 Banteo is not around, so I have to start myself. Okay, so today it's going to be more recreative, but still with no films, which is very bad, because the subject lends itself to lots of beautiful films, but we won't have them. I think that discipline is getting lost. People are sleeping more. Yes. It's like marriage. Okay, stop. We start. So remember that I insisted this a thousand times. There are two elements to statistical mechanics. The first, that the books emphasize a lot, is that you're going to study systems with many actors, and you don't want to know the story of the life of each one of them. You want to know the mass, the statistical mechanics. The second element that I insisted throughout is time reversal, which is the property of your system or the bath to take the system to equilibrium to have a dynamics that satisfies the fluctuation, dissipation, and on saga relation. It also gives you the... The violation of this allows you, as we did yesterday, to derive the fluctuation theorem, but this is because we control exactly that the violation of the time reversal is precisely the work we are doing. They are divided by the temperature. Good. Now we're going to discuss systems that do not have this. So they are macroscopic, but they don't have a form of time reversal. So they will not lead you to equilibrium. You will not have the Gibbs measure or the micro canonical measure, but you have a stationary state or you can have a stationary state. What can you say about it? As we said, not very much. You have to study them case by case. There are many more systems that don't have detailed balance than those who do. And okay, so let me tell you two or three kinds, which I'm not going to discuss too much. If you have a system and you subject it to strong alternating field, it has to be in contact with a bath, because if I shake something, it will heat up. So you need a bath to keep it. And then this system can reach a situation where it is not stationary, but it is periodic with your field. Nowadays, because of the influence of quantum people, this is called floquet system. And it means that I apply an enormous field and for example, I measure a property of the system, let's say the magnetization. So at the beginning, the system does something, but then it reaches a periodic state. And a floquet theory is when you look at it stroboscopically at precisely the intervals. So from that point of view, it is stationary. These systems, if the driving is strong, there's nothing you can do. I mean, you have to solve the dynamics. I had planned to give you a model system of which you can solve it, but not easily. But that will remain for the next time. Okay. This is one kind of system. But we're not going to look at this kind of system. We're going to look at more fashionable, well, this is fashionable, but in quantum mechanics. In classical, it's not so fashionable. What is fashionable in classical mechanics is what is called active matter. I talked a lot about this. I think that it all started with a model of V-shek. You can find it in Wikipedia in the 90s. And it's a very simple model. You have a little things, particles, that have their velocity. And they move according to the line of the arrow. So they are like little rockets. And they have an angle. And then every now and then, you update the angle. How? You look around your particle up to a radius that depends. You choose it, but we fix it. And you measure the average angle of the guys you have around you. Here it would be that. But you allow yourself for a little bit of error. So a random number. So you will average, but with a bit of error. And then your particle is simply going to fly at that speed. Let me get the notation right. Velocity is the velocity with which it flies. And then, what am I missing? Of course, the angle. We call it the little vector that is your angle. So this is a model of birds, for example. More or less. OK. So lots have been done. So you see, the velocity doesn't really change. It changes only direction. And so we don't need a bath for it because it won't heat up by construction. Or if you want an implicit friction. And there is a parameter, which is called the persistence, which is how often do you change the angle and the velocity with which you fly. OK. This model was studied a lot in the last 30 years. And I should have done a more multimedia talk. There is lots, you will find lots of films in it. And what happens is that, you see, there is a tendency to flock. So the little particles, because of this interaction, want to fly together. Because there are errors, every now and then, this one changes a bit and it transmits it to its neighbors and you get flocks that fly like this. Or turn around. The phenomenology is rich. And this has been studied a lot. How do you treat this? Well, one is to put these equations in the computer and look what it happens. What more? Can you do a stat mech? You see, here you could have a lot, a lot, a lot of particles. And the argument that you have a lot of particles goes through. But there's nothing like a partition function or a thermodynamics of this because we don't have these principles that allow us to get rid of time. There is no fluctuation, dissipation, et cetera, et cetera, et cetera. This is a model. Then there is, let me... So these are the active particles. Another model which we're