1 00:00:01,133 --> 00:00:04,133 Let's go for 2 00:00:05,266 --> 00:00:08,266 super 820. 3 00:00:10,166 --> 00:00:10,500 All right. 4 00:00:10,500 --> 00:00:14,100 So thank you all for being here on a Saturday morning. 5 00:00:14,100 --> 00:00:17,100 So this morning I'll tell you about nonlinear 6 00:00:17,100 --> 00:00:20,100 dynamics of, active particles. 7 00:00:20,133 --> 00:00:23,300 So I've got, two key phrases there in my talk. 8 00:00:23,633 --> 00:00:26,633 We have nonlinear dynamics and active particles. 9 00:00:26,700 --> 00:00:31,333 So what I'll do is I'll start by giving you an overview of each of these topics. 10 00:00:31,566 --> 00:00:35,333 So first I will tell you something about active particles and active matter. 11 00:00:35,733 --> 00:00:41,200 And then I'll tell you some ideas from nonlinear dynamical systems. 12 00:00:41,700 --> 00:00:46,700 And after that, we'll go over a couple of examples from my own work 13 00:00:47,033 --> 00:00:51,300 where we try to understand the dynamics of these active particles 14 00:00:51,300 --> 00:00:54,300 using these ideas from nonlinear dynamical systems. 15 00:00:54,833 --> 00:00:58,000 So we'll we'll explore two simple theoretical models. 16 00:00:58,266 --> 00:01:01,433 So the first one will be an active particle, 17 00:01:01,666 --> 00:01:04,833 flowing in a fluid flow along a channel. 18 00:01:05,400 --> 00:01:07,933 And the second would be, 19 00:01:07,933 --> 00:01:11,033 I'll introduce you to walking and super walking droplets. 20 00:01:11,400 --> 00:01:14,566 And then after the talk, as you make your head towards the coffee, 21 00:01:14,700 --> 00:01:16,233 I have a little demo of these super 22 00:01:16,233 --> 00:01:18,300 walking droplets so you can have a bit of a play around 23 00:01:18,300 --> 00:01:19,566 with these droplets on your own. 24 00:01:22,100 --> 00:01:22,466 All right. 25 00:01:22,466 --> 00:01:27,000 So so let me start by telling you what I mean by an active particle. 26 00:01:27,766 --> 00:01:30,933 So the best way to do this is by showing you some examples. 27 00:01:30,933 --> 00:01:33,766 But before I do that, let me give you a working definition 28 00:01:33,766 --> 00:01:36,733 that is generally used by researchers in this area. 29 00:01:36,733 --> 00:01:42,266 So an active particle is an entity which consumes energy from the environment 30 00:01:42,266 --> 00:01:46,633 and converts it into some sort of directed and persistent motion. 31 00:01:47,433 --> 00:01:47,733 Right. 32 00:01:47,733 --> 00:01:50,400 So in doing this, it also dissipates some energy. 33 00:01:50,400 --> 00:01:53,400 So these are out of thermodynamic equilibrium. 34 00:01:53,533 --> 00:01:55,966 So these are non-equilibrium entities. 35 00:01:55,966 --> 00:01:58,166 And they come in all shapes and sizes. 36 00:01:58,166 --> 00:02:02,300 And they spend a wide range of scales all the way from nanometer 37 00:02:02,300 --> 00:02:05,633 and micrometer to the scales of several meters. 38 00:02:05,633 --> 00:02:06,100 Right. 39 00:02:06,100 --> 00:02:10,766 So at the nano and microscale, you have molecular motors 40 00:02:10,766 --> 00:02:17,033 that do processes that maintain, life within these subcellular scale processes. 41 00:02:17,200 --> 00:02:20,366 And then you can think of cell itself as an active particle 42 00:02:20,466 --> 00:02:24,000 and other microorganisms as well, such as bacteria. 43 00:02:24,533 --> 00:02:27,166 So here is a video of a type 44 00:02:27,166 --> 00:02:30,166 of white blood cell, chasing a bacteria. 45 00:02:30,566 --> 00:02:31,133 Right. 46 00:02:31,133 --> 00:02:36,100 So it consumes energy, by eating food in the form of chemical energy. 47 00:02:36,100 --> 00:02:39,100 And it's converting it into this crawling sort of motion here. 48 00:02:40,033 --> 00:02:43,766 Another example of a microorganism are these E coli bacteria. 49 00:02:44,033 --> 00:02:46,566 So these are immersed in a fluid medium. 50 00:02:46,566 --> 00:02:51,300 And as you see they are self propelling using their helical tails. 51 00:02:51,300 --> 00:02:54,300 And they are doing running and tumble kind of motion. 52 00:02:54,300 --> 00:02:58,700 So they will run in one direction for a while and then reorient themselves. 53 00:02:59,233 --> 00:03:02,233 So, so these were some of examples from the from the living world 54 00:03:02,300 --> 00:03:03,233 of active particles. 55 00:03:03,233 --> 00:03:07,600 You can also have synthetic systems, and synthetic active particles as well. 56 00:03:08,000 --> 00:03:11,100 So so one example here are these millimeter size walking 57 00:03:11,100 --> 00:03:14,100 droplets, which we'll explore in detail a bit more later. 58 00:03:14,400 --> 00:03:15,800 But to to give you an overview. 59 00:03:15,800 --> 00:03:19,933 Now these are droplets that walk horizontally while bouncing 60 00:03:20,100 --> 00:03:21,733 on a vibrating liquid. But 61 00:03:23,266 --> 00:03:24,366 and then another 62 00:03:24,366 --> 00:03:28,066 synthetic system are these, millimeter sized robots. 63 00:03:28,200 --> 00:03:28,466 Right. 64 00:03:28,466 --> 00:03:32,033 So when you have robots, doing some sort of self propulsion, right. 65 00:03:32,033 --> 00:03:36,500 Either preprogramed or sensing its environment, it is an active particle. 66 00:03:36,800 --> 00:03:41,100 And then at the scale of, meters, we have big mammals, right? 67 00:03:41,100 --> 00:03:43,833 Birds, fish and even other cells. Right. 68 00:03:43,833 --> 00:03:46,566 So we are in some sense active particles as well. 69 00:03:46,566 --> 00:03:50,700 We eat our food and then convert that into some sort of motion all the way around. 70 00:03:51,633 --> 00:03:51,900 Right. 71 00:03:51,900 --> 00:03:54,566 So so that's active particles. 72 00:03:54,566 --> 00:03:57,666 Now active matter is matter 73 00:03:57,833 --> 00:04:01,333 composed of a large collection of these active particles. 74 00:04:01,800 --> 00:04:02,100 Right. 75 00:04:02,100 --> 00:04:06,900 So like you have your passive phases of matter solids liquids and gas 76 00:04:06,900 --> 00:04:11,366 where the constituent elements are atoms and subatomic structures. 77 00:04:11,700 --> 00:04:15,133 For these active matter, the constituent elements 78 00:04:15,133 --> 00:04:18,133 are active particles that I showed you on the last slide. 79 00:04:18,500 --> 00:04:22,666 So since these active particles are out of equilibrium, 80 00:04:22,666 --> 00:04:26,266 the phases you get of matter when you have lots of these active particles, 81 00:04:26,600 --> 00:04:29,566 they are also non-equilibrium phases. 82 00:04:29,566 --> 00:04:33,066 So let me show you some examples here of active matter. 83 00:04:33,666 --> 00:04:37,366 So a classic example is flocking of birds. 84 00:04:37,366 --> 00:04:37,600 Right. 85 00:04:37,600 --> 00:04:41,100 So when you have a large collection of birds such as starlings 86 00:04:41,366 --> 00:04:44,366 they form these beautiful murmuration patterns. 87 00:04:44,466 --> 00:04:48,533 And early on in this area, there were quite a few models 88 00:04:48,533 --> 00:04:51,533 developed to, to describe some of these behaviors using, 89 00:04:51,866 --> 00:04:54,600 these ideas of active particles and active matter. 90 00:04:55,733 --> 00:04:56,100 You can 91 00:04:56,100 --> 00:04:59,866 get this, this type of flocking behavior also at the scale of cells. 92 00:05:00,033 --> 00:05:03,833 So what you are seeing in that video, are epithelial cells. 93 00:05:03,833 --> 00:05:06,400 So these are the scales that you have on your skin. 94 00:05:06,400 --> 00:05:08,800 And if you have a cat, let's say, on your skin, 95 00:05:08,800 --> 00:05:11,833 then you might see something like this, a motion like this, where 96 00:05:11,833 --> 00:05:15,233 all the cells collectively try to migrate and heal the wound. 97 00:05:15,600 --> 00:05:15,866 Right. 98 00:05:15,866 --> 00:05:20,700 So so you can have collective behaviors arising at the scale of cells, as well. 99 00:05:21,900 --> 00:05:25,800 A common feature that arises when you have these active particles 100 00:05:26,100 --> 00:05:29,833 is you get, motility induced phase separation. 101 00:05:30,166 --> 00:05:34,733 So here the interaction between the particles is purely repulsive. 102 00:05:34,900 --> 00:05:37,733 And yet you see these clustered phases forming. 103 00:05:37,733 --> 00:05:42,166 And the gist of it is that if two active particles come near each other, 104 00:05:42,600 --> 00:05:43,333 these are active. 105 00:05:43,333 --> 00:05:46,366 So they don't have to obey energy or momentum conservation. 106 00:05:46,366 --> 00:05:47,833 They are out of equilibrium. 107 00:05:47,833 --> 00:05:51,600 What happens is that they can hang around each other for longer 108 00:05:51,766 --> 00:05:54,933 unless one of them decides to turn away and move, right. 109 00:05:55,133 --> 00:05:59,200 So because of this, active particles slow down when they come near each other, 110 00:05:59,400 --> 00:06:01,666 and this gives rise to a positive feedback 111 00:06:01,666 --> 00:06:04,433 where because they have slowed down, more of them will aggregate. 112 00:06:04,433 --> 00:06:07,833 And you get these clustered phases, forming inactive matter 113 00:06:08,066 --> 00:06:12,033 just purely due to its persistent motion with repulsive interactions. 114 00:06:13,800 --> 00:06:14,166 And then 115 00:06:14,166 --> 00:06:18,366 lastly you get these spatial temporal chaotic behavior. 116 00:06:18,433 --> 00:06:22,600 So here you are seeing a video of a large collection of bacteria. 117 00:06:22,900 --> 00:06:27,000 And you see it's, it's behaving in a in a very chaotic fashion. 118 00:06:27,033 --> 00:06:29,633 So so we call this active turbulence. 119 00:06:29,633 --> 00:06:32,666 And this is different from the the fluid turbulence. 120 00:06:32,700 --> 00:06:35,700 You see for example in the atmosphere because they're 121 00:06:35,700 --> 00:06:38,700 the turbulence is due to the inertia of the fluid. 122 00:06:38,833 --> 00:06:43,433 While here at the scale of these bacteria, the fluid flow for them is very viscous. 123 00:06:43,766 --> 00:06:44,000 Right. 