 Okay, show me a start. Our first talk is will be presented by a professor at the University of Seoul National University. Please welcome him. Thank you for the kind introduction. And also I'd like to thank the organizers, especially for giving me an opportunity to speak here. And so today, I noticed that this session is like the first talk was about quite relevant biology. The second talk was more theoretical in general. The third talk was again very biological. And now as you can predict, this talk is going to be a very theoretical speculative. So I'm sorry if you feel that this is not really relevant to biology, but I think what I'm going to talk about today might be happening in a system which looks like this. So like let's say you have some kind of swimming bacteria in some kind of solution, and then you have a very symmetrically shaped object, or a tracer particle, which is being pushed here and there by these bacteria. This is swimming bacteria. And I'm going to talk about some kind of very, which appears like a very direct mechanism for some kind of symmetry breaking phenomena associated with the mortality of this tracer particle. In a dilute active fluid, where you can treat these bacteria almost like some kind of ideal gas particles. So this work was done in collaboration with my students, Gi-won Kim and Yoon-sik Choi. Gi-won Kim is sitting somewhere here, I think. If not... I'll talk with him later. Let's begin. So I think I'm the first speaker who is really talking about active matter. So let me first introduce the concepts. So here's the very representative example of active matter, the E. coli. So as you can see here, E. coli, maybe many of you have already seen this movie, but it swims by moving its flagella at the end. It's shaped like an oblong ellipsoid, and it keeps moving in a straight line for some time, and then at some given rate it then starts rotating, tumbling in the same position, and then it starts running again. So basically the motion of this kind of bacteria can be simplified like this. So basically you understand this to be like a series of straight runs punctuated by random tumbles at a given rate to a random direction. So this is called one-end tumbling dynamics, understandably. So this is a very simple example of active matter, active particle. So how is it different from non-active particle, passive particle? So here's a schematic diagram showing the difference. So in the case of past particle doing the usual thermal Brownian motion, you can see this particle with some kind of mechanical energy, exchanging energy with the surroundings to form a hit or work. And you typically model the dynamics using some kind of long-run equation. So here you have the drag force, the friction force, and you have some kind of potential conceptual interactions with the surroundings, and also you can have some kind of thermal noise. But for this kind of Brownian particle, what happens is that the particle satisfies fluctuation with patient relation. So the way the particle is paged is energy to the environment, and the way the particle absorbs energy from the environment has some kind of time-reversal symmetry. So therefore the fluctuations, the sort of fluctuations in the environment and the way you lose energy to the environment by this patient is actually related by temperature. So this is called, this fluctuation of this patient relation ensures a time-reversal symmetry of the heat dissipation and heat absorption. And so this kind of particle can reach equilibrium at some given temperature. But in contrast, so for the active particle like this bacteria, so naturally you can see that this particle has some kind of separate source of energy other than the heat from the surroundings. So it has an ambient or stored energy in the form, for example, the chemical fuel, and then it converts that to mechanical energy and then in turn that is dissipated into the environment. So because of this extra source of energy, in addition to the usual terms appearing in the long-term equation for the Brownian particle, we have some additional things here, for example like this propulsion force, which is highlighted by POP here. So there is some kind of long-conservative self-propulsion force which actually is in addition to the summer interaction, but this additional term is not necessarily related to the dissipation by any fluctuation-dispassion relation. And all this more is like this force is actually termed, the direction of this force is termed by the shape of the particle, for example the body shape of this bacteria, and then like it takes some time usually until the direction changes. So these are like the time-reversed metric waking and the direction of the body are like very important properties of active particles to matter. So you can find these kinds of active particles at various length scales and time scales as shown here, like from nanometer scale with molecular motors, a serial bacteria and insects to the order rate of the birds at the middle scale. So the current properties of these things, these particles, all of them converge toward the ambient energy into systematic motion and they stay far from equilibrium by breaking time-reversed metric, the fluctuation-dispassion relation, and then each of them has some kind of direction of motion determined by their body shape. So remembering this, let me talk about how you can distinguish a system consisting of such active particles from a system consisting only of passive particles. So let's say you drop some asymmetric object that looks like this, this wedge inside some fluid. So provided that the fluid is only consisting of passive particles, then the pressure acting on this object is the same everywhere and therefore this object will not really travel anywhere, except for some kind of summer diffusion. So there's not going to be any persistent motion. But if these particles, the circular particles are active particles, then what happens is that they try to swim in the same direction for some time. So what happens is that the active particles coming from the left like collide into the concave wall and then they cannot actually escape quickly because to escape they really need to turn their direction by a large angle and that takes time. On the other hand, the particles which are coming from the right can just slide by the object without really having to turn their direction. So therefore this leads to a surplus of density on the concave side of this wedge-like object and that means there are more particles pushing this wedge-like