1 00:00:18,160 --> 00:00:27,140 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, 2 00:00:27,140 --> 00:00:32,890 my father was on sabbatical at Oxford and we lived in White America, which is just a few miles away. 3 00:00:32,890 --> 00:00:42,280 I'm hoping to return there tomorrow and in the backyard. I used to play soccer all the time with my brothers. 4 00:00:42,280 --> 00:00:46,060 And I think this is the first scientific problem. I thought about it. 5 00:00:46,060 --> 00:00:51,820 How does this happen? Right? I know. Are there air fluctuations? 6 00:00:51,820 --> 00:00:58,390 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? 7 00:00:58,390 --> 00:01:07,880 Interesting. Don't know. That's another. 8 00:01:07,880 --> 00:01:10,850 Another problem I thought about was this one is everyone would. 9 00:01:10,850 --> 00:01:17,750 And here we see an example of an early hydrodynamic quantum analogue, more of these later. 10 00:01:17,750 --> 00:01:24,770 And so I thought, Well, maybe I'll start. I wasn't interested in the melting point of caesium or the standard model. 11 00:01:24,770 --> 00:01:29,330 I was interested in trying to understand what I saw around me, so I thought, maybe I'll go study physics. 12 00:01:29,330 --> 00:01:35,520 So I studied physics, and during my physics, four years of physics, I took zero course in fluid dynamics. 13 00:01:35,520 --> 00:01:39,860 So I had and still had no idea why a soccer ball bent why a stone skipped. 14 00:01:39,860 --> 00:01:45,890 But I took six courses in quantum mechanics where we really weren't encouraged to try and understand what we saw around us. 15 00:01:45,890 --> 00:01:52,310 Instead, we were invited to wallow in the mysteries of quantum mechanics. So that drove me to fluid mechanics. 16 00:01:52,310 --> 00:02:00,380 And it turns out that in the states, fluid mechanics is represented primarily in engineering departments or happily applied maths departments. 17 00:02:00,380 --> 00:02:06,410 And if this is your guiding light, if you dig deeply enough into any physical system, you find mathematics. 18 00:02:06,410 --> 00:02:12,410 For example, here in the heart of the sunflower, if you count the number of scrolls going clockwise and counterclockwise, 19 00:02:12,410 --> 00:02:19,970 you always find adjacent numbers in the Fibonacci sequence, which is one one two three five eight 13. 20 00:02:19,970 --> 00:02:31,640 And so you can, of course, find mathematics anywhere, whether it's in the heart of the sunflower on the pavement, stone outside you. 21 00:02:31,640 --> 00:02:36,230 You're therefore pretty much free to study anything you want, so I sort of specialise in surface tension. 22 00:02:36,230 --> 00:02:38,310 So this is a feature of an interface. 23 00:02:38,310 --> 00:02:46,580 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. 24 00:02:46,580 --> 00:02:53,870 We can think of surface tension as imparting an effective surface pressure. 25 00:02:53,870 --> 00:02:58,550 So a tensile force per unit length along the interface. And as a result, you can. 26 00:02:58,550 --> 00:03:02,930 Small objects can float on the surface of water, even if they're heavy relative to the water. 27 00:03:02,930 --> 00:03:06,140 So here we have a paperclip suspended by surface tension. 28 00:03:06,140 --> 00:03:15,470 Likewise for water striders, these small insects and the if they're gradients in surface tension because it's a force per unit length, 29 00:03:15,470 --> 00:03:21,470 you generate interfacial forces, which can drag things along. So this is a cocktail boat full of alcohol. 30 00:03:21,470 --> 00:03:25,320 Alcohol leaks at the back turns out, surface tension is a function of alcohol. 31 00:03:25,320 --> 00:03:33,020 It decreases with alcohol concentration, so the boat is then pulled along by the surface tension gradient. 32 00:03:33,020 --> 00:03:41,420 OK? And so these are the two main physical features of surface tension and mathematically and fluid dynamics. 33 00:03:41,420 --> 00:03:48,680 You're basically dissolving Newton's laws, so mass times acceleration equals the sum of the forces in your butt. 34 00:03:48,680 --> 00:03:54,620 You're writing this down forever for every infinitesimal blob within the fluid. 35 00:03:54,620 --> 00:03:59,660 So this is your mass times acceleration term. This is the inertial term and the short. 36 00:03:59,660 --> 00:04:07,220 The so a blob of fluid moves in response to pressure gradients, gravity and viscous stresses and surface tension enters into the boundary condition. 37 00:04:07,220 --> 00:04:13,940 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. 38 00:04:13,940 --> 00:04:19,250 And again, we can generate tangential stresses if there are gradients and surface tension 39 00:04:19,250 --> 00:04:23,900 as arise through chemical gradients as we've seen or temperature gradients. 40 00:04:23,900 --> 00:04:30,450 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. 41 00:04:30,450 --> 00:04:40,610 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? 42 00:04:40,610 --> 00:04:44,570 I went through, of course, my undergraduate actually went into fluid mechanics. 43 00:04:44,570 --> 00:04:49,400 I went through geophysics, and so after my share, I really didn't know what surface tension was. 44 00:04:49,400 --> 00:04:52,230 But this is because it's important on a small scale. 45 00:04:52,230 --> 00:04:58,190 OK, so it's if you have a blob of fluid, we know that we take a bucket of fluid reporter on the ground. 46 00:04:58,190 --> 00:05:04,010 We get a thin puddle. Basically, its extent is prescribed by surface tension. 47 00:05:04,010 --> 00:05:09,200 We have a small drop. It'll just sit there. So you basically have gravity trying to drive you outwards. 48 00:05:09,200 --> 00:05:13,460 So there's a hydrostatic pressure, which is Rogue A, if that is the characters. 49 00:05:13,460 --> 00:05:17,780 The scale of our blob surface tension wants to minimise the surface area. 50 00:05:17,780 --> 00:05:22,850 So it's acting. It wants to maintain the sphere city of the blob of fluid. 51 00:05:22,850 --> 00:05:27,230 So there's a balance between these two, and when you have a large volume of fluid, 52 00:05:27,230 --> 00:05:34,010 it will spread again out to a puddle with small surface tension winds, and it maintains its felicity. 53 00:05:34,010 --> 00:05:34,490 So basically, 54 00:05:34,490 --> 00:05:42,170 the relative magnitudes of these two forces gravity acting to cause spreading surface tension to resist it is prescribed by the bond number. 55 00:05:42,170 --> 00:05:50,650 And so if there's a critical length, which is so-called capillary length below which surface tension winds relative to gravity? 56 00:05:50,650 --> 00:05:53,150 OK, and so then again, we're in this scenario. 57 00:05:53,150 --> 00:05:59,830 And so this capillary length is about two millimetres, and coincidentally, this is the scale which sets the size of raindrops. 58 00:05:59,830 --> 00:06:05,760 OK, OK. So surface tension dominates the world of insects and microfluidics. 59 00:06:05,760 --> 00:06:10,770 Why it's been enjoying a resurgence in the engineering community. 60 00:06:10,770 --> 00:06:15,180 So the first thing I'm going to talk about is walking on water in the biological sphere. 61 00:06:15,180 --> 00:06:23,140 So it turns out many creatures have learnt to walk on water for various reasons, in particular for foraging on the surface, avoiding predators. 62 00:06:23,140 --> 00:06:31,980 So this is true. So you have insects, spiders, reptiles, birds and even mammals that can support themselves on the water surface. 