So it's a homecoming of sorts for me, although I haven't spent as much time at Oxford as I would like when 1970 to 73,
my father was on sabbatical at Oxford and we lived in White America, which is just a few miles away.
I'm hoping to return there tomorrow and in the backyard. I used to play soccer all the time with my brothers.
And I think this is the first scientific problem. I thought about it.
How does this happen? Right? I know. Are there air fluctuations?
How could this? Possibly because sometimes it hit a ball with no spin whatsoever. It will flutter around and say, Oh, how does that work?
Interesting. Don't know. That's another.
Another problem I thought about was this one is everyone would.
And here we see an example of an early hydrodynamic quantum analogue, more of these later.
And so I thought, Well, maybe I'll start. I wasn't interested in the melting point of caesium or the standard model.
I was interested in trying to understand what I saw around me, so I thought, maybe I'll go study physics.
So I studied physics, and during my physics, four years of physics, I took zero course in fluid dynamics.
So I had and still had no idea why a soccer ball bent why a stone skipped.
But I took six courses in quantum mechanics where we really weren't encouraged to try and understand what we saw around us.
Instead, we were invited to wallow in the mysteries of quantum mechanics. So that drove me to fluid mechanics.
And it turns out that in the states, fluid mechanics is represented primarily in engineering departments or happily applied maths departments.
And if this is your guiding light, if you dig deeply enough into any physical system, you find mathematics.
For example, here in the heart of the sunflower, if you count the number of scrolls going clockwise and counterclockwise,
you always find adjacent numbers in the Fibonacci sequence, which is one one two three five eight 13.
And so you can, of course, find mathematics anywhere, whether it's in the heart of the sunflower on the pavement, stone outside you.
You're therefore pretty much free to study anything you want, so I sort of specialise in surface tension.
So this is a feature of an interface.
If you imagine an air water interface, we know that within the bulk of a fluid, there's a force per unit area, so-called pressure.
We can think of surface tension as imparting an effective surface pressure.
So a tensile force per unit length along the interface. And as a result, you can.
Small objects can float on the surface of water, even if they're heavy relative to the water.
So here we have a paperclip suspended by surface tension.
Likewise for water striders, these small insects and the if they're gradients in surface tension because it's a force per unit length,
you generate interfacial forces, which can drag things along. So this is a cocktail boat full of alcohol.
Alcohol leaks at the back turns out, surface tension is a function of alcohol.
It decreases with alcohol concentration, so the boat is then pulled along by the surface tension gradient.
OK? And so these are the two main physical features of surface tension and mathematically and fluid dynamics.
You're basically dissolving Newton's laws, so mass times acceleration equals the sum of the forces in your butt.
You're writing this down forever for every infinitesimal blob within the fluid.
So this is your mass times acceleration term. This is the inertial term and the short.
The so a blob of fluid moves in response to pressure gradients, gravity and viscous stresses and surface tension enters into the boundary condition.
So when you have an interface, there's a jump in pressure, which is proportional to the surface tension and the curvature of the interface.
And again, we can generate tangential stresses if there are gradients and surface tension
as arise through chemical gradients as we've seen or temperature gradients.
OK, so that's the basic maths. These equations, the beauty of fluid mechanics is these are the only equations you ever have to worry about.
OK, they're a mess. That's the downside. And so why is it that we go through our lives and many of us never even know what surface tension is?
I went through, of course, my undergraduate actually went into fluid mechanics.
I went through geophysics, and so after my share, I really didn't know what surface tension was.
But this is because it's important on a small scale.
OK, so it's if you have a blob of fluid, we know that we take a bucket of fluid reporter on the ground.
We get a thin puddle. Basically, its extent is prescribed by surface tension.
We have a small drop. It'll just sit there. So you basically have gravity trying to drive you outwards.
So there's a hydrostatic pressure, which is Rogue A, if that is the characters.
The scale of our blob surface tension wants to minimise the surface area.
So it's acting. It wants to maintain the sphere city of the blob of fluid.
So there's a balance between these two, and when you have a large volume of fluid,
it will spread again out to a puddle with small surface tension winds, and it maintains its felicity.
So basically,
the relative magnitudes of these two forces gravity acting to cause spreading surface tension to resist it is prescribed by the bond number.
And so if there's a critical length, which is so-called capillary length below which surface tension winds relative to gravity?
OK, and so then again, we're in this scenario.
And so this capillary length is about two millimetres, and coincidentally, this is the scale which sets the size of raindrops.
OK, OK. So surface tension dominates the world of insects and microfluidics.
Why it's been enjoying a resurgence in the engineering community.
So the first thing I'm going to talk about is walking on water in the biological sphere.
So it turns out many creatures have learnt to walk on water for various reasons, in particular for foraging on the surface, avoiding predators.
So this is true. So you have insects, spiders, reptiles, birds and even mammals that can support themselves on the water surface.