124 00:06:44,000 --> 00:06:48,000 So the fluid Reynolds number is very small, and it's the activity 125 00:06:48,000 --> 00:06:51,800 of these bacteria that's giving rise to these turbulent behaviors. 126 00:06:52,133 --> 00:06:52,300 Right. 127 00:06:52,300 --> 00:06:58,033 So you have these, emergent behaviors, that arise in these active systems. 128 00:06:58,233 --> 00:07:02,133 And I just like to point out it's it's in a sense very similar to the 129 00:07:02,133 --> 00:07:06,000 the condensed matter conundrum of physics, which was beautifully explained 130 00:07:06,000 --> 00:07:09,100 by Phil Anderson in his essay More is Different, right? 131 00:07:09,266 --> 00:07:12,933 So the properties and the behaviors you get emerging at a collective scale, 132 00:07:13,333 --> 00:07:15,166 they need different tools to be understood. 133 00:07:15,166 --> 00:07:16,400 And you can't just predict them from 134 00:07:16,400 --> 00:07:18,433 what's happening with the single particle level. 135 00:07:18,433 --> 00:07:20,633 And same is true in these active matter systems. Right? 136 00:07:20,633 --> 00:07:24,000 So you have these active particles, but then when you have lots of them, 137 00:07:24,000 --> 00:07:27,800 you can get these collective emergent behaviors at larger scales emerging. 138 00:07:30,233 --> 00:07:30,566 All right. 139 00:07:30,566 --> 00:07:35,266 So having introduced active particles and active matters, let me tell you something 140 00:07:35,266 --> 00:07:39,600 about, nonlinear dynamical systems before we move on to the two examples. 141 00:07:40,700 --> 00:07:44,833 So as theoretical physicist and also applied mathematicians, 142 00:07:45,066 --> 00:07:48,466 the dynamics that we see out there in the real world, 143 00:07:48,700 --> 00:07:52,366 we try to make sense of it by creating mathematical models. 144 00:07:52,633 --> 00:07:55,566 And much of the artistry of this process 145 00:07:55,566 --> 00:07:58,566 lies in identifying the relevant variables. 146 00:07:58,700 --> 00:08:02,700 That would model the system, or the process we are interested 147 00:08:02,700 --> 00:08:07,400 in, and write down equation that describe the evolution of these variables. 148 00:08:07,800 --> 00:08:08,200 Right. 149 00:08:08,200 --> 00:08:12,400 And more often than not, these equations, they turn out to be nonlinear. 150 00:08:12,666 --> 00:08:17,633 So if we have, a list of variables that are described by this vector x 151 00:08:17,800 --> 00:08:21,600 that we are interested in, then a dynamical system is a system 152 00:08:21,600 --> 00:08:24,500 that models the evolution of these variables in time. Right. 153 00:08:24,500 --> 00:08:27,566 So the dots there on top of the x is a time derivative. 154 00:08:27,933 --> 00:08:29,333 And many times 155 00:08:29,333 --> 00:08:33,066 these function on the right hand side they can be non-linear functions. 156 00:08:33,500 --> 00:08:33,733 Right. 157 00:08:33,733 --> 00:08:35,333 So you can have product of variables 158 00:08:35,333 --> 00:08:38,833 or the variables inside the argument of a non-linear function. 159 00:08:39,333 --> 00:08:43,100 So it is often useful to go away from the picture 160 00:08:43,100 --> 00:08:44,766 of what these variables are describing. 161 00:08:44,766 --> 00:08:48,966 And consider an abstract space which is made up of all these variables. 162 00:08:48,966 --> 00:08:52,533 It's usually called the state space or the phase space of the system. 163 00:08:52,700 --> 00:08:55,766 So it's the space of all possible states of the system. 164 00:08:56,100 --> 00:08:58,800 And the evolution of a system would correspond 165 00:08:58,800 --> 00:09:01,933 to a trajectory evolving in this state space. 166 00:09:04,000 --> 00:09:05,666 Now, there are many ways you 167 00:09:05,666 --> 00:09:09,200 can classify dynamical systems and nonlinear dynamical systems. 168 00:09:09,466 --> 00:09:13,133 But the, the way the one which would be useful for today's talk 169 00:09:13,133 --> 00:09:16,233 is the classification into conservative 170 00:09:16,233 --> 00:09:19,233 and dissipative dynamical systems. 171 00:09:19,500 --> 00:09:22,800 So conservative dynamical systems are systems 172 00:09:23,200 --> 00:09:26,133 where the volume in phase space stays conserved. 173 00:09:26,133 --> 00:09:29,133 So if you start in this phase space with a box 174 00:09:29,300 --> 00:09:33,366 and as the system evolves, this box will stretch and can do complicated things, 175 00:09:33,600 --> 00:09:36,866 but its volume will stay conserved in a conservative systems. 176 00:09:37,266 --> 00:09:41,300 So, classical Hamiltonian systems, 177 00:09:41,300 --> 00:09:44,400 which are time independent, fall into this category of system. 178 00:09:44,400 --> 00:09:47,100 So generally when you have a closed system where you have some 179 00:09:47,100 --> 00:09:51,233 some constants of motion, you usually have, conservative system. 180 00:09:51,400 --> 00:09:54,700 And a classic example is of a pendulum. 181 00:09:54,700 --> 00:09:55,200 Right. 182 00:09:55,200 --> 00:09:58,766 So if you when the pendulum is doing back and forth motion, if we go to the phase 183 00:09:58,766 --> 00:10:02,300 space picture and look at the position and the velocity coordinate, 184 00:10:02,566 --> 00:10:05,733 then what you get is what you are seeing here as this, this closed loop. 185 00:10:05,733 --> 00:10:06,533 Right. 186 00:10:06,533 --> 00:10:10,800 So back and forth motion would correspond to a closed loop in phase space. 187 00:10:10,800 --> 00:10:14,700 And if I take a patch in phase space, it will just go around this closed loop. 188 00:10:16,500 --> 00:10:19,500 Now if the system turns out to be non-linear, 189 00:10:19,500 --> 00:10:22,500 then you can get more complex and interesting things going on. 190 00:10:22,633 --> 00:10:25,733 And again, a classic example is your double pendulum. 191 00:10:26,066 --> 00:10:29,733 So with a double pendulum you have a four dimensional phase space. 192 00:10:29,733 --> 00:10:32,700 You have two coordinates and two momenta. 193 00:10:32,700 --> 00:10:33,966 And so what you're seeing here 194 00:10:33,966 --> 00:10:37,466 is for different projections of this phase space in 2D. 195 00:10:37,900 --> 00:10:41,733 And what you see is that in the initial patch you can see it's stretching. 196 00:10:41,733 --> 00:10:44,433 It's bending. It's folding in complicated ways. 197 00:10:44,433 --> 00:10:45,900 But again it's a conservative system. 198 00:10:45,900 --> 00:10:49,066 So the overall volume of that patch will stay conserved. 199 00:10:49,266 --> 00:10:50,900 And there is a very well established 200 00:10:50,900 --> 00:10:54,666 mathematical framework of understanding chaos in this type of systems. 201 00:10:54,666 --> 00:10:58,233 And conservative chaos or Hamiltonian chaos. 202 00:10:59,566 --> 00:11:03,800 Now, in contrast to these conservative systems we have dissipative system. 203 00:11:03,900 --> 00:11:06,600 So in this space the speed of systems, sorry, 204 00:11:06,600 --> 00:11:09,600 the phase space volume, it will shrink with time. 205 00:11:09,766 --> 00:11:10,066 Right. 206 00:11:10,066 --> 00:11:12,800 So the phase space volume shrinks here with time. 207 00:11:12,800 --> 00:11:17,900 And to give you again a simple example, if we add friction to our pendulum right 208 00:11:18,166 --> 00:11:22,133 then it will go back and forth, but then it will eventually come to a steady state. 209 00:11:22,133 --> 00:11:25,700 And in phase space, this would correspond to a spiraling motion 210 00:11:26,000 --> 00:11:27,533 towards a stable point. 211 00:11:29,000 --> 00:11:31,633 Now, dissipative system does 212 00:11:31,633 --> 00:11:33,900 not always mean that everything will just come to a stop. 213 00:11:33,900 --> 00:11:37,933 You can still get periodic motion and chaotic motion in dissipative systems 214 00:11:37,933 --> 00:11:38,866 as well. 215 00:11:38,866 --> 00:11:43,533 And again, a classic example is the the Lorenz chaotic system where 216 00:11:43,533 --> 00:11:46,766 where chaos was kind of first, reported 217 00:11:47,033 --> 00:11:50,100 and again, it's it's a dissipative system. 218 00:11:50,100 --> 00:11:53,466 But as you see, all initial conditions, 219 00:11:53,466 --> 00:11:58,500 they kind of converge on this fractal shaped attractor and a volume in phase 220 00:11:58,500 --> 00:12:02,500 space would again stretch, fold bending, different ways. 221 00:12:02,700 --> 00:12:06,033 Even though the, the total volume, would be shrinking with time. 222 00:12:06,866 --> 00:12:07,133 Right. 223 00:12:07,133 --> 00:12:11,033 So, so using these, ideas from conservative systems 224 00:12:11,033 --> 00:12:14,600 and dissipative system, what we'll see that, the examples will explore today. 225 00:12:15,033 --> 00:12:16,033 One of them will be, 226 00:12:16,033 --> 00:12:19,266 will form a conservative system and one will form a dissipative system. 227 00:12:19,500 --> 00:12:22,000 And with dissipative system, usually if you have an open system 228 00:12:22,000 --> 00:12:25,100 where you have energy injection and energy dissipation, usually 229 00:12:25,100 --> 00:12:28,200 they will fall into this category of dissipative dynamical systems. 230 00:12:30,466 --> 00:12:30,800 All right. 231 00:12:30,800 --> 00:12:34,600 So with that, let's move on to the, the first example, 232 00:12:34,600 --> 00:12:38,533 which is an active particle, flowing in a unit 233 00:12:38,533 --> 00:12:41,533 direction of the fluid, slow. 234 00:12:44,200 --> 00:12:48,366 So going back to our active particles that I introduced in the first slide, 235 00:12:48,366 --> 00:12:49,633 if you have an active particle 236 00:12:49,633 --> 00:12:53,400 which is immersed in a fluid medium, it's called a micro swimmer. 237 00:12:53,400 --> 00:12:56,666 And again, you can have a, living, example 238 00:12:56,666 --> 00:12:59,733 or an artificial or a synthetic system. 239 00:12:59,966 --> 00:13:00,166 Right. 240 00:13:00,166 --> 00:13:04,900 So a bacteria, or other motile cells or even a micro robot 241 00:13:04,900 --> 00:13:08,633 immersed in a fluid medium would be, considered a micro swimmer. 242 00:13:09,666 --> 00:13:12,233 Now, micro swimmer, they in many situations, 243 00:13:12,233 --> 00:13:15,900 they experience fluid flows in confined environment. 