63 00:06:31,980 --> 00:06:36,360 And so I thought to try and classify how they actually do it. 64 00:06:36,360 --> 00:06:44,940 So again, surface tension is important for small creatures, so you can have static weight support if the object is small enough. 65 00:06:44,940 --> 00:06:54,720 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. 66 00:06:54,720 --> 00:07:02,460 So again, there's a ratio of the weight of the object relative to the surface tension force generated by deflecting the interface. 67 00:07:02,460 --> 00:07:08,790 And if this is less than once, the object can stand at rest on the interface, as does this water strider, 68 00:07:08,790 --> 00:07:15,300 which is about a centimetre long, large object such as the lizard can't reside at rest. 69 00:07:15,300 --> 00:07:19,300 They would sink unless they hammer the surface with their feet. 70 00:07:19,300 --> 00:07:25,620 OK, so you can basically divide creatures into small and large again, small being. 71 00:07:25,620 --> 00:07:34,260 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. 72 00:07:34,260 --> 00:07:40,380 So small creatures reside, it can reside at rest. The larger creatures rely on dynamic weight support. 73 00:07:40,380 --> 00:07:44,700 OK, so we can then say, OK, aesthetics is easy. How do they actually propel themselves? 74 00:07:44,700 --> 00:07:49,470 And I think this kind of constructive. So this is the mathematicians view of a water walking creature. 75 00:07:49,470 --> 00:07:58,140 It's a sphere with characteristic size, a cross-sectional area capital A. And it's hitting the surface at speed you so 76 00:07:58,140 --> 00:08:02,880 you can ask what forces it can use to contribute to its lateral propulsion. 77 00:08:02,880 --> 00:08:10,440 So you have a buoyancy term simply because there is a hydrostatic pressure across which it can push it when it generates a cavity. 78 00:08:10,440 --> 00:08:14,970 Then there's a form drag, and this is the force you feel when you stick your hand out of a cardboard, 79 00:08:14,970 --> 00:08:19,200 so it's proportional to your speed squared times the area of your hand. 80 00:08:19,200 --> 00:08:23,950 Then there is a basically an added mass term. You have a wiskus term, then you have the surface tension terms. 81 00:08:23,950 --> 00:08:31,260 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. 82 00:08:31,260 --> 00:08:35,550 And then we have the possibility of surface tension gradients propelling bugs. 83 00:08:35,550 --> 00:08:42,120 And so if you if all of these terms are important, things are going to be a mess. 84 00:08:42,120 --> 00:08:49,350 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. 85 00:08:49,350 --> 00:08:53,580 I think what's interesting if you ask the question which force are used by which creature? 86 00:08:53,580 --> 00:08:55,710 Every single force is used by some creature. 87 00:08:55,710 --> 00:09:05,010 So there's some sort of principle suggesting any mechanism that can promote life will be exploited by nature. 88 00:09:05,010 --> 00:09:09,630 OK, so here are some of the big guys that don't. 89 00:09:09,630 --> 00:09:18,450 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. 90 00:09:18,450 --> 00:09:21,720 But they're slapping the surface, generating a cavity, pushing across the back. 91 00:09:21,720 --> 00:09:28,480 So they're using form, drag and hydrostatic pressures to drive themselves along. 92 00:09:28,480 --> 00:09:32,930 And here's a new one, by the way, how do I make this OK? 93 00:09:32,930 --> 00:09:38,710 He doesn't mute. There's some override. OK, so here's another one. 94 00:09:38,710 --> 00:09:50,690 They can walk on water. Perfect. 95 00:09:50,690 --> 00:09:55,960 Yeah, right. OK, so there we go. The skittering frog, so that's sort of at the cusp. 96 00:09:55,960 --> 00:09:59,980 It's too big for a surface tension, but it uses it to a certain extent. 97 00:09:59,980 --> 00:10:09,870 And now we have smaller objects. So these water striders stroke the surface and throw water backwards in the form of vortices. 98 00:10:09,870 --> 00:10:13,710 And so there was a paradox saying that these things couldn't move again. 99 00:10:13,710 --> 00:10:17,910 Another theme of this talk is paradoxes. When you see paradoxes in a field, 100 00:10:17,910 --> 00:10:24,660 you are put off the problem because people explain to you in your field how to think about something in the wrong way. 101 00:10:24,660 --> 00:10:33,360 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. 102 00:10:33,360 --> 00:10:37,810 And then I have subsequently seen some of them resolved. I said, Well, why didn't I look at that? 103 00:10:37,810 --> 00:10:43,080 So I really think paradoxes should be an invitation to work on a problem, not the converse in biology. 104 00:10:43,080 --> 00:10:48,190 There are lots of paradoxes. For example, there was, I think it was Grey's Paradox, 105 00:10:48,190 --> 00:10:52,730 which says that a dolphin can't swim as quickly as it does and say, Well, what was that based on? 106 00:10:52,730 --> 00:10:58,590 And so they did an experiment in which they took a more plaster cast of a dolphin 107 00:10:58,590 --> 00:11:03,030 and dragged it behind a boat and found that it had anomalously high drag. 108 00:11:03,030 --> 00:11:07,170 So of course, one might conclude that flexibility is important. 109 00:11:07,170 --> 00:11:11,850 So in other paradoxes, saying that a bumblebee can't fly? Of course we know it can. 110 00:11:11,850 --> 00:11:16,770 So I think working in biology was useful in pointing out when there paradoxes in a field, 111 00:11:16,770 --> 00:11:21,630 it's because people in that field are looking at things the wrong way. OK, so this is it. 112 00:11:21,630 --> 00:11:28,440 So this is a photo of these water striders, which you just saw in this sort of vertical weight wake of the water strider. 113 00:11:28,440 --> 00:11:34,950 So we basically sprinkle the surface with a blue dye. And this is the pattern it leaves behind. 114 00:11:34,950 --> 00:11:39,780 It turns out that we gave it a yellow brush to give it a little bit of Van Gogh. 115 00:11:39,780 --> 00:11:43,950 All right. And so here's another creature that uses surface tension for propulsion. 116 00:11:43,950 --> 00:11:50,880 It basically arches its back in order to match the curvature of the meniscus, which it's trying to climb. 117 00:11:50,880 --> 00:11:58,170 So it's trying to climb a meniscus from right to left, and it does so again just by arching its back. 118 00:11:58,170 --> 00:12:08,220 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. 119 00:12:08,220 --> 00:12:14,580 So we basically distorts the interface, pulls up on the interface and climbs upwards. 120 00:12:14,580 --> 00:12:20,430 So again, it's acting to minimise the surface energy of the system. OK? 121 00:12:20,430 --> 00:12:26,370 And here's one that uses the Marigny forces basically dumps a lipid onto the surface. 122 00:12:26,370 --> 00:12:28,920 This changes the surface tension and propels it towards the shore. 123 00:12:28,920 --> 00:12:36,600 So it's not a very efficient mechanism, but this is used by something which which doesn't habitually walk on water. 124 00:12:36,600 --> 00:12:40,530 It's getting an emergency propulsion device. 125 00:12:40,530 --> 00:12:48,270 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 126 00:12:48,270 --> 00:12:52,350 mechanisms available to to water walking creatures are used by some creature, 127 00:12:52,350 --> 00:12:58,620 and which mechanism they use is determined by their size. 128 00:12:58,620 --> 00:13:07,560 OK, so so I was happily working on this sort of thing. And then suddenly you could air introduced me to the following problem. 