going to see is... Well, sorry, before I say another model, if you look at the papers, if you're interested of Cavagna, Giardina, and Parisi, they have thousands of... A lot of papers. You will find true flocks. What they did is they took three photographic cameras or film cameras, and they filmed, I don't know if you've seen the birds above the starlings, about Rome. In Rome you can see it, and apparently in France too, but I've never seen them. These waves of birds that draw pictures in the sky, they fly together like this. If you're standing near the central station in Rome, you can see them in the sky. And they wanted to understand what is it that makes them fly together. These are groups of a lot of birds, and they fly together like this. It's very beautiful pictures, and it's very beautiful to see too. So they took the films of these guys, and they wanted to know what was going on. Why do they fly together? One explanation would be that there is a boss that tells you, okay, follow me, and there they go. The answer is no. The cameras, what they did, the difficult stuff is that when you are filming two birds that cross like this at different planes, it's easy for you to get it wrong and think that this bird is this one and so on. So with the cameras and with very interesting software, they managed to follow with very little error individual birds. This is the work of many years. And then they discovered a lot of things, but basically the nicest is that, contrary to what you would think, a bird just looks around, more or less its six nearest neighbors, and that's what they do. And this, which with no leader at all, this creates this order because everybody, those are looking at others and so on and so forth. And then you can go on and on. And you see it's not very different from when it's richer, more biologically, if you want, but it's of the same taste as the Beechek model. And it's very nice. And why is it something that captures the imagination of people? Because here you have a system that is complex in the sense that it has many actors that self-organizes. There is no boss. And that it's a very beautiful example of how when you have motions that are collective, you need not suppose that there is a Martian that is obliging everybody. You know, it's very common for the man in the street to think that economy or politics or things, it's because there are nasty guys that are ordering people to do something. Here what you see is a system that self-organizes. It's an example that everybody can understand. And it's a nice metaphor also for people who do economics or sociology or whatever. Again, I'm insisting a lot on this, but it's a system that has many actors from that point of view. It is statistical, but at the same time, you don't have the principles of equilibrium that make your life easier. So, the last theoretical model I will do is the following. I hope I don't get the notation wrong. It's very similar to V-shek in a way, but there are potentials. We are in three dimensions, let's say. There is also a velocity, but now the velocity is random. Alpha and beta are the coordinates, x, y, z, or x and y, if you're in two dimensions. And the average of this, of t of zero, so this works like a noise. And the noise, as we did many times, is uncorrelated between different particles, uncorrelated between different coordinates of a particle, and it has a memory kernel, just like the one we had. But now, in this case, I'm going to tell you what the kernel is, so that the memory kernel, the noise correlation, if you want, is an exponential with tau. And this is called an active Brownian particle. Now, remember you don't need to copy it, but remember that we wrote many times already the generalized Langevin equation, which, if I am over-damped, I'm not going to put this term, but this is the one we wrote. I'm going to put a hat on this one. And eta eta, well, I'm doing it in one dimension, but this is just to remind you, this was gamma of t minus t prime, and there was a temperature. And the trick was that in order that this is equilibrium, the memory here has to be the same as the one here. And we call this fluctuation dissipation of the first kind. And this is the relation that tells you that you have a good, nice, decent bar. This model is the same, you see, because V is playing the role of eta, and gamma is gamma, the correlation of the noise. So up to here, we are good. The only difference is that here, instead of having a memory term, the memory term is like the Barcovian limit we did so that gamma hat, sorry, here, of course, I forgot as usual, remember that we did it for this one. This one is the delta, or let's say, half a delta, very peaked function. Let me just put it as a very peaked function of t. So that if gamma, the one here, is like this, then you get the Marcovian limit. Yes? The index. Which one? Here? So i is the name of the particle. Alpha is x, y, and eventually z, the components. Okay? So remember what we did when we went to the generalized Langevin equation to the Marcovian one? Well, we first said, okay, the gammas are the same and we make them very peaked, we make them deltas. Here you're doing something that is no good for equilibrium. You are making the gamma of the noise still wide with a tau amplitude, but you're making the gamma of the memory different, and that one you're making