244 00:13:16,233 --> 00:13:16,500 Right. 245 00:13:16,500 --> 00:13:20,533 So if, if a micro swimmer is flowing in a confined environment 246 00:13:20,533 --> 00:13:24,333 and you have fluid flow along one direction, that is a common situation. 247 00:13:24,333 --> 00:13:28,000 So for example, sperm cells moving in fallopian tubes, 248 00:13:28,266 --> 00:13:31,300 micro robots for targeted drug delivery 249 00:13:31,300 --> 00:13:34,300 application and even pathogens moving in blood vessels. 250 00:13:34,700 --> 00:13:37,166 So what we are going to do is we are going to 251 00:13:37,166 --> 00:13:42,600 look at a very simple model as theoretical physicist, and try to see 252 00:13:42,600 --> 00:13:46,833 what sort of complexity and dynamics does this give rise to in this situation. 253 00:13:47,233 --> 00:13:49,066 So so the model we have is 254 00:13:49,066 --> 00:13:53,266 as you see in the schematic, we have a three dimensional pipe, right? 255 00:13:53,266 --> 00:13:54,566 A three dimensional channel. 256 00:13:54,566 --> 00:13:57,766 And we have a fluid flow going from left to right. 257 00:13:58,100 --> 00:14:00,166 So fluid flow is uni directional. 258 00:14:00,166 --> 00:14:01,600 It's in one direction. 259 00:14:01,600 --> 00:14:03,733 However it's not uniform. Right. 260 00:14:03,733 --> 00:14:08,566 Because we have walls, the fluid right next to the wall will stick to the wall. 261 00:14:08,566 --> 00:14:12,066 So there will be a variation in the fluid flow profile across the channel. 262 00:14:12,266 --> 00:14:14,600 So you get this parabolic flow profile. 263 00:14:15,900 --> 00:14:18,900 And we are going to consider an active particle 264 00:14:18,900 --> 00:14:22,233 which is very unintelligent a very simple active particle. 265 00:14:22,500 --> 00:14:24,766 It moves at a constant speed. 266 00:14:24,766 --> 00:14:27,800 V naught in some direction ahead. Right. 267 00:14:27,800 --> 00:14:29,500 So that's all the active particles are doing. 268 00:14:29,500 --> 00:14:32,400 It's persistent motion in some direction. 269 00:14:32,400 --> 00:14:34,500 And we are going to make two key assumptions. 270 00:14:34,500 --> 00:14:36,166 The first is we are saying 271 00:14:36,166 --> 00:14:39,166 that the active particle is very small compared to the channel. 272 00:14:39,266 --> 00:14:42,900 So the disturbance is caused by the active particles are small, 273 00:14:42,900 --> 00:14:45,900 and it only experiences the fluid flowing along the channel. 274 00:14:46,200 --> 00:14:48,900 And also we are going to neglect any direct interaction 275 00:14:48,900 --> 00:14:51,300 between the particle and the walls. Right. 276 00:14:51,300 --> 00:14:54,133 So we are saying that the particle only experiences 277 00:14:54,133 --> 00:14:57,133 the fluid flow and the gradients in those fluid flow. 278 00:14:57,966 --> 00:15:01,166 So if you try to write down equations of motion for this particle, 279 00:15:01,166 --> 00:15:05,700 you get these two equations and I'll explain what they mean, step by step. 280 00:15:06,066 --> 00:15:10,800 So the first equation is telling you the total velocity of the particle. 281 00:15:11,100 --> 00:15:14,833 And we are saying it's the addition of its active velocity, 282 00:15:14,833 --> 00:15:17,633 which is a constant velocity in some direction. 283 00:15:17,633 --> 00:15:20,400 Plus this background's fluid flow. 284 00:15:20,400 --> 00:15:23,100 And you can do this linear superposition because we are in the lower 285 00:15:23,100 --> 00:15:23,900 Reynolds numbers. 286 00:15:23,900 --> 00:15:26,100 So the fluid flow equations are linear. 287 00:15:27,400 --> 00:15:29,366 And the second equation 288 00:15:29,366 --> 00:15:32,966 tells you how the orientation of the active particle evolves with time. 289 00:15:33,200 --> 00:15:36,900 Now because the fluid flow is not constant, different layers of fluid 290 00:15:36,900 --> 00:15:38,700 are moving at different velocities. 291 00:15:38,700 --> 00:15:42,966 So if you immerse something in in a in a flow like that, it will start rotating. 292 00:15:42,966 --> 00:15:44,700 The particles will start rotating. 293 00:15:44,700 --> 00:15:47,600 So the direction of motion of my particle 294 00:15:47,600 --> 00:15:50,600 will rotate because of the shear in the flow. 295 00:15:50,833 --> 00:15:51,066 Right. 296 00:15:51,066 --> 00:15:53,100 And that's what the second equation is describing 297 00:15:53,100 --> 00:15:56,766 that because of this local spinning motion, local vorticity of the fluid, 298 00:15:56,933 --> 00:15:59,933 you can get tumbling of the direction of motion. 299 00:15:59,933 --> 00:16:00,133 Right. 300 00:16:00,133 --> 00:16:02,366 So we have these two simple equations. 301 00:16:02,366 --> 00:16:05,133 Now let's try to write this in component forms. 302 00:16:05,133 --> 00:16:06,866 We are in three dimensions. 303 00:16:06,866 --> 00:16:09,300 So in component form we will get six. 304 00:16:09,300 --> 00:16:13,233 Equation three for the position of the particle three for its orientation. 305 00:16:13,866 --> 00:16:17,700 Now if you look at this equations you can see that the z equation 306 00:16:17,700 --> 00:16:21,333 which describes the motion along the channel the z variable. 307 00:16:21,333 --> 00:16:24,000 It decouples from the rest of the system. 308 00:16:24,000 --> 00:16:26,133 So we can integrate that separately. 309 00:16:26,133 --> 00:16:30,966 So effectively we only have five equations that are coupled to each other. 310 00:16:31,666 --> 00:16:31,900 Right. 311 00:16:31,900 --> 00:16:34,900 So we have a five dimensional face space here. 312 00:16:35,100 --> 00:16:38,766 Now it turns out that in this system you end up getting constants of motion. 313 00:16:38,766 --> 00:16:42,566 So quantities that will not change with time as the system evolves. 314 00:16:42,900 --> 00:16:46,300 And an obvious one, which you might have noticed 315 00:16:46,300 --> 00:16:49,900 is that the orientation vector has to be a unit vector. 316 00:16:50,300 --> 00:16:50,533 Right. 317 00:16:50,533 --> 00:16:53,800 So that constrains, the orientation variables. 318 00:16:53,966 --> 00:16:57,000 So the orientation vector can only lie on the surface of a sphere. 319 00:16:57,866 --> 00:17:01,500 And it turns out that you also get another constant of motion in the system 320 00:17:01,766 --> 00:17:03,633 which is described by this C. 321 00:17:03,633 --> 00:17:07,800 And it relates to the the fluid flow velocity U, 322 00:17:08,066 --> 00:17:12,066 and also the z component of the active particle velocity. 323 00:17:12,300 --> 00:17:13,500 One thing I forgot to mention 324 00:17:13,500 --> 00:17:16,500 is that, in these equations, which you see in component form, 325 00:17:16,633 --> 00:17:19,633 I've rescaled the active particle velocity, 326 00:17:19,700 --> 00:17:22,733 and it's kind of embedded in the fluid flow. 327 00:17:22,733 --> 00:17:24,633 So the active particle velocity is one 328 00:17:24,633 --> 00:17:27,866 and the fluid flow profile is modified to this U by. 329 00:17:28,566 --> 00:17:32,400 So we have five equations two constants of motion, which means that 330 00:17:32,400 --> 00:17:36,166 the effective dynamics of the system take place in three dimensions. 331 00:17:37,566 --> 00:17:39,000 And it turns out 332 00:17:39,000 --> 00:17:42,000 that this is an example of a conservative dynamical system. 333 00:17:42,000 --> 00:17:43,833 And you can show this mathematically 334 00:17:43,833 --> 00:17:46,833 by taking the the divergence of the dynamical flow. 335 00:17:47,600 --> 00:17:50,533 And surprisingly, and more importantly, 336 00:17:50,533 --> 00:17:54,433 it turns out you can map this system on to a Hamiltonian system. 337 00:17:54,833 --> 00:17:58,500 So you can do a 1 to 1 mapping of the system and write it in 338 00:17:58,500 --> 00:18:00,366 terms of Hamilton's equation. 339 00:18:00,366 --> 00:18:05,700 And in this mapping it turns out that, the system obeys Hamilton's equation. 340 00:18:05,700 --> 00:18:08,466 And your Hamiltonian looks like this. Right. 341 00:18:08,466 --> 00:18:11,833 So you have a kinetic energy part and the potential energy part. 342 00:18:12,200 --> 00:18:14,733 And the momenta which are in the kinetic energy part, 343 00:18:14,733 --> 00:18:17,666 turn out to be basically the velocity of the active particle 344 00:18:17,666 --> 00:18:21,300 in the cross-section of the channel and the potential energy part. 345 00:18:21,666 --> 00:18:24,666 It basically is related to the fluid flow U, 346 00:18:24,833 --> 00:18:28,433 and there is also a C in there which is a constant of motion. 347 00:18:28,433 --> 00:18:31,566 So your initial conditions of the system will tell you what the value is. 348 00:18:31,566 --> 00:18:31,866 There. 349 00:18:33,000 --> 00:18:33,233 Right. 350 00:18:33,233 --> 00:18:35,366 So after these equations 351 00:18:35,366 --> 00:18:40,233 let's see what the, what the system does when we try to simulate the system. 352 00:18:41,166 --> 00:18:43,933 So you see three different plots here. 353 00:18:43,933 --> 00:18:45,900 And I'm going to start a video in a second. 354 00:18:45,900 --> 00:18:50,266 But the plot on the left side is the top view of the 3D channel. 355 00:18:50,266 --> 00:18:55,600 So in the in the y z plane, the the plot in the middle 356 00:18:55,633 --> 00:18:59,233 is the cross-sectional view and the plot on the right. 357 00:18:59,233 --> 00:19:02,233 The panel on the right is showing the orientation of the particle. 358 00:19:02,800 --> 00:19:06,133 So if we start the particle near the center of the channel, 359 00:19:06,366 --> 00:19:09,000 then it will flow from left to right with the flow. 360 00:19:09,000 --> 00:19:12,300 However, its orientation is against the flow 361 00:19:12,600 --> 00:19:15,400 and it keeps on oscillating as you see there. 362 00:19:15,400 --> 00:19:15,600 Right? 363 00:19:15,600 --> 00:19:18,266 So you get this periodic motion of the particle. 364 00:19:18,266 --> 00:19:21,466 However, it stays near the center of this channel. 