129 00:13:07,560 --> 00:13:15,120 So this was so part two that was just a preamble, an excuse to show some nice videos. 130 00:13:15,120 --> 00:13:24,350 But so the problem this is nice and that it took me back to my roots in physics and my frustration with physics. 131 00:13:24,350 --> 00:13:32,600 OK, so we're taught in as undergraduates that the universe is sort of in homogeneous on a in a philosophical sense, 132 00:13:32,600 --> 00:13:39,620 that is to say the macroscopic world is deterministic. You have particles following trajectories described by Newton's laws. 133 00:13:39,620 --> 00:13:43,160 If you have the initial conditions, you can predict the outcome. 134 00:13:43,160 --> 00:13:49,700 So quantum physics, we are taught, conversely, that the microscopic world is intrinsically probabilistic. 135 00:13:49,700 --> 00:13:53,360 So particle trajectories are not described only the statistics. 136 00:13:53,360 --> 00:13:59,130 So you can only predict the probability of a particular outcome in a quantum system. 137 00:13:59,130 --> 00:14:05,750 OK, and of course, there are those that dissented. Einstein in particular. 138 00:14:05,750 --> 00:14:09,050 His most famous dictum is this and God does not play dice. 139 00:14:09,050 --> 00:14:15,510 But what he really objected to was the concept of quantum non-local avi, which is to say super luminal signalling. 140 00:14:15,510 --> 00:14:24,810 So of course, having derived developed relativity, this was anathema to him, and that will be another theme of this talk today. 141 00:14:24,810 --> 00:14:32,370 OK, so so we're going to see how non-local behaviour can be mis inferred from a local hereditary system. 142 00:14:32,370 --> 00:14:41,370 OK. Let me dig into that a little bit. So hidden variable theories or theories which seek a rational dynamics underlying quantum statistics. 143 00:14:41,370 --> 00:14:46,770 So we know that quantum mechanics correctly predicts the statistical behaviour of microscopic systems. 144 00:14:46,770 --> 00:14:50,670 But what's the dynamics? That's really still an outstanding question. 145 00:14:50,670 --> 00:15:00,060 So if we were to characterise these hidden variables, so that's to say those variables that would describe the trajectory of microscopic particles, 146 00:15:00,060 --> 00:15:05,340 we could then restore our rationality to the microscopic physics. 147 00:15:05,340 --> 00:15:08,640 OK, so a brief history. 148 00:15:08,640 --> 00:15:15,300 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 149 00:15:15,300 --> 00:15:22,590 based on the notion of a pilot wave theory where you have a particle moving in concert with its with the guiding wave, 150 00:15:22,590 --> 00:15:31,500 OK? And so to play in the nineteen twenty six proposed a double wave pilot wave theory of quantum dynamics. 151 00:15:31,500 --> 00:15:37,560 So again, so as particle moving in resonance with the guiding wave and then this sort of sense. 152 00:15:37,560 --> 00:15:44,640 And on the basis of which she drives several things in particular particles h hbk, a number of the cornerstones of quantum mechanics, 153 00:15:44,640 --> 00:15:50,550 the Einstein deployed relation, which will come to you later, later and on the basis of his physical picture. 154 00:15:50,550 --> 00:15:54,930 Schrodinger derived shorting his equation and also Klein Gordon. 155 00:15:54,930 --> 00:16:02,070 And so really, it motivated the development of the mathematical formulation of quantum mechanics. 156 00:16:02,070 --> 00:16:06,540 Nevertheless, this physical picture was discarded in favour of nothing. 157 00:16:06,540 --> 00:16:14,100 OK, so and this is as quite sensible view suffered a setback when von Neumann in nineteen thirty two, 158 00:16:14,100 --> 00:16:17,910 was alleged to have proven that there can be no hidden variable theory. 159 00:16:17,910 --> 00:16:24,990 And this bomb then and that stood for twenty years until David Bohm presented a single wave pilot wave theory, 160 00:16:24,990 --> 00:16:33,180 which was consistent with the predictions of quantum mechanics. It has some difficulties again, some of which were pointed out by Joe Keller. 161 00:16:33,180 --> 00:16:38,760 But this boom's contribution was important because it attracted John Bell, 162 00:16:38,760 --> 00:16:46,590 who subsequently went through and discredited the impossibility proofs, and which led to the following conclusion. 163 00:16:46,590 --> 00:16:51,300 OK, so we're going to come back and talk a little about Bell's there at the end. 164 00:16:51,300 --> 00:16:59,670 OK, so fast forward now. Almost a century in two thousand five, if there discovered a hydrodynamic pilot wave system, 165 00:16:59,670 --> 00:17:04,050 it's really a macroscopic realisation of the sort of mechanics that the BCCI had imagined. 166 00:17:04,050 --> 00:17:12,330 OK. And lo and behold, it exhibits several features of quantum systems that were thought to be exclusive to the microscopic quantum realm. 167 00:17:12,330 --> 00:17:21,470 In particular, the statistical behaviour of this macroscopic classical system looks very much like that of quantum systems in certain cases. 168 00:17:21,470 --> 00:17:27,010 OK, so the questions raised, what are the key dynamical features responsible for this quantum like behaviour? 169 00:17:27,010 --> 00:17:31,380 What are the potential limitations of this high dynamic system as a quantum analogue? 170 00:17:31,380 --> 00:17:38,160 And the big question that I want to address today is my memory account for quantum known locality. 171 00:17:38,160 --> 00:17:49,650 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. 172 00:17:49,650 --> 00:17:53,820 If you have the governing equations and the initial conditions, you can predict its outcome. 173 00:17:53,820 --> 00:17:59,430 It's like a record being played. We know what's going to happen for all time. 174 00:17:59,430 --> 00:18:05,460 We realise that predictability in classical systems, if they're sufficiently complex, 175 00:18:05,460 --> 00:18:12,570 is limited because complex systems are their behaviour is sensitive to initial conditions. 176 00:18:12,570 --> 00:18:21,600 OK, so there are practical limitations to predictability even in classical systems, more complicated, still hereditary systems. 177 00:18:21,600 --> 00:18:28,560 So these are systems in which. Which are in which the system evolution is dependent on its history. 178 00:18:28,560 --> 00:18:37,440 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. 179 00:18:37,440 --> 00:18:40,980 But well, we'll see how this comes up in the hydrodynamic case. 180 00:18:40,980 --> 00:18:46,870 But basically, in order to predict the future, you have to know not only the initial conditions, but the system's past. 181 00:18:46,870 --> 00:18:55,590 OK, so this is a very rich class of dynamical systems, and we see in this particular one the bouncing droplet system, 182 00:18:55,590 --> 00:19:01,620 which we'll come to now local hereditary mechanics as a spatially local territory. 183 00:19:01,620 --> 00:19:06,300 Mechanics can give rise to apparently non-local behaviour. OK, so this is the system. 184 00:19:06,300 --> 00:19:12,180 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. 185 00:19:12,180 --> 00:19:15,870 So we have a bath about this big. We drive it around 50 hertz. 186 00:19:15,870 --> 00:19:25,920 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. 187 00:19:25,920 --> 00:19:35,430 And so when the vibrational acceleration exceeds a value, which is around four g, you get a standing field of sub harmonic Faraday waves. 