it a delta. You could have made two different amplitudes, but the important point here is not so much that this one became a delta, but the fact that the two gammas are different. So you broke something important in the bath. You broke the detailed balance eventually. You broke the equilibrium nature of the bath. You know why? Because when we derived the bath, we didn't have the choice. We found that the gammas of memory have to be the same. Okay, this model is fantastic because it interpolates between what we have seen and the world of active matter, and it's perhaps one of the best studied models. And this article here, that if you're interested, it's interesting. Where did I do one? Ah, doing T zero, yes, yes, sorry. Okay, you can do it like this if you want. It's the same, okay? Because it is time translation and invariance, sometimes in the literature they put this one to zero, which is the same. We are going to try to discuss this one a lot, a bit. But okay, is it okay? So, no, no, no, the dynamics is of the arts. There are many particles that interact with one another through this potential, sorry, I was going to say it now, through a potential, a mutual potential, and they're in contact with this bath where we have destroyed the interactions. They chose as a potential, this is what I'm missing, I don't know for what reason, they chose this function. I honestly don't know why, but this is some function that is, you know, some potential that depends on the distance. But okay, so we have, in this model we have, it doesn't matter at all, any, they say it, any repulsive force between the particles will do the job. Okay, so what do we have? We have particles that are, I use V because this is a notation of active matter, but in fact V is like a noise. So your particle is like this, and it's motorized by this V, which as I say is a random variable, but it's a random variable that is correlated in time in this way. This is called an Einstein-Ulembeck form, just in case you ever hear, this exponential law is called an Einstein-Ulembeck form. Okay, and but, so it's very similar to when we were doing this thing, overdamped, but there's only one difference that is important is that this gamma here, which is the memory of the friction that should be the same as the memory of the noise, they are not. So we broke something important in the, if you want to think of it as a thermal bath, in this case it's not mimicking a thermal bath, it is mimicking something, okay, we will, I will tell you now the phenomenology of this, but let us talk a little bit more about Brownian particles, sorry, about active particles. So two or three experimental things. Do you know the nano-hex bugs, okay, Google it, they are like little cockroaches with legs like this and they vibrate and they move around. They are this size and they cost, well, I don't know, but if you had children at the right age at the right time, you should know them. You might also know them by the name of Bristlebots. What? Bristlebots. I don't know. How do you write that? Because you take basically a toothbrush and you cut out the handle, you take a toothbrush, you cut out the handle, take a cell phone vibrator, stick it on top and it will go. Exactly. This is a bit like the edge of a toothbrush. They are inert, these things, they are just flexible and there is a little motor inside that is eccentric and it makes them vibrate and so they run. I should show you. They are fun because they look exactly like cockroaches and sometimes they behave interestingly because you put many in a box and because they crash against one another, they start going together as if they had a mind. I was travelling in Korea and I discovered them and of course bought some for home and then I went to Sergio Ciliberto, an eminent experimentalist, and I said, I found these things and he said, I've just bought 600. He had them in a box and I think he published a paper about the motion of these guys, of these nano-hex bugs. Then there is another example. I already gave you the Starlings, the things that fly over Rome, at least Rome. There is another one that I discovered today that is very old. It's called a Janus particle. You have little particles, tiny little particles, which Janus is because it has two faces and you make them, for example, I think one half is gold and one half is platinum. Why platinum? Because if you put this in peroxidized water, the one you use for your wounds, when the peroxidized water sees the platinum for reasons that I don't know, it catalyzes the reaction and little bubbles come out. And because of this, this acts as a motor for the particle that pulls you away from the platinum side. It's very cute and they are tiny, tiny and you can put thousands of them and do things with them. And then there are others that react to light and when you turn on the light, they are motorized and when you turn off the light, they stop being motorized. There's lovely, lovely experimental work. If you google active particles, you will see all sorts of lovely stuff. As to theory, we will only discuss this model because I think that it's very nice that it interpolates very nicely between what we've seen and the world of active particles and also because the article I recommended highly is very good. Okay, so now let's turn to what I call the why not questions. Part of the literature