365 00:19:22,866 --> 00:19:23,766 On the other hand, 366 00:19:23,766 --> 00:19:27,133 if I start the particle away from the center of this channel, 367 00:19:27,600 --> 00:19:31,100 then you can see that it's doing this irregular behavior. 368 00:19:31,333 --> 00:19:32,433 It's still going from left 369 00:19:32,433 --> 00:19:36,266 to right with the flow, but its orientation is going all crazy. 370 00:19:36,466 --> 00:19:39,300 And it's also staying away from the center of the channel. 371 00:19:39,300 --> 00:19:41,733 Right. It's never visiting the center of the channel. 372 00:19:41,733 --> 00:19:44,400 So, so to make sense of why we see these different types 373 00:19:44,400 --> 00:19:47,733 of behaviors, let's go back to our Hamiltonian mapping. 374 00:19:47,733 --> 00:19:50,400 The Hamiltonian picture of the system. 375 00:19:50,400 --> 00:19:54,200 So in the Hamiltonian picture this is what the potential energy looked like. 376 00:19:54,533 --> 00:19:58,933 So if you look at the constant c in there is the potential energy. 377 00:19:59,200 --> 00:20:03,600 So the potential energy depends on two things the fluid flow profile and the c. 378 00:20:03,800 --> 00:20:07,666 The constant C is determined by the initial conditions of the system. 379 00:20:08,033 --> 00:20:11,566 So what that means is that depending on where the particle starts, 380 00:20:11,733 --> 00:20:15,766 it sees a different potential landscape available to explore. 381 00:20:16,200 --> 00:20:19,200 So if the particle starts near the center of the channel, 382 00:20:19,333 --> 00:20:22,433 then the potential landscape that is available to explore for 383 00:20:22,433 --> 00:20:25,433 the particle is something you see there on the top, left, right. 384 00:20:25,433 --> 00:20:28,500 So it only has this region, near 385 00:20:28,500 --> 00:20:32,100 the center of the channel to explore, and that's the corresponding trajectory. 386 00:20:32,566 --> 00:20:36,733 Whereas if the particle starts away from the channel, then the potential landscape 387 00:20:36,733 --> 00:20:39,633 it has available to explore has this annular shape. 388 00:20:39,633 --> 00:20:42,500 And so its trajectory is confined in this region. 389 00:20:42,500 --> 00:20:42,700 Right. 390 00:20:42,700 --> 00:20:47,133 So this, this mapping allows us to kind of understand why this particle 391 00:20:47,133 --> 00:20:51,000 is only exploring certain regions of the, of the channel. 392 00:20:52,433 --> 00:20:53,400 Now, to get a full 393 00:20:53,400 --> 00:20:56,400 appreciation of the complexity of behavior you get, 394 00:20:56,500 --> 00:20:59,700 here is a plot showing all the different behaviors you get 395 00:21:00,000 --> 00:21:03,400 depending on where the particles starts in the cross-section. 396 00:21:03,700 --> 00:21:05,033 So the top plot you see, 397 00:21:05,033 --> 00:21:08,800 with all the different colors are the qualitatively different types 398 00:21:08,800 --> 00:21:12,533 of trajectories that are obtained based on where the particle starts. 399 00:21:12,833 --> 00:21:15,433 So if the particle starts near the center of the channel, 400 00:21:15,433 --> 00:21:18,633 you have that green trajectory but stays near the center of the channel. 401 00:21:19,066 --> 00:21:23,533 If it starts in one of those light blue or dark blue regions, it will stay confined 402 00:21:23,533 --> 00:21:27,100 within one of those slabs or vertical or horizontal slab. 403 00:21:27,700 --> 00:21:30,133 If you start in one of these special yellow regions, 404 00:21:30,133 --> 00:21:33,133 you stay confined near the corner of the channel, 405 00:21:33,300 --> 00:21:36,300 in these purple regions, as we saw in the video, 406 00:21:36,300 --> 00:21:39,200 you will stay confined away from the center of the channel. 407 00:21:39,200 --> 00:21:42,900 And then these red trajectories is you can basically wander anywhere 408 00:21:43,166 --> 00:21:44,700 in the cross-section. 409 00:21:44,700 --> 00:21:47,133 Now the the bottom plot there is important. 410 00:21:47,133 --> 00:21:50,833 And this is a bit more interesting what you are seeing in the bottom plot. 411 00:21:50,833 --> 00:21:55,300 There is a measure of what's called the largest Lyapunov exponent. 412 00:21:55,300 --> 00:21:57,466 It's a measure of chaos in the system. 413 00:21:57,466 --> 00:22:01,800 So whenever you see a violet region there, the motion is regular, 414 00:22:01,933 --> 00:22:04,000 periodic or quasi periodic. 415 00:22:04,000 --> 00:22:06,766 And wherever you see this orangish regions, 416 00:22:06,766 --> 00:22:08,966 that's where the motion is chaotic. 417 00:22:08,966 --> 00:22:12,033 So what you see that is if you start near the center of the channel, 418 00:22:12,300 --> 00:22:15,300 the particle does periodic motion, regular motion. 419 00:22:15,300 --> 00:22:19,133 But it's as you go away from the center, you start getting these chaotic regions 420 00:22:19,133 --> 00:22:20,233 coming in right. 421 00:22:20,233 --> 00:22:23,766 And this transition from periodic to chaotic behavior can again 422 00:22:23,766 --> 00:22:27,533 be rationalized in terms of these nearly integrable Hamiltonian systems. 423 00:22:27,933 --> 00:22:28,166 Right? 424 00:22:28,166 --> 00:22:31,733 So that was just to give you a flavor of the the complexity of behavior 425 00:22:32,066 --> 00:22:36,000 that you can get for a very simple setup of an active particle interacting 426 00:22:36,000 --> 00:22:40,500 with a unidirectional flow, you get this very nonlinear dynamical system. 427 00:22:40,633 --> 00:22:41,933 But we can rationalize, 428 00:22:43,100 --> 00:22:45,466 why we get these different behaviors in terms 429 00:22:45,466 --> 00:22:48,466 of our understanding of these, nonlinear systems. 430 00:22:50,633 --> 00:22:51,000 All right. 431 00:22:51,000 --> 00:22:53,433 So let me move on to the second example. 432 00:22:53,433 --> 00:22:56,800 Now, of super walking droplets. 433 00:22:59,500 --> 00:22:59,766 All right. 434 00:22:59,766 --> 00:23:03,500 So if you take a container filled with liquid 435 00:23:04,066 --> 00:23:06,700 and you vibrate it vertically, 436 00:23:06,700 --> 00:23:11,633 then above some critical amplitude of vibration, the free surface of the liquid 437 00:23:11,966 --> 00:23:15,033 will no longer remain slide and it will become unstable. 438 00:23:15,033 --> 00:23:16,000 And you will get these 439 00:23:16,000 --> 00:23:19,000 these standing waves which are known as Faraday standing waves. 440 00:23:19,333 --> 00:23:21,000 And that's because these were first 441 00:23:21,000 --> 00:23:24,066 discovered by Michael Faraday back in 1831. 442 00:23:24,700 --> 00:23:24,900 Right. 443 00:23:24,900 --> 00:23:29,233 So you have a critical threshold below which your free surface 444 00:23:29,233 --> 00:23:32,933 remains flat, above which it becomes unstable to standing waves. 445 00:23:33,600 --> 00:23:37,933 Now, if we are just below this instability and we create a droplet 446 00:23:37,933 --> 00:23:42,066 of the same liquid as the bath, then this is what happens, 447 00:23:43,300 --> 00:23:43,600 right? 448 00:23:43,600 --> 00:23:48,966 So back in 2005, Eves, Cooper and colleagues in Paris discovered 449 00:23:48,966 --> 00:23:52,300 that if you take a droplet, of silicon oil, 450 00:23:52,633 --> 00:23:56,100 on a vertically vibrating bath of the same liquid, 451 00:23:56,400 --> 00:23:58,133 then the droplet can bounce 452 00:23:58,133 --> 00:24:02,133 and walk on the oscillating surface of the liquid, right? 453 00:24:02,133 --> 00:24:05,533 So as you see in those top videos, each time 454 00:24:05,533 --> 00:24:09,266 the droplet bounces, it creates a little wave around itself. 455 00:24:09,633 --> 00:24:12,666 And the droplet then interacts with these waves on subsequent 456 00:24:12,666 --> 00:24:15,666 bounces to propel itself horizontally. 457 00:24:16,300 --> 00:24:19,833 So three key features to note about this system. 458 00:24:20,233 --> 00:24:24,900 The first is that the particle and its underlying wave, 459 00:24:25,166 --> 00:24:28,300 they co-exist as a wave particle entity. 460 00:24:28,466 --> 00:24:32,333 So if I were to make the droplet disappear by poking it, the underlying 461 00:24:32,333 --> 00:24:34,233 liquid surface would remain flat. 462 00:24:34,233 --> 00:24:36,733 It would eventually decay to a flat surface, right. 463 00:24:36,733 --> 00:24:39,733 So the droplet and the wave are couple two ways. 464 00:24:40,200 --> 00:24:45,266 Second is that there is memory in the system, and that is because the 465 00:24:45,266 --> 00:24:48,733 the waves created by the droplet, they decay very slowly in time. 466 00:24:49,066 --> 00:24:53,200 So the droplet is not only influenced by its most recent wave that it created, 467 00:24:53,433 --> 00:24:57,166 but also by the waves it created, let's say 10 or 20 bounces before. 468 00:24:57,566 --> 00:24:57,800 Right? 469 00:24:57,800 --> 00:25:00,800 So we have this idea of memory in the system. 470 00:25:01,200 --> 00:25:04,033 And the third, it's an active system, right. 471 00:25:04,033 --> 00:25:08,100 So even though the whole container is being vibrated up and down, the droplet 472 00:25:08,100 --> 00:25:11,466 locally extracts that energy through these waves it generates 473 00:25:11,466 --> 00:25:14,500 and it converts it into this directed persistent motion. 474 00:25:15,066 --> 00:25:15,766 Right. 475 00:25:15,766 --> 00:25:19,800 And, usually you get two different kind of walking droplets here. 476 00:25:20,000 --> 00:25:22,866 So if you drive the path at a single frequency, 477 00:25:22,866 --> 00:25:26,533 then you get slower and smaller droplets, which are called workers. 478 00:25:26,700 --> 00:25:29,633 If you drive it at multiple frequencies, 479 00:25:29,633 --> 00:25:33,433 then you can get inertia dominated, bigger droplets that move more faster. 480 00:25:33,433 --> 00:25:35,433 And we'll see them, later today. 481 00:25:38,166 --> 00:25:38,500 All right. 482 00:25:38,500 --> 00:25:41,733 So, what we'll try to do is let's let us now 483 00:25:41,733 --> 00:25:44,866 try to write down a simple model of the system, 484 00:25:44,866 --> 00:25:49,566 and then we'll see the type of dynamics that we can capture using this model. 