188 00:19:35,430 --> 00:19:43,680 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. 189 00:19:43,680 --> 00:19:51,450 Benjamin and Arsal. OK, so and it turns out if you have this vibrating bath, you can levitate drops. 190 00:19:51,450 --> 00:20:00,990 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. 191 00:20:00,990 --> 00:20:05,880 And effectively, the interface behaves like a trampoline because of surface tension. 192 00:20:05,880 --> 00:20:12,180 And there's no coalescence because the thin air layer doesn't have time to drain during the impact. 193 00:20:12,180 --> 00:20:17,550 OK. So this thing again, bouncing 50 times per second. 194 00:20:17,550 --> 00:20:24,900 And remarkably, what could air if discovered by accident actually is that there's this corner of 195 00:20:24,900 --> 00:20:30,990 parameter space where these drops bouncing drops become unstable translational motion. 196 00:20:30,990 --> 00:20:34,860 So if you look carefully, you can see that this thing is actually landing on the side of its wave. 197 00:20:34,860 --> 00:20:40,140 So each time it lands, it gets a horizontal impulse. It's of being slowly nudged from left to right. 198 00:20:40,140 --> 00:20:45,150 OK, so a key feature of this system is the resonance between the particle and the wave. 199 00:20:45,150 --> 00:20:57,750 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. 200 00:20:57,750 --> 00:21:03,090 And so this concept of a particle moving in its own way field is throughout physics. 201 00:21:03,090 --> 00:21:09,120 But the theoretical treatment thereof is generally limited, even in electromagnetism. 202 00:21:09,120 --> 00:21:15,480 The Lawrence Dirac equation, which which tries to describe a charge moving in its own electromagnetic field, has runaway solutions. 203 00:21:15,480 --> 00:21:22,270 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. 204 00:21:22,270 --> 00:21:26,280 OK. And so again, the system is non-market Jovian. 205 00:21:26,280 --> 00:21:35,310 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. 206 00:21:35,310 --> 00:21:40,070 So it's a local local in space. The whole system, of course, is local. 207 00:21:40,070 --> 00:21:48,700 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. 208 00:21:48,700 --> 00:21:53,070 So this is, we can think in terms of path, memory as you've called it. 209 00:21:53,070 --> 00:22:01,320 And so the extent of the hereditary dynamics of the importance of memory is determined by how close you are to the farraday threshold. 210 00:22:01,320 --> 00:22:04,440 So at the farraday threshold, again, you excite waves. 211 00:22:04,440 --> 00:22:10,650 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. 212 00:22:10,650 --> 00:22:14,340 But it locally generates Faraday waves, which are damped in time. 213 00:22:14,340 --> 00:22:19,470 As you approach the very threshold, these waves are more persistent. 214 00:22:19,470 --> 00:22:24,190 And so you have to integrate further backwards in time to predict the behaviour of the system. 215 00:22:24,190 --> 00:22:32,100 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, 216 00:22:32,100 --> 00:22:36,420 and this is exactly the physical picture that the boy had, he said. Particles move along. 217 00:22:36,420 --> 00:22:46,860 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. 218 00:22:46,860 --> 00:22:52,490 OK, so we drive the maths of the equations describing the dynamics, 219 00:22:52,490 --> 00:22:59,220 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. 220 00:22:59,220 --> 00:23:06,840 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. 221 00:23:06,840 --> 00:23:13,710 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. 222 00:23:13,710 --> 00:23:21,510 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. 223 00:23:21,510 --> 00:23:27,360 So as the this is, the vibrational acceleration is that approaches that this thing gets large, OK? 224 00:23:27,360 --> 00:23:32,700 And that's basically a damping time, and that's the very period. 225 00:23:32,700 --> 00:23:42,540 OK, so again, you have this trajectory equation and now in the limit where the vertical dynamics is fast relative the horizontal dynamics, 226 00:23:42,540 --> 00:23:50,340 you can approximate this infinite sum by an integral and you have a nice integral differential equation that you can analyse. 227 00:23:50,340 --> 00:23:55,770 In particular, you can look for a static solution. 228 00:23:55,770 --> 00:24:00,900 You can look for circular orbital solutions, steady walking solutions and assess their stability. 229 00:24:00,900 --> 00:24:12,450 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, 230 00:24:12,450 --> 00:24:18,840 so we've looked at the stability of these states, the pairs just came out and we're looking at lattices. 231 00:24:18,840 --> 00:24:23,970 So it's really quite rich. Mathematically, there's a lot to be done. 232 00:24:23,970 --> 00:24:31,800 And also, they're dynamic bound states where the drops are joined by their common pilot wave field. 233 00:24:31,800 --> 00:24:37,980 So here we see them locking into circular orbits. And here we see them moving in this fashion. 234 00:24:37,980 --> 00:24:43,590 So if this looks completely bizarre because we have strobe and we strobe slightly off the bouncing frequencies, 235 00:24:43,590 --> 00:24:46,890 so they appear to be sort of gliding in and out. 236 00:24:46,890 --> 00:24:56,490 OK, so and here's a serious something done by Stuart Thompson, who is a recent graduate of of this department. 237 00:24:56,490 --> 00:25:03,180 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, 238 00:25:03,180 --> 00:25:07,500 the farraday threshold, a critical dynamic threshold. 239 00:25:07,500 --> 00:25:12,600 You generate a solitary wave which propagates around the ring. 240 00:25:12,600 --> 00:25:18,260 And this is really a realisation of a total L.A., which is a model of crystal vibrations. 241 00:25:18,260 --> 00:25:24,950 So you get some sort of some nice mathematical connexions to solid state physics. 242 00:25:24,950 --> 00:25:30,530 This is now a free ring, so the drops are free to move anywhere in certain regimes and move readily. 243 00:25:30,530 --> 00:25:39,320 Here we see this, but this binary oscillation of two lattices. And so what's going to happen here? 244 00:25:39,320 --> 00:25:44,920 10 drops, 20 drops and where's your money, where's your money? 245 00:25:44,920 --> 00:25:57,860 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, 246 00:25:57,860 --> 00:26:03,620 if you look at their motion in various force fields, so now we have a drop moving in a rotating frame. 247 00:26:03,620 --> 00:26:08,900 So this is first looked at in Eve's group, you expect four drop moves. 248 00:26:08,900 --> 00:26:14,060 So if a particle moves at uniform speed in a rotating frame that it will follow an inertial orbit 249 00:26:14,060 --> 00:26:20,930 where you have a balance between the centripetal outwards force and the inwards Coriolis force. 250 00:26:20,930 --> 00:26:25,250 And this is what so you basically expect the radius, the orbit to decrease with rotation rate. 251 00:26:25,250 --> 00:26:31,340 That's what happens at low memory when the wave field is not particularly important as you go to high memory, 252 00:26:31,340 --> 00:26:35,510 as is the case here, you see the drops has to navigate its own way field. 253 00:26:35,510 --> 00:26:42,380 So it's basically exciting potential and it's only stable in the troughs of this wave field. 254 00:26:42,380 --> 00:26:52,930 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. 