of active particles consists of being surprised that it's not equilibrium. Let me give you an example. Imagine that I have a tank and I have this thing. This is a fish trap. Well, it has spherical symmetry. There is a hole here and when the fish comes inside, because of the way fish swim, it cannot find the hole because fish has tendency, as you all have seen, to swim near the boundary of the fish pond. So the fish get inside. This is easy for them to do, but they cannot get outside so you just leave it there and then you get a lot of trapped fish. This I discovered in the Latin languages has a name which, for example, in Italian is like this. In Spanish I discovered at my age that it has a name and in French it's like this and in English it's fish trap. I don't know, for some reason it seems that kids in France play with this, but not where I come from, although there are fish, but as far as I know, kids don't play with this. At least when I was a kid I didn't. Okay. So the fish get trapped. Fine. Now imagine that instead of fish, fish are active matter, of course. Now imagine that instead of fish, you had a Brownian particle. So particles that move under the influence of the noise produced by the molecules of water. Will the fish trap work? Will it work? The answer is no In contact with a thermal bath, you are going to go to equilibrium. Equilibrium means that the density of particles inside and the density outside is the same. There is only just a wall. So a fish trap for, let's say, bacteria. You have tiny bacteria, viruses that are in the water. If they are alive, maybe it works. I don't know, but it's very probable that for bacteria that swim, the trap works. But if they are dead and they're moving under Brownian motion, because now you have an equilibrium bath, a bath that corresponds to a temperature, equilibrium tells you that the trap cannot work because what is the equilibrium measure for this fish trap? Just the uniform one. So if you are in equilibrium, this is your fish trap, the density is uniform. Remember that when we gave the example of the Sinai billiard, it was you explore democratically. Here, an equilibrium particle explores democratically everything. So it is the fact that fish are active matter, that they break equilibrium and that they have the psychology they have that makes them get trapped. So if I always joke, if you throw a few drops of alcohol and you make the fish drunk, they will free themselves paradoxically because they will swim randomly. This is quite remarkable, isn't it? So you see that equilibrium is no joke. It tells you, for example, that a fish trap for inert equilibrium particles doesn't work. But if you have active particles, any of these models, there is no symmetry, there is no reason that tells you that inside will be the same outside. You don't have an equilibrium measure. So either the fish trap works as a fish trap, sorry, an active particle trap, or on the contrary, it has less than outside. It has more or it has less, more specific reason why the density should be constant if you don't have equilibrium. So you see how powerful the notion of equilibrium is. Okay, good. Let me give you another example that is also shocking. So I'm trying to make you surprised of the thing that we've been doing all days. So you have a ratchet, which is something like this, a wheel like this, and you put it in water. It has an axis so it can rotate. And now the water is at equilibrium at temperature T. And of course the molecules of the water are crashing against this and producing a noise. Is the ratchet going to turn? No, because equilibrium, as we saw already is time reversal, a hidden form of time reversal, but is a form of time reversal. And that's one way of saying it. Another way of saying it is that if it did work, I would be able to extract work from an equilibrium water and life would be easy, but life is not. Okay, so it doesn't work, which is very, very, very strange, because it's clearly asymmetric. Molecules that crash coming from here and molecules that crash coming from here crash in different ways. And still there is some form of compensation. The form of compensation is what we saw when we saw fluctuation, dissipation. Somehow you get more kicks that want to turn you in that direction, but you also get more friction in that direction. That's how it works. Very good. So now instead of putting inert particles, I put bacteria or better, Janus particles here. Now immediately the wheels start turning because these are not equilibrium systems and the wheels start turning. And you ask about the fish trap and about this one, why do they start turning and the correct answer to that question is why not? What is surprising is equilibrium. The rest is obvious. If you ask me in which direction it will turn, then let me depend on the model, but the fact that it's exactly zero is because of equilibrium. There is a lovely experiment and now I don't remember. I think it's somebody in Rome who had particles that activate when you put light, shine light on them. So the particles are inert and they are brownian, but when you shine light on them, that motorizes them for some chemical reaction. So you see they have this fantastic film. You can Google it. It's a ratchet and it's without light and you see it standing. They turn the light on and it starts working and then they turn