485 00:25:50,100 --> 00:25:52,300 Now the the video, you see that at the top. 486 00:25:52,300 --> 00:25:54,533 It's the same video from the experiments. 487 00:25:54,533 --> 00:25:58,000 However, it has been stroked at the bouncing frequency. 488 00:25:58,233 --> 00:25:58,766 So if you take 489 00:25:58,766 --> 00:26:02,900 only one image of the droplet per bounce, then you don't get the bouncing motion. 490 00:26:02,900 --> 00:26:05,033 You only get the horizontal walking motion. 491 00:26:05,033 --> 00:26:06,866 And that's what you're seeing in the video. 492 00:26:06,866 --> 00:26:09,600 And this is essentially what the model is doing. 493 00:26:09,600 --> 00:26:11,733 So it's called the strobes Copic model. 494 00:26:11,733 --> 00:26:12,700 And what it does 495 00:26:12,700 --> 00:26:16,466 is you average over the vertical periodic bouncing fast time scale. 496 00:26:16,700 --> 00:26:20,733 And you look at what happens in the in the horizontal direction equation of motion. 497 00:26:21,000 --> 00:26:24,600 And we are going to look at a very simple, model where you get 498 00:26:24,600 --> 00:26:27,933 you are only allowed to move horizontally in one dimension left or right, right. 499 00:26:27,933 --> 00:26:30,200 And there is no bouncing motion. 500 00:26:30,200 --> 00:26:34,466 So if you try to write down an equation of motion, for this droplet, 501 00:26:34,766 --> 00:26:38,400 it's Newton's second law as equals, Ma. 502 00:26:38,700 --> 00:26:42,000 So on the left side, and this is in dimensionless form. 503 00:26:42,000 --> 00:26:46,300 So on the left side you have a term which looks like mass times acceleration. 504 00:26:46,533 --> 00:26:50,400 So you can think of kappa as a dimensionless mass parameter. 505 00:26:51,133 --> 00:26:55,133 The second term, it kind of models the dissipation in the system. 506 00:26:55,133 --> 00:26:57,566 So because these droplets are moving through air 507 00:26:57,566 --> 00:27:00,266 and they also lose momentum when they impact the liquid, 508 00:27:00,266 --> 00:27:03,800 you have dissipation in the system, which is modeled proportional to the velocity. 509 00:27:04,266 --> 00:27:04,500 Right. 510 00:27:04,500 --> 00:27:09,000 So you have a dissipation term there, the horrible looking turbo on the right. 511 00:27:09,300 --> 00:27:13,600 That's the kick that the droplet receives from these self-generated waves. 512 00:27:13,800 --> 00:27:14,166 So. Right. 513 00:27:14,166 --> 00:27:16,900 So so let's try to understand where is that coming from. 514 00:27:16,900 --> 00:27:19,600 So first of all you have an integral there and not a sum. 515 00:27:19,600 --> 00:27:21,000 Because in the model 516 00:27:21,000 --> 00:27:24,100 we are kind of averaging over the bouncing dynamics. 517 00:27:24,100 --> 00:27:27,133 So we are saying that the droplet is continuously emitting these waves 518 00:27:27,133 --> 00:27:28,500 as it moves. 519 00:27:28,500 --> 00:27:31,900 And how are how do we calculate this force. 520 00:27:32,133 --> 00:27:36,900 So we say that at each instant of time the droplet generates a wave of shape 521 00:27:37,200 --> 00:27:40,966 A of X, and the force that the droplet receives 522 00:27:41,000 --> 00:27:44,133 is found by adding up all these waves, integrating through 523 00:27:44,133 --> 00:27:47,300 all these waves that it has generated in the entire history. 524 00:27:47,400 --> 00:27:50,466 So you get an overall waves yield, and the gradient of that 525 00:27:50,466 --> 00:27:53,466 ratio is what gives you the push right. 526 00:27:53,633 --> 00:27:57,866 So what you have here, this f is the gradient of the individual waves. 527 00:27:58,100 --> 00:28:01,300 And you can integrate that over the entire history of the droplet. 528 00:28:01,566 --> 00:28:03,700 And these waves are decaying exponentially in time. 529 00:28:03,700 --> 00:28:05,800 So you have an exponential term there. Right. 530 00:28:05,800 --> 00:28:08,533 So you have this integral term on the right hand side 531 00:28:08,533 --> 00:28:09,800 which will eventually tell you 532 00:28:09,800 --> 00:28:13,300 what is the force on the droplet from these self-generated waves. 533 00:28:14,366 --> 00:28:16,000 Now this is harder to solve. 534 00:28:16,000 --> 00:28:18,366 It's an integral differential equation. 535 00:28:18,366 --> 00:28:21,100 However, if we make some simplifications 536 00:28:21,100 --> 00:28:24,366 we can make some progress into converting this into. 537 00:28:24,366 --> 00:28:25,466 Integrate essentially equation A 538 00:28:25,466 --> 00:28:28,466 into something which is more analytically tractable. 539 00:28:29,233 --> 00:28:33,333 So if I say that the the shape of the waves that the droplet 540 00:28:33,333 --> 00:28:39,066 generates, it's, it's so usually in experiments the waves that the droplet 541 00:28:39,066 --> 00:28:42,100 generates has both spatial oscillations and spatial decay. 542 00:28:42,400 --> 00:28:43,933 If you neglect the spatial decay 543 00:28:43,933 --> 00:28:48,033 and say that the droplet generates simple sinusoidal waves, right, 544 00:28:48,033 --> 00:28:52,800 then the wave field would be a cosine and the gradient would be a sine function. 545 00:28:52,800 --> 00:28:55,300 And you get this integral differential equation. 546 00:28:55,300 --> 00:28:58,266 And it turns out that you can map the system. 547 00:28:58,266 --> 00:29:01,266 To a system of ordinary differential equations. 548 00:29:01,300 --> 00:29:06,900 And I don't know if any of you recognize that these are the the classic celebrated 549 00:29:06,900 --> 00:29:10,566 Lorenz chaotic system that you can map this system to. 550 00:29:11,066 --> 00:29:11,300 Right. 551 00:29:11,300 --> 00:29:15,566 So the capital X variable maps to the velocity of the particle. 552 00:29:15,966 --> 00:29:20,066 The capital Y variable maps to this horrible integral force 553 00:29:20,066 --> 00:29:21,533 term on the right hand side. 554 00:29:21,533 --> 00:29:25,833 And the capital z variable is also related to this, memory force. 555 00:29:25,833 --> 00:29:27,700 And it kind of is a measure of the 556 00:29:27,700 --> 00:29:30,700 the amplitude of the waves where the droplet is. 557 00:29:31,000 --> 00:29:31,200 Right. 558 00:29:31,200 --> 00:29:35,166 So you can get this 1 to 1 correspondence between the motion of the droplet 559 00:29:35,633 --> 00:29:38,633 and the dynamics of the Lorenz system. 560 00:29:38,900 --> 00:29:39,166 Right. 561 00:29:39,166 --> 00:29:41,833 And again this is if you make this simplification 562 00:29:41,833 --> 00:29:45,533 that he behaves that sinusoidal, then you can do this nice conversion. 563 00:29:45,533 --> 00:29:48,400 And you can try to understand what the dynamics are. 564 00:29:49,500 --> 00:29:52,466 And also the parameters of the system 565 00:29:52,466 --> 00:29:56,266 map to the, the you can map it to the parameters in the Lorenz system. 566 00:29:56,700 --> 00:30:00,566 And as we saw before, the Lorenz system, that's an example of a dissipative 567 00:30:00,566 --> 00:30:01,600 dynamical system. Right. 568 00:30:01,600 --> 00:30:04,600 So here we have a dissipative dynamical system. 569 00:30:06,766 --> 00:30:07,100 All right. 570 00:30:07,100 --> 00:30:09,966 So these are the same equations as on the previous slide. 571 00:30:09,966 --> 00:30:12,700 Same Lorenz equations I've just rescale them 572 00:30:12,700 --> 00:30:15,633 so they are more meaningful for the walking droplet. 573 00:30:15,633 --> 00:30:17,766 So we see two dimensionless parameters. 574 00:30:17,766 --> 00:30:20,533 Here we have r and we have tau. 575 00:30:20,533 --> 00:30:21,900 So you can think of 576 00:30:21,900 --> 00:30:26,033 as a dimensionless amplitude of the waves generated by the droplet. 577 00:30:26,400 --> 00:30:30,600 And tau is the rate at which the waves decay, which are generated by the droplet. 578 00:30:30,766 --> 00:30:35,033 So tau think of Tao as the memory of the system or the memory of these waves. 579 00:30:36,966 --> 00:30:37,233 Right? 580 00:30:37,233 --> 00:30:38,966 So what we are going to do is we are going to see 581 00:30:38,966 --> 00:30:42,900 what sort of dynamics does this give rise to as we increase this memory parameter. 582 00:30:43,000 --> 00:30:43,633 Right. 583 00:30:43,633 --> 00:30:46,933 So let's start with very small values of memory parameter. 584 00:30:47,166 --> 00:30:49,166 And I'm showing you two plots there. 585 00:30:49,166 --> 00:30:52,300 The one on the left is this phase space plot. 586 00:30:52,633 --> 00:30:54,733 But it's projected in two dimensions. 587 00:30:54,733 --> 00:30:58,200 So you are seeing the capital X and the capital Z projection 588 00:30:58,433 --> 00:31:01,433 of the phase space of this three dimensional Lorenz system. 589 00:31:01,700 --> 00:31:05,366 And the plot on the right is showing you how does the position of the droplet 590 00:31:05,366 --> 00:31:07,033 change with time. 591 00:31:07,033 --> 00:31:10,133 So at low values of tau, when you solve 592 00:31:10,133 --> 00:31:13,833 the steady states of the Lorenz system, you find that there is one stable point, 593 00:31:14,233 --> 00:31:17,833 where capital X is equal to zero, which is this black dot. 594 00:31:18,100 --> 00:31:21,900 So capital X, if you remember, was the velocity of my particle. 595 00:31:22,200 --> 00:31:25,066 So this means that my velocity of the particle is zero. 596 00:31:25,066 --> 00:31:27,200 So my particle stays stationary. 597 00:31:27,200 --> 00:31:29,400 And we have a stationary state of the droplet. 598 00:31:29,400 --> 00:31:30,533 And this is something you see 599 00:31:30,533 --> 00:31:33,733 in experiments when you have low amplitudes of vibration. 600 00:31:34,466 --> 00:31:39,466 Now as I keep on increasing the memory you get bifurcations in the Lorenz system. 601 00:31:39,900 --> 00:31:42,800 So these are the transitions between these stable states. 602 00:31:42,800 --> 00:31:46,366 And it turns out that the stationary state which was that stable before 603 00:31:46,566 --> 00:31:49,200 it becomes unstable. So it's now gray here. 604 00:31:49,200 --> 00:31:52,333 And you get a pair of new stable points in the system. 605 00:31:52,666 --> 00:31:53,100 Right. 