255 00:26:52,930 --> 00:26:59,980 OK, and so owing to the identical form of the Coriolis force acting on a mass M in a uniformly 256 00:26:59,980 --> 00:27:06,520 rotating frame and the Laurence Force acting on a Charge Q and a uniform magnetic field, 257 00:27:06,520 --> 00:27:11,890 you can draw the analogy between these inertial quantised inertial orbits and larmer levels. 258 00:27:11,890 --> 00:27:20,850 Here, the quantisation length is again the Faraday wavelength that's playing the role of the display length in the quantum system. 259 00:27:20,850 --> 00:27:30,270 OK. I've also looked at the motion of a drop in a central, simple harmonic oscillator potential. 260 00:27:30,270 --> 00:27:36,490 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. 261 00:27:36,490 --> 00:27:42,480 So it basically gets dragged back towards the centre or so the various possibilities and turns out. 262 00:27:42,480 --> 00:27:48,000 So this is the radius of the orbit, which is a proxy for the energy of the system. 263 00:27:48,000 --> 00:27:53,790 Then you have the angular momentum and so you actually get a double quantisation, which is reminiscent of that in quantum mechanics. 264 00:27:53,790 --> 00:28:00,630 So here we see circular orbits. Here we have these lamis gates, which have zero angular momentum. 265 00:28:00,630 --> 00:28:04,800 Then we have more elaborate tree foils and so forth. 266 00:28:04,800 --> 00:28:13,590 And so you see as this we've seen in this and other systems, you have these quantised periodic orbits. 267 00:28:13,590 --> 00:28:23,460 And when the system becomes chaotic, they basically switch between them, giving rise to sort of multi-modal quantum statistics. 268 00:28:23,460 --> 00:28:28,380 And so I'm not going to dwell on the orbital dynamics, but I think it's quite well understood. 269 00:28:28,380 --> 00:28:35,560 And you see that the statistics of chaotic pilot wave systems can look very much like those of quantum mechanics. 270 00:28:35,560 --> 00:28:43,950 OK, so we see that the quantisation arises because of the dynamic constraint imposed on the droplet by its monochromatic wave field. 271 00:28:43,950 --> 00:28:51,930 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. 272 00:28:51,930 --> 00:28:58,860 And again, when the system becomes chaotic, it switches between unstable periodic orbits. 273 00:28:58,860 --> 00:29:07,260 OK. And this notion of a drop surfing, it's self-generated potential is important, and we'll come back to that later. 274 00:29:07,260 --> 00:29:11,980 OK, so my first contribution on this. 275 00:29:11,980 --> 00:29:20,330 Was to address the one problem I remembered from my undergraduate course in quantum courses, in quantum mechanics, which is particle in a box. 276 00:29:20,330 --> 00:29:25,060 OK, so it was wonderful to come back to this problem because they've now been done experimentally. 277 00:29:25,060 --> 00:29:31,030 The other thing I didn't like about my training in quantum mechanics is they didn't compare their theory to experiments. 278 00:29:31,030 --> 00:29:39,370 Here's one where they have. So these are electrons zipping along on the surface of metal and the electrons see inside. 279 00:29:39,370 --> 00:29:49,450 So there are some 20 or 30 electrons. I think inside are confined by this, by these atoms. 280 00:29:49,450 --> 00:29:54,110 So they're bouncing around in the waves. You see, there are basically the probability density function. 281 00:29:54,110 --> 00:30:03,620 So these you basically solve Helmholtz equation or stationary shorting equation and you solve for the modes of the cavity. 282 00:30:03,620 --> 00:30:08,020 OK. And the wavelength you see there is the wavelength. OK. 283 00:30:08,020 --> 00:30:11,980 And so I said, Oh, can we do that with walkers? Surely not. 284 00:30:11,980 --> 00:30:16,810 OK. And the student who did this, Dan Harris is outstanding experimentalists. 285 00:30:16,810 --> 00:30:23,650 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, 286 00:30:23,650 --> 00:30:28,100 and you actually see speed variations on the Faraday wavelength. 287 00:30:28,100 --> 00:30:33,760 OK, so this is sort of classic feature of the system, which were the importance of which we're beginning to explore now. 288 00:30:33,760 --> 00:30:35,260 So you let the thing run and run and run. 289 00:30:35,260 --> 00:30:44,920 It's a very chaotic motion, but you see the emergence of statistics which look very much like those in the quantum system. 290 00:30:44,920 --> 00:30:54,640 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. 291 00:30:54,640 --> 00:31:01,210 A recurring theme is how coherent wave like statistics can emerge from chaotic pilot wave dynamics. 292 00:31:01,210 --> 00:31:06,100 OK, and so here we have we've revisited this with an elliptical cross of Pedro. 293 00:31:06,100 --> 00:31:11,050 Sainz has done this, and it's a very robust result. 294 00:31:11,050 --> 00:31:17,590 Here we see a speed map and you see that there. Basically, the speed, the mean speed is a function of position, 295 00:31:17,590 --> 00:31:24,580 and it's because of this dependence that the one has the emerging quantum light statistics. 296 00:31:24,580 --> 00:31:32,030 And in this study, we noticed something interesting, which is that the we actually measured the mean wave field. 297 00:31:32,030 --> 00:31:39,580 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. 298 00:31:39,580 --> 00:31:45,880 So this is the particle histogram. This is the average wage field, and you can see they take precisely the same form. 299 00:31:45,880 --> 00:31:53,350 So then we proved that this average Wakefield can be expressed as the convolution of the bounce away field. 300 00:31:53,350 --> 00:31:57,340 So that's the way field you get if the drop dispensing place and the histogram. 301 00:31:57,340 --> 00:32:02,860 So this particle is effectively navigating the mean way field of the cavity. 302 00:32:02,860 --> 00:32:08,120 In this mean way field of the cavity depends on the statistical behaviour, so it's a very strange potential. 303 00:32:08,120 --> 00:32:14,870 Right. So you have this drop again surfing a background wave whose form depends on its statistics. 304 00:32:14,870 --> 00:32:21,160 OK, and this is a sort of confusion which arises in Bromium mechanics, which we can see our way around now. 305 00:32:21,160 --> 00:32:29,680 OK. And this and this potential, if you like again, it is non-local. 306 00:32:29,680 --> 00:32:38,500 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. 307 00:32:38,500 --> 00:32:45,100 OK, and so we can then see in standard quantum mechanics you solve for the modes of the 308 00:32:45,100 --> 00:32:49,690 cavity and what do they actually do in comparing to experiments they just choose? 309 00:32:49,690 --> 00:32:55,180 They put the it's a superposition of modes and they are free to choose the coefficients on those modes. 310 00:32:55,180 --> 00:33:00,960 So in our system, we say, OK, we could do the same or we can say there's actually an underlying dynamics, 311 00:33:00,960 --> 00:33:12,670 and we can think of this underlying dynamics as b as being a trajectory in which which can be decomposed into various component parts, 312 00:33:12,670 --> 00:33:18,730 in particular periodic sub trajectories and the thing the drop is actually jumping between them. 313 00:33:18,730 --> 00:33:27,940 And most importantly, we see here that the notion of trajectory is not inconsistent with the emergent emergence of quantum statistics in this system. 314 00:33:27,940 --> 00:33:40,280 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. 315 00:33:40,280 --> 00:33:45,700 So we have the times of wave generation, which is the time scale of vibration of the particle effectively. 