And so I thought to try and classify how they actually do it.
So again, surface tension is important for small creatures, so you can have static weight support if the object is small enough.
So it turns out all biological creatures are slightly heavy relative to water. So would if submerged sink, but the surface tension to stay afloat.
So again, there's a ratio of the weight of the object relative to the surface tension force generated by deflecting the interface.
And if this is less than once, the object can stand at rest on the interface, as does this water strider,
which is about a centimetre long, large object such as the lizard can't reside at rest.
They would sink unless they hammer the surface with their feet.
OK, so you can basically divide creatures into small and large again, small being.
Their world is dominated by surface tension, so these small creatures are above the sort of this is the point at which you fall through the interface.
So small creatures reside, it can reside at rest. The larger creatures rely on dynamic weight support.
OK, so we can then say, OK, aesthetics is easy. How do they actually propel themselves?
And I think this kind of constructive. So this is the mathematicians view of a water walking creature.
It's a sphere with characteristic size, a cross-sectional area capital A. And it's hitting the surface at speed you so
you can ask what forces it can use to contribute to its lateral propulsion.
So you have a buoyancy term simply because there is a hydrostatic pressure across which it can push it when it generates a cavity.
Then there's a form drag, and this is the force you feel when you stick your hand out of a cardboard,
so it's proportional to your speed squared times the area of your hand.
Then there is a basically an added mass term. You have a wiskus term, then you have the surface tension terms.
OK, so these guys again, the interface behaves like a trampoline, so it's conceivable that insects could use this to bounce off the surface.
And then we have the possibility of surface tension gradients propelling bugs.
And so if you if all of these terms are important, things are going to be a mess.
And so really, what do you want to do is simplify things to the point that you can get some insight into what's going on.
I think what's interesting if you ask the question which force are used by which creature?
Every single force is used by some creature.
So there's some sort of principle suggesting any mechanism that can promote life will be exploited by nature.
OK, so here are some of the big guys that don't.
You can't use surface tension. So these guys are about that long and they have to run at about 15 metres per second to do this.
But they're slapping the surface, generating a cavity, pushing across the back.
So they're using form, drag and hydrostatic pressures to drive themselves along.
And here's a new one, by the way, how do I make this OK?
He doesn't mute. There's some override. OK, so here's another one.
They can walk on water. Perfect.
Yeah, right. OK, so there we go. The skittering frog, so that's sort of at the cusp.
It's too big for a surface tension, but it uses it to a certain extent.
And now we have smaller objects. So these water striders stroke the surface and throw water backwards in the form of vortices.
And so there was a paradox saying that these things couldn't move again.
Another theme of this talk is paradoxes. When you see paradoxes in a field,
you are put off the problem because people explain to you in your field how to think about something in the wrong way.
So when I was a student, I was my supervisor. Explain to me why I should never work on certain problems and fluid dynamics because they were so hard.
And then I have subsequently seen some of them resolved. I said, Well, why didn't I look at that?
So I really think paradoxes should be an invitation to work on a problem, not the converse in biology.
There are lots of paradoxes. For example, there was, I think it was Grey's Paradox,
which says that a dolphin can't swim as quickly as it does and say, Well, what was that based on?
And so they did an experiment in which they took a more plaster cast of a dolphin
and dragged it behind a boat and found that it had anomalously high drag.
So of course, one might conclude that flexibility is important.
So in other paradoxes, saying that a bumblebee can't fly? Of course we know it can.
So I think working in biology was useful in pointing out when there paradoxes in a field,
it's because people in that field are looking at things the wrong way. OK, so this is it.
So this is a photo of these water striders, which you just saw in this sort of vertical weight wake of the water strider.
So we basically sprinkle the surface with a blue dye. And this is the pattern it leaves behind.
It turns out that we gave it a yellow brush to give it a little bit of Van Gogh.
All right. And so here's another creature that uses surface tension for propulsion.
It basically arches its back in order to match the curvature of the meniscus, which it's trying to climb.
So it's trying to climb a meniscus from right to left, and it does so again just by arching its back.
And so it's basically releasing surface energy as it climbs. And here's another one which uses the Cheerios Vela effect as it's come to be known.
So we basically distorts the interface, pulls up on the interface and climbs upwards.
So again, it's acting to minimise the surface energy of the system. OK?
And here's one that uses the Marigny forces basically dumps a lipid onto the surface.
This changes the surface tension and propels it towards the shore.
So it's not a very efficient mechanism, but this is used by something which which doesn't habitually walk on water.
It's getting an emergency propulsion device.
OK, so those are the things that walk on water. And so we basically classified all such creatures and could see again all of the
mechanisms available to to water walking creatures are used by some creature,
and which mechanism they use is determined by their size.
OK, so so I was happily working on this sort of thing. And then suddenly you could air introduced me to the following problem.
