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

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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.