Could such a setup be used to create a microfluidic “in-a-droplet” chemical reactor mixer ?

Perrard, S., Labousse, M., Miskin, M., Fort, E., & Couder, Y. Effets de quantification d’une association onde-particule soumise à une force centrale.Résumés des exposés de la 16e Rencontre du Non-Linéaire Paris 2013, 68.

http://nonlineaire.univ-lille1.fr/SNL/media/2012/CR/Perrard.pdf

eigenstates in circular cavity

Where it all started …

http://www.feynmanlectures.info/docroot/III_01.html

Bach, R., Pope, D., Liou, S. H., & Batelaan, H. (2013). Controlled double-slit electron diffraction. New Journal of Physics, 15(3), 033018.

http://iopscience.iop.org/1367-2630/15/3/033018/pdf/1367-2630_15_3_033018.pdf

And some movies of the interference pattern build-up :

http://iopscience.iop.org/1367-2630/15/3/033018/media

The famous Feynman thought experiment reproduced ! ((cf. Feynman Lectures on Physics, vol III, figures 1–3,))

Crommie, M. F., Lutz, C. P., & Eigler, D. M. (1993). Confinement of electrons to quantum corrals on a metal surface. Science, 262(5131), 218-220.

http://users.phys.psu.edu/~rick/MATH/LECTURES/LECT13/PDF/eigler_circular_corral.pdf

Harris, D. M., & Bush, J. W. (2014). Droplets walking in a rotating frame: from quantized orbits to multimodal statistics. Journal of Fluid Mechanics, 739, 444-464.

We present the results of an experimental investigation of a droplet walking on the
surface of a vibrating rotating fluid bath. Particular attention is given to demonstrating
that the stable quantized orbits reported by Fort et al. (Proc. Natl Acad. Sci.,
vol. 107, 2010, pp. 17515–17520) arise only for a finite range of vibrational
forcing, above which complex trajectories with multimodal statistics arise. We first
present a detailed characterization of the emergence of orbital quantization, and
then examine the system behaviour at higher driving amplitudes. As the vibrational
forcing is increased progressively, stable circular orbits are succeeded by wobbling
orbits with, in turn, stationary and drifting orbital centres. Subsequently, there is a
transition to wobble-and-leap dynamics, in which wobbling of increasing amplitude
about a stationary centre is punctuated by the orbital centre leaping approximately
half a Faraday wavelength. Finally, in the limit of high vibrational forcing, irregular
trajectories emerge, characterized by a multimodal probability distribution that reflects
the persistent dynamic influence of the unstable orbital states.

http://math.mit.edu/~bush/wordpress/wp-content/uploads/2014/01/HB-JFM-2014.pdf

On this video, you’ll see :

– How 2 dotwaves can synchronize on 2 orbits
– How a dotwave can change his orbit (with a little help from the experimentator)

Video shot with a 30 Hz camera, at a forcing frequency of 60 Hz, hence not much stroboscopic flickering

The bath is excited just at the Faraday llevel, or slighly upper.

Robert Brady, Ross Anderson. Jan 16, 2014.

http://arxiv.org/pdf/1401.4356v1.pdf

review of all analogies between dotwaves mechanics and quantum mechanics.

On this video, the walking droplet is confined in a small cavity (Approx. 6 to 8 times the faraday wavelength)

We observe the emergence of several normal modes of vibration of a walking droplet in a circular corral.

At first, the movement seems “random”, then, the whole wave-particle system synchronizes, and starts turning alltogether.

Differrent modes are possible, with different radius for the droplet trajectory : on this video we see a mode with the droplet on a large radius trajectory, and also a mode where the droplet is on a short radius trajectory

Its seems that thoses radius coincide with with the knots ofs thefaraday standing waves of the cavity, hence we have a quantization of the radius of the different trajectories.

A Transparent plate is mounted  (via magnets) to three loud peakers serial-connected to an amplifier delivering a sinus signal.

Silicon oil ( with viscosuity 20 Cst)  is put on the transparent plate.

A  point source light is collimated, goes through the oil

A mirror placed below redirects the light beam to tha camera.

NB : it is not easy to set up this light system : the light source muste precisely tuned at the focal point of the lens.

When a walking droplet is trapped in a small box, the waves that it has generated extends to the whole frame.

Thus those waves  can resonate  and form normal modes of vibration inside the cavity.

The droplet is then forced to move according to this normal wave field

A Dotwave after it has bounced on a obstacle.

Two DotWaves interact at a distance of  15 faraday wavelength

Sbitnev, V. I. (2013). Droplets moving on a fluid surface: interference pattern from two slits. arXiv preprint arXiv:1307.6920.

http://arxiv.org/pdf/1307.6920.pdf

Feynman path integral approach

Droplet / Antidroplet

Soliton diffraction

Bohm, D. (1952). A suggested interpretation of the quantum theory in terms of” hidden” variables. I. Physical Review, 85(2), 166.

http://www.psiquadrat.de/downloads/bohm52a.pdf

Hidden variable theory foundation

Oza, A. U., Rosales, R. R., & Bush, J. W. (2013). A trajectory equation for walking droplets: hydrodynamic pilot-wave theory. Journal of Fluid Mechanics,737, 552-570.

http://math.mit.edu/~bush/wordpress/wp-content/uploads/2013/12/ORB-JFM.pdf

Integro-differential equation describing the horizontal motion of a walking droplet

Stability to perturbations

Moláček, J., & Bush, J. W. (2013). Drops bouncing on a vibrating bath. Journal of Fluid Mechanics, 727, 582-611.

http://math.mit.edu/~bush/wordpress/wp-content/uploads/2013/07/MB1-2013.pdf

Linear and logarithmic  spring model of a boucing droplet

Wind-Willassen, Ø., Moláček, J., Harris, D. M., & Bush, J. W. (2013). Exotic states of bouncing and walking droplets. Physics of Fluids, 25, 082002.

http://windw.dk/2013Bouncing.pdf

Phase diagram refinement : the different styles of bouncing

Moláček, J., & Bush, J. W. (2013). Drops walking on a vibrating bath: towards a hydrodynamic pilot-wave theory. Journal of Fluid Mechanics, 727, 582-617.

http://math.mit.edu/~bush/wordpress/wp-content/uploads/2013/07/MB2-2013.pdf

The most advanced theoretical model