 So we start this second week, okay, so, Jorge? Okay, we start. Good. So, let me recap a bit. We're going to go back to the Fokker Planck-Smoluchowski equation. FB stands for Fokker Planck. Okay, so one thing I didn't mention the other days, you can see that this is the divergence of something. So this something can be interpreted as a current of probability. Okay, so now, to make it clear, the notation I'm going to use is the one of quantum mechanics because I think it's worthwhile to take advantage of things one knows. So if I start in a point Q, which could be in any number of dimensions, and I end up in a point Q prime, the interpretation is that in time t, you start it in Q and you end it in Q time and this is the probability, okay? So the first point, and I'm going to repeat this a few times, notice that although this resembles a bit Schrodinger, a bit. It has a second derivative, but it has a first derivative, which Schrodinger doesn't have. There is no i here, and that's very important. No Schrodinger has an i there. Okay, there is a lot of literature. So those of you who want to get into a bit the relation between this and Schrodinger, one place that I recommend is Parisi's book, which is called, it's the only book, in fact, of the Memories. It's called Statistical Field Theory. Parisi was one of the very first to give some titles of, some status to things like dynamical equations, even Monte Carlo simulations. Before that, the books had almost nothing of it. And what he cites there is at the time of, not the beginnings, but let's say the 50s, there were people who were trying to do foundations of quantum mechanics, and some of them wanted to make, boom, most notably, wanted to make some relation between stochastic and quantum mechanics, where deep down some stochastic process. I don't know how far this got. It's more philosophical than anything at this stage. Okay, so now we will do an important exercise, which is the following. Let me take this and apply it and compose it with v of q. And I'm going to try to commute it to that side. So I will use the following obvious relation, that every time I see a d, dq, you act on this, or you act on whatever you have there. So when you act on this, you get this, and this is the result. Okay, so this thing here, applying this little trick twice, tick, tick, commuting here, I will do it without the f for the moment. I get, it's a simple exercise. The first time that you get this, it cancels this term and it corresponds to this derivative, and then you apply it again and you get that. Okay, so collecting here, I can see it this way. Pass this to this side. Please, Professor. Yes? I have the, I don't understand very well this equation. This thing? Yes. Okay. So, okay, let's do it slowly. So you want to see how this operator, let's say, acts on a thing side. Okay, so let's see how it acts. Let me first attacks on this. So, and then there is psi here. And then the second term, just the chain rule. Okay? So this is what? This is D, DQI, I'm just rewriting it. Okay? And now the magic trick is that because this is true for every psi, I can, I'm not going to do it, I'm going to erase the psi's and it would be true at the level of what it stands for. This one, sorry, I have to erase. Okay, and this, you recognize here, compare this one, which is the Fokker Planck, with this one, so without the F for the moment. So imagine I transpose this. Remember how you transpose. The transposing is like integrating by parts. It changes the sign of the derivatives, and it changes the order of terms. So if I transpose this one, this sign is going to change and this term is going to go there. And if I transpose this one, the sign is going to change and it's going to go here. And what I'm going to get is exactly as an operator. And we're going to discuss this ad nauseam because this is the very much promised time reversal property that an equilibrium or a system in contact with a bath has, in the case of Fokker Planck. We will work it out and see all the consequences it has. But the important thing that you have to remember is that such symmetries are at the heart of all the special properties that equilibrium has. Yes, before answering Mahesh's question, let me insist once again. I'm using the word equilibrium and we all use the word equilibrium in several contradict non-equivalent ways. One thing is, is my bath in equilibrium and when I connect it to the system, will it allow for equilibrium to happen? This sometimes is called equilibrium. But even if the bath is a perfectly decent bath, when you connect your system, the system takes some time to equilibrate itself. So equilibrium is used in two different senses and the word is used in two different senses and I will not make this ambiguity smaller. Equilibrium is either means you are in contact with a decent nice equilibrium bath at temperature T or sometimes it's used to say, and on top of it I was able to equilibrate as a system. So having said this, now I'm going to answer Mahesh's question. This very precise thing that happened, it happened because I had this derivative term, this piece which became this piece. But if this term would have been here, check it, this wouldn't happen. So this kind of symmetry is only true if the forces that don't derive from a potential are zero. If there are forces that do not derive from a potential, this symmetry does not exist. And your system, although it is in contact with a