Let's go for super 820. All right. So thank you all for being here on a Saturday morning. So this morning I'll tell you about nonlinear dynamics of, active particles. So I've got, two key phrases there in my talk. We have nonlinear dynamics and active particles. So what I'll do is I'll start by giving you an overview of each of these topics. So first I will tell you something about active particles and active matter. And then I'll tell you some ideas from nonlinear dynamical systems.
And after that, we'll go over a couple of examples from my own work where we try to understand the dynamics of these active particles using these ideas from nonlinear dynamical systems. So we'll we'll explore two simple theoretical models. So the first one will be an active particle, flowing in a fluid flow along a channel. And the second would be, I'll introduce you to walking and super walking droplets.
And then after the talk, as you make your head towards the coffee, I have a little demo of these super walking droplets so you can have a bit of a play around with these droplets on your own. All right. So so let me start by telling you what I mean by an active particle. So the best way to do this is by showing you some examples. But before I do that, let me give you a working definition that is generally used by researchers in this area.
So an active particle is an entity which consumes energy from the environment and converts it into some sort of directed and persistent motion. Right. So in doing this, it also dissipates some energy. So these are out of thermodynamic equilibrium. So these are non-equilibrium entities. And they come in all shapes and sizes. And they spend a wide range of scales all the way from nanometer and micrometer to the scales of several meters. Right.
So at the nano and microscale, you have molecular motors that do processes that maintain, life within these subcellular scale processes. And then you can think of cell itself as an active particle and other microorganisms as well, such as bacteria. So here is a video of a type of white blood cell, chasing a bacteria. Right. So it consumes energy, by eating food in the form of chemical energy. And it's converting it into this crawling sort of motion here.
Another example of a microorganism are these E coli bacteria. So these are immersed in a fluid medium. And as you see they are self propelling using their helical tails. And they are doing running and tumble kind of motion. So they will run in one direction for a while and then reorient themselves. So, so these were some of examples from the from the living world of active particles. You can also have synthetic systems, and synthetic active particles as well.
So so one example here are these millimeter size walking droplets, which we'll explore in detail a bit more later. But to to give you an overview. Now these are droplets that walk horizontally while bouncing on a vibrating liquid. But and then another synthetic system are these, millimeter sized robots. Right. So when you have robots, doing some sort of self propulsion, right. Either preprogramed or sensing its environment, it is an active particle.
And then at the scale of, meters, we have big mammals, right? Birds, fish and even other cells. Right. So we are in some sense active particles as well. We eat our food and then convert that into some sort of motion all the way around. Right. So so that's active particles. Now active matter is matter composed of a large collection of these active particles. Right.
So like you have your passive phases of matter solids liquids and gas where the constituent elements are atoms and subatomic structures. For these active matter, the constituent elements are active particles that I showed you on the last slide. So since these active particles are out of equilibrium, the phases you get of matter when you have lots of these active particles, they are also non-equilibrium phases. So let me show you some examples here of active matter.
So a classic example is flocking of birds. Right. So when you have a large collection of birds such as starlings they form these beautiful murmuration patterns. And early on in this area, there were quite a few models developed to, to describe some of these behaviors using, these ideas of active particles and active matter. You can get this, this type of flocking behavior also at the scale of cells. So what you are seeing in that video, are epithelial cells.
So these are the scales that you have on your skin. And if you have a cat, let's say, on your skin, then you might see something like this, a motion like this, where all the cells collectively try to migrate and heal the wound. Right. So so you can have collective behaviors arising at the scale of cells, as well. A common feature that arises when you have these active particles is you get, motility induced phase separation. So here the interaction between the particles is purely repulsive.
And yet you see these clustered phases forming. And the gist of it is that if two active particles come near each other, these are active. So they don't have to obey energy or momentum conservation. They are out of equilibrium. What happens is that they can hang around each other for longer unless one of them decides to turn away and move, right.
So because of this, active particles slow down when they come near each other, and this gives rise to a positive feedback where because they have slowed down, more of them will aggregate. And you get these clustered phases, forming inactive matter just purely due to its persistent motion with repulsive interactions. And then lastly you get these spatial temporal chaotic behavior. So here you are seeing a video of a large collection of bacteria.
And you see it's, it's behaving in a in a very chaotic fashion. So so we call this active turbulence. And this is different from the the fluid turbulence. You see for example in the atmosphere because they're the turbulence is due to the inertia of the fluid. While here at the scale of these bacteria, the fluid flow for them is very viscous. Right. So the fluid Reynolds number is very small, and it's the activity of these bacteria that's giving rise to these turbulent behaviors. Right.
