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。 扩散通过分子碰撞将奶油分散在一杯咖啡中;混乱的平流或搅拌产生薄薄的咖啡和奶油层，然后合并;湍流通过强烈搅拌或剧烈搅拌将咖啡和奶油混合。 然而，剪切和洗牌的简单技术，就像洗牌一副牌或混合魔方的颜色一样，也是一种常见的混合手段。 然而，在工程系统的背景下，如混合药物，塑料树脂和化学品的粉末和颗粒，它还没有得到很好的理解。 该奖项支持开发一个正式的数学框架，通过计算和实验结果验证的切割和洗牌来预测混合。 该理论的灵感来自抽象的数学概念，但其动机是用于混合和混合散装颗粒或颗粒的系统，这是化学和制药行业中的常见过程。 这项研究将基础数学和实际应用联系起来，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，但通过切割和洗牌进行的混合还没有得到很好的探索或理解。 这个奖项集中在一个相对较新的数学方法，描述了切割和洗牌称为分段等距（PWI）。 我们的想法是将PWI概念应用于混合，并将PWI与传统的动力系统方法连接起来。 我们的目标是开发和利用PWI的动态系统框架内预测混合切割和洗牌在2D和3D的几何形状，最终应用到实际的混合系统。 3D动力系统框架的发展将通过颗粒材料的混合来获得信息，从而形成一个受抽象数学概念启发的综合理论，但与切割和洗牌的实际应用有关。PWI的数学将被集成到动力系统工具包中，而补充实验和模拟将用于确认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导致理解和预测混合的全新范式。

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.