So this was so part two that was just a preamble, an excuse to show some nice videos.
But so the problem this is nice and that it took me back to my roots in physics and my frustration with physics.
OK, so we're taught in as undergraduates that the universe is sort of in homogeneous on a in a philosophical sense,
that is to say the macroscopic world is deterministic. You have particles following trajectories described by Newton's laws.
If you have the initial conditions, you can predict the outcome.
So quantum physics, we are taught, conversely, that the microscopic world is intrinsically probabilistic.
So particle trajectories are not described only the statistics.
So you can only predict the probability of a particular outcome in a quantum system.
OK, and of course, there are those that dissented. Einstein in particular.
His most famous dictum is this and God does not play dice.
But what he really objected to was the concept of quantum non-local avi, which is to say super luminal signalling.
So of course, having derived developed relativity, this was anathema to him, and that will be another theme of this talk today.
OK, so so we're going to see how non-local behaviour can be mis inferred from a local hereditary system.
OK. Let me dig into that a little bit. So hidden variable theories or theories which seek a rational dynamics underlying quantum statistics.
So we know that quantum mechanics correctly predicts the statistical behaviour of microscopic systems.
But what's the dynamics? That's really still an outstanding question.
So if we were to characterise these hidden variables, so that's to say those variables that would describe the trajectory of microscopic particles,
we could then restore our rationality to the microscopic physics.
OK, so a brief history.
It turns out that all the most of the attempts I shouldn't say all attempts to come up with the rational dynamics have been
based on the notion of a pilot wave theory where you have a particle moving in concert with its with the guiding wave,
OK? And so to play in the nineteen twenty six proposed a double wave pilot wave theory of quantum dynamics.
So again, so as particle moving in resonance with the guiding wave and then this sort of sense.
And on the basis of which she drives several things in particular particles h hbk, a number of the cornerstones of quantum mechanics,
the Einstein deployed relation, which will come to you later, later and on the basis of his physical picture.
Schrodinger derived shorting his equation and also Klein Gordon.
And so really, it motivated the development of the mathematical formulation of quantum mechanics.
Nevertheless, this physical picture was discarded in favour of nothing.
OK, so and this is as quite sensible view suffered a setback when von Neumann in nineteen thirty two,
was alleged to have proven that there can be no hidden variable theory.
And this bomb then and that stood for twenty years until David Bohm presented a single wave pilot wave theory,
which was consistent with the predictions of quantum mechanics. It has some difficulties again, some of which were pointed out by Joe Keller.
But this boom's contribution was important because it attracted John Bell,
who subsequently went through and discredited the impossibility proofs, and which led to the following conclusion.
OK, so we're going to come back and talk a little about Bell's there at the end.
OK, so fast forward now. Almost a century in two thousand five, if there discovered a hydrodynamic pilot wave system,
it's really a macroscopic realisation of the sort of mechanics that the BCCI had imagined.
OK. And lo and behold, it exhibits several features of quantum systems that were thought to be exclusive to the microscopic quantum realm.
In particular, the statistical behaviour of this macroscopic classical system looks very much like that of quantum systems in certain cases.
OK, so the questions raised, what are the key dynamical features responsible for this quantum like behaviour?
What are the potential limitations of this high dynamic system as a quantum analogue?
And the big question that I want to address today is my memory account for quantum known locality.
OK. OK, so if we zoom a little in a little deeper into classical physics, we had the lower class in view of the universe.
If you have the governing equations and the initial conditions, you can predict its outcome.
It's like a record being played. We know what's going to happen for all time.
We realise that predictability in classical systems, if they're sufficiently complex,
is limited because complex systems are their behaviour is sensitive to initial conditions.
OK, so there are practical limitations to predictability even in classical systems, more complicated, still hereditary systems.
So these are systems in which. Which are in which the system evolution is dependent on its history.
OK, so a classic example. It's elastic solids where you have the state of instantaneous state of stress is determined by the history of strain.
But well, we'll see how this comes up in the hydrodynamic case.
But basically, in order to predict the future, you have to know not only the initial conditions, but the system's past.
OK, so this is a very rich class of dynamical systems, and we see in this particular one the bouncing droplet system,
which we'll come to now local hereditary mechanics as a spatially local territory.
Mechanics can give rise to apparently non-local behaviour. OK, so this is the system.
We have a vibrating bath of fluid. Again, if you go to my web page, you can see how to do this with $60 for $60.
So we have a bath about this big. We drive it around 50 hertz.
The amplitude of vibration is around a millimetre. So you have the vibrational acceleration and the control parameter is the ratio of that to gravity.
And so when the vibrational acceleration exceeds a value, which is around four g, you get a standing field of sub harmonic Faraday waves.
That is to say they have half the frequency of the driving. And the theory of this was done by Brooke Benjamin in a beautiful paper.
Benjamin and Arsal. OK, so and it turns out if you have this vibrating bath, you can levitate drops.