So you have these, emergent behaviors, that arise in these active systems. And I just like to point out it's it's in a sense very similar to the the condensed matter conundrum of physics, which was beautifully explained by Phil Anderson in his essay More is Different, right? So the properties and the behaviors you get emerging at a collective scale, they need different tools to be understood. And you can't just predict them from what's happening with the single particle level.
And same is true in these active matter systems. Right? So you have these active particles, but then when you have lots of them, you can get these collective emergent behaviors at larger scales emerging. All right. So having introduced active particles and active matters, let me tell you something about, nonlinear dynamical systems before we move on to the two examples.
So as theoretical physicist and also applied mathematicians, the dynamics that we see out there in the real world, we try to make sense of it by creating mathematical models. And much of the artistry of this process lies in identifying the relevant variables. That would model the system, or the process we are interested in, and write down equation that describe the evolution of these variables. Right. And more often than not, these equations, they turn out to be nonlinear.
So if we have, a list of variables that are described by this vector x that we are interested in, then a dynamical system is a system that models the evolution of these variables in time. Right. So the dots there on top of the x is a time derivative. And many times these function on the right hand side they can be non-linear functions. Right. So you can have product of variables or the variables inside the argument of a non-linear function.
So it is often useful to go away from the picture of what these variables are describing. And consider an abstract space which is made up of all these variables. It's usually called the state space or the phase space of the system. So it's the space of all possible states of the system. And the evolution of a system would correspond to a trajectory evolving in this state space. Now, there are many ways you can classify dynamical systems and nonlinear dynamical systems.
But the, the way the one which would be useful for today's talk is the classification into conservative and dissipative dynamical systems. So conservative dynamical systems are systems where the volume in phase space stays conserved. So if you start in this phase space with a box and as the system evolves, this box will stretch and can do complicated things, but its volume will stay conserved in a conservative systems.
So, classical Hamiltonian systems, which are time independent, fall into this category of system. So generally when you have a closed system where you have some some constants of motion, you usually have, conservative system. And a classic example is of a pendulum. Right. So if you when the pendulum is doing back and forth motion, if we go to the phase space picture and look at the position and the velocity coordinate, then what you get is what you are seeing here as this, this closed loop.
Right. So back and forth motion would correspond to a closed loop in phase space. And if I take a patch in phase space, it will just go around this closed loop. Now if the system turns out to be non-linear, then you can get more complex and interesting things going on. And again, a classic example is your double pendulum. So with a double pendulum you have a four dimensional phase space. You have two coordinates and two momenta.
And so what you're seeing here is for different projections of this phase space in 2D. And what you see is that in the initial patch you can see it's stretching. It's bending. It's folding in complicated ways. But again it's a conservative system. So the overall volume of that patch will stay conserved. And there is a very well established mathematical framework of understanding chaos in this type of systems. And conservative chaos or Hamiltonian chaos.
Now, in contrast to these conservative systems we have dissipative system. So in this space the speed of systems, sorry, the phase space volume, it will shrink with time. Right. So the phase space volume shrinks here with time. And to give you again a simple example, if we add friction to our pendulum right then it will go back and forth, but then it will eventually come to a steady state. And in phase space, this would correspond to a spiraling motion towards a stable point.
Now, dissipative system does not always mean that everything will just come to a stop. You can still get periodic motion and chaotic motion in dissipative systems as well. And again, a classic example is the the Lorenz chaotic system where where chaos was kind of first, reported and again, it's it's a dissipative system. But as you see, all initial conditions, they kind of converge on this fractal shaped attractor and a volume in phase space would again stretch, fold bending, different ways.
Even though the, the total volume, would be shrinking with time. Right. So, so using these, ideas from conservative systems and dissipative system, what we'll see that, the examples will explore today. One of them will be, will form a conservative system and one will form a dissipative system. And with dissipative system, usually if you have an open system where you have energy injection and energy dissipation, usually they will fall into this category of dissipative dynamical systems.
All right. So with that, let's move on to the, the first example, which is an active particle, flowing in a unit direction of the fluid, slow. So going back to our active particles that I introduced in the first slide, if you have an active particle which is immersed in a fluid medium, it's called a micro swimmer. And again, you can have a, living, example or an artificial or a synthetic system. Right.
So a bacteria, or other motile cells or even a micro robot immersed in a fluid medium would be, considered a micro swimmer. Now, micro swimmer, they in many situations, they experience fluid flows in confined environment. Right. So if, if a micro swimmer is flowing in a confined environment and you have fluid flow along one direction, that is a common situation.
