Probability Distribution of a Pilot Wave Droplet

Transcription

1 WJP, PHY382 (2015) Wabash Journal of Physics v3.3, p.1 Probability Distribution of a Pilot Wave Droplet Badger, Bradon, Caddick, Jacob, and J. Brown Department of Physics, Wabash College, Crawfordsville, IN (Dated: March 16, 2015) It is known that droplets can walk on a vibrating silicon oil bath along a random path near a resonance of the surface wave and the droplet. Modifying the frequency and amplitude, we tracked a droplet s behavior using a vibrating bath setup filled with silicon oil. We were able to track walking droplets and form a probability distribution.

2 WJP, PHY382 (2015) Wabash Journal of Physics v3.3, p.2 I. INTRODUCTION The Pilot Wave Theory, proposed by Louis De Broglie and David Bohm is an attempt at a hidden variable theory in which the momentum and position of a particle is hidden to an observer[1]. This contrasts with Quantum Mechanics, which suggests that nature is inherently probabilistic as the wave equations used have no physical quantities, only probabilities, until they are observed or measured. The basic ideas of this Pilot Wave Theory can be replicated with fluid dynamics, where if one vibrates an oil bath at a specific amplitude and frequency, a walking droplet driven by a pilot wave acts as an analogous quantum particle. Studying the behavior of this particle could lead to a simpler model for explaining Pilot Wave Theory. This phenomenon of walking droplets has been studied for decades. It was first discovered by Walker in 1978, through fluid dynamics that a droplet could bounce on a vibrating bath at a wide range of frequencies[6]. Research shows that the hydrodynamic force on the droplet is responsible for the oil droplet generating a pilot wave when bouncing on the bath[2]. When the frequency and amplitude are set just right, they droplet and wave interact in such a way to make the droplet walk. This gives the quantum-like properties that were thought to be unique to quantum particles. II. MODEL The Faraday Threshold is described as the frequency and amplitude combination at which a vibrating bath experiences resonance. It is known that at a point just below this threshold, a droplet can be made to walk. Above this threshold, the bath becomes unstable and chaotic and is unable to support a walking droplet[5]. One could calculate the Faraday Threshold if desired, but for purposes of this experiment, we are able to adjust the amplitude or frequency to a proper value so that we can produce walking droplets. A team at MIT found that droplets in a 50 cst silicon oil bath are most likely to walk when the diameter of the droplet is 1.0 mm and the value of the ratio (unitless) between the peak bath acceleration and the gravitational acceleration, Γ is between 4.5 and 5.5[4]. It should be noted that Γ = ω2 0A g, (1) where ω 0 = 2πf is the frequency of oscillation, A is the amplitude of the corral, and g

3 WJP, PHY382 (2015) Wabash Journal of Physics v3.3, p.3 is the gravitational acceleration constant (9.8 m/s 2 ). It is not possible to correlate these values for the dimensionless peak bath acceleration directly to the behavior of droplets in arbitrary fluids[5]. To clarify, the fluid of the droplet is always separated from the fluid of the bath by a thin layer of air, as one can see in Fig.(1). This is a high pressure air packet that provides an upward reaction force once the droplet comes in contact with the surface. Researchers at the University of Liege find that if one examines the fundamental forces at play, one can begin to understand the wavelike statistics that appear in the pilotwave scenario. There are two forces that help us explain this: the hydrodynamic force and a quantum force. Although our system is not quantum mechanical, the system can be paralleled to an analogous quantum particle driven by a quantum force derived by its wave equation. In reality, the force interacting with the droplet from the fluid is Newtonian. The hydrodynamic force is proportional to the slope of the wave at the point that the droplet interacts with the wave[3]. Therefore, at points with a high slope, or points in between the peaks and troughs of the waves, the hydrodynamic force will be relatively large and point toward the trough of the wave, causing it to walk. Ultimately, we want to show that a droplet prefers some portions of the corral to others and therefore creates a distinct and familiar probability distribution, mimicking probability distributions from quantum mechanics. III. SETUP Our setup consists of a vibrating driver with a circular corral attached to it, 50 cst silicon oil, an amplifier, a DC power supply, and a function generator with range 0 to 10,000 Hz, which can be seen in Fig. (2a). Since viscosity is defined as the resistance to motion of a fluid, our system works better with an oil viscosity below 100 cst to allow for better interaction between the wave and droplet. The entire system must be level in order to reduce the variations in depth (and therefore variations in wave speed), so we adjusted the system by placing our driver on a plywood board with three points of adjustable height, creating a plane. We then slipped two small wooden blocks with an elevation of 16 degrees underneath the driver. The three points on the plywood board were then adjusted until the driver created concentric circles around the center of the corral. To reduce wind interference, we placed four sheets of plexiglass standing upright around the sides of the corral. Overhead, we placed a Microsoft LifeCam webcam (30fps) in a fixed position downward onto the corral

4 WJP, PHY382 (2015) Wabash Journal of Physics v3.3, p.4 Droplet Reaction Force Gravitational Force FIG. 1: Figure shows a droplet on a vibrating silicon oil bath. Notice the fluid of the droplet never touches the fluid of the wave. Before the droplet can touch the bath, a high pressure air pad provides a reaction force that sends it back into the air. Once the frequency of the fluid in the corral reaches a point just below the Faraday Threshold, the droplet begins to move horizontally, and can be considered a walking droplet.