object to the right compared to the particles pushing this object to the left. So that leads to the net motion of this object to the right and in response the net current of active particles to the left. Similar thing happens in a situation on the right-hand side here. As you can see, this is some kind of surface some asymmetry from the right to the left and the active particles which are actually like A can easily slide by but active particles which are actually like B, they just talk in the chat. So this means there's going to be an overall force from the right to the left exerted by the active particles on this surface. So in general these active particles like break time with this metric, they are the equilibrium and then if that is combined with some kind of asymmetric spatial geometry then that leads to some kind of force on the body or the net current of the active particles which we can utilize to obtain some kind of active motion or do some kind of useful work. So this is the main idea of utilizing active particles to build some kind of like micro motors or very small scale of engines which was briefly discussed by Hyun-Joo Park in the talk before this. Okay, so these were like a schematic drawings but I mean you can really see that these happen. So I mean these are actually very known experiments using a rich-like object and a geol-like object and as you can see these objects are just moving even though these active particles are really like dumb particles. They are just trying to move in the same direction and consequently they give rise to this kind of rectified motion. Okay, so now here's the question. So do we always need such asymmetric objects in just like current or mortality in this system? So that's what we are going to think about here and the short answer is no. Actually we were not too forced to say this answer. Actually it's actually quite well known that there are some situations where even symmetric situation can lead to the overall current or jacket motion. So for example let's say you drop some kind of flexible polymer in a fluid-fated active particles, active fluid and then as you can see here because of some kind of fluctuation the polymer can have some kind of induced asymmetry and then in response to induced asymmetry the active particles will form some kind of large-scale current and also push the polymer in the opposite direction. So therefore that leads to very persistent motion and also depending on the length scale of the polymer you can have persistent motion or persistent rotation in the same place or even if the length is not long enough then you will not get really persistent motion like it's shown in the left lower corner. So if the object can deform then of course you can get this kind of induced asymmetry and then constantly you can get directed motion. There is more complicated example so I'm not going to really explain what all these equations mean but this is a system which consists of these small filaments like actin and then some motor proteins which are grabbing these filaments and what these motor proteins do is they tend to align these protein filaments because of their shape, tend to align together and then because of the presence of these motor proteins they tend to get aligned with the same polarity and also they tend to get some kind of contractile stress from the motor protein. And let's say we gather these kind of filaments together and motor proteins which are formed what we call the active gel so here are some like a droplet of active gel inside some Newtonian fluid and then what happens is that they tend to form this kind of polar order because of the interactions I talked about but if you really strengthen the strengths of the motor protein there then the contractile stress will get larger and larger and that will give rise to some kind of spray instability which means if you follow the grain of these filaments then they form a shape like this spontaneously and then once this kind of instability happens then there's an imbalance of stress across this droplet and then this droplet will be propelled in some direction so as you increase the contractile activity of the molecular motors then you can see this droplet transitioning between some kind of stationary symmetric compuration to some asymmetric moving compuration which can be in both directions depending on how you break the I mean how does plane stability happens but still okay so these are examples of some examples of how symmetric breaking mortality can happen in active fluid but all of these like had some kind of some kind of strict requirement like for example in the first example the object didn't really have like fixed geometry it was like it can be deformed very easily and secondly there was I mean the active fluid had to be like ordered beforehand like for example in the case of active gel it had to have this like a polar order but in our study we are actually going to point out that even in an active fluid without any order so it's just like an active ideal gas without any order still the object immersed in it can be worked out through some kind of symmetry breaking mechanism induced by just the motion of the object and negative drag which means the drag force applied in the same direction as the velocity of the object plays a very crucial role here and this continuation is controversial and just happened so here is a very schematic diagram showing what is really happening there but I'm going to hang it in here just as a trailer actually I think I have just like how much time do I have? 15 minutes so I'm going to describe a very simplistic statistical model that we solved to reach our conclusion and then we are going to discuss what negative mobility means there and then we are also going to discuss what kind of phase changes happened there so here is a very simplistic model of a one dimensional system with the object immersed within some kind of active fluid so here we are considering a 1D system with periodic boundary condition for convenience actually we are going to send this L to infinity later and so there are in this one dimensional fluid there are this one entombed particles since this is one dimensional active particle either travels to the right or travels to the left and there is the same velocity U, same speed U and then it can come to its direction with some fixed rate alpha over 2 and then so I do a circuit to represent the periodic system and then these arrows are running on particles and this green triangle at the top is a really schematic drawing of how an