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

对于一个适当振动的流体浴，由于与先前的液滴-浴碰撞产生的波的推进相互作用，同一流体的小液滴将“行走”在表面上。当浴液以足够大的（但非临界的）振幅垂直振动时，液滴将周期性地反弹。弹跳的液滴不会与浴槽接触，而是被推回到空气中，这是由于在浴槽和液滴之间的空气润滑层的缓冲作用，只有在微观尺度上才能看到。每次撞击时，液滴都会触发一个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，并在水平方向上受到“踢”。这导致液滴沿着表面移动。增加强迫振动将增加法拉第波的衰减时间，从而从先前的跌落冲击中产生更长的路径“记忆”。法拉第导波的主要研究对象是由液滴（粒子）和相关的法拉第波对组成的导波。它们的动力学是复杂的（混沌的），在时间上是非局部的（即系统中存在记忆）。本研究的目的是建立混合流体-随机模型来描述法拉第导波在长记忆状态下的动力学。这将使我们能够理解已经在实验中发现的某些流体力学量子类似物，例如当粒子被限制在畜栏中时的波状统计，当粒子被谐波势限制时的双量子化，穿越障碍的隧道，以及单缝和双缝衍射。特别地，一个重要的开放问题是控制粒子概率分布的方程。研究将涉及：1.；2.法拉第导频波场的建模；2 .系统的一维和三维分析；3 .数值波模拟；4 .系统的随机建模；5 .随机微分方程（SDE）分析，包括McKean-Vlasov过程；随机模拟;监督小组由流体和连续过程的物理建模专家Paul Milewski教授和随机微分方程和随机建模专家Tim Rogers教授组成。他们都在各自的领域拥有计算专业知识。它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.