606 00:31:53,100 --> 00:31:57,033 One at a positive value of x and the other at a negative value of x. 607 00:31:57,466 --> 00:31:57,933 Right. 608 00:31:57,933 --> 00:32:01,033 And what this corresponds to is a positive constant 609 00:32:01,033 --> 00:32:04,266 velocity and a constant velocity going in the other direction. 610 00:32:04,500 --> 00:32:08,600 So these are again steady walking states which are again observed in experiments 611 00:32:08,866 --> 00:32:12,000 droplet moving at a constant speed in a given direction. 612 00:32:13,200 --> 00:32:13,900 Now what 613 00:32:13,900 --> 00:32:16,900 happens if you keep on continue increasing memory? 614 00:32:17,133 --> 00:32:20,600 Well, it turns out that these innocent looking Lorenz equations, 615 00:32:20,933 --> 00:32:24,933 the exhibit quite profound complexity of behavior. 616 00:32:25,566 --> 00:32:29,366 There is a whole book written on the, the Lorenz equations 617 00:32:29,366 --> 00:32:31,833 and the kinds of dynamics that it exhibits. 618 00:32:31,833 --> 00:32:33,233 And it's non-trivial. 619 00:32:33,233 --> 00:32:36,500 So here, here's a cartoon showing the sequence of bifurcations, 620 00:32:37,000 --> 00:32:39,533 that you get as you keep on varying these parameter. 621 00:32:39,533 --> 00:32:42,533 And I'll not even attempt to explain it today. 622 00:32:42,600 --> 00:32:43,533 But, 623 00:32:43,533 --> 00:32:47,066 the important thing is that eventually, as you keep on increasing tau, eventually, 624 00:32:47,066 --> 00:32:52,200 after all these complex bifurcations, you get chaos in the Lorenz system, right? 625 00:32:52,533 --> 00:32:55,233 Which is something you might have seen as the butterfly effect 626 00:32:55,233 --> 00:32:57,433 or this Lorenz attractor. Right. 627 00:32:57,433 --> 00:33:00,933 So you get this Lorenz chaotic attractor in phase space. 628 00:33:00,933 --> 00:33:02,866 And if we now try to understand what does that mean 629 00:33:02,866 --> 00:33:07,800 for the droplet motion, it means that the droplet is moving back and forth. 630 00:33:07,800 --> 00:33:10,533 So when you are on one wing of the Lorenz attractor, 631 00:33:10,533 --> 00:33:12,200 you are moving in one direction. 632 00:33:12,200 --> 00:33:15,066 And when you're on the other, when you're moving in the opposite direction. 633 00:33:15,066 --> 00:33:18,300 So your particle is essentially doing kind of like a random walk between left 634 00:33:18,300 --> 00:33:22,066 and right, as you have chaotic motion between the two wings of the attractor. 635 00:33:22,433 --> 00:33:25,433 Now, this is something that's not seen in experiments yet, 636 00:33:25,633 --> 00:33:29,866 because to get into this very high memory regime, you need to be very close to 637 00:33:29,866 --> 00:33:33,933 that instability that I mentioned before without triggering that instability. 638 00:33:34,233 --> 00:33:34,666 Right. 639 00:33:34,666 --> 00:33:36,233 And and that's something that people 640 00:33:36,233 --> 00:33:38,400 have not been able to get a handle on experiment 641 00:33:38,400 --> 00:33:41,866 when the droplet is moving in free space, if you can't find the droplet, 642 00:33:41,866 --> 00:33:44,533 then you can get chaotic motion, but not in free space. 643 00:33:47,200 --> 00:33:47,533 All right. 644 00:33:47,533 --> 00:33:49,866 So let me show you, 645 00:33:49,866 --> 00:33:53,500 a couple more slides, and then, I will try to wrap up my talk. 646 00:33:53,766 --> 00:33:55,233 So what happens? 647 00:33:55,233 --> 00:33:57,000 A natural question is what happens if you have more 648 00:33:57,000 --> 00:34:00,000 than one of these walking droplets interacting with each other, right. 649 00:34:00,066 --> 00:34:04,900 So let me show you some of, some videos from some experiments during my PhD where 650 00:34:04,900 --> 00:34:07,966 we are trying to look at interactions between these walking droplets. 651 00:34:09,533 --> 00:34:12,000 So if you have two of these droplets 652 00:34:12,000 --> 00:34:15,000 at low memory or low vibration amplitudes, 653 00:34:15,033 --> 00:34:19,633 then they will stick to each other and they will work together as a pair. 654 00:34:19,666 --> 00:34:23,733 So you have a doublet of droplets walking together as a constant velocity. 655 00:34:24,333 --> 00:34:26,866 If you have more than two, you can get a staggered 656 00:34:26,866 --> 00:34:29,866 configuration with three droplets and they'll work together again. 657 00:34:30,366 --> 00:34:32,900 You can also get quadruplets, as well. 658 00:34:32,900 --> 00:34:34,966 Moving together. Right. 659 00:34:34,966 --> 00:34:37,966 So this is at low amplitudes of vibration or low memory. 660 00:34:38,233 --> 00:34:41,400 Now if we increase the the memory in the system 661 00:34:41,400 --> 00:34:44,066 and look at that pair of doublet, 662 00:34:44,066 --> 00:34:47,866 then what you see is that they will still stay bound and work together. 663 00:34:47,866 --> 00:34:50,866 But you get these sideways oscillations between the droplet. 664 00:34:51,100 --> 00:34:54,100 So so this is what's called in the literature from an air. 665 00:34:54,366 --> 00:34:54,600 Right. 666 00:34:54,600 --> 00:34:57,600 So you get these droplets oscillating sideways as they work. 667 00:34:57,700 --> 00:35:01,033 If they are not of the same size, if one is bigger, one is smaller, 668 00:35:01,566 --> 00:35:05,666 then you get this chasing mode where the bigger droplet will kind of drag 669 00:35:05,666 --> 00:35:08,166 the smaller droplet in its wake, and they'll work together. 670 00:35:10,300 --> 00:35:10,900 Right. 671 00:35:10,900 --> 00:35:13,300 And you can also get orbiting states. 672 00:35:13,300 --> 00:35:16,800 So a very big droplet cannot survive on its own. 673 00:35:17,066 --> 00:35:20,800 But if it's accompanied by a smaller droplet as a satellite, 674 00:35:21,166 --> 00:35:24,400 then the smaller droplet will keep on orbiting the bigger droplet. 675 00:35:24,400 --> 00:35:26,300 And that will somehow sustain the bigger droplet. 676 00:35:26,300 --> 00:35:29,300 And we don't understand why this happens. 677 00:35:29,500 --> 00:35:29,666 Right. 678 00:35:29,666 --> 00:35:32,666 So you get these interesting orbiting states, as well. 679 00:35:32,933 --> 00:35:34,666 And then now next question. 680 00:35:34,666 --> 00:35:36,600 What happens if you have lots of them. Right. 681 00:35:36,600 --> 00:35:38,433 So if you have lots of them again at low memory 682 00:35:38,433 --> 00:35:41,866 you get a nice crystal like lattice structure forming. 683 00:35:42,266 --> 00:35:46,733 As you increase the memory that the crystal melts, 684 00:35:46,733 --> 00:35:49,700 but it still stays bounded and you get this jiggling state, 685 00:35:49,700 --> 00:35:53,266 a liquid like state, and then eventually the cluster will disintegrate 686 00:35:53,266 --> 00:35:57,033 and you get a gas of droplets just hitting each other like billiard balls. 687 00:35:57,833 --> 00:35:58,033 Right. 688 00:35:58,033 --> 00:35:59,566 So you can get these again, 689 00:35:59,566 --> 00:36:01,100 you get when you have lots of them, can get these 690 00:36:01,100 --> 00:36:04,233 interesting collective behaviors that emerge in this system, 691 00:36:04,233 --> 00:36:07,400 which is something that we are trying to understand now. 692 00:36:08,700 --> 00:36:11,300 So lastly, I wanted to 693 00:36:11,300 --> 00:36:15,200 mention, why this system is interesting from a physics perspective. 694 00:36:15,533 --> 00:36:16,100 Right. 695 00:36:16,100 --> 00:36:19,866 So, there have been, attempts, 696 00:36:19,933 --> 00:36:25,366 or people have tried to, to mimic quantum like behavior in this system. 697 00:36:25,366 --> 00:36:28,966 So a disclaimer this system is not a quantum system. 698 00:36:28,966 --> 00:36:32,066 It's a classical system, but it's a classical system 699 00:36:32,066 --> 00:36:33,600 which is out of equilibrium. 700 00:36:33,600 --> 00:36:35,266 So it's an active system, 701 00:36:35,266 --> 00:36:36,333 and it seems like you can get 702 00:36:36,333 --> 00:36:39,400 some really interesting dynamical behaviors in the system. 703 00:36:39,400 --> 00:36:42,400 And then if you look at the statistics of that chaotic dynamics, 704 00:36:42,533 --> 00:36:45,266 you get seem to get wave like statistics in the system. 705 00:36:45,266 --> 00:36:47,666 So I'll show you some examples. 706 00:36:47,666 --> 00:36:52,000 So here, is an experimental video, showing 707 00:36:52,500 --> 00:36:57,200 tracking of a droplet confined in a cavity and the size of the cavity 708 00:36:57,433 --> 00:37:01,266 is of the same order as the wavelength of the waves generated by the droplet. 709 00:37:01,566 --> 00:37:05,400 So the droplet does this apparently chaotic looking motion inside the cavity. 710 00:37:05,700 --> 00:37:07,100 And if you track the motion 711 00:37:07,100 --> 00:37:10,100 for a few hours, it's chaotic and nothing much is going on. 712 00:37:10,366 --> 00:37:14,100 But if you get enough statistics and you plot the probability distribution 713 00:37:14,100 --> 00:37:15,700 of the droplets position, 714 00:37:15,700 --> 00:37:19,333 then you get this wave like distribution in the position of the droplet. 715 00:37:19,600 --> 00:37:21,600 And this is not too different from what you would find 716 00:37:21,600 --> 00:37:25,000 if you had an electron confined in a in a ring of copper atoms. 717 00:37:25,000 --> 00:37:27,500 And you look at the probability distribution of the of the electron. 718 00:37:27,500 --> 00:37:27,966 Right. 719 00:37:27,966 --> 00:37:32,000 So the underlying chaotic dynamics here seem to be giving rise to 720 00:37:32,166 --> 00:37:33,500 wave like statistics. 721 00:37:35,100 --> 00:37:36,200 Another 722 00:37:36,200 --> 00:37:40,400 example is an analog of tunneling. 723 00:37:40,533 --> 00:37:44,066 So if you have these walking droplets and you put 724 00:37:44,500 --> 00:37:47,200 barriers below the liquid surface, 725 00:37:47,200 --> 00:37:50,500 then these droplets will typically reflect from these barriers. 726 00:37:50,933 --> 00:37:54,800 However, occasionally it turns out that these droplets again 727 00:37:54,800 --> 00:37:59,066 due to these non-linear interactions can unpredictably cross these barriers. 