So for example, sperm cells moving in fallopian tubes, micro robots for targeted drug delivery application and even pathogens moving in blood vessels. So what we are going to do is we are going to look at a very simple model as theoretical physicist, and try to see what sort of complexity and dynamics does this give rise to in this situation. So so the model we have is as you see in the schematic, we have a three dimensional pipe, right? A three dimensional channel.
And we have a fluid flow going from left to right. So fluid flow is uni directional. It's in one direction. However it's not uniform. Right. Because we have walls, the fluid right next to the wall will stick to the wall. So there will be a variation in the fluid flow profile across the channel. So you get this parabolic flow profile. And we are going to consider an active particle which is very unintelligent a very simple active particle. It moves at a constant speed.
V naught in some direction ahead. Right. So that's all the active particles are doing. It's persistent motion in some direction. And we are going to make two key assumptions. The first is we are saying that the active particle is very small compared to the channel. So the disturbance is caused by the active particles are small, and it only experiences the fluid flowing along the channel. And also we are going to neglect any direct interaction between the particle and the walls. Right.
So we are saying that the particle only experiences the fluid flow and the gradients in those fluid flow. So if you try to write down equations of motion for this particle, you get these two equations and I'll explain what they mean, step by step. So the first equation is telling you the total velocity of the particle. And we are saying it's the addition of its active velocity, which is a constant velocity in some direction. Plus this background's fluid flow.
And you can do this linear superposition because we are in the lower Reynolds numbers. So the fluid flow equations are linear. And the second equation tells you how the orientation of the active particle evolves with time. Now because the fluid flow is not constant, different layers of fluid are moving at different velocities. So if you immerse something in in a in a flow like that, it will start rotating. The particles will start rotating.
So the direction of motion of my particle will rotate because of the shear in the flow. Right. And that's what the second equation is describing that because of this local spinning motion, local vorticity of the fluid, you can get tumbling of the direction of motion. Right. So we have these two simple equations. Now let's try to write this in component forms. We are in three dimensions. So in component form we will get six.
Equation three for the position of the particle three for its orientation. Now if you look at this equations you can see that the z equation which describes the motion along the channel the z variable. It decouples from the rest of the system. So we can integrate that separately. So effectively we only have five equations that are coupled to each other. Right. So we have a five dimensional face space here. Now it turns out that in this system you end up getting constants of motion.
So quantities that will not change with time as the system evolves. And an obvious one, which you might have noticed is that the orientation vector has to be a unit vector. Right. So that constrains, the orientation variables. So the orientation vector can only lie on the surface of a sphere.
And it turns out that you also get another constant of motion in the system which is described by this C. And it relates to the the fluid flow velocity U, and also the z component of the active particle velocity. One thing I forgot to mention is that, in these equations, which you see in component form, I've rescaled the active particle velocity, and it's kind of embedded in the fluid flow. So the active particle velocity is one and the fluid flow profile is modified to this U by.
So we have five equations two constants of motion, which means that the effective dynamics of the system take place in three dimensions. And it turns out that this is an example of a conservative dynamical system. And you can show this mathematically by taking the the divergence of the dynamical flow. And surprisingly, and more importantly, it turns out you can map this system on to a Hamiltonian system. So you can do a 1 to 1 mapping of the system and write it in terms of Hamilton's equation.
And in this mapping it turns out that, the system obeys Hamilton's equation. And your Hamiltonian looks like this. Right. So you have a kinetic energy part and the potential energy part. And the momenta which are in the kinetic energy part, turn out to be basically the velocity of the active particle in the cross-section of the channel and the potential energy part. It basically is related to the fluid flow U, and there is also a C in there which is a constant of motion.
So your initial conditions of the system will tell you what the value is. There. Right. So after these equations let's see what the, what the system does when we try to simulate the system. So you see three different plots here. And I'm going to start a video in a second. But the plot on the left side is the top view of the 3D channel. So in the in the y z plane, the the plot in the middle is the cross-sectional view and the plot on the right.
The panel on the right is showing the orientation of the particle. So if we start the particle near the center of the channel, then it will flow from left to right with the flow. However, its orientation is against the flow and it keeps on oscillating as you see there. Right? So you get this periodic motion of the particle. However, it stays near the center of this channel.