the light off and it stops. So the question, the answer to most of the questions I'm going to, the correct answer to most of the questions I'm going to ask now is why not? This is just to remind you how specific equilibrium is and how this comes from detail balance. Let me go one more example. There are literally thousands of applications like this, but there are two that are particularly nice. Now I have a bath of things, sorry, and I'm going to connect, let's say several thermometers that are different, these are old-fashioned glass thermometers. And now I am in equilibrium and what temperature do I measure? This is water. Well, I measure the same temperature, of course. Why not? To make the ideas clear, I can model a thermometer in this way. Instead of a thermometer at the bottom of the tank, I put a little piston and I make a harmonic oscillator. And I look at how much energy this can move, and here is the water. This is a perfectly good thermometer and how does it work? Well, I look at the average energy of my oscillator and the average energy over time is twice a half kT. kBoltzmann, now somebody had criticized that I didn't put the Boltzmann constant, now I'm putting it. KBT. So this is a thermometer, but it has a typical frequency, no, it's an oscillator. This is technically speaking a thermometer. So now I am going to, for example, make these thermometers just to fix ideas. Three different oscillators with omega 1, big mass, omega 2 and omega 3. Do I measure the same temperature? In other words, am I measuring the same energy on average in the three oscillators? They can move, because of what, this is a piston, a perfect piston that moves it. It transmits its energy to the oscillator, or a membrane if you want. A membrane that allows it to move. Okay. Do they measure the same temperature? Yes, fortunately. And if you want, this is due to equipartition. The energy of these three oscillators has to be the same. Good, excellent. Equipartition tells me that considered as thermometers, they measure the same temperature. Okay. Now, instead of water, I have water with Janus particles or bacteria or active matter. So the crashes against these guys are going to be also crashes of the active particles. Will these... And I let it go on for a long time so that everything is stationary. Do I measure the same temperature in the three of them? And the answer is no. And when you ask me why not, I answer, why not? Why should it? Why should it measure this? The miracle was that three things measure... that are different measures the same temperature in equilibrium. When you don't have equilibrium, there is no miracle. That's the only thing to say. So a lot of active matter is just discovering like here that what was equilibrium was the exception and that the rule was the same. Okay. This one, I don't think anybody wrote a paper like this, but there is a lovely paper instead by Thayer, Kates, Kardar, and I think this is it, but maybe there are more of us. But it's a bit more abstract. So I chose this example first. It's exactly the same idea, but now we will measure pressure. Okay. So, for example, pressure could be that here I have a membrane and the pressure depends on how the membrane distorts. Okay. And here I have another one, et cetera. Now, pressure is a bit like temperature in the sense that temperature is the counterpart of energy and pressure is the counterpart of volume. Mathematically, they arise in very similar manner. Okay. So what these guys did is, okay, they said, okay, obviously what you measure as a pressure here is due to the impact of your particles that are impacting on your thing. So the membrane, let me consider it here enlarged, has a potential that repels the particles once they reach the membrane. So let me make it in one case, a potential that falls very rapidly so that a particle that is approaching here feels a potential like this and then it crashes and it comes back. In another case, in another of the proteins, I will do the same, but the potential I will choose it a bit softer, but still the particles will bang and come back. You see, a wall is a potential that repels you, but you can make it soft the wall and make it repel you like a spring or you can make it very good. So now you do an equilibrium calculation like the ones they did but you do it and because of the same things as usual what are the pressures that you measure here in equilibrium, they are the same. Fortunately, because if not people when they tell you that the pressure today is so many bars, they didn't tell you how they measured it because it doesn't matter because presumably we are more or less in equilibrium, at least the air is in equilibrium, we are not. Okay. So they said, okay, in these two cases with the potential being shortish range and longish range the particles of course come and in this case they reflect like this and in this case they reflect more like this, but still you can do the calculation and you see that you measure the same pressure. Okay, now instead of particles of equilibrium matter, I make active particles like a gas of Janus particles or of bacteria. Do I measure the same pressure? No. They did explicitly the calculation and because active matter feels the walls differently they made the precise calculation, for example you could do it with a model like this one put it in the computer it's