Hydrodynamic Quantum Analogs

Hollywood and Black Hole Analogs

The Big Bang Theory’s end-of-season cliff-hanger referred to a similarity between the equations of hydrodynamics and the equations of black holes, and the usefulness of hydrodynamic simulations to understand black holes.  Leonard joined a team put together by Stephen Hawking to search for the equivalent of Unruh radiation in water (at sea).  Here he is telling Penny that he will be on an ocean research vessel for four months:

Hawking and Unruh: Radiation from the Vacuum

One of Hawking’s many contributions to our understanding of black holes is his prediction of Hawking radiation.  By combining concepts and math from General Relativity and Quantum Mechanics, Hawking showed that black holes have a surface temperature and radiate particles.  The possibility of actually detecting the equivalent of Unruh radiation in the ocean may have been an exaggeration.  But, as is typical for The Big Bang series, the physics that appears is based on actual physics and is inspired by current events in science.  See, for example, Black Hole Analogue Discovered in South Atlantic Ocean”  and Coherent Lagrangian Vortices: Our Oceans Have Their Own Kind Of Black Holes.”

According to the equivalence principle, physics in a uniform gravitational field should be the same as that in a uniformly accelerating reference frame.  So, a particle or object undergoing uniform acceleration should also emit (thermal) radiation, analogous to the surface of a black hole.  This is essentially what Unruh radiation is.  Unruh radiation has the same mathematical relationship as Hawking radiation, except it is proportional to the uniform acceleration rather than gravity.  To reach detectable levels, the acceleration needs to be pretty drastic.  Experimentalists hope to use intense lasers to accelerate electrons sufficiently to detect Unruh radiation.  Unruh radiation is different from the usual radiation emitted by accelerated charged particles.  It is independent of the particle’s mass and charge and also has a different frequency distribution and angular distribution, features that will be used to identify its presence.

Hydrodynamic Quantum Analogs

Now, on to the subject of hydrodynamic quantum analogs.  In a previous post, I mentioned experiments with silicon droplets that were mimicking quantum physics:  The Folly of Physics: Interpretations of Quantum Physics, Part 1:  De Broglie-Bohm mechanics at work?”  If you missed it, take a look at this amazing clip from the Science Channel’s Through the Wormhole:

This Science Channel video shows how the results of the canonical double slit experiment can be reproduced by a silicon droplet (the “particle”) riding on an actual, physical wave (a “pilot-wave”, reminiscent of de Broglie-Bohm pilot-wave theory).  Physicists and mathematicians continue to explore this rich environment to further advance our understanding of nature.

Getting into the Experimental Details of Hydrodynamic Quantum Analogs

You can take a look at the MIT web page of John Bush for highlights of some of his group’s work on hydrodynamic quantum analogs.   Their work is further discussed in “Wavelike statistics from pilot-wave dynamics in a circular corral”, which is also available here“Exotic states of bouncing and walking droplets” (also available at this location), explains the experimental setup in more detail and digs deeper into the theory and math.

A silicon-filled tray is placed on a vibrating table.  The depth and geometry of the tray are chosen to enable studying the desired behavior or phenomenon.  The intensity of the vibration is adjusted to just below the threshold at which waves would be generated on the surface of the fluid by the vibrations.  When a droplet of silicon is then placed on the surface of the vibrating fluid, a cushion of air between the drop and the fluid bath prevents the drop from coalescing. The droplet bounces and “walks” on the vibrating surface.  This bouncing causes a wave field to be generated on the surface of the bath, similar to skipping a rock on a pond.  The wave field becomes more and more complex as waves from subsequent bounces interfere with each other and reflect off of the boundaries of the surface (or off of other obstacles placed in the fluid bath).

The motion of the particle depends on its current location as well as its history, due to the complex wave field generated by previous bounces.  The motion also depends on the environment; the geometry and depth of the tray, depth changes, boundaries and obstacles, etc.  In addition to a vertical component, there is a horizontal component to the force on the droplet.  This is due to the droplet landing on a sloping part of a wave.  Under the right conditions, the droplet achieves resonance with its self-generated wave field and is propelled horizontally along the surface.  This two-dimensional motion displays properties of a microscopic quantum system.  The trajectories that are observed, and the probability distributions mapped out by the areas in which the droplet spends the majority of its time, are equivalent to the results of quantum physics experiments with microscopic particles.

Visualizing Hydrodynamic Quantum Analogs

Take a look at this YouTube video, provided by MIT, to help visualize what is going on.  It is important to note that in the images where you see the droplet walking across the surface, the camera is being strobed in synch with the bouncing – so you just see the horizontal motion, not the vertical bouncing.

If the silicon bath is rotating, in addition to vibrating vertically, the droplet will lock into an orbit determined by the troughs of its self-generated wave pattern.  This is precisely a demonstration of “quantization” of the allowed orbitals for a subatomic particle confined in a potential.

Similar experiments have demonstrated other behaviors that are typically assumed to be exclusive to the quantum realm.  These phenomenon include diffraction, tunneling, quantized orbits, orbital level splitting, and more (see “Wavelike statistics from pilot-wave dynamics in a circular corral” and references therein).  To mimic tunneling, for example, a walking droplet approaching a barrier that it will on most occasions simply bounce off of, will once in a while receive enough energy from the wave enabling it to jump over the barrier.

Interpreting Hydrodynamic Quantum Analogs

The authors of “Wavelike statistics from pilot-wave dynamics in a circular corral” state that:

“Our study indicates that this hydrodynamic system is closely related to the physical picture of quantum dynamics envisaged by de Broglie, in which rapid oscillations originating in the particle give rise to a guiding wave field.”