OK, so this drop is about one millimetre across. We're driving around 50 hertz, so it's a fluid which is a little more viscous than water.
And effectively, the interface behaves like a trampoline because of surface tension.
And there's no coalescence because the thin air layer doesn't have time to drain during the impact.
OK. So this thing again, bouncing 50 times per second.
And remarkably, what could air if discovered by accident actually is that there's this corner of
parameter space where these drops bouncing drops become unstable translational motion.
So if you look carefully, you can see that this thing is actually landing on the side of its wave.
So each time it lands, it gets a horizontal impulse. It's of being slowly nudged from left to right.
OK, so a key feature of this system is the resonance between the particle and the wave.
OK, and and this object, which you've called a walker, which we've now called the Cudo Walker in his honour, is a particle dressed in a wave.
And so this concept of a particle moving in its own way field is throughout physics.
But the theoretical treatment thereof is generally limited, even in electromagnetism.
The Lawrence Dirac equation, which which tries to describe a charge moving in its own electromagnetic field, has runaway solutions.
Here we have a particle dressed in a way we can see the particle, we can see the wave, we can see the dynamics on the scale of the vibration.
OK. And so again, the system is non-market Jovian.
So say I just introduced the term hereditary. It's the instantaneous force which gets to impact depends on the local slope or the way you feel.
So it's a local local in space. The whole system, of course, is local.
It's classical. But in order to get that slope, you have to integrate backwards in time to take into account all of the waves generated in the past.
So this is, we can think in terms of path, memory as you've called it.
And so the extent of the hereditary dynamics of the importance of memory is determined by how close you are to the farraday threshold.
So at the farraday threshold, again, you excite waves.
Even in the absence of the drop, we're always doing experiments below the Ferriday threshold, so there would be no waves in the absence of the drop.
But it locally generates Faraday waves, which are damped in time.
As you approach the very threshold, these waves are more persistent.
And so you have to integrate further backwards in time to predict the behaviour of the system.
OK. OK, so now it's more clear, if we strobe this, we grab one frame per bounce and can see it being pushed along by its guiding wave,
and this is exactly the physical picture that the boy had, he said. Particles move along.
They move through a resident interaction with their own way field along a line of constant phase, and that's precisely what we have here.
OK, so we drive the maths of the equations describing the dynamics,
so we have a mass times acceleration term that is a drag term induced by flight and impact on the surface, and this is the interesting term.
So this is the way for this term and this is this. So this depends. It's there's a force proportional to the gradient of the fluid depth.
And so the way field we can get by summing up the waves generated by each impact and these look something like damped Bessel functions.
The form isn't too important, but you see how memory the memory parameter is basically an indication of the proximity to the farraday threshold.
So as the this is, the vibrational acceleration is that approaches that this thing gets large, OK?
And that's basically a damping time, and that's the very period.
OK, so again, you have this trajectory equation and now in the limit where the vertical dynamics is fast relative the horizontal dynamics,
you can approximate this infinite sum by an integral and you have a nice integral differential equation that you can analyse.
In particular, you can look for a static solution.
You can look for circular orbital solutions, steady walking solutions and assess their stability.
So we've done this in various settings. So if you go back and look at the static state so you can form crystals of these bouncing drops,
so we've looked at the stability of these states, the pairs just came out and we're looking at lattices.
So it's really quite rich. Mathematically, there's a lot to be done.
And also, they're dynamic bound states where the drops are joined by their common pilot wave field.
So here we see them locking into circular orbits. And here we see them moving in this fashion.
So if this looks completely bizarre because we have strobe and we strobe slightly off the bouncing frequencies,
so they appear to be sort of gliding in and out.
OK, so and here's a serious something done by Stuart Thompson, who is a recent graduate of of this department.
There's currently an instructor in our department, so you looked at an annual ring of bouncers and as you crossed the fair or not,
the farraday threshold, a critical dynamic threshold.
You generate a solitary wave which propagates around the ring.
And this is really a realisation of a total L.A., which is a model of crystal vibrations.
So you get some sort of some nice mathematical connexions to solid state physics.
This is now a free ring, so the drops are free to move anywhere in certain regimes and move readily.
Here we see this, but this binary oscillation of two lattices. And so what's going to happen here?
10 drops, 20 drops and where's your money, where's your money?
Oh. OK, so this we haven't predicted that one yet, now may never, but OK, so if we go back to single drops now,
if you look at their motion in various force fields, so now we have a drop moving in a rotating frame.
So this is first looked at in Eve's group, you expect four drop moves.
So if a particle moves at uniform speed in a rotating frame that it will follow an inertial orbit
where you have a balance between the centripetal outwards force and the inwards Coriolis force.
And this is what so you basically expect the radius, the orbit to decrease with rotation rate.