not much better than you can do and you will find that the pressure is different, yes. I just want to come back in the previous example, you said that if we change the the water by bacteria we are not going to get the same I mean the same temperature so I went to one theorem, I mean the theorem of a recurrent of Poincaré theorem who said that in closed system after a certain time the system will return to its position or closer, so I would like to know whether that is applied to that same system. Okay, I think that's not what Poincaré proved let me repeat the question Poincaré theorem says that if I have a system of n particles and I wait very, very long they will come back to their position now with any precision I want if I wait long enough long means really long okay, it's a theoretical theorem because in practice it's the age of the universe but in theory it's important because it concerns the question of irreversibility etc. Okay does it apply, he proved it for classical dynamics for Hamilton. Does it apply for active matter? I would say yes probably because chaos, there is chaos and so if the system is finite I would say it applies to of course as far as I know nobody has proven it probably they could probably it's a nice question probably if you prove it because this is a fashionable subject as you can imagine so there is lots of attention from papers on this subject understandably because you see that it's all quite fascinating so yeah they might return to their original positions in the coordinates but the trajectories in phase should not intersect if it is a truly chaotic system yeah but this is not phase space this is a configuration space I suppose that if you wait enough these theorems of recurrence are explained to you in books in some way or another but in a modern world you probably all have used random number generators in the computer and you know that they are not random numbers you know that they are just a function that is so nasty that it seems random and chaos is the same and recurrence with random number generators so I think that again I'm too lazy to do it but if somebody wrote a modern book on statistical mechanics they should say come on irreversibility is just like random number generators not much more than that and people of this day not of 40 years ago but people of now would understand them more easily okay so very good so when they did the calculation and now they did some calculation or maybe it was they found that the pressures are not the same when your pistons or diaphragms or whatever have a different composition so again you get the same point that the pressure is not uniquely something it's not a state variable and it doesn't make any sense to talk about let me say going back that because you don't measure the same temperature for an active system you cannot talk of temperature of an active system because there is no such thing as a single thing that is the temperature temperature is a thing related to equilibrium if you're not in equilibrium you don't have any right to talk about temperature the bath but not of you you may wonder why do we talk of temperature of ourselves because we are not in equilibrium but that's another story let me ask a nasty question which irritates a lot of my friends and colleagues what is special about active matter other than the fact that it is just another non-equilibrium if we are studying non-equilibrium stratomic it is called active matter not just study all of non-equilibrium stratomic the answer you should be I get is oh the forcing is different because in the old days we used to think about forcing at the scale of the system whereas here it is at the microscopic constituent level does that really change the physics at the model level let me translate it in one example extra example first question before you could get ferrofluid for example little particles of little magnets that are floating and you applied an alternate magnetic field and this energised them and nobody called this active matter and now we call active matter bacteria genus particle but what is the essential difference between one and the other and of course this is a rhetorical question because he knows that there isn't one so there isn't one but the point here okay but now maybe what we have is a much more stimulus to studying these things yes my question is about how active has the as I said the material has to be the system has to be active matter because we know that in our daily life the air that we measure the temperature is not exactly just air and there are some impurities but that usually doesn't matter so they do things that are active I mean they put I don't know exactly in each case but there are lovely experiments with genus particles and they behave it's true that equilibrium the perfect sense doesn't exist and it's always a good question to ask but because also I think this concept of being in equilibrium is not that new because even in the study of thermodynamic people know like the quasi-study the way of the system that has to be assumed when you formulate those theories yes equilibrium is never perfect active matter is very imperfect equilibrium if you want so by the way a very nice paper is what it's called a ratchet and it's important for people who do is the following let me suppose that I have this model so this model when we come back to it