728 00:37:59,333 --> 00:37:59,566 Right. 729 00:37:59,566 --> 00:38:04,233 So you have an analog of tunneling where these droplets can unpredictably 730 00:38:04,233 --> 00:38:07,266 cross these, barriers, in the liquid. 731 00:38:09,500 --> 00:38:12,100 Now, if you had a walking 732 00:38:12,100 --> 00:38:15,333 droplet interacting with these barriers, it can get deflected, right? 733 00:38:15,333 --> 00:38:18,600 So people tried what happens if you put a single slit or a double slit 734 00:38:18,600 --> 00:38:21,700 and let these droplets pass through them? 735 00:38:21,700 --> 00:38:22,766 And what happens? 736 00:38:22,766 --> 00:38:27,000 And surprisingly, what turns out that you if you do this experiments 737 00:38:27,000 --> 00:38:31,366 and you fire droplet one by one, either at a single slit or a double slit, 738 00:38:31,666 --> 00:38:33,000 then even though the droplet 739 00:38:33,000 --> 00:38:36,033 is going through one of the two slits, the wave that is guiding it, 740 00:38:36,200 --> 00:38:39,566 it's going through both the slits and you get some interesting patterns. 741 00:38:39,566 --> 00:38:41,833 If you get look at the distribution of droplets. 742 00:38:41,833 --> 00:38:44,700 Now, it's not the exact same pattern, diffraction 743 00:38:44,700 --> 00:38:47,266 pattern or interference pattern that you would get in a quantum system. 744 00:38:47,266 --> 00:38:49,800 And I wouldn't expect that to happen as well anyways. 745 00:38:49,800 --> 00:38:53,433 But you you do get some interesting wave like patterns in the distribution of 746 00:38:53,433 --> 00:38:54,700 the position of the droplet. 747 00:38:55,666 --> 00:38:57,766 And then lastly, if you can find one of these droplets 748 00:38:57,766 --> 00:39:02,800 in a harmonic potential, then it turns out that it can't do any random motion. 749 00:39:02,800 --> 00:39:05,800 Its motion collapses onto a set of these limit 750 00:39:05,800 --> 00:39:08,800 cycle orbits, so its motion is constrained. 751 00:39:08,800 --> 00:39:11,233 So if you kind of plot the radius of curvature 752 00:39:11,233 --> 00:39:14,333 and the angular momentum of the droplet, and you get these 753 00:39:14,333 --> 00:39:18,400 quantized states of motion, because the motion is no longer 754 00:39:18,400 --> 00:39:22,333 chaotic, it's collapsing into these, well-defined periodic states. 755 00:39:23,100 --> 00:39:23,333 Right. 756 00:39:23,333 --> 00:39:27,666 So, so these are some of the examples, where the system exhibits quantum 757 00:39:27,933 --> 00:39:30,133 like behavior. And again, it's not a quantum system. 758 00:39:30,133 --> 00:39:33,133 It's a classical system, but it's out of equilibrium. 759 00:39:34,666 --> 00:39:35,000 All right. 760 00:39:35,000 --> 00:39:39,266 So so just to summarize, so so I introduced a bit about active particles. 761 00:39:39,266 --> 00:39:41,100 So these are non-equilibrium entities 762 00:39:41,100 --> 00:39:44,366 that consume energy and convert it into some form of self-propelled. 763 00:39:45,000 --> 00:39:49,633 And when you have lots of these, you get these emergent behaviors, in the system, 764 00:39:49,866 --> 00:39:53,600 such as flocking, active turbulence and all sorts of other features. 765 00:39:54,133 --> 00:39:58,500 And then we looked at two simple, models of active particles, 766 00:39:58,800 --> 00:40:02,400 individual active particles interacting with some form of environment. 767 00:40:02,400 --> 00:40:02,933 Right. 768 00:40:02,933 --> 00:40:06,800 And we saw that even the simple models exhibit very rich behavior. 769 00:40:06,966 --> 00:40:10,533 And we tried to understand or rationalize some of these in terms of using these 770 00:40:10,533 --> 00:40:14,700 ideas of conservative and dissipative nonlinear dynamical systems. 771 00:40:15,600 --> 00:40:16,466 Thank you for listening. 772 00:40:30,600 --> 00:40:31,600 Yeah. No. 773 00:40:31,600 --> 00:40:32,066 Thank you. 774 00:40:32,066 --> 00:40:33,900 Very fascinating question. 775 00:40:33,900 --> 00:40:36,266 So, just to repeat the question. 776 00:40:36,266 --> 00:40:39,333 So the question was, I presented, toy 777 00:40:39,333 --> 00:40:42,766 models, numerical simulations as well as experiments. 778 00:40:42,766 --> 00:40:43,666 And then what? 779 00:40:43,666 --> 00:40:45,900 Wait, what should I give to to each of them? 780 00:40:45,900 --> 00:40:48,833 So, these simple woody 781 00:40:48,833 --> 00:40:53,700 type toy models, they are mainly useful in kind of exploring the, 782 00:40:53,966 --> 00:40:58,700 the parameter space of the system quite rapidly and to, to get, 783 00:40:58,700 --> 00:41:00,033 gist of the different types 784 00:41:00,033 --> 00:41:03,833 of qualitative behaviors that can, that the system can exhibit. 785 00:41:04,433 --> 00:41:07,433 Now, if you want to accurately capture what's going on in experiments 786 00:41:07,433 --> 00:41:10,433 that have been, a plethora of models that have been developed 787 00:41:10,466 --> 00:41:12,066 with increasing complexity. 788 00:41:12,066 --> 00:41:13,233 So you have to resolve the 789 00:41:13,233 --> 00:41:16,866 the bouncing motion of the droplet as well as the evolution of the surface waves. 790 00:41:17,100 --> 00:41:18,333 But then the downside is that 791 00:41:18,333 --> 00:41:21,800 because you have all the fast time scale, they are very inefficient. 792 00:41:22,300 --> 00:41:25,100 Now in some of these active matter models, when you look at, 793 00:41:25,100 --> 00:41:28,100 effects at a collective scale there, 794 00:41:28,266 --> 00:41:31,933 the motion of the individual particles may not become the details. 795 00:41:31,933 --> 00:41:33,700 Motion may not become that important. 796 00:41:33,700 --> 00:41:37,700 So when you have a collection of these, these active particles, theories have 797 00:41:37,700 --> 00:41:41,466 been developed, to which coarse grained, over this small time scale. 798 00:41:41,466 --> 00:41:42,500 And you look at a continuum 799 00:41:42,500 --> 00:41:46,466 model of equations, so a continuum model that describes the system. 800 00:41:46,466 --> 00:41:51,566 So, for example, people have borrowed, theory of liquid crystals. 801 00:41:51,866 --> 00:41:56,100 So theory of rod like particles in a defining a continuum. 802 00:41:56,100 --> 00:42:00,233 And so Julia, in the department, she and her group, they have developed, 803 00:42:00,766 --> 00:42:03,800 theory of active liquid crystals now are active pneumatics. 804 00:42:03,933 --> 00:42:08,000 So you can describe, large collection of these bacteria or cells 805 00:42:08,233 --> 00:42:11,233 using these coarse grained equations, which would describe 806 00:42:11,233 --> 00:42:14,333 what happens at the collective scale rather than the individual scale. 807 00:42:14,433 --> 00:42:15,966 So I think it's a complex thing. 808 00:42:15,966 --> 00:42:18,966 And it's, it's like depending on what scale you are 809 00:42:19,033 --> 00:42:21,900 interested in, you would have different models. 810 00:42:21,900 --> 00:42:24,000 And because of these emergent features, 811 00:42:24,000 --> 00:42:25,600 you might need, again, different models 812 00:42:25,600 --> 00:42:27,633 to describe things happening at different scales. 813 00:42:27,633 --> 00:42:31,533 But the models I presented today, are more of very simple toy models of. 814 00:42:31,733 --> 00:42:35,033 But just to illustrate the complexity, even those models can give rise to. 815 00:42:35,800 --> 00:42:37,833 So yeah. Yeah, yeah. 816 00:42:37,833 --> 00:42:38,966 Very good question. 817 00:42:38,966 --> 00:42:42,400 So if you see a raindrop falling on a puddle, it coalesces. 818 00:42:42,400 --> 00:42:42,866 Right. 819 00:42:42,866 --> 00:42:44,800 So the reason the droplets don't coalesce 820 00:42:44,800 --> 00:42:47,333 here is because the underlying liquid is vibrating. 821 00:42:47,333 --> 00:42:50,300 So there is not enough time for the droplet to merge 822 00:42:50,300 --> 00:42:53,466 with the liquid because the air layer which separates them, 823 00:42:53,833 --> 00:42:57,400 it kind of acts like a spring and the droplet keeps on bouncing. 824 00:42:57,633 --> 00:43:00,833 So you, the way you create these droplets, which you will see, is that 825 00:43:00,833 --> 00:43:04,000 I just use a toothpick, put in the liquid and then like, hit it and then, 826 00:43:05,733 --> 00:43:06,800 Hiroko. 827 00:43:06,800 --> 00:43:07,633 Yeah, right. 828 00:43:07,633 --> 00:43:09,900 So think of it like, like jumping on a trampoline. 829 00:43:09,900 --> 00:43:12,366 That's what the droplet is doing because of the surface tension. 830 00:43:12,366 --> 00:43:14,400 It can just, like, maintain this bouncing motion 831 00:43:14,400 --> 00:43:17,400 without coalescing with the liquid. 832 00:43:18,433 --> 00:43:18,800 Sorry. 833 00:43:18,800 --> 00:43:21,066 What was the question at the end? It was the question. 834 00:43:21,066 --> 00:43:23,200 Okay. Right. 835 00:43:23,200 --> 00:43:24,166 Yeah. 836 00:43:24,166 --> 00:43:24,533 Yeah. 837 00:43:24,533 --> 00:43:27,666 No, the thing is, like, I don't enough understand 838 00:43:27,666 --> 00:43:30,666 enough of the pilot wave or other models. 839 00:43:31,333 --> 00:43:34,266 So I wanted to stay a bit away from that, but that there are, 840 00:43:34,266 --> 00:43:37,466 there is, a research group at MIT led by John Bush. 841 00:43:37,466 --> 00:43:41,466 They are trying to actually make these analogies, 842 00:43:41,466 --> 00:43:45,800 more, let's say, towards the quantum side to see how far you can take them. 843 00:43:46,066 --> 00:43:48,966 And yeah, 844 00:43:48,966 --> 00:43:51,100 there have been some interesting developments, 845 00:43:51,100 --> 00:43:54,100 but it's not clear how close it is. 846 00:43:56,566 --> 00:43:56,833 Yeah. 847 00:43:56,833 --> 00:43:59,200 No, that's a that's a very interesting question. 848 00:43:59,200 --> 00:44:02,766 So I think part of this motivation of this whole area of active matter, 849 00:44:02,766 --> 00:44:07,200 which has only really begun to kind of flourish in the last 20 years 850 00:44:07,200 --> 00:44:12,200 or so, is that, is the kind of like biological inspirations, right, 851 00:44:12,200 --> 00:44:16,133 that you get, in biology, things happening in coordinated 852 00:44:16,133 --> 00:44:19,633 and organized fashion at all scales, a different hierarchy of scales. 