object or wherever object interacts with the running on particles so let's have a look at it in detail so as the distance between the location of the active particle changes which is indicated by this X here if the object, if the active particle is on the right side of this object then there will be like repulsive force to the right and if the active particle is on the left side of the object then it's going to have like some constant repulsive force to the left so the repulsion is like just on and off and it's always like F or minus F so that's why I drew a triangular potential representing the interaction so that this object also exerts force on the active particles so the equation motion can be written in this manner, there's the interaction described by the potential V the triangular potential V that I showed before and there's this set propulsion force following the run and tumble dynamics and the object itself moves and well, I'm just ignoring all the kind of possible noise here I just assume that the object is an overlapped particle and when the active particle is on the object then the object is pushed in the corresponding direction so what is important actually here is that this is a one-dimensional system and the object and the active particles can interact when they're like kind of overlapping so I'm assuming that this object is somewhat like a soft object but the potential shape is not really deformed by the interaction so the question is how this kind of object will move in the non-nuclear steady state so I'm going to assume that the first of all to solve this problem I'm going to use some kind of main field approach so I'm going to assume that the object is moving at a constant velocity so it's active fluid so then I can fix my frame of reference to a frame which is moving together with the object and then since this is an overdamped system you don't really have the Galilean imbalance you actually have some kind of medium around the object therefore moving with the object means there's an overall constant force in the opposite direction to the speed of the object so you take that tape that like fixes force into account and then you can basically set up some kind of master equation corresponding to this frame of reference and then you try to solve it for the steady state so I skip the detail these are all just mathematics the steady state solution can be calculated actually exactly for any potential in fact and then you can calculate the drag force based on it and then assuming that the speed of the object is small then at linear order then you get direct coefficient and you can basically obtain the direct coefficient as a function of the persistence length objective particle which means how straight the active particle trace how far the straight particle travels in a straight line and also f which is like the the repulsion between the active particle and the object so you get this kind of diagram so the axis is the persistence length and the vertical axis is the interaction strength and when the like I mean as you can see here in this red region in the region to the lower right corner you get like a negative drag which means the drag force is in the direction of the motion so okay and that happens when the capacitance length is substantially larger compared to the object size and when the real repulsion is sufficiently small so what exactly is happening here so let me get some exact solutions for the like shape of the density profile of the active particles at like 0th order of the velocity and at linear of the order of the velocity okay so this is actually this is a case for the passive brinyan particle so when there's this like a potential which looks like this triangle drawn by the blue line here then in response the the density of the brinyan particles will be like this inverted triangle okay actually not really triangle okay but anyway yeah like lower density within the object now what happens if you move this object to the right then there's some kind of like a fictitious drift to the opposite direction in the combing frame and then like there are more passive particles in front of the object compared to behind and like this change of density basically means you are going to have a drag force which is opposing the body motion so this is like the like a pressure some kind of friction like induced by pressure difference so this is typically what happens in equilibrium state like for passive particles but for active particles something else can happen so first of all let's again start from an object at rest in an active fluid and what would the density of the active particles look like so as I said active particles tend to travel in the same direction for some time and therefore when it meets a wall then it tries to push against the wall for some time before turning away okay so that means the active particles will kind of tend to accumulate at the wall as if there was some kind of like a traction that's only caused by the persistence of time okay so now what happens when we start moving this object then what happens is that the active particles actually like piled up with the higher density behind the object so that leads to some drag force which actually aids the motion so this is what we call the mechanism of negative mobility negative drag so let's try to get a bit of intuition about what is happening here so let's say we have some kind of static body over there the blue triangle now as I said when we have active particles around this triangular object triangular potential then these active particles move as if they are filling some kind of effective attraction towards the boundaries of this object okay and they are also trying to go over the object but since the right-hand side and left-hand side are symmetric the tendencies are balanced but when we start to move this object then what happens is that like for the right-hand side the repulsion of the object is kind of a bit cancelled by the fictitious force due to the motion of the object so on the left-hand side the repulsion from the object is kind of strengthened by the fictitious force in the moving frame so the object actually becomes a bit asymmetric like that and then what that gives rise to is that like the active particles on the right-hand side actually find it easier to go over the object and go to the left while those on the left-hand side find it more difficult to cross this barrier so therefore there's going to be a higher number of active particles behind the object compared to the front so this is a very hand-raving