is a model where the bath is imperfect because the gamma of I have two gammas the gamma of memory is almost instantaneous but the gamma of noise is different and this already breaks equilibrium very good what happens if I make a ratchet with this so I have particles that move with this law they are in a bath but then I do this the ratchet will it turn in some direction I don't know the answer is yes or to put it in another way if I have a surface this is a wall that is asymmetric and you have on this in a channel and you have particles here does this asymmetry make the particles go in one direction if you are in equilibrium no because if not we have discovered the perfect pump and why not well because of detail balance once again because of time reversal now instead I use active particles or to go back to Mahesh's question I apply an alternating field of some kind to the system both do the same thing will it move yes and the answer is when you ask why would it move well why not this is an asymmetric ratchet there is no reason that there is symmetry and there is no reason that this doesn't pump why doesn't it work because of the ordinary equilibrium water because there is detail balance that protects me the principle of time reversal protects this and tells you this is not going to pump water but if you break detail balance if you break the fluctuation dissipation of your bath if you break your equilibrium properties this will work and this is no joke because this thing which are called ratchets is important because this is a mechanism inside your cell that makes the different organelles of the cell motorize so it is used by the biological systems and what is it kicking the motion because in equilibrium it doesn't work what kicks the motion is ATP burning or whatever inside the cell and this again about this there are 2000 or 3000 papers because if we take the tau in the memory kernel going to zero we get again the what good I'm getting there but is the transition to equilibrium sharp or like okay so if you read this paper this is precisely the question they ask this is exactly the question so as you see the gammas are different but if I the tau is the typical time characteristic time here here it is zero let's say and here it is tau if I make the tau smaller and smaller and smaller I get to a point where these two are the same and when they are the same I'm in equilibrium in fact the paper is called if I find it well I don't find it but the paper is called how out of equilibrium is exactly your question when tau goes to zero you have an equilibrium bath and when tau is bigger than zero no there is no sharp transition as soon as your tau is non-zero then you are a bit out of equilibrium exactly this question in fact they make an expansion in small tau in the paper let us think what it means and with this we will end so there are two gammas one is this which is the memory of the friction which is super sharp and there is another one that is not sharp and it's the thing of the noise I can interpret this so the one of the noise I called it gamma and this one I called it gamma so this is friction the dissipation and this is noise so it's the energizing that you get from the kicks and now we turn on its head the argument of Einstein for the Brownian particles and remember that there was for a Brownian particle this delicate equilibrium between the crashes of the fluctuations of your molecules around you and what you lose because of the friction because you have to move around molecules in order to move and what Einstein realized in his Brownian motion paper is that these two things cannot be independent because the pollen particle has to be in equilibrium with the system so what it gets and what it gives has to be tuned so that the energy of a pollen particle goes to the good equilibrium of its temperature, a half kT okay but now let's turn it on its head now our pollen particle is swimming in a sea of active particles like that one, like this one okay so it gets crashes that are correlated with these times scales because this is the memory of the noise so it means that the crashes occur on different on a certain time scale very correlated in time so you are crashed if you want once in a while while the friction is immediate okay the model is constructed in this way and it's normal because why should these two things be the same if the system is active we found that they are the same for systems that are very specific systems etc okay so how can you think of this well you are getting your energy in the low frequencies if I do the Fourier transform of this because this is because this is wide in Fourier space it's relatively low frequencies and you are losing it in the high frequencies because a sharp thing corresponds to high frequencies so it's as if you had kind of like two temperatures one that is hot in the low frequencies and one that is not so for example imagine that I decide arbitrarily to break these functions into two one that corresponds to this one and another one that does not all the rest okay this one with this one would be equilibrium of a certain temperature and all this is alone in the world alone in the world and it's a noise so it's giving you an extra energy without any friction on the other side because we don't have it here so this is telling you that you are receiving it's not like having