养下一代科学家和工程师。虽然分子扩散、混沌平流和湍流作为混合机制已经研究了很长时间，但切割和洗牌混合尚未得到很好的探索或理解。该奖项的中心是一种相对较新的数学方法，它描述了被称为分段等距（PWI）的切割和洗牌。其想法是将PWI概念应用于混合，并将PWI与传统的动力系统方法联系起来。目标是在动态系统框架内开发和利用PWI，通过在2D和3D几何形状中切割和洗牌来预测混合，并最终应用于实际混合系统。三维动力系统框架的发展将受到颗粒材料混合的影响，导致一个由抽象数学概念启发的综合理论，但与切割和洗刷的实际应用有关。PWI的数学将被整合到动力系统工具包中，而互补的实验和模拟将被用来证实理论方法对模型颗粒混合系统的适用性。新的数学、动力系统方法和物理应用的融合可能会产生一个全新的理解和预测混合的范例。

Engineers and scientists generally think in terms of three means to mix fluids, which can be easily explained in terms of mixing cream in coffee. Diffusion disperses the cream in a cup of coffee via molecular collisions; chaotic advection, or stirring, generates thin layers of coffee and cream that merge; turbulence mixes the coffee and cream by strong agitation or vigorous stirring. However, the simple technique of cutting and shuffling, much like what takes place in shuffling a deck of cards or mixing the colors of a Rubik's cube, is also a means of mixing that is commonplace. However, it is not well understood in the context of engineering systems such as mixing powders and granules for pharmaceuticals, plastic resins, and chemicals. This award supports the development of a formal mathematical framework to predict mixing by cutting and shuffling that is validated by computational and experimental results. The theory is inspired by abstract mathematical concepts, but is motivated by systems for blending and mixing bulk particles or granules, a common process in the chemical and pharmaceutical industries. The research links fundamental mathematics and practical applications while training the next generation of scientists and engineers.While molecular diffusion, chaotic advection, and turbulence as mixing mechanisms have long been studied, mixing by cutting and shuffling is not well explored or understood. This award centers on a relatively new mathematical approach that describes cutting and shuffling known as Piecewise Isometries (PWI). The idea is to apply PWI concepts to mixing as well as to connect PWI with traditional dynamical systems approaches. The objective is to develop and utilize PWI within a dynamical systems framework to predict mixing by cutting and shuffling in 2D and 3D geometries with eventual applications to practical mixing systems. The development of the 3D dynamical systems framework will be informed by the mixing of granular materials, leading to an integrated theory inspired by abstract mathematical concepts but connected to practical applications of cutting and shuffling. The mathematics of PWI will be integrated into the dynamical systems toolkit, while complementary experiments and simulations will be used to confirm the applicability of the theoretical approaches to a model granular mixing system. The merging of new mathematics, dynamical systems approaches, and physical applications could result in an entirely new paradigm for understanding and predicting mixing.