On the other hand, if I start the particle away from the center of this channel, then you can see that it's doing this irregular behavior. It's still going from left to right with the flow, but its orientation is going all crazy. And it's also staying away from the center of the channel. Right. It's never visiting the center of the channel. So, so to make sense of why we see these different types of behaviors, let's go back to our Hamiltonian mapping. The Hamiltonian picture of the system.
So in the Hamiltonian picture this is what the potential energy looked like. So if you look at the constant c in there is the potential energy. So the potential energy depends on two things the fluid flow profile and the c. The constant C is determined by the initial conditions of the system. So what that means is that depending on where the particle starts, it sees a different potential landscape available to explore.
So if the particle starts near the center of the channel, then the potential landscape that is available to explore for the particle is something you see there on the top, left, right. So it only has this region, near the center of the channel to explore, and that's the corresponding trajectory. Whereas if the particle starts away from the channel, then the potential landscape it has available to explore has this annular shape. And so its trajectory is confined in this region. Right.
So this, this mapping allows us to kind of understand why this particle is only exploring certain regions of the, of the channel. Now, to get a full appreciation of the complexity of behavior you get, here is a plot showing all the different behaviors you get depending on where the particles starts in the cross-section. So the top plot you see, with all the different colors are the qualitatively different types of trajectories that are obtained based on where the particle starts.
So if the particle starts near the center of the channel, you have that green trajectory but stays near the center of the channel. If it starts in one of those light blue or dark blue regions, it will stay confined within one of those slabs or vertical or horizontal slab. If you start in one of these special yellow regions, you stay confined near the corner of the channel, in these purple regions, as we saw in the video, you will stay confined away from the center of the channel.
And then these red trajectories is you can basically wander anywhere in the cross-section. Now the the bottom plot there is important. And this is a bit more interesting what you are seeing in the bottom plot. There is a measure of what's called the largest Lyapunov exponent. It's a measure of chaos in the system. So whenever you see a violet region there, the motion is regular, periodic or quasi periodic. And wherever you see this orangish regions, that's where the motion is chaotic.
So what you see that is if you start near the center of the channel, the particle does periodic motion, regular motion. But it's as you go away from the center, you start getting these chaotic regions coming in right. And this transition from periodic to chaotic behavior can again be rationalized in terms of these nearly integrable Hamiltonian systems. Right?
So that was just to give you a flavor of the the complexity of behavior that you can get for a very simple setup of an active particle interacting with a unidirectional flow, you get this very nonlinear dynamical system. But we can rationalize, why we get these different behaviors in terms of our understanding of these, nonlinear systems. All right. So let me move on to the second example. Now, of super walking droplets. All right.
So if you take a container filled with liquid and you vibrate it vertically, then above some critical amplitude of vibration, the free surface of the liquid will no longer remain slide and it will become unstable. And you will get these these standing waves which are known as Faraday standing waves. And that's because these were first discovered by Michael Faraday back in 1831. Right.
So you have a critical threshold below which your free surface remains flat, above which it becomes unstable to standing waves. Now, if we are just below this instability and we create a droplet of the same liquid as the bath, then this is what happens, right?
So back in 2005, Eves, Cooper and colleagues in Paris discovered that if you take a droplet, of silicon oil, on a vertically vibrating bath of the same liquid, then the droplet can bounce and walk on the oscillating surface of the liquid, right? So as you see in those top videos, each time the droplet bounces, it creates a little wave around itself. And the droplet then interacts with these waves on subsequent bounces to propel itself horizontally.
So three key features to note about this system. The first is that the particle and its underlying wave, they co-exist as a wave particle entity. So if I were to make the droplet disappear by poking it, the underlying liquid surface would remain flat. It would eventually decay to a flat surface, right. So the droplet and the wave are couple two ways. Second is that there is memory in the system, and that is because the the waves created by the droplet, they decay very slowly in time.
So the droplet is not only influenced by its most recent wave that it created, but also by the waves it created, let's say 10 or 20 bounces before. Right? So we have this idea of memory in the system. And the third, it's an active system, right. So even though the whole container is being vibrated up and down, the droplet locally extracts that energy through these waves it generates and it converts it into this directed persistent motion. Right.
And, usually you get two different kind of walking droplets here. So if you drive the path at a single frequency, then you get slower and smaller droplets, which are called workers. If you drive it at multiple frequencies, then you can get inertia dominated, bigger droplets that move more faster. And we'll see them, later today. All right.
So, what we'll try to do is let's let us now try to write down a simple model of the system, and then we'll see the type of dynamics that we can capture using this model. Now the the video, you see that at the top. It's the same video from the experiments. However, it has been stroked at the bouncing frequency. So if you take only one image of the droplet per bounce, then you don't get the bouncing motion. You only get the horizontal walking motion. And that's what you're seeing in the video.