That's what happens at low memory when the wave field is not particularly important as you go to high memory,
as is the case here, you see the drops has to navigate its own way field.
So it's basically exciting potential and it's only stable in the troughs of this wave field.
As a result, there's a quantisation of the orbit. OK, so there you have quantised orbits, where the quantisation length is the Ferriday wavelength.
OK, and so owing to the identical form of the Coriolis force acting on a mass M in a uniformly
rotating frame and the Laurence Force acting on a Charge Q and a uniform magnetic field,
you can draw the analogy between these inertial quantised inertial orbits and larmer levels.
Here, the quantisation length is again the Faraday wavelength that's playing the role of the display length in the quantum system.
OK. I've also looked at the motion of a drop in a central, simple harmonic oscillator potential.
So basically the spring force, so the drop wants to move in a straight line, but it's also being constrained by the spring forth.
So it basically gets dragged back towards the centre or so the various possibilities and turns out.
So this is the radius of the orbit, which is a proxy for the energy of the system.
Then you have the angular momentum and so you actually get a double quantisation, which is reminiscent of that in quantum mechanics.
So here we see circular orbits. Here we have these lamis gates, which have zero angular momentum.
Then we have more elaborate tree foils and so forth.
And so you see as this we've seen in this and other systems, you have these quantised periodic orbits.
And when the system becomes chaotic, they basically switch between them, giving rise to sort of multi-modal quantum statistics.
And so I'm not going to dwell on the orbital dynamics, but I think it's quite well understood.
And you see that the statistics of chaotic pilot wave systems can look very much like those of quantum mechanics.
OK, so we see that the quantisation arises because of the dynamic constraint imposed on the droplet by its monochromatic wave field.
So the fact that you have a resonance between the drop in the wave is key because that ensures the monochromatic nature of its wave field.
And again, when the system becomes chaotic, it switches between unstable periodic orbits.
OK. And this notion of a drop surfing, it's self-generated potential is important, and we'll come back to that later.
OK, so my first contribution on this.
Was to address the one problem I remembered from my undergraduate course in quantum courses, in quantum mechanics, which is particle in a box.
OK, so it was wonderful to come back to this problem because they've now been done experimentally.
The other thing I didn't like about my training in quantum mechanics is they didn't compare their theory to experiments.
Here's one where they have. So these are electrons zipping along on the surface of metal and the electrons see inside.
So there are some 20 or 30 electrons. I think inside are confined by this, by these atoms.
So they're bouncing around in the waves. You see, there are basically the probability density function.
So these you basically solve Helmholtz equation or stationary shorting equation and you solve for the modes of the cavity.
OK. And the wavelength you see there is the wavelength. OK.
And so I said, Oh, can we do that with walkers? Surely not.
OK. And the student who did this, Dan Harris is outstanding experimentalists.
I want you to note one thing as the thing moves along, you see its speed varying so that the trajectory is colour coded according to speed,
and you actually see speed variations on the Faraday wavelength.
OK, so this is sort of classic feature of the system, which were the importance of which we're beginning to explore now.
So you let the thing run and run and run.
It's a very chaotic motion, but you see the emergence of statistics which look very much like those in the quantum system.
OK, so we see here and this looks here, the PDF looks like the amplitude of the Faraday mode of the cavity, but you see more again, sort of.
A recurring theme is how coherent wave like statistics can emerge from chaotic pilot wave dynamics.
OK, and so here we have we've revisited this with an elliptical cross of Pedro.
Sainz has done this, and it's a very robust result.
Here we see a speed map and you see that there. Basically, the speed, the mean speed is a function of position,
and it's because of this dependence that the one has the emerging quantum light statistics.
And in this study, we noticed something interesting, which is that the we actually measured the mean wave field.
So if the instantaneous way you feel that any instant doesn't look like the wave mode of the cavity, but the average way field does.
So this is the particle histogram. This is the average wage field, and you can see they take precisely the same form.
So then we proved that this average Wakefield can be expressed as the convolution of the bounce away field.
So that's the way field you get if the drop dispensing place and the histogram.
So this particle is effectively navigating the mean way field of the cavity.
In this mean way field of the cavity depends on the statistical behaviour, so it's a very strange potential.
Right. So you have this drop again surfing a background wave whose form depends on its statistics.
OK, and this is a sort of confusion which arises in Bromium mechanics, which we can see our way around now.
OK. And this and this potential, if you like again, it is non-local.
It is basically imposed by the statistics, so it's non-local in the same way that the quantum potential is non-local and in mechanics.
OK, and so we can then see in standard quantum mechanics you solve for the modes of the
cavity and what do they actually do in comparing to experiments they just choose?
They put the it's a superposition of modes and they are free to choose the coefficients on those modes.
So in our system, we say, OK, we could do the same or we can say there's actually an underlying dynamics,
and we can think of this underlying dynamics as b as being a trajectory in which which can be decomposed into various component parts,
in particular periodic sub trajectories and the thing the drop is actually jumping between them.