argument but I think this is what is like roughly speaking happening in this situation and again we notice that this happens only when the person's length Lp is sufficiently larger compared to the size of the object this is normalized by the size of the object as you can see the negative drag happens only when Lp is actually less larger than 0.4 and this might be because for this kind of mechanism to really occur and the active particle has to actually sense both sides of the object without like losing any memory so the active particle can travel from the left-hand side to the right-hand side and see that really because the motion of this object has effectively become asymmetric now if the active particle is changing direction very quickly then before even crossing this object you will soon forget about what it felt before so this I mean you have no information about how asymmetric this potential has effectively become in the moving frame and that's actually kind of the equivalent to the equivalent situation now also we we observed that the negative drag can happen only when the whirlpool is actually small here and that basically means when the whirlpool is relatively strong then there will be like a few active particles really crossing the object so this basically most active particles will not be able to sense both sides of the object if the whirlpool is too high so again here the force is miscarried by the self-propelled force so as you can see the whole region is below the self-propelled force actually the whirlpool less than the self-propelled force so negative drag can happen only in that region so everything here is related to whether the active particle can sense both sides of the object that's the I think that's the important intuition we can gain from here okay so basically so far I showed you a very specific example of like a triangular potential and but in principle we can serve the equations for any kind of potential and we've seen similar things happening for like sinusoidal potential and harmonic potential and then for a very general kind of potential actually we can do the perturbative analysis assuming that the potential strength is very small and I'm not actually going to really explain the detail but assuming that the V-self is very small then you can calculate the drag coefficient here as shown in this last formula and as you can see when the run length is sufficiently large or when the potential so the repulsion of the substance is strong then you can get negative drag coefficient okay for a general potential in the weak potential limit so far I've shown you that like in the linear response regime the fictions felt by this object in active fluid can be like in the directional velocity but of course to get the steady state velocity we actually need to go to the nonlinear and then try to serve some kind of a self-constituted equation between the drag force and like friction from other kind of friction from the environment okay so this is basically what being fed up function does so we calculate the mean for the approximation and for the steady so for the self-constituted equation okay I think I cannot really explain this diagram in a short while I'm just going to tell you that like the what is the axis like with carrier repulsion the vertical axis is like the object velocity and the lines here are like the solutions where the force from the environment the force applied by the active particles on the object and then what you see is that depending on like the situation we can get both continuous phase transition of the velocity and like this quick jump of the velocity steady state velocity as you vary the like the representations while fixing the persistence length so now you can see the phase diagram based on the result so again on this repulsion and persistence length plane you get many kind of different kinds of possible steady state behaviors so for example we have this immobile regime where for whatever the velocity is you never really have like positive force in the direction of the motion so that's what this I represents then there's the mobile regime where for small velocity you get some like meta tip drag overall and then as soon as you increase the velocity the force again the net force again becomes zero so in that case the steady state solution will be like mean zero so the mobile state will be the only multi state will be the only stable solution it's blue regime in the middle there's also a case where like the mobile state and the immobile state can coexist which suggests that in those regimes there will be like this conscious phase transitions between the two states and what is even more interesting there can be like situations where like in this case there can be like some kind of non-climbing system but anyway we can like find the regions in this space where both mobile state and immobile states there can be like situations where multiple mobile states can coexist so some tracer particles can travel slowly and other tracer particles can travel more rapidly and those regions are very very small but still they are there so so far all we have seen is like just a theoretical result based on kind of main field approximation but can you really trust them with that question but we did numerical simulations of the system they did model using these equations and then what you see here is that like now when again this is f and the velocity plane and what you can see here is that the main field provision indicated by this red line is actually quite close to what the system really does in the steady state indicated by this color the the sum of plot so provided the object mobility is sufficiently small then mean field predictions are actually quite accurate and you can even see that there seem to be like really two different kinds of transitions like the one corresponding to the continuous transition and the other corresponding to a discontinuous transition so the as you can see there let's try to close a look at what is happening for the continuous transition so here I'm trying to change the like the number of active particles the system while keeping the density at a constant value and also a little bit of cave material I'm also rescaling the mobility of the traveling object inversely proportional to the number of active particles here so the situation in this case becomes a bit similar to the case where you have like Ising spins interacting with each other ultra interaction with each other which can be likened to this active particles some interacting with each other