And this is essentially what the model is doing. So it's called the strobes Copic model. And what it does is you average over the vertical periodic bouncing fast time scale. And you look at what happens in the in the horizontal direction equation of motion. And we are going to look at a very simple, model where you get you are only allowed to move horizontally in one dimension left or right, right. And there is no bouncing motion.
So if you try to write down an equation of motion, for this droplet, it's Newton's second law as equals, Ma. So on the left side, and this is in dimensionless form. So on the left side you have a term which looks like mass times acceleration. So you can think of kappa as a dimensionless mass parameter. The second term, it kind of models the dissipation in the system.
So because these droplets are moving through air and they also lose momentum when they impact the liquid, you have dissipation in the system, which is modeled proportional to the velocity. Right. So you have a dissipation term there, the horrible looking turbo on the right. That's the kick that the droplet receives from these self-generated waves. So. Right. So so let's try to understand where is that coming from. So first of all you have an integral there and not a sum.
Because in the model we are kind of averaging over the bouncing dynamics. So we are saying that the droplet is continuously emitting these waves as it moves. And how are how do we calculate this force. So we say that at each instant of time the droplet generates a wave of shape A of X, and the force that the droplet receives is found by adding up all these waves, integrating through all these waves that it has generated in the entire history.
So you get an overall waves yield, and the gradient of that ratio is what gives you the push right. So what you have here, this f is the gradient of the individual waves. And you can integrate that over the entire history of the droplet. And these waves are decaying exponentially in time. So you have an exponential term there. Right. So you have this integral term on the right hand side which will eventually tell you what is the force on the droplet from these self-generated waves.
Now this is harder to solve. It's an integral differential equation. However, if we make some simplifications we can make some progress into converting this into. Integrate essentially equation A into something which is more analytically tractable. So if I say that the the shape of the waves that the droplet generates, it's, it's so usually in experiments the waves that the droplet generates has both spatial oscillations and spatial decay.
If you neglect the spatial decay and say that the droplet generates simple sinusoidal waves, right, then the wave field would be a cosine and the gradient would be a sine function. And you get this integral differential equation. And it turns out that you can map the system. To a system of ordinary differential equations. And I don't know if any of you recognize that these are the the classic celebrated Lorenz chaotic system that you can map this system to.
Right. So the capital X variable maps to the velocity of the particle. The capital Y variable maps to this horrible integral force term on the right hand side. And the capital z variable is also related to this, memory force. And it kind of is a measure of the the amplitude of the waves where the droplet is. Right. So you can get this 1 to 1 correspondence between the motion of the droplet and the dynamics of the Lorenz system. Right.
And again this is if you make this simplification that he behaves that sinusoidal, then you can do this nice conversion. And you can try to understand what the dynamics are. And also the parameters of the system map to the, the you can map it to the parameters in the Lorenz system. And as we saw before, the Lorenz system, that's an example of a dissipative dynamical system. Right. So here we have a dissipative dynamical system. All right. So these are the same equations as on the previous slide.
Same Lorenz equations I've just rescale them so they are more meaningful for the walking droplet. So we see two dimensionless parameters. Here we have r and we have tau. So you can think of as a dimensionless amplitude of the waves generated by the droplet. And tau is the rate at which the waves decay, which are generated by the droplet. So tau think of Tao as the memory of the system or the memory of these waves. Right?
So what we are going to do is we are going to see what sort of dynamics does this give rise to as we increase this memory parameter. Right. So let's start with very small values of memory parameter. And I'm showing you two plots there. The one on the left is this phase space plot. But it's projected in two dimensions. So you are seeing the capital X and the capital Z projection of the phase space of this three dimensional Lorenz system.
And the plot on the right is showing you how does the position of the droplet change with time. So at low values of tau, when you solve the steady states of the Lorenz system, you find that there is one stable point, where capital X is equal to zero, which is this black dot. So capital X, if you remember, was the velocity of my particle. So this means that my velocity of the particle is zero. So my particle stays stationary. And we have a stationary state of the droplet.
And this is something you see in experiments when you have low amplitudes of vibration. Now as I keep on increasing the memory you get bifurcations in the Lorenz system. So these are the transitions between these stable states. And it turns out that the stationary state which was that stable before it becomes unstable. So it's now gray here. And you get a pair of new stable points in the system. Right. One at a positive value of x and the other at a negative value of x. Right.