And most importantly, we see here that the notion of trajectory is not inconsistent with the emergent emergence of quantum statistics in this system.
OK. And so. At a more general level, the system has three scales at least are actually more, but we can think of it in terms of three times a year.
So we have the times of wave generation, which is the time scale of vibration of the particle effectively.
Then you have this pilot wave dynamics, which is revealed by strobing at the bouncing frequency today.
And then you have this long term statistical behaviour emerging.
There's actually so again, if you look at some products there,
there's a huge range of timescales in this problem because there's a timescale of emergence of the main pilot wave.
That's the timescale of emergence of the statistics. And, of course, the dynamical timescales as well.
OK, so what does this remind us of? Well, it turns out it's very similar to the brain's mechanics.
So this is what Frank will take calls a poem in two lines.
This is the Einstein, the relation. So relativity requires this.
Quantum mechanics requires this if you equate the two.
You see that a particle of mass m must have a natural frequency.
This is the so-called Compton frequency, it's the frequency of the Zetterberg, as they call it, in the early days of quantum mechanics.
And he said that this was the frequency of oscillation of a particle and there was at this energy.
At this frequency, there's an exchange between rest, mass energy and field energy.
Again, this basically being rest mass energy. This being field energy. So he said particles or oscillators with this high frequency,
they're then generating a wave and riding that wave and then somehow you it will give rise to quantum statistics, OK?
But so notice that there are two waves in his theory, there's the wave centred on the particle, which is pushing it around.
And then there's the emergent statistical form. OK. And that's very much like ours.
And so we have the PSI, which is the standard wave in quantum mechanics.
But then you have the particles entered the high wave, OK?
And this is generated by the particle vibration at this constant frequency.
And he said that the waves are solutions of the Klein Gordon equation.
So that's just the relativistic form of shorting.
This equation is actually very nice because it looks very much like the water wave equation.
And so from this formulation, you said particles move perpendicular to the surfaces of constant phase.
So that and that from that, if you have a monochromatic wave, it gives you this peak was a spark.
So the particles riding along a line of constant phase. And the other thing that he stressed in his theory was this harmony of phases.
So the particle oscillates in resonance with its guiding wave.
So there you have to consider the fact you have a moving clock, but its frequency is prescribed by its mass.
And so the two cancel beautifully, and he thought that he called this and gone weidler not to.
He thought this is a very important point, and it's critical in our system the resonance between the particle and its wave.
OK. And so really, he imagined something very close to what we have now in the lab.
You have high frequency isolation at the constant frequency.
You have an intermediate pilot wave dynamics in which the particle surfs along its gliding wave and explores its background potential.
And then you have the emergent long term statistics described by standard quantum theory so well,
he was unable to do was to show how this would emerge from this.
And of course, the question arises What is the pilot wave?
And so there are those who sort of, I would say, the modern extensions of the body's mechanics have looked to the electromagnetic quantum vacuum.
This is nice in that it it has a spectral form which has a bar in it.
So the energy in. So basically, you just have electromagnetic background noise, which is then interacts with the particle.
But so the energy in a mode with frequency omega is h by omega over two.
So the idea then, is that the object interacts with this background field generates a pilot wave and off you go.
And you can think of it in terms of if you have a turbulent fluid and you throw a spring into the turbulent fluid.
It will start oscillating in its natural frequency or pump energy into the fluid at that frequency.
So in some sense, the particle only talks with that component of the field, which has the right frequency, the constant frequency in this case.
OK, so then you can do a map and notice.
I mean, it has the right number of variables, which is beautiful if you draw the map between the body's mechanics,
complemented by a steady and the walker system.
So again, the balancing is playing the role of this Zetterberg on this high frequency oscillation in the brain's mechanics.
You have this the harmony of phases as assured by this residents commission in our system.
Surface tension basically plays the role of each park because we have capillary waves.
This is the way parameter and quantum mechanics the and all of the analogues we've seen.
The Faraday wavelength is playing the role of the deployed wavelength.
But there's a second wavelength in the plays mechanics, which is the Compton wavelength.
So this is basically the scale of vibration. And so in our system, it's basically this length.
So the length, the step length in the bouncing. And of course, if you strobe over it, that thing's gone.
Just like all consideration of the Compton, length is gone. If you go from relativistic mechanics, quantum mechanics to non relativistic.
OK, so we can play a few games here, so we know that this system is driven.
So basically. So in terms of, I should mention,
the energetic so this system is of course driven and that's then playing the role of the quantum vacuum in deploys mechanics.
But we also know that there is state a steady state.
So there's steady walking states, their periodic states in which the driving the dissipation balance in our system.
So we can imagine that there might be some invested like description, some Hamiltonian like description of our system.
So let's imagine let's look at the system and pretend we don't know that it's driven this.