853 00:44:19,966 --> 00:44:20,266 Right. 854 00:44:20,266 --> 00:44:21,700 And then can you, 855 00:44:21,700 --> 00:44:26,033 can you understand these features using some of these, physical models? 856 00:44:26,266 --> 00:44:27,533 And I think, as you're saying, like, 857 00:44:27,533 --> 00:44:30,633 it might be that things that happen at a collective scale, 858 00:44:30,966 --> 00:44:34,466 you might be able to describe them not going back to these microscopic 859 00:44:34,466 --> 00:44:38,366 picture and just having some sort of a statistical picture of what goes on 860 00:44:38,366 --> 00:44:38,966 at that scale. 861 00:44:38,966 --> 00:44:41,666 And people in the area use both tools. 862 00:44:41,666 --> 00:44:45,066 I think tools are being used even from equilibrium statistical physics 863 00:44:45,366 --> 00:44:48,366 to to rationalize some of these, large scale behaviors. 864 00:44:48,433 --> 00:44:52,466 But then in some regions, those, tools fall down and you get truly 865 00:44:52,466 --> 00:44:55,700 non-equilibrium behaviors, and then you need to kind of modify 866 00:44:55,700 --> 00:45:00,066 the equations of, statistical physics and look at more non-equilibrium pictures. 867 00:45:00,300 --> 00:45:02,766 But yeah. No, it's it's a it's an important question. 868 00:45:02,766 --> 00:45:07,566 And I think it's, it's, it's slowly being addressed with this type of framework. 869 00:45:07,566 --> 00:45:11,400 And I think biology is, is a very good example where you get these 870 00:45:11,400 --> 00:45:14,466 emergent behaviors happening at all scales and how they are talking to each other 871 00:45:14,700 --> 00:45:15,533 at different scales. 872 00:45:16,933 --> 00:45:17,233 Right. 873 00:45:17,233 --> 00:45:21,300 So, so the only experiments I did were during my PhD with these droplets 874 00:45:21,300 --> 00:45:24,333 because we have a tabletop, I haven't done any other experiments. 875 00:45:24,333 --> 00:45:28,433 So most of the other time is, again doing this mathematical modeling. 876 00:45:28,433 --> 00:45:33,033 And then also when you can solve these equations or approximate them 877 00:45:33,033 --> 00:45:36,900 in a nice way, you results, you resort to numerics 878 00:45:36,900 --> 00:45:40,533 and try to simulate these systems, which can sometimes be quicker. 879 00:45:40,600 --> 00:45:42,266 Sometimes it can take a few days. 880 00:45:42,266 --> 00:45:42,800 Yeah. 881 00:45:42,800 --> 00:45:46,166 So right now it's mainly mainly mathematical modeling side 882 00:45:46,166 --> 00:45:49,166 and then simulation side of things. 883 00:45:49,333 --> 00:45:49,700 Right. 884 00:45:49,700 --> 00:45:54,266 So so there are people even in engineering, departments, 885 00:45:54,266 --> 00:45:57,633 who are quite excited about this whole idea of active matter 886 00:45:57,900 --> 00:46:02,800 and it's being used in the sense that people have started looking into, 887 00:46:02,800 --> 00:46:06,066 a swarm or a collection of these robots, these 888 00:46:06,233 --> 00:46:08,566 which can be modeled as these active particles 889 00:46:08,566 --> 00:46:12,133 and trying to get some useful work, done out of them. 890 00:46:12,300 --> 00:46:17,400 So there are, in a sense, more practical problems 891 00:46:17,400 --> 00:46:21,833 that people are trying to tackle using these, ideas from active matter. 892 00:46:21,833 --> 00:46:25,366 So it's it's it's it has started recently, but it's, it's it's ongoing. 893 00:46:26,066 --> 00:46:26,900 Yeah. 894 00:46:26,900 --> 00:46:32,033 No, it's, it's I don't think it's like, I don't know how seriously to take it, 895 00:46:32,033 --> 00:46:35,633 but it's it's like there's this system which is a particle. 896 00:46:36,033 --> 00:46:37,500 It's coupled to its waves. 897 00:46:37,500 --> 00:46:39,766 And then it moves together. 898 00:46:39,766 --> 00:46:44,100 And because it moves together, it doesn't interact directly with its surroundings. 899 00:46:44,333 --> 00:46:45,000 It's indirect. 900 00:46:45,000 --> 00:46:48,766 Through these guiding waves, and you get some interesting interference 901 00:46:48,766 --> 00:46:49,800 like effects. 902 00:46:49,800 --> 00:46:52,766 But you're right that that in, in other 903 00:46:52,766 --> 00:46:57,066 classical pictures, you could probably form similar analogies. 904 00:46:57,300 --> 00:46:58,200 Perfectly as well. 905 00:46:59,266 --> 00:47:00,666 I don't know. 906 00:47:00,666 --> 00:47:05,266 And I'm not sure how, how much people agree within the field as well. 907 00:47:05,433 --> 00:47:08,433 So that's why I kind of said it's, it's like a working definition. 908 00:47:08,733 --> 00:47:14,066 But yeah, the, the key feature seems to be that it can somehow consume 909 00:47:14,066 --> 00:47:18,833 or absorb energy and convert it into some form of persistent motion. 910 00:47:19,100 --> 00:47:21,900 And if if an entity can do that, 911 00:47:21,900 --> 00:47:24,900 then roughly it would fall into this category. 912 00:47:25,133 --> 00:47:28,966 And it's like also depends on what sort of things you are interested in. 913 00:47:29,200 --> 00:47:32,133 In a sense, if you have a very complex 914 00:47:32,133 --> 00:47:34,966 animal, it's doing 915 00:47:34,966 --> 00:47:38,466 even though it's like doing locomotion, it might be very different reasons. 916 00:47:38,466 --> 00:47:41,700 It might be sensing its environment and, and lots of different things going on. 917 00:47:41,700 --> 00:47:42,200 Right. 918 00:47:42,200 --> 00:47:43,666 But then if you're just interested in on 919 00:47:43,666 --> 00:47:46,666 how a large collection of them behave, then maybe you might be able to use 920 00:47:46,700 --> 00:47:47,700 some of these ideas 921 00:47:47,700 --> 00:47:51,800 to have a more simplified picture of what happens at a larger scale, 922 00:47:52,966 --> 00:47:53,300 right? 923 00:47:53,300 --> 00:47:53,966 Yeah. So yeah. 924 00:47:53,966 --> 00:47:56,566 So it's not it's not carrying its own energy source. 925 00:47:56,566 --> 00:47:58,666 But yeah, you are driving it from outside. 926 00:47:58,666 --> 00:48:02,400 So so there is again, this kind of hazy line between non-equilibrium systems. 927 00:48:02,400 --> 00:48:04,466 And when would you call them active. Right. 928 00:48:04,466 --> 00:48:06,100 So in general, like when you see turbulence 929 00:48:06,100 --> 00:48:07,100 in the atmosphere 930 00:48:07,100 --> 00:48:10,933 and other non-equilibrium processes, they are also out of equilibrium. 931 00:48:10,933 --> 00:48:14,000 But they're the thing that forces them out of equilibrium 932 00:48:14,200 --> 00:48:16,133 is that the scale of the system. 933 00:48:16,133 --> 00:48:17,300 Whereas here what you see 934 00:48:17,300 --> 00:48:20,833 is that even though I'm injecting energy everywhere in the liquid surface, 935 00:48:21,200 --> 00:48:25,000 it's being locally extracted by the droplet and converted into motion. 936 00:48:25,133 --> 00:48:28,133 So that's kind of the idea that these active particles. 937 00:48:28,933 --> 00:48:29,233 Yeah. 938 00:48:29,233 --> 00:48:31,333 So, so the the models which I haven't presented were 939 00:48:31,333 --> 00:48:33,633 where we looked at in terms of modeling these droplets. 940 00:48:33,633 --> 00:48:36,633 We assumed that these droplets are spherical rigid 941 00:48:36,633 --> 00:48:40,500 objects, whereas you have a deforming quite a lot as you saw in this videos. 942 00:48:40,666 --> 00:48:44,900 So I think our models are inadequate to capture those deformations. 943 00:48:44,900 --> 00:48:48,166 And the interaction with the, liquid surface. 944 00:48:48,433 --> 00:48:50,066 And yeah, it's not clear. 945 00:48:50,066 --> 00:48:52,800 So because the earlier between the droplet and the liquid, it 946 00:48:52,800 --> 00:48:55,800 somehow needs to be sustained, right, for the droplet to stay. 947 00:48:55,966 --> 00:48:58,633 And if it's by itself, it would just coalesce. 948 00:48:58,633 --> 00:49:01,166 But here, I don't know, some of the bouncing motion of the droplet 949 00:49:01,166 --> 00:49:05,800 is somehow replenishing the air layer, which is giving rise to its persistence. 950 00:49:05,800 --> 00:49:09,366 Or it can stay alive, while the other one is orbiting itself. 951 00:49:10,466 --> 00:49:12,266 That's a very, very fascinating question. 952 00:49:12,266 --> 00:49:16,800 I, I don't know, but I'll tell you something. 953 00:49:16,800 --> 00:49:22,866 Which happens when physicist or when we try to model, biological processes. 954 00:49:22,866 --> 00:49:23,400 Right. 955 00:49:23,400 --> 00:49:26,400 So in modeling biological processes, you get these complex spatial 956 00:49:26,400 --> 00:49:29,400 temporal behavior that you are trying to create a model for. 957 00:49:29,633 --> 00:49:34,133 And that could be multiple models which would fit that observation. 958 00:49:34,133 --> 00:49:36,733 And then this is which one is what. 959 00:49:36,733 --> 00:49:38,400 It's what the biology is doing. 960 00:49:38,400 --> 00:49:43,200 And that takes like very careful experiments to do and sometimes does. 961 00:49:43,200 --> 00:49:45,666 The data is very noisy even in biology. 962 00:49:45,666 --> 00:49:46,733 So yeah. 963 00:49:46,733 --> 00:49:50,966 So you're right that one question is like, can you use these models to 964 00:49:51,533 --> 00:49:54,800 produce some pattern which you see and then the patterns which you do see how 965 00:49:54,833 --> 00:49:57,900 you can be sure that these what this model is, what is the mechanism 966 00:49:58,200 --> 00:49:58,800 that's going on. 967 00:49:58,800 --> 00:50:02,100 So that's even in biology when you it's hard to poke things around. 968 00:50:02,400 --> 00:50:05,466 It's difficult to to clearly say that this is actually 969 00:50:05,466 --> 00:50:08,666 what's going on or it's, one of the possible mechanisms, 970 00:50:13,233 --> 00:50:14,000 Okay. 971 00:50:14,000 --> 00:50:16,566 With another question. So thanks very much. Again.