two temperatures in two extremes as if each particle had two temperatures to talk to receiving a lot of energy in the low frequencies because it's noise without a friction at that frequencies and giving it out at the high frequencies so the system doesn't know at which temperature it is one last thing, yes we have time okay let me tell you what happened with this model so they did the expansion and everything and so on how does this model behave so as I told you I think already many times you're on your own you cannot use a Gibbs measure you have to simulate the dynamics and what this model does is amazing unfortunately I don't have here to project these are particles that have this rule and look something amazing happens you normally with a normal bath no problem you just interact with the potential it doesn't matter which potential just repulsion and that will be it when you apply it with this bath with a slight modification of the Langevin force what happens is that the particles cluster they get together and then on outside this so here there is a high density region and outside there is a low density why because the particles are motorized with a time scale so they move straight for a longish time corresponding to this no and because they are all moving like this they crash against the walls of this crowd of particles and they stay there for a while and so well this is words you have to put it in the computer and check it you get this and this is called MIPS motility induced phase separation phase separation because you have a phase that is like a solid or a dense liquid and outside you have a phase that is low and this happens spontaneously amazing no and you have in equilibrium also the same you have seen it many times in your whiskey the ice and water ice at zero, water at zero so in equilibrium such things exist and you can calculate them using the method of statistical mechanics no problem but this one is an essentially non-equilibrium phenomenon and quite, quite universal I mean whatever active matter model you do often they have this they used to be something called MIPS about 30 years ago million instructions per second I have so many questions for that model this one that one yes first of all the boundaries are fixed or what do they do periodic boundary conditions because it's in the computer the size depends on the number of particles yes and the parameters no the tau if it goes to zero you're in equilibrium so this stops and then is there a phase transition is there not it's all in many papers and also the trajectory this thing will move very slowly because it loses some and gets some and so but because it's big it's sort of heavy to move and that both does it act as a random walker or something yeah probably so active matter if they cluster they become a random walker basically maybe yes yes I think so have a look at papers if you look MIPS you will find I don't know a thousand papers this was discovered by these guys basically arm and there is a lot of study does a model have it does it not but again as I said there is no statistical mechanics of it you have to put the equations in the computer and see what they do as I say always if you don't have detail balance you're on your own I mean you there is nothing clever so just to conclude my lessons in general I finish with this message do we have principles that guy what is the problem of non-equilibrium what is it that we want well personally I would like a theory that tells me this without having to solve the entire dynamics the theory of phase transitions in statistical mechanics is a robust theory where you don't need to always solve completely a problem to know what it will do and you have inequalities you have entropy productions that have to be positive you have a lot of things out of equilibrium you're a bit lost you still can simulate the things so the argument I don't want to know what each individual particle does is still true I don't want to know but do I have a guiding principle that allows me to have such principles that I have in equilibrium no the only things that are generic are the fluctuations here and for example we saw yesterday which is for out of equilibrium you could use it here but it's a very poor consolation as you have noticed it's not something that really the fluctuations here that it's what you would like to have known it's you like it because you're poor and when they give you something you're accepted but honestly it's not a lot so this gives you a bit of perspective I think but all sorts of as I try to convince you here all sorts of lovely things happen but many questions are questions whose answer is why not because it's simply the absence of equilibrium that you're discovering you're discovering how strange it was to be in equilibrium or like that you were talking like the man who discovered that he was talking in prose this is exactly the same thing okay thank you and everything even non-equilibrium ah let me end up with a quote of Madhma Gandhi here in honor of Mahesh ah we have other representatives of the Indian there are others yes I see Mahatma Gandhi was asked what he thought of occidental culture you probably know the quote and he thought a little bit and he said it would be an excellent idea and I think that the same thing applies to auto-equilibrium statistical mechanics okay you know the quote