What of what we infer for the mass of the Walker? Well, we don't expect it to be exactly the mass of the drop because it's dressed in this way field.
And for that matter, what trajectory equation would we infer?
OK, so if you look at the limit of weak droplet acceleration, you get the following equation.
So basically you do a asymptotic expansion of the wave force and it contributes to two to two terms on the left hand side.
So basically, this is some classical potential, but the way force is giving rise to basically something equivalent to a boost factor.
So there's an added mass associated with the wave field.
And there's also a nonlinear drag which always drives the walker towards its free walking speed.
OK, so this is the free walking speed here.
It's moving at that speed, then the drag goes to zero in that limit, which is the case when you have steady walking or circular motion.
You get the Envisat mechanics of a particle whose mass depends on its speed, which just looks, of course, like relativity.
OK, so let's use this now. So this is an experiment in which a draw up approaches a submerged pillar you expected just to scatter,
but it doesn't scatter, it scatters, but in this very strange way.
So it's actually follows a logarithmic spiral. OK.
And so it turns out. So of course, the only reason that the drop is being deflected is because its pilot wave is being distorted by this object.
But let's let's again follow this conceit where we say, Oh, we don't know what's driving this a bit of pilot wave system.
Let's just infer the effective force required to cause this walker to move along this logarithmic spiral.
And let's imagine that that force is generated by this pillar. OK, so what is that force?
We can infer it and it takes the form, so we use our booster equation again.
We can use it because the thing happens to be moving at uniform speed. So we're justified in doing this and we infer this lift force.
So it's a force which is proportional to its velocity time cross with its instantaneous angular velocity around the post.
So if we use this analogue between again the Coriolis force and the Lorenz force that we saw before, you can see this.
This looks like a an example of self induction.
So it's like a charged feeling the magnetic field associated with its own current.
OK, so this and more importantly, so this is still puzzling me, and I think there's something very deep there.
I've been puzzling over over it for years, as Darren knows when we first saw this, but.
You can see how this wave mediated force gives rise to spooky action at a distance.
If you don't know that there's a wave field, you're going to infer a non-local force.
OK? Right, so here's another example.
So, Fidel oscillations is something that arises in quantum mechanics if you have an impurity
again in your electron sea of electrons zipping around on the surface of a metal.
So we have a localised disturbance there, and they basically solve this with scattering theory.
And this is effectively the duprey wavelength in their system.
And so let's try it in our. So now we're going to operate in shallow water.
Turns out these things, all these things can walk in half a millimetre.
And we have here a deep well. OK, so that's a region of high excitability.
It turns out these things spiral in towards the well, and as they exit the well, their speed is modulated.
OK. So if we let the thing run and run and run, these are actually experiments.
So the drop is attracted towards the well, waves are excited in the well and the waves modulated speed.
So I noted before, there's this tendency for the drop to oscillate in speed along its direction of motion with the Faraday wavelength.
So this is a sort of roots of a statistical signature with the Faraday wavelength.
OK, and so you see then the this is the speed map consists of concentric circles, and of course,
that gives rise to a PDF, which looks very much like that in the analogue quantum problem.
The wavelength here is, of course, the Faraday wavelength.
OK. So once again, we can conclude that the fidele oscillations,
which is the quantum phenomenon or is not these are not inconsistent with the notion of particle trajectories.
OK, so we're kind of building up this class of analogues, and, you know, some of them are quantitative, such as the ones in the crawl,
in the final isolation, some of their more more qualitative and of course, their limitations because this is a hydrogen amick system.
And we, of course, expect it to be quite different from that in quantum mechanics.
So there are significant difference. Significant differences of measurement process is not intrusive.
Details of the wave behaviour are different.
Spin states are unstable in our system, but this is just invite us to look at a more generalised class of pilot wave theories.
So if we, for example, if we take our Strober scalloping model and we know dimensionalize it, they've just two dimensional as groups.
One basically prescribes the amplitude of the waves, and this is the particle inertia relative to the waves.
And so we can ask. So it turns out that this parameter is limited between 0.6 and 1.4 in the experiments.
What happens if it's 10 to the minus six? What happens if it's 10 to 16?
It turns out if if we set this to zero actually, or even make it say half the value we see in the lab,
we can stabilise spin states, which is a drop zipping around its own way.
OK. And so we're free to explore this generalised pi wave framework and act and ask
which quantum features can we capture in which regions of parameter space?
And really, what we're seeing is the quantum limit is small particle inertia, if not zero, and notice if you go to small particle inertia,
you go to the debris limit where you have gradient driven motion and then large wave amplitude.
OK, and so they're further extensions. You can change the form of the waves, of course, which you'd have to do if you went to 3D.
You can add a stochastic forcing, but that would seem cheating to me.
And so we're basically just developing this growing catalogue of quantum analogues and and trying to make connexions with the quantum mechanics.
