Cutting and Shuffling: A New Dynamical Systems Paradigm for Mixing

工程师和科学家通常从混合液体的三种方式来考虑，这可以很容易地用在咖啡中混合奶油来解释。扩散通过分子碰撞分散咖啡中的奶油；混乱的平流或搅拌产生薄层咖啡和奶油融合；湍流通过强烈的搅拌或猛烈的搅拌将咖啡和奶油混合。然而，简单的剪裁和洗牌技术，就像洗一副牌或混合魔方的颜色一样，也是一种常见的混合方法。然而，在诸如药品、塑料树脂和化学品的混合粉末和颗粒的工程系统的背景下，它并没有被很好地理解。该奖项支持开发一个正式的数学框架，通过切割和洗牌来预测混合，并通过计算和实验结果进行验证。该理论的灵感来自抽象的数学概念，但受到混合和混合散装颗粒或颗粒的系统的启发，这是化学和制药行业的常见过程。这项研究将基础数学和实际应用联系在一起，同时培养下一代科学家和工程师。尽管人们早就研究了分子扩散、混沌平流和湍流作为混合机制，但通过切割和洗牌进行混合还没有得到很好的探索和理解。该奖项集中在一种相对较新的数学方法上，该方法描述了切割和洗牌，称为分段等距线(PWI)。我们的想法是将PWI的概念应用于混合，并将PWI与传统的动力系统方法联系起来。其目标是在动力系统框架内开发和利用PWI，通过在2D和3D几何图形中切割和洗牌来预测混合，并最终应用于实际混合系统。3D动力系统框架的发展将以颗粒材料的混合为基础，导致一个受抽象数学概念启发但与切割和洗牌的实际应用相联系的综合理论。PWI的数学将被集成到动力系统工具包中，同时将使用补充的实验和模拟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.