through this very slow object in the middle so what happens is that taking that kind of simplicity limit you can show that the system exhibits the Ising Ising model like criticality with like very critical exponents we type 1 over 2 and 2 so this is the continuous transition so we do have a continuous transition as supported by this evidence and then like there's another transition on the left hand side which looks discontinuous but let's make sure this is continuous so we first measure the how the distribution of the density of the traveling stage changes as we increase the system as the number of particles sorry for a bit of the inconstancy in the order but as we increase the number of particles we clearly see this multi-group peaks developing and if we measure the window cumulative which is typically used as a measure of this constant transition you can see that it's been the cumulative develops a huge dip here infinity which is usually interpreted as the distribution of the parameter having some kind of two different peaks at the transition point so it seems like the system is really capable of this constant transition so I'm a bit over time so let me quickly summarize so we found a simple mechanism by which an object with fixed geometry in a disorder state can become more tall we are symmetry breaking and this is because of the negative drag and with substitute long-person length and weak object particle interaction this mechanism can happen and both kinds of continuous inconstantions are possible now here's some a bit of cable to note here so in addition to this drag force exerted by the active particles there's of course another kind of drag force exerted by the surrounding fluid the ambient fluid and if the drag force from the ambient fluid is even stronger than the effect of active particles then of course this kind of spontaneous mutality will not happen so we actually need to assume that the surrounding fluid is also like exerting friction compared to this drag force exerted by the active particles but still there was some like another research in a similar vein by like the group in the Nexel and they found that this kind of this kind of magnetic drag mechanism even if it is too small in this motion can still get rise to some kind of ominous fluctuations and when the system is very close to a critical line like for example here then we can expect that because of the critical fluctuations also with this isling-like transition there's gonna be the some kind of ominous diffusion of the object immersed in this active fluid and also we can expect that I mean considering how these lines look like some kind of re-entered behavior as we cut the diagram in this direction we can start to expect that this also has something to do with the well-known experiment observation of the non-monotonic dependence of the effective diffusion coefficient on the tracer size compared to the the person's length of the active particle now so now the real question is whether I mean all of this here is nice but of course the real thing is whether we can observe this in any experiment well so I'm not aware of any real experiment or observation of this yet but there's a very similar some similar situation where you have this kind of disk made of substance called camphor which weakens the surface tension of water so if this starts to move a bit then they create some dancing balance of camphor in the moving frame of this object and then that imbalance again releases a certain effective negative drag and then this guy starts to move of course in this case the disk is not in an active fluid its surface is an active particle but but like this kind of situation actually gives me hope that what we have predicted might be sort of possible if you get the right kind of parameters it's a very simple mechanism okay so I'd like to thank my students Gi-Won Kim and Yoon-Sik Choi for their theory and their remarks respectively and also this work was basically a product of discussions with Patrick Kae-Jung-Kang and Erich Kaffee okay thank you for your attention and I'd like to answer any questions or comments you have questions thanks a lot for the interesting talks so so you introduced the MIMPIA theory and the analytic solution to the the friction coefficient gamma in the frame so could you extend that argument and theory to higher dimension to higher dimension so that's actually quite challenging one dimension is special in that you can actually solve this problem for any potential effect that's mainly because you don't have the shape of the circulating current field is very simple in one dimension I mean but if you do circulation it has to be like across the system but in two dimension there's going to be circulating current which looks like I wonder whether such a negative drag phenomenon is so present in higher dimension I mean empirical experimentally so yeah actually that's a good point and so basically what I believe to the main intuition here is that if the active particles tend to kind of stick to the boundary of this object then if there's some kind of motion here then the tendency of this active particle sticking to the boundary is kind of altered in a way which aids the motion so even in two dimension it can make the active particles kind of stick to the boundary and I think it's natural to suspect a similar thing can happen and actually we are working towards a building some ambient simulation which can check that and Yoon-Jik is doing or trying hard to do that also you can construct some kind of cause I wonder my situation and then maybe it can check this more easily any other question? if I remember correctly you used the first of the differential equation for describing this object not the second order is there any reason to use that instead of the Newtonian second order differential equation Newtonian second order equation okay first of all the reason is because it's simpler so the first of the equation in principle I think you can do this for example I mean you are thinking about considering the effect of inertia here for the object because it's likely to be actually much more massive so maybe experiment the situation closer to that so that's a good question maybe we can do that I assume that you don't have to yeah we are all assuming that you didn't assume any drag on this object so we are just assuming that the drag is actually much much bigger but yeah energy balance okay I'm not sure about the question maybe we can discuss later any other question okay it's not all let's just start here let's put this back