Walking droplet interactions and stability

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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.