316 00:33:45,700 --> 00:33:52,030 Then you have this pilot wave dynamics, which is revealed by strobing at the bouncing frequency today. 317 00:33:52,030 --> 00:33:56,470 And then you have this long term statistical behaviour emerging. 318 00:33:56,470 --> 00:33:59,290 There's actually so again, if you look at some products there, 319 00:33:59,290 --> 00:34:04,210 there's a huge range of timescales in this problem because there's a timescale of emergence of the main pilot wave. 320 00:34:04,210 --> 00:34:10,210 That's the timescale of emergence of the statistics. And, of course, the dynamical timescales as well. 321 00:34:10,210 --> 00:34:17,230 OK, so what does this remind us of? Well, it turns out it's very similar to the brain's mechanics. 322 00:34:17,230 --> 00:34:22,090 So this is what Frank will take calls a poem in two lines. 323 00:34:22,090 --> 00:34:28,060 This is the Einstein, the relation. So relativity requires this. 324 00:34:28,060 --> 00:34:31,930 Quantum mechanics requires this if you equate the two. 325 00:34:31,930 --> 00:34:36,050 You see that a particle of mass m must have a natural frequency. 326 00:34:36,050 --> 00:34:44,890 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. 327 00:34:44,890 --> 00:34:51,970 And he said that this was the frequency of oscillation of a particle and there was at this energy. 328 00:34:51,970 --> 00:34:55,990 At this frequency, there's an exchange between rest, mass energy and field energy. 329 00:34:55,990 --> 00:35:04,900 Again, this basically being rest mass energy. This being field energy. So he said particles or oscillators with this high frequency, 330 00:35:04,900 --> 00:35:13,660 they're then generating a wave and riding that wave and then somehow you it will give rise to quantum statistics, OK? 331 00:35:13,660 --> 00:35:20,230 But so notice that there are two waves in his theory, there's the wave centred on the particle, which is pushing it around. 332 00:35:20,230 --> 00:35:24,910 And then there's the emergent statistical form. OK. And that's very much like ours. 333 00:35:24,910 --> 00:35:29,830 And so we have the PSI, which is the standard wave in quantum mechanics. 334 00:35:29,830 --> 00:35:33,820 But then you have the particles entered the high wave, OK? 335 00:35:33,820 --> 00:35:38,440 And this is generated by the particle vibration at this constant frequency. 336 00:35:38,440 --> 00:35:42,410 And he said that the waves are solutions of the Klein Gordon equation. 337 00:35:42,410 --> 00:35:47,050 So that's just the relativistic form of shorting. 338 00:35:47,050 --> 00:35:51,730 This equation is actually very nice because it looks very much like the water wave equation. 339 00:35:51,730 --> 00:35:58,030 And so from this formulation, you said particles move perpendicular to the surfaces of constant phase. 340 00:35:58,030 --> 00:36:03,880 So that and that from that, if you have a monochromatic wave, it gives you this peak was a spark. 341 00:36:03,880 --> 00:36:10,700 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. 342 00:36:10,700 --> 00:36:15,700 So the particle oscillates in resonance with its guiding wave. 343 00:36:15,700 --> 00:36:21,400 So there you have to consider the fact you have a moving clock, but its frequency is prescribed by its mass. 344 00:36:21,400 --> 00:36:27,330 And so the two cancel beautifully, and he thought that he called this and gone weidler not to. 345 00:36:27,330 --> 00:36:33,060 He thought this is a very important point, and it's critical in our system the resonance between the particle and its wave. 346 00:36:33,060 --> 00:36:40,360 OK. And so really, he imagined something very close to what we have now in the lab. 347 00:36:40,360 --> 00:36:42,880 You have high frequency isolation at the constant frequency. 348 00:36:42,880 --> 00:36:52,840 You have an intermediate pilot wave dynamics in which the particle surfs along its gliding wave and explores its background potential. 349 00:36:52,840 --> 00:36:58,090 And then you have the emergent long term statistics described by standard quantum theory so well, 350 00:36:58,090 --> 00:37:04,300 he was unable to do was to show how this would emerge from this. 351 00:37:04,300 --> 00:37:08,440 And of course, the question arises What is the pilot wave? 352 00:37:08,440 --> 00:37:16,240 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. 353 00:37:16,240 --> 00:37:22,060 This is nice in that it it has a spectral form which has a bar in it. 354 00:37:22,060 --> 00:37:30,460 So the energy in. So basically, you just have electromagnetic background noise, which is then interacts with the particle. 355 00:37:30,460 --> 00:37:36,370 But so the energy in a mode with frequency omega is h by omega over two. 356 00:37:36,370 --> 00:37:45,790 So the idea then, is that the object interacts with this background field generates a pilot wave and off you go. 357 00:37:45,790 --> 00:37:51,910 And you can think of it in terms of if you have a turbulent fluid and you throw a spring into the turbulent fluid. 358 00:37:51,910 --> 00:37:58,660 It will start oscillating in its natural frequency or pump energy into the fluid at that frequency. 359 00:37:58,660 --> 00:38:06,310 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. 360 00:38:06,310 --> 00:38:09,160 OK, so then you can do a map and notice. 361 00:38:09,160 --> 00:38:15,700 I mean, it has the right number of variables, which is beautiful if you draw the map between the body's mechanics, 362 00:38:15,700 --> 00:38:19,750 complemented by a steady and the walker system. 363 00:38:19,750 --> 00:38:27,010 So again, the balancing is playing the role of this Zetterberg on this high frequency oscillation in the brain's mechanics. 364 00:38:27,010 --> 00:38:34,630 You have this the harmony of phases as assured by this residents commission in our system. 365 00:38:34,630 --> 00:38:38,860 Surface tension basically plays the role of each park because we have capillary waves. 366 00:38:38,860 --> 00:38:45,760 This is the way parameter and quantum mechanics the and all of the analogues we've seen. 367 00:38:45,760 --> 00:38:48,730 The Faraday wavelength is playing the role of the deployed wavelength. 368 00:38:48,730 --> 00:38:53,650 But there's a second wavelength in the plays mechanics, which is the Compton wavelength. 369 00:38:53,650 --> 00:38:59,170 So this is basically the scale of vibration. And so in our system, it's basically this length. 370 00:38:59,170 --> 00:39:04,160 So the length, the step length in the bouncing. And of course, if you strobe over it, that thing's gone. 371 00:39:04,160 --> 00:39:14,420 Just like all consideration of the Compton, length is gone. If you go from relativistic mechanics, quantum mechanics to non relativistic. 372 00:39:14,420 --> 00:39:20,540 OK, so we can play a few games here, so we know that this system is driven. 373 00:39:20,540 --> 00:39:22,820 So basically. So in terms of, I should mention, 374 00:39:22,820 --> 00:39:30,680 the energetic so this system is of course driven and that's then playing the role of the quantum vacuum in deploys mechanics. 375 00:39:30,680 --> 00:39:33,350 But we also know that there is state a steady state. 376 00:39:33,350 --> 00:39:39,680 So there's steady walking states, their periodic states in which the driving the dissipation balance in our system. 377 00:39:39,680 --> 00:39:46,640 So we can imagine that there might be some invested like description, some Hamiltonian like description of our system. 