Walking droplet interactions and stability

宏观物理学是确定性的，其可预测性仅受混沌入侵的限制。相反，微观物理学被广泛认为本质上是概率性的，量子力学的统计预测是对微观世界的完整描述。十年前，伊夫·库德（Yves Couder）和他在巴黎的同事发现了第一个量子模拟系统。该系统由毫米大小的液滴组成，这些液滴沿着振动液浴的表面自我推进，代表了路易斯·德布罗意在20世纪20年代为微观量子粒子提出的导波力学的宏观实现。这种流体动力系统表现出类似量子的统计行为，这一事实提出了一个问题，即微观世界和宏观世界之间是否真的存在质的区别。研究人员通过研究这种流体动力系统作为量子模拟物的潜力和局限性来探索这个问题。研究生也参与了该项目的工作。行走液滴系统提出了许多引人入胜的新数学问题，涉及波粒系统的动力学和统计学。在波粒系统中，粒子产生波场，然后响应波场而运动。研究者和共同研究者已经开发了一个理论框架来描述这个系统，这是一个越来越复杂的数学模型的层次结构，已经仔细地对实验进行了基准测试。该项目旨在通过分析、扩展和调整这些模型来合理化单液滴和多液滴静态和动态状态的稳定性。具体地，我们分析了模型方程的稳定解和周期解的稳定性。特别关注的是合理化大小相等或不等的多个液滴、对和晶格之间的相互作用，这些液滴已知表现出各种新的动态束缚态和不稳定性。通过能量分析，通过评估模型方程可达解的全球能量的相对大小，寻求单个和多个液滴行为的新原理。理论发展受到麻省理工学院应用数学实验室并行进行的支持性实验项目的指导和约束。

BushDMS-1614043 Macroscopic physics is deterministic, with predictability limited only by the intrusion of chaos. Conversely, microscopic physics is widely thought to be intrinsically probabilistic, the statistical predictions of quantum mechanics a complete description of the microscopic world. A decade ago, Yves Couder and co-workers in Paris discovered the first quantum analog system. The system, consisting of millimeter-sized droplets self-propelling along the surface of a vibrating fluid bath, represents a macroscopic realization of the pilot-wave mechanics proposed by Louis de Broglie in the 1920s for microscopic quantum particles. The fact that this hydrodynamic system exhibits quantum-like statistical behavior raises the question as to whether there really is a qualitative difference between the microscopic and macroscopic worlds. The investigators explore this question by studying the potential and limitations of this hydrodynamic system as a quantum analogue. Graduate students are involved in the work of the project. The walking-droplet system raises a number of fascinating new mathematical questions concerning the dynamics and statistics of wave-particle systems in which a particle generates a wave field, then moves in response to it. The investigator and co-investigator have developed a theoretical framework for describing this system, a hierarchy of mathematical models of increasing sophistication that have been carefully benchmarked against experiment. The project is directed towards rationalizing the stability of single- and multiple-droplet static and dynamic states by analyzing, extending and adapting these models. Specifically, we analyze the stability of both steady and periodic solutions to the model equations. Particular attention is given to rationalizing the interactions of multiple droplets of equal or unequal size, pairs and lattices, which are known to exhibit a variety of novel dynamical bound states and instabilities. New rationale for the behavior of single and multiple droplets is sought through an energetic analysis, by assessing the relative magnitudes of the global energies of the accessible solutions of the model equations. Theoretical developments are guided and constrained by a supporting experimental program conducted in parallel in MIT's Applied Math Laboratory.