Louis de Broglie is perhaps best recognized for postulating in his PhD thesis that all matter (not just photons) has wave properties.  He received the Nobel Prize in Physics in 1929, “for his discovery of the wave nature of electrons”.  Clinton Davisson and George Paget were jointly awarded the Nobel Prize in Physics in 1937, “for their experimental discovery of the diffraction of electrons by crystals”.

Louis de Broglie; hydrodynamic quantum analogs image

Louis de Broglie generalized Einstein’s theory of the photon to propose that all matter has wave-like behaviors. The story of his pilot-wave theory is one that is still being written. (image from Wikipedia)

de Broglie presented his theory of pilot waves at the famous Solvay conference in 1927.  However, his idea lost out to the personalities of Bohr, Heisenberg, and others, in favor of the Copenhagen Interpretation (CI) of Quantum Mechanics.  There is a substantial debate in the literature over whether the adoption of the CI was the result of personalities, politics, and personal ambitions, rather than a deliberative and unbiased review of the available alternatives.  See, for example, “Quantum Theory at the Crossroads.  My personal opinion is that the CI was adopted prematurely and went unquestioned by the bulk of the physics community for far too long. As a result, experimental and theoretical progress towards a fundamental conceptual understanding of the universe has been delayed.  I plan to address this issue in more detail, from a historical and current events perspective, eventually; either in this blog or in a book.  Nonetheless, it is intriguing to consider what conclusions would have come out of the Solvay Conference if de Broglie could have shown the above video.

Pilot-wave mechanics was abandoned until David Bohm independently re-discovered something very similar to it in the 1950’s.  The theory has subsequently become known as Bohmian Mechanics, or de Broglie-Bohm Pilot-Wave Theory.  According to this model, particles are objective point-like objects with deterministic trajectories.  These trajectories are guided by pilot waves, which also objectively exist.  The pilot waves are described by the wave function.  Wave function collapse never happens (contrary to the assumption of the CI).  Hence, pilot-wave theory removes the measurement paradox.  It also provides a mechanism for explaining and visualizing wave-particle duality.  It is easy to see how the movement of a particle can appear to be determined by the interference of waves, because it is directly!

The mathematics used to describe damped harmonic oscillators and RLC circuits are equivalent.  Variables from one regime (such as displacement, mass, spring constant, and damping coefficient) can be mapped to the other regime (charge, inductance, capacitance, and resistance).  However, this does not mean that an RLC circuit is a mass on a spring oscillating in some viscous damping medium.  It just means that the two systems share similar dynamical properties.  It also means that you can use one system to study or visualize the other.  However, beyond the similarities, there remain significant differences between electromagnetism and classical mechanics.

Implications of Hydrodynamic Quantum Analogs

Nonetheless, the equivalence between the motions mapped out by these classical droplet-wave systems and quantum mechanics is jaw-dropping.  And there is certainly a lot that we can learn from them.  These recent findings should help revive the question of whether there is a more fundamental dynamics in quantum physics.  Whether the correct conclusion is that the illusion of quantum mechanics is just that, and the quantum realm is nothing new (compared to classical systems) is yet to be seen.  There are certain phenomenon in quantum experiments, such as (apparently) discontinuous particle trajectories, for which the classical analog is not yet clear.  Additionally, in de Broglie-Bohm mechanics, there is no dynamic mechanism for the particle to influence the wave field as in the case of hydrodynamic quantum analogs.  But perhaps an extension of de Broglie-Bohm mechanics should account for this feedback?

Randomness is an intrinsic feature of the quantum world.  After reading these papers and watching the videos, it can be tempting to attribute this (apparent?) randomness to chaos theory.  Chaos theory applies to dynamical systems that are extremely sensitive to initial conditions.  Tiny differences in initial conditions lead to huge differences in future outcomes.  The idea to apply it to quantum theory would essentially involve assuming that there is some hidden information about a particle’s initial state that we cannot know well enough to enable a precise prediction for the future.  Hence, the best we can do is predict probabilities.  What bothers me about this idea, however, is that the intrinsic and unavoidable randomness in quantum mechanics is closely tied to non-locality.  Without the intrinsic and unavoidable randomness, problems with causality and relativity quickly show up. On the other hand, if it were true, that there is an underlying explanation for the intrinsic randomness in QM using chaos theory and hidden variables, the technological and conceptual breakthroughs would be astounding, I’m sure!

There are a lot of details that go into these experiments, including how the apparatus is set up and how it is filmed.  So they are definitely not a proof or refutation of any particular interpretation of QM at this point.  However, they are intriguing, and they offer an irresistible visualization that begs further investigation.  Quantum physics is typically presented as a mystical and bizarre subject, involving multiple universes, superimposed cats, and conscious minds deciding reality.  These experiments should push us to recognize that a belief in the mystical aspects of quantum mechanics is a choice and not a necessity.

About Warren Huelsnitz

Former submariner, now high energy physicist. Neutrino physics and astrophysics, quantum physics and other mysteries of the universe. Engaged in life-long learning and pursuit of knowledge. Blogging about quantum mechanics and other issues at www.thefunisreal.com.
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It's true that damped harmonic oscillators and RLC circuits are different systems even though they yield similar dynamic equations. However both systems can be derived from fundamental definitions and are similarly related with variables for voltage (V) and potential energy (U). In quantum mechanics the Schrödinger Equation is not derivable from fundamental principals but these hydrodynamic systems can be determined from basic principals. If an equivalent to the Schrödinger Equation is found for hydrodynamic systems, than it could certainly lead to insight about the fundamental principals that guide quantum mechanics.


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  3. […] Hollywood and Black Hole Analogs The Big Bang Theory’s end-of-season cliff-hanger referred to a similarity between the equations of hydrodynamics and the equations of black holes, and the usefulness of hydrodynamic simulations to understand black …  […]