378 00:39:46,640 --> 00:39:51,400 So let's imagine let's look at the system and pretend we don't know that it's driven this. 379 00:39:51,400 --> 00:39:58,190 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. 380 00:39:58,190 --> 00:40:01,970 And for that matter, what trajectory equation would we infer? 381 00:40:01,970 --> 00:40:11,670 OK, so if you look at the limit of weak droplet acceleration, you get the following equation. 382 00:40:11,670 --> 00:40:19,790 So basically you do a asymptotic expansion of the wave force and it contributes to two to two terms on the left hand side. 383 00:40:19,790 --> 00:40:27,170 So basically, this is some classical potential, but the way force is giving rise to basically something equivalent to a boost factor. 384 00:40:27,170 --> 00:40:31,640 So there's an added mass associated with the wave field. 385 00:40:31,640 --> 00:40:36,510 And there's also a nonlinear drag which always drives the walker towards its free walking speed. 386 00:40:36,510 --> 00:40:39,050 OK, so this is the free walking speed here. 387 00:40:39,050 --> 00:40:45,830 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. 388 00:40:45,830 --> 00:40:54,360 You get the Envisat mechanics of a particle whose mass depends on its speed, which just looks, of course, like relativity. 389 00:40:54,360 --> 00:41:02,370 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, 390 00:41:02,370 --> 00:41:06,330 but it doesn't scatter, it scatters, but in this very strange way. 391 00:41:06,330 --> 00:41:10,950 So it's actually follows a logarithmic spiral. OK. 392 00:41:10,950 --> 00:41:20,070 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. 393 00:41:20,070 --> 00:41:25,710 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. 394 00:41:25,710 --> 00:41:32,730 Let's just infer the effective force required to cause this walker to move along this logarithmic spiral. 395 00:41:32,730 --> 00:41:37,980 And let's imagine that that force is generated by this pillar. OK, so what is that force? 396 00:41:37,980 --> 00:41:42,180 We can infer it and it takes the form, so we use our booster equation again. 397 00:41:42,180 --> 00:41:48,870 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. 398 00:41:48,870 --> 00:41:56,790 So it's a force which is proportional to its velocity time cross with its instantaneous angular velocity around the post. 399 00:41:56,790 --> 00:42:04,620 So if we use this analogue between again the Coriolis force and the Lorenz force that we saw before, you can see this. 400 00:42:04,620 --> 00:42:08,310 This looks like a an example of self induction. 401 00:42:08,310 --> 00:42:12,070 So it's like a charged feeling the magnetic field associated with its own current. 402 00:42:12,070 --> 00:42:19,290 OK, so this and more importantly, so this is still puzzling me, and I think there's something very deep there. 403 00:42:19,290 --> 00:42:26,710 I've been puzzling over over it for years, as Darren knows when we first saw this, but. 404 00:42:26,710 --> 00:42:32,230 You can see how this wave mediated force gives rise to spooky action at a distance. 405 00:42:32,230 --> 00:42:36,850 If you don't know that there's a wave field, you're going to infer a non-local force. 406 00:42:36,850 --> 00:42:40,920 OK? Right, so here's another example. 407 00:42:40,920 --> 00:42:45,240 So, Fidel oscillations is something that arises in quantum mechanics if you have an impurity 408 00:42:45,240 --> 00:42:49,140 again in your electron sea of electrons zipping around on the surface of a metal. 409 00:42:49,140 --> 00:42:54,300 So we have a localised disturbance there, and they basically solve this with scattering theory. 410 00:42:54,300 --> 00:42:59,970 And this is effectively the duprey wavelength in their system. 411 00:42:59,970 --> 00:43:04,080 And so let's try it in our. So now we're going to operate in shallow water. 412 00:43:04,080 --> 00:43:08,970 Turns out these things, all these things can walk in half a millimetre. 413 00:43:08,970 --> 00:43:13,320 And we have here a deep well. OK, so that's a region of high excitability. 414 00:43:13,320 --> 00:43:18,630 It turns out these things spiral in towards the well, and as they exit the well, their speed is modulated. 415 00:43:18,630 --> 00:43:24,840 OK. So if we let the thing run and run and run, these are actually experiments. 416 00:43:24,840 --> 00:43:32,580 So the drop is attracted towards the well, waves are excited in the well and the waves modulated speed. 417 00:43:32,580 --> 00:43:41,340 So I noted before, there's this tendency for the drop to oscillate in speed along its direction of motion with the Faraday wavelength. 418 00:43:41,340 --> 00:43:48,120 So this is a sort of roots of a statistical signature with the Faraday wavelength. 419 00:43:48,120 --> 00:43:56,760 OK, and so you see then the this is the speed map consists of concentric circles, and of course, 420 00:43:56,760 --> 00:44:03,390 that gives rise to a PDF, which looks very much like that in the analogue quantum problem. 421 00:44:03,390 --> 00:44:07,200 The wavelength here is, of course, the Faraday wavelength. 422 00:44:07,200 --> 00:44:11,970 OK. So once again, we can conclude that the fidele oscillations, 423 00:44:11,970 --> 00:44:19,550 which is the quantum phenomenon or is not these are not inconsistent with the notion of particle trajectories. 424 00:44:19,550 --> 00:44:28,640 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, 425 00:44:28,640 --> 00:44:36,740 in the final isolation, some of their more more qualitative and of course, their limitations because this is a hydrogen amick system. 426 00:44:36,740 --> 00:44:42,280 And we, of course, expect it to be quite different from that in quantum mechanics. 427 00:44:42,280 --> 00:44:48,070 So there are significant difference. Significant differences of measurement process is not intrusive. 428 00:44:48,070 --> 00:44:52,540 Details of the wave behaviour are different. 429 00:44:52,540 --> 00:45:00,490 Spin states are unstable in our system, but this is just invite us to look at a more generalised class of pilot wave theories. 430 00:45:00,490 --> 00:45:07,060 So if we, for example, if we take our Strober scalloping model and we know dimensionalize it, they've just two dimensional as groups. 431 00:45:07,060 --> 00:45:12,850 One basically prescribes the amplitude of the waves, and this is the particle inertia relative to the waves. 432 00:45:12,850 --> 00:45:20,710 And so we can ask. So it turns out that this parameter is limited between 0.6 and 1.4 in the experiments. 433 00:45:20,710 --> 00:45:25,100 What happens if it's 10 to the minus six? What happens if it's 10 to 16? 434 00:45:25,100 --> 00:45:32,440 It turns out if if we set this to zero actually, or even make it say half the value we see in the lab, 435 00:45:32,440 --> 00:45:36,280 we can stabilise spin states, which is a drop zipping around its own way. 436 00:45:36,280 --> 00:45:42,040 OK. And so we're free to explore this generalised pi wave framework and act and ask 437 00:45:42,040 --> 00:45:46,540 which quantum features can we capture in which regions of parameter space? 438 00:45:46,540 --> 00:45:55,120 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, 439 00:45:55,120 --> 00:45:59,800 you go to the debris limit where you have gradient driven motion and then large wave amplitude. 440 00:45:59,800 --> 00:46:06,700 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. 441 00:46:06,700 --> 00:46:10,180 You can add a stochastic forcing, but that would seem cheating to me. 442 00:46:10,180 --> 00:46:21,700 And so we're basically just developing this growing catalogue of quantum analogues and and trying to make connexions with the quantum mechanics. 