And what this corresponds to is a positive constant velocity and a constant velocity going in the other direction. So these are again steady walking states which are again observed in experiments droplet moving at a constant speed in a given direction. Now what happens if you keep on continue increasing memory? Well, it turns out that these innocent looking Lorenz equations, the exhibit quite profound complexity of behavior.
There is a whole book written on the, the Lorenz equations and the kinds of dynamics that it exhibits. And it's non-trivial. So here, here's a cartoon showing the sequence of bifurcations, that you get as you keep on varying these parameter. And I'll not even attempt to explain it today. But, the important thing is that eventually, as you keep on increasing tau, eventually, after all these complex bifurcations, you get chaos in the Lorenz system, right?
Which is something you might have seen as the butterfly effect or this Lorenz attractor. Right. So you get this Lorenz chaotic attractor in phase space. And if we now try to understand what does that mean for the droplet motion, it means that the droplet is moving back and forth. So when you are on one wing of the Lorenz attractor, you are moving in one direction. And when you're on the other, when you're moving in the opposite direction.
So your particle is essentially doing kind of like a random walk between left and right, as you have chaotic motion between the two wings of the attractor. Now, this is something that's not seen in experiments yet, because to get into this very high memory regime, you need to be very close to that instability that I mentioned before without triggering that instability. Right.
And and that's something that people have not been able to get a handle on experiment when the droplet is moving in free space, if you can't find the droplet, then you can get chaotic motion, but not in free space. All right. So let me show you, a couple more slides, and then, I will try to wrap up my talk. So what happens? A natural question is what happens if you have more than one of these walking droplets interacting with each other, right.
So let me show you some of, some videos from some experiments during my PhD where we are trying to look at interactions between these walking droplets. So if you have two of these droplets at low memory or low vibration amplitudes, then they will stick to each other and they will work together as a pair. So you have a doublet of droplets walking together as a constant velocity. If you have more than two, you can get a staggered configuration with three droplets and they'll work together again.
You can also get quadruplets, as well. Moving together. Right. So this is at low amplitudes of vibration or low memory. Now if we increase the the memory in the system and look at that pair of doublet, then what you see is that they will still stay bound and work together. But you get these sideways oscillations between the droplet. So so this is what's called in the literature from an air. Right. So you get these droplets oscillating sideways as they work.
If they are not of the same size, if one is bigger, one is smaller, then you get this chasing mode where the bigger droplet will kind of drag the smaller droplet in its wake, and they'll work together. Right. And you can also get orbiting states. So a very big droplet cannot survive on its own. But if it's accompanied by a smaller droplet as a satellite, then the smaller droplet will keep on orbiting the bigger droplet. And that will somehow sustain the bigger droplet.
And we don't understand why this happens. Right. So you get these interesting orbiting states, as well. And then now next question. What happens if you have lots of them. Right. So if you have lots of them again at low memory you get a nice crystal like lattice structure forming.
As you increase the memory that the crystal melts, but it still stays bounded and you get this jiggling state, a liquid like state, and then eventually the cluster will disintegrate and you get a gas of droplets just hitting each other like billiard balls. Right. So you can get these again, you get when you have lots of them, can get these interesting collective behaviors that emerge in this system, which is something that we are trying to understand now.
So lastly, I wanted to mention, why this system is interesting from a physics perspective. Right. So, there have been, attempts, or people have tried to, to mimic quantum like behavior in this system. So a disclaimer this system is not a quantum system. It's a classical system, but it's a classical system which is out of equilibrium. So it's an active system, and it seems like you can get some really interesting dynamical behaviors in the system.
And then if you look at the statistics of that chaotic dynamics, you get seem to get wave like statistics in the system. So I'll show you some examples. So here, is an experimental video, showing tracking of a droplet confined in a cavity and the size of the cavity is of the same order as the wavelength of the waves generated by the droplet. So the droplet does this apparently chaotic looking motion inside the cavity.
And if you track the motion for a few hours, it's chaotic and nothing much is going on. But if you get enough statistics and you plot the probability distribution of the droplets position, then you get this wave like distribution in the position of the droplet. And this is not too different from what you would find if you had an electron confined in a in a ring of copper atoms. And you look at the probability distribution of the of the electron. Right.
So the underlying chaotic dynamics here seem to be giving rise to wave like statistics. Another example is an analog of tunneling. So if you have these walking droplets and you put barriers below the liquid surface, then these droplets will typically reflect from these barriers. However, occasionally it turns out that these droplets again due to these non-linear interactions can unpredictably cross these barriers. Right.