And again, I think a very interesting direction. So these are these are spin lattices, and this works very nicely.
You get transitions which you can prompt by rotation quantitative comparisons with the equations of stochastic electrodynamics,
the Lorentz Dirac equation, which is the charge moving its own field.
So trying to see what the origins of the shortcomings of those trajectory equations might be and most exciting for me right now is we're now.
Looking at what we call hydrodynamic quantum field theory, so this is revisiting the brain's mechanics informed by the high dynamic system.
OK, so we're basically doing now what the brain would have done if he'd had MATLAB.
OK, so he said Clean, Gordon. So, Klein, Gordon is nice, he just has a it has a resonance in it, which is the constant frequency.
And so you have a particle which is exciting at that frequency, you're going to generate a pilot wave, OK?
So it's really very much like a system. We treat a particle as a perturbation at the resident frequency of the equation localised,
and we look at gradient driven motions were suggested by the play.
And when you do this, the thing spontaneously starts moving in a straight line at peak was HBK.
OK. So if nothing else, I feel that we understand the free particle and quantum mechanics.
But when you study the free particle in quantum mechanics, this is their treatment pickles.
Okay, so now we see if you treat a particle as an oscillation in a field.
You get that for free. And moreover, the particle actually oscillates about that means speed.
So that's its mean speed. It's oscillating at the company, frequency at the wavelength.
So that gives you then a mechanism for a statistical signature with the typical wavelength, as we saw in our experiments and various others,
the fidele oscillations, you have oscillations with the wavelength of the pilot wave that will give rise to statistics.
OK. And so moreover, this so we're just getting started here.
Notice, by the way, if you have this these oscillations with the degree wavelength,
then if you have circular orbits, they have to satisfy the more Sommerfeld condition.
All right. So we're just warming up here, but there's much to be done.
And again, I think this this sort of dynamics suggests that the uncertainty relations are really indicative of an unresolved dynamics on the comp,
which scale the scale of these fast oscillations. OK.
And so the question which is normally. As to me, so I've decided to be pre-emptive is.
What about Bell's theorem, so. Bell driving inequality, which people interpret as saying there can be no hidden variable theory,
so closer inspection suggests this doesn't have any bearing on systems in which particles interact with the background field.
OK. And I think more importantly, so even if you don't believe that Bell himself,
who drive does inequality and saw it violated by the experiments of Alan Asprey,
came to the conclusion that there must be some pilot wave dynamics underlying quantum mechanics.
And he sort of sided with the bohemian interpretation, which I haven't had time for.
But that has problems in that it has this non-local potential and we're beginning to see in our system how you can get around that.
So really, we're trying to rationalise quantum non locality via local hereditary dynamics and so avert the need for spooky action at a distance.
So these are just examples of where you could missing fur non quantum non locality from this from our system.
So, for example, wave function collapse.
If we insisted that this was the complete description of our physical state of our physical system, then the act of observation,
which would reveal the drop to be at one particular point would cause this wave function to collapse instantaneously to a point.
OK, that's a bit trivial, but here we've seen how action at a distance can be inferred if you deny the fact that there is a pilot wave.
Likewise, we've looked at the slit in double slit experiments.
So it turns out in the double slit, the drop through will feel the second slit by virtue of its spatially extended pilot wave.
And we've seen this mean pilot wave potential, which is effectively a non-local potential.
And also Andre Nash, Aspen has looked at these drops interacting.
These are done numerically through a sort of resonant cavity.
So these drops talk to each other by virtue of their common way field so they can become either perfectly synchronised or chaotic,
but with identical statistical signatures. So we're we're now looking at entanglement measures in hydrodynamic quantum analogue.
So to conclude, we've seen that this system provides a means to explore the boundaries between classical and quantum systems.
And it certainly extends the range of classical mechanics to include systems which have statistical behaviour reminiscent of quantum systems.
And we see really how hereditary mechanics can give rise to apparently non-local behaviour.
OK, and again, we've seen the very close connexion with the play's mechanics,
and we're now developing a hydrodynamic quantum field theory informed by this hydrodynamic system and for those doubters out there.
I am often, of course, we're going to have the question period over drinks, which will make it much more civil,
I'm sure, but people say, well, these paradoxes have been around for a century.
What makes you think they'll ever be resolved? And I think it's worth reminding people of the paradoxes in fluid mechanics.
So Delaware's paradox stood for a hundred and fifty years until the resolution of the wiskus boundary layer by parental.
And this walking droplet system for me suggests that there's an unresolved dynamics
on the Compton scale resolution of which will put Spade to the quantum paradoxes.
OK, so I wanted to thank all of my research group, I now there are people working on this all over the place.
And of course, I wanted to thank in particular Yves Crowder, who sadly passed a couple of months ago.
He was a brilliant man and a wonderful. Wonderful scientists, so thank you all for your attention.