443 00:46:21,700 --> 00:46:27,250 And again, I think a very interesting direction. So these are these are spin lattices, and this works very nicely. 444 00:46:27,250 --> 00:46:35,830 You get transitions which you can prompt by rotation quantitative comparisons with the equations of stochastic electrodynamics, 445 00:46:35,830 --> 00:46:39,280 the Lorentz Dirac equation, which is the charge moving its own field. 446 00:46:39,280 --> 00:46:49,980 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. 447 00:46:49,980 --> 00:46:57,690 Looking at what we call hydrodynamic quantum field theory, so this is revisiting the brain's mechanics informed by the high dynamic system. 448 00:46:57,690 --> 00:47:04,430 OK, so we're basically doing now what the brain would have done if he'd had MATLAB. 449 00:47:04,430 --> 00:47:13,070 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. 450 00:47:13,070 --> 00:47:18,230 And so you have a particle which is exciting at that frequency, you're going to generate a pilot wave, OK? 451 00:47:18,230 --> 00:47:25,610 So it's really very much like a system. We treat a particle as a perturbation at the resident frequency of the equation localised, 452 00:47:25,610 --> 00:47:30,500 and we look at gradient driven motions were suggested by the play. 453 00:47:30,500 --> 00:47:37,400 And when you do this, the thing spontaneously starts moving in a straight line at peak was HBK. 454 00:47:37,400 --> 00:47:42,390 OK. So if nothing else, I feel that we understand the free particle and quantum mechanics. 455 00:47:42,390 --> 00:47:46,530 But when you study the free particle in quantum mechanics, this is their treatment pickles. 456 00:47:46,530 --> 00:47:51,470 Okay, so now we see if you treat a particle as an oscillation in a field. 457 00:47:51,470 --> 00:47:57,830 You get that for free. And moreover, the particle actually oscillates about that means speed. 458 00:47:57,830 --> 00:48:03,200 So that's its mean speed. It's oscillating at the company, frequency at the wavelength. 459 00:48:03,200 --> 00:48:10,490 So that gives you then a mechanism for a statistical signature with the typical wavelength, as we saw in our experiments and various others, 460 00:48:10,490 --> 00:48:17,090 the fidele oscillations, you have oscillations with the wavelength of the pilot wave that will give rise to statistics. 461 00:48:17,090 --> 00:48:21,860 OK. And so moreover, this so we're just getting started here. 462 00:48:21,860 --> 00:48:27,140 Notice, by the way, if you have this these oscillations with the degree wavelength, 463 00:48:27,140 --> 00:48:31,160 then if you have circular orbits, they have to satisfy the more Sommerfeld condition. 464 00:48:31,160 --> 00:48:35,780 All right. So we're just warming up here, but there's much to be done. 465 00:48:35,780 --> 00:48:44,530 And again, I think this this sort of dynamics suggests that the uncertainty relations are really indicative of an unresolved dynamics on the comp, 466 00:48:44,530 --> 00:48:47,930 which scale the scale of these fast oscillations. OK. 467 00:48:47,930 --> 00:48:57,660 And so the question which is normally. As to me, so I've decided to be pre-emptive is. 468 00:48:57,660 --> 00:49:08,000 What about Bell's theorem, so. Bell driving inequality, which people interpret as saying there can be no hidden variable theory, 469 00:49:08,000 --> 00:49:16,870 so closer inspection suggests this doesn't have any bearing on systems in which particles interact with the background field. 470 00:49:16,870 --> 00:49:21,830 OK. And I think more importantly, so even if you don't believe that Bell himself, 471 00:49:21,830 --> 00:49:27,080 who drive does inequality and saw it violated by the experiments of Alan Asprey, 472 00:49:27,080 --> 00:49:32,150 came to the conclusion that there must be some pilot wave dynamics underlying quantum mechanics. 473 00:49:32,150 --> 00:49:36,920 And he sort of sided with the bohemian interpretation, which I haven't had time for. 474 00:49:36,920 --> 00:49:44,440 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. 475 00:49:44,440 --> 00:49:53,770 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. 476 00:49:53,770 --> 00:50:02,570 So these are just examples of where you could missing fur non quantum non locality from this from our system. 477 00:50:02,570 --> 00:50:04,390 So, for example, wave function collapse. 478 00:50:04,390 --> 00:50:12,190 If we insisted that this was the complete description of our physical state of our physical system, then the act of observation, 479 00:50:12,190 --> 00:50:17,890 which would reveal the drop to be at one particular point would cause this wave function to collapse instantaneously to a point. 480 00:50:17,890 --> 00:50:26,200 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. 481 00:50:26,200 --> 00:50:29,350 Likewise, we've looked at the slit in double slit experiments. 482 00:50:29,350 --> 00:50:37,420 So it turns out in the double slit, the drop through will feel the second slit by virtue of its spatially extended pilot wave. 483 00:50:37,420 --> 00:50:43,180 And we've seen this mean pilot wave potential, which is effectively a non-local potential. 484 00:50:43,180 --> 00:50:47,380 And also Andre Nash, Aspen has looked at these drops interacting. 485 00:50:47,380 --> 00:50:51,880 These are done numerically through a sort of resonant cavity. 486 00:50:51,880 --> 00:50:58,960 So these drops talk to each other by virtue of their common way field so they can become either perfectly synchronised or chaotic, 487 00:50:58,960 --> 00:51:06,910 but with identical statistical signatures. So we're we're now looking at entanglement measures in hydrodynamic quantum analogue. 488 00:51:06,910 --> 00:51:15,550 So to conclude, we've seen that this system provides a means to explore the boundaries between classical and quantum systems. 489 00:51:15,550 --> 00:51:24,520 And it certainly extends the range of classical mechanics to include systems which have statistical behaviour reminiscent of quantum systems. 490 00:51:24,520 --> 00:51:32,090 And we see really how hereditary mechanics can give rise to apparently non-local behaviour. 491 00:51:32,090 --> 00:51:36,130 OK, and again, we've seen the very close connexion with the play's mechanics, 492 00:51:36,130 --> 00:51:45,950 and we're now developing a hydrodynamic quantum field theory informed by this hydrodynamic system and for those doubters out there. 493 00:51:45,950 --> 00:51:51,320 I am often, of course, we're going to have the question period over drinks, which will make it much more civil, 494 00:51:51,320 --> 00:51:58,370 I'm sure, but people say, well, these paradoxes have been around for a century. 495 00:51:58,370 --> 00:52:06,100 What makes you think they'll ever be resolved? And I think it's worth reminding people of the paradoxes in fluid mechanics. 496 00:52:06,100 --> 00:52:13,760 So Delaware's paradox stood for a hundred and fifty years until the resolution of the wiskus boundary layer by parental. 497 00:52:13,760 --> 00:52:17,480 And this walking droplet system for me suggests that there's an unresolved dynamics 498 00:52:17,480 --> 00:52:24,640 on the Compton scale resolution of which will put Spade to the quantum paradoxes. 499 00:52:24,640 --> 00:52:31,600 OK, so I wanted to thank all of my research group, I now there are people working on this all over the place. 500 00:52:31,600 --> 00:52:39,220 And of course, I wanted to thank in particular Yves Crowder, who sadly passed a couple of months ago. 501 00:52:39,220 --> 00:53:08,378 He was a brilliant man and a wonderful. Wonderful scientists, so thank you all for your attention.