So you have an analog of tunneling where these droplets can unpredictably cross these, barriers, in the liquid. Now, if you had a walking droplet interacting with these barriers, it can get deflected, right? So people tried what happens if you put a single slit or a double slit and let these droplets pass through them? And what happens?
And surprisingly, what turns out that you if you do this experiments and you fire droplet one by one, either at a single slit or a double slit, then even though the droplet is going through one of the two slits, the wave that is guiding it, it's going through both the slits and you get some interesting patterns. If you get look at the distribution of droplets. Now, it's not the exact same pattern, diffraction pattern or interference pattern that you would get in a quantum system.
And I wouldn't expect that to happen as well anyways. But you you do get some interesting wave like patterns in the distribution of the position of the droplet. And then lastly, if you can find one of these droplets in a harmonic potential, then it turns out that it can't do any random motion. Its motion collapses onto a set of these limit cycle orbits, so its motion is constrained.
So if you kind of plot the radius of curvature and the angular momentum of the droplet, and you get these quantized states of motion, because the motion is no longer chaotic, it's collapsing into these, well-defined periodic states. Right. So, so these are some of the examples, where the system exhibits quantum like behavior. And again, it's not a quantum system. It's a classical system, but it's out of equilibrium. All right.
So so just to summarize, so so I introduced a bit about active particles. So these are non-equilibrium entities that consume energy and convert it into some form of self-propelled. And when you have lots of these, you get these emergent behaviors, in the system, such as flocking, active turbulence and all sorts of other features. And then we looked at two simple, models of active particles, individual active particles interacting with some form of environment. Right.
And we saw that even the simple models exhibit very rich behavior. And we tried to understand or rationalize some of these in terms of using these ideas of conservative and dissipative nonlinear dynamical systems. Thank you for listening. Yeah. No. Thank you. Very fascinating question. So, just to repeat the question. So the question was, I presented, toy models, numerical simulations as well as experiments. And then what? Wait, what should I give to to each of them?
So, these simple woody type toy models, they are mainly useful in kind of exploring the, the parameter space of the system quite rapidly and to, to get, gist of the different types of qualitative behaviors that can, that the system can exhibit. Now, if you want to accurately capture what's going on in experiments that have been, a plethora of models that have been developed with increasing complexity.
So you have to resolve the the bouncing motion of the droplet as well as the evolution of the surface waves. But then the downside is that because you have all the fast time scale, they are very inefficient. Now in some of these active matter models, when you look at, effects at a collective scale there, the motion of the individual particles may not become the details. Motion may not become that important.
So when you have a collection of these, these active particles, theories have been developed, to which coarse grained, over this small time scale. And you look at a continuum model of equations, so a continuum model that describes the system. So, for example, people have borrowed, theory of liquid crystals. So theory of rod like particles in a defining a continuum.
And so Julia, in the department, she and her group, they have developed, theory of active liquid crystals now are active pneumatics. So you can describe, large collection of these bacteria or cells using these coarse grained equations, which would describe what happens at the collective scale rather than the individual scale. So I think it's a complex thing. And it's, it's like depending on what scale you are interested in, you would have different models.
And because of these emergent features, you might need, again, different models to describe things happening at different scales. But the models I presented today, are more of very simple toy models of. But just to illustrate the complexity, even those models can give rise to. So yeah. Yeah, yeah. Very good question. So if you see a raindrop falling on a puddle, it coalesces. Right. So the reason the droplets don't coalesce here is because the underlying liquid is vibrating.
So there is not enough time for the droplet to merge with the liquid because the air layer which separates them, it kind of acts like a spring and the droplet keeps on bouncing. So you, the way you create these droplets, which you will see, is that I just use a toothpick, put in the liquid and then like, hit it and then, Hiroko. Yeah, right. So think of it like, like jumping on a trampoline. That's what the droplet is doing because of the surface tension.
It can just, like, maintain this bouncing motion without coalescing with the liquid. Sorry. What was the question at the end? It was the question. Okay. Right. Yeah. Yeah. No, the thing is, like, I don't enough understand enough of the pilot wave or other models. So I wanted to stay a bit away from that, but that there are, there is, a research group at MIT led by John Bush.
They are trying to actually make these analogies, more, let's say, towards the quantum side to see how far you can take them. And yeah, there have been some interesting developments, but it's not clear how close it is. Yeah. No, that's a that's a very interesting question.
So I think part of this motivation of this whole area of active matter, which has only really begun to kind of flourish in the last 20 years or so, is that, is the kind of like biological inspirations, right, that you get, in biology, things happening in coordinated and organized fashion at all scales, a different hierarchy of scales. Right. And then can you, can you understand these features using some of these, physical models?
