Fluid-stochastic models to describe the dynamics of the Faraday pilot waves in the long memory regime

对于适当振动的流体浴，相同流体的小液滴将由于与由先前液滴浴冲击产生的波的推进相互作用而“行走”穿过表面。当熔池以足够大的（亚临界的）振幅垂直振动时，液滴将周期性地反弹。弹跳的液滴不与浴接触，而是由于滞留在浴和液滴之间的空气润滑层的缓冲作用而被推回到空气中，仅在微观尺度上可见。在每次撞击时，液滴触发由传播波和法拉第波组成的波场。随着强迫振幅的增加，触发波的振幅增加，液滴不稳定，并在水平方向上接收“踢”。这将导致液滴沿曲面沿着移动。增加强迫振动将增加法拉第波的衰减时间，从而产生对先前液滴撞击的较长路径“记忆”。研究的主要对象，法拉第导频波是由液滴（粒子）和相关的法拉第波组成的对。它们的动力学是复杂的（混沌的）和非本地的时间（即在系统中有记忆）。本研究的目标是开发混合流体随机模型来描述长记忆机制下的法拉第导频波的动力学行为。这将使人们能够理解某些实验上发现的流体动力学量子类似物，例如当粒子被限制在围栏中时的类波统计，当粒子被谐波势限制时的双量子化，穿过障碍的隧道效应，以及单缝和双缝衍射。特别是，一个重要的开放问题是控制粒子概率分布的方程。研究将包括：1.模拟法拉第导频波场;2.系统的一维和三维分析; 3.数值波模拟; 4.系统的随机模拟; 5.随机微分方程分析，包括McKean-Vlasov过程; 6.随机模拟;监督小组由Paul Milewski教授和Tim Rogers教授组成，Paul Milewski教授是流体和连续过程物理建模以及波动动力学方面的专家，Tim Rogers教授是随机微分方程和随机建模方面的专家。两人都在各自的领域拥有计算专业知识。他们一起跨越了项目所需的专业知识。

For a suitably vibrating bath of fluid, a small droplet of the same fluid will "walk" across the surface due to the propulsive interactions with the waves generated by the previous droplet-bath impacts. As the bath vertically vibrates at a sufficiently large (yet subcritical) amplitude, the droplet will bounce periodically. The bouncing droplet does not make contact with the bath, it is instead propelled back into the air due to the cushioning effect of the lubrication layer of air trapped between the bath and droplet visible only on a microscopic scale. At each impact the droplet triggers a wavefield consisting of propagating and Faraday waves. As the forcing amplitude increases, the triggered waves increase in amplitude and the droplet destabilises and receives a "kick" in the horizontal direction. This results in the drop walking along the surface. Increasing the forcing vibration will increase the Faraday wave's decay time yielding a longer path "memory" from previous drop impacts.The main object of study, a Faraday pilot wave is the pair consisting of the droplet (particle) and the associated Faraday wave. Their dynamics are complex (chaotic) and non-local in time (i.e. there is memory in the system). The goal of the present research is to develop hybrid fluid-stochastic models to describe the dynamics of the Faraday pilot waves in the long memory regime. This will enable the understanding of certain hydrodynamic quantum analogues that have been experimentally discovered, such as the wavelike statistics when particles are confined to a corral, double-quantization when particles are confined by a harmonic potential, tunnelling across barriers, and single- and double-slit diffraction. In particular, an important open problem is the equation governing the probability distribution for particles.The research will involve:1.Modelling the Faraday pilot wavefield;2.One-dimensional and three-dimensional analysis of the system;3.Numerical wave simulations;4.Stochastic modelling of the system;5.Stochastic Differential Equation (SDE) analysis including McKean-Vlasov processes;6.Stochastic simulations;The supervisory team consists of Prof. Paul Milewski who is an expert on physical modelling of fluid and continuum processes and on wave dynamics, and Prof. Tim Rogers who is an expert on stochastic differential equations and stochastic modelling. Both have computational expertise in their fields. Together they span the expertise needed for the project.