5 WJP, PHY382 (2015) Wabash Journal of Physics v3.3, p.5 to track the position of the droplet. Our complete setup can be seen in Fig. (2b). IV. DATA To track the droplet in real-time, the computer read real-time frames from our camera and tracked the largest contour in the threshold difference image by taking the absolute difference between sequential frames. The coordinates of the largest difference is then presumed to be a moving droplet, so its coordinates and time from the start of tracking are recorded. We observed a circular path displayed by Fig. (3). As one can see, this droplet traveled in a circle around the corral at a radius from the center with little deviation. From inspection, this closely resembles the results that were gathered by the MIT group stated earlier. However, the MIT group finds necessity in tracking the particle above the Faraday Threshold, where movement is more random[3]. It is believed that this random motion helps to better show that the particle favors certain areas of the corral as opposed to what we have. One can see from a radial histogram (Fig. (4)), that the droplet clearly favors a certain region of the corral as opposed to anywhere else. This can be taken as a probability distribution for the position of the particle relative to the center of the corral. V. CONCLUSION AND FUTURE WORK After tracking a droplet, we can say from its radial histogram that its position is more probable to lie in a certain region of the corral. However, it is of interest to investigate the droplet s response to values of amplitude that correspond with the favorable Γ values we have stated in the model section. Also, in order to better conclude that a droplet favors an area, it would be worthwhile to measure the speed of the droplet in the corral as it experiences random walking. As Harris and Bush have found, the speed of the droplet inversely correlates with being located in a favorable region. Finally, since the pilot-wave experiment is analogous to quantum results, it would be ideal to replicate results of quantum experiments such as single and double slit diffraction. For example, one could place two barriers close together in a bath to resemble a single slit. Extrapolating from position and

6 WJP, PHY382 (2015) Wabash Journal of Physics v3.3, p.6 (a) 2a (b) 2b FIG. 2: Figure (a) shows a circuit diagram for our experiment setup. We have an oscilloscope set up to measure the amplitude and frequency of the final signal going to our bass speaker which drives our oil bath. A function generator controls the frequency and amplitude of the electrical signal sent to the vibrating driver (a commercial sub-woofer driven with a 400 W variable gain audio amplifier with a low-pass filter). Figure (b) is a schematic showing the driver- a speaker, the corral, and the camera we used to track the motion of walking droplets. The diameter of our corral is d = ( ±.096) mm (95% CI). The camera above the driver was used for tracking the position of a droplet relative to the center.

7 WJP, PHY382 (2015) Wabash Journal of Physics v3.3, p.7 y HmmL x HmmL (a) 3a y HmmL x HmmL (b) 3b FIG. 3: Figure 3a represents the path of a walking droplet. The blue data points represent the position of the particle for certain frames. As one can see, it takes a circular path around the inside of the corral, hardly diverging from its trajectory. The red circle represents the inner corral. One should note that the path is not random and is not considered to be in a true walking state. As the frequency is brought closer to the Faraday Threshold, the trajectory of the droplet is better resembled by Figure 3b, which is slightly more erratic in nature and deviates from its circular path.

8 WJP, PHY382 (2015) Wabash Journal of Physics v3.3, p.8 FIG. 4: Histogram of the position of a droplet in terms of the radius (distance in mm) from the center of the corral. The droplet appears to spend the most time about 44 mm from the center. The distance from the center is between 35 mm and 50 mm.

9 WJP, PHY382 (2015) Wabash Journal of Physics v3.3, p.9 movement data, one could determine the trajectory through the slit and onto a screen. [1] Brady R., Anderson R., Why bouncing droplets are a pretty good model of quantum mechanics, University of Cambridge, (2014). [2] Harris D.M., Bush J.W.M., The Pilot Wave Dynamics of Walking Droplets, Phys. Fluids, 25, 737, (2013). [3] Harris D.M., Mouktar J., Fort E., Couder Y., Bush J.W.M., Wavelike statistics from pilotwave dynamics in a circular corral, Phys. Rev. E, 88, (2013). [4] Molacek J., Bush J.W.M., Drops Bouncing on a Vibrating Bath, Cambridge University, J. Fluid Mech., 727, , (2013). [5] Oza A., Rosales R.R., and Bush J.W.M., A trajectory equation for walking droplets: Hydrodynamic pilot-wave theory, J. Fluid Mech., 737, , (2013). [6] Richardson C.D., Schlagheck P., Martin J., Vandewalle Thierry Bastin, On the analogy of quantum wave-particle duality with bouncing droplets, Departement de Physique, University of Liege, 4000 Liege, Bel- gium, (2014)