And I think, as you're saying, like, it might be that things that happen at a collective scale, you might be able to describe them not going back to these microscopic picture and just having some sort of a statistical picture of what goes on at that scale. And people in the area use both tools. I think tools are being used even from equilibrium statistical physics to to rationalize some of these, large scale behaviors.
But then in some regions, those, tools fall down and you get truly non-equilibrium behaviors, and then you need to kind of modify the equations of, statistical physics and look at more non-equilibrium pictures. But yeah. No, it's it's a it's an important question. And I think it's, it's, it's slowly being addressed with this type of framework.
And I think biology is, is a very good example where you get these emergent behaviors happening at all scales and how they are talking to each other at different scales. Right. So, so the only experiments I did were during my PhD with these droplets because we have a tabletop, I haven't done any other experiments. So most of the other time is, again doing this mathematical modeling.
And then also when you can solve these equations or approximate them in a nice way, you results, you resort to numerics and try to simulate these systems, which can sometimes be quicker. Sometimes it can take a few days. Yeah. So right now it's mainly mainly mathematical modeling side and then simulation side of things. Right.
So so there are people even in engineering, departments, who are quite excited about this whole idea of active matter and it's being used in the sense that people have started looking into, a swarm or a collection of these robots, these which can be modeled as these active particles and trying to get some useful work, done out of them. So there are, in a sense, more practical problems that people are trying to tackle using these, ideas from active matter.
So it's it's it's it has started recently, but it's, it's it's ongoing. Yeah. No, it's, it's I don't think it's like, I don't know how seriously to take it, but it's it's like there's this system which is a particle. It's coupled to its waves. And then it moves together. And because it moves together, it doesn't interact directly with its surroundings. It's indirect. Through these guiding waves, and you get some interesting interference like effects.
But you're right that that in, in other classical pictures, you could probably form similar analogies. Perfectly as well. I don't know. And I'm not sure how, how much people agree within the field as well. So that's why I kind of said it's, it's like a working definition. But yeah, the, the key feature seems to be that it can somehow consume or absorb energy and convert it into some form of persistent motion. And if if an entity can do that, then roughly it would fall into this category.
And it's like also depends on what sort of things you are interested in. In a sense, if you have a very complex animal, it's doing even though it's like doing locomotion, it might be very different reasons. It might be sensing its environment and, and lots of different things going on. Right.
But then if you're just interested in on how a large collection of them behave, then maybe you might be able to use some of these ideas to have a more simplified picture of what happens at a larger scale, right? Yeah. So yeah. So it's not it's not carrying its own energy source. But yeah, you are driving it from outside. So so there is again, this kind of hazy line between non-equilibrium systems. And when would you call them active. Right.
So in general, like when you see turbulence in the atmosphere and other non-equilibrium processes, they are also out of equilibrium. But they're the thing that forces them out of equilibrium is that the scale of the system. Whereas here what you see is that even though I'm injecting energy everywhere in the liquid surface, it's being locally extracted by the droplet and converted into motion. So that's kind of the idea that these active particles.
Yeah. So, so the the models which I haven't presented were where we looked at in terms of modeling these droplets. We assumed that these droplets are spherical rigid objects, whereas you have a deforming quite a lot as you saw in this videos. So I think our models are inadequate to capture those deformations. And the interaction with the, liquid surface. And yeah, it's not clear.
So because the earlier between the droplet and the liquid, it somehow needs to be sustained, right, for the droplet to stay. And if it's by itself, it would just coalesce. But here, I don't know, some of the bouncing motion of the droplet is somehow replenishing the air layer, which is giving rise to its persistence. Or it can stay alive, while the other one is orbiting itself. That's a very, very fascinating question. I, I don't know, but I'll tell you something.
Which happens when physicist or when we try to model, biological processes. Right. So in modeling biological processes, you get these complex spatial temporal behavior that you are trying to create a model for. And that could be multiple models which would fit that observation. And then this is which one is what. It's what the biology is doing. And that takes like very careful experiments to do and sometimes does. The data is very noisy even in biology. So yeah.
So you're right that one question is like, can you use these models to produce some pattern which you see and then the patterns which you do see how you can be sure that these what this model is, what is the mechanism that's going on. So that's even in biology when you it's hard to poke things around. It's difficult to to clearly say that this is actually what's going on or it's, one of the possible mechanisms, Okay. With another question. So thanks very much. Again.