decent nice bath of temperature T, it's forced and cannot reach equilibrium because itself, again, a third sense of the word equilibrium, non-equilibrium equilibrium. So let me insist again, we use the word equilibrium in three different possible meanings. One is, am I in contact with a decent bath, a decent equilibrium bath? Active matter you will see is not. Second, is my system itself respecting equilibrium? Well, for this it needs to not have either forces that don't derive from a potential or depend on time. Also applying an AC field kicks a system. And the third meaning is even though I have all these conditions, it takes some time for my system to equilibrate. It needs these conditions, but on top of it it takes time for my system to equilibrate and reach the Boltzmann measure. Normally the context allows you to guess, which is the meaning, but okay. Okay, so this one we're going to discuss a lot now, but let me just now say one thing that is interesting. If I, well, you will see, I only put a half here and a half here, you can see. So we discover one thing that already is interesting. If I look at this guy here, it's the same as this guy here, and it is a change of basis of my operator, this one. So this means that the Fokker Planck operator, although it doesn't seem permission from here, there is a change of basis that makes it into the Hermitian form. In particular, this means that it will be diagonalizable and it will have real eigenvalues. And we're going to see in a minute that they're not only real, but they're positive, semi-positive. If you want more details on this, this is the connection, the best connection with Schrodinger. You start with a diffusion problem, convert it into a Hermitian problem, and there it becomes something that resembles Schrodinger a lot. But the important point you have to remember is there is no I. Okay, and now we're going to make a bit of philosophy about the I. I am half tempted to write what this guy is. Well, I will, but just for curiosity, this guy here do the exercise, reads like this. This is just in case somebody is curious. It's a nice exercise to do. It's not a hard exercise, but it's tedious. Okay, the summation is... So this is for those of you who are curious. This thing here works as if it were a potential, and if you call this a potential, you see this is exactly Schrodinger, with the temperature playing the role of H-bar. If you want more details on this, go to Paris's book. I do a bit in my lectures, too. What do you have to take out as a message is the Fokker Planck operator, once you transform it, it becomes something quite close to Schrodinger, except that not with the original potential, but with this horrible function, which is calculated on the basis of the potential, but it's not the potential that you started with. Okay, so now here it's clear that it's a remission. And this underlines also the fact that the temperature plays a role of H-bar, which is very important. So low temperature, little noise, semi-classical. We will come back to this. This is clear. Why T plays the role of H-bar? Because I wrote it last time in Schrodinger, there is a part that it says, and all the magic of H-bar, well, here there is a 2M, H-bar, you see, is playing the same role as the temperature here. So, questions. This is a bit technical, and it's more or less as technical as we will get, but today is the peak of technicality. But before I let you go, there will be more technicalities. Okay, so the I that is missing is very important because remember that this is the evolution of a probability cloud that we are studying, a probability of all the trajectories. There is no interference. In Schrodinger, the I is crucial to get interference between your terms. And then you take the modulus square. So, the big difference between this and Schrodinger is not so much the form of the equation because as I show you here, you can transform it into something that resembles Schrodinger a lot. However, in the evolution there is no I, and also the rule to calculate probabilities is this one, which is not the one you use in quantum mechanics when you take the modulus square. So two things, you don't take any modulus square, this is the way to do it, and there is no I. And so this is why stochastic dynamics is not as fun, as much fun as quantum dynamics because quantum dynamics has all sorts of funny interference effects that are beautiful, and that we don't have. So what the physical interpretation of this would it be something akin to a weighted average across all the possible paths that you might take? Exactly. At time t, for every noise realization, you get to a different point, and then you consider you do this in your computer, and then you record the positions at time t, and you get a cloud of probability, a histogram, and the Fokker-Planck equation as you directly the history of the histogram without going to the individual trajectories. This is the meaning it has. Okay. And now, for example, from here, you can do anything you want. For example, imagine I want to calculate what is the expectation of A. It could be Q squared, whatever, any function, A of Q at time t. So I'm going to do the histogram I was describing to Mahesh just now, and I will measure, on average, what is some observable. So what I have to do is, let's say I start from somewhere, let's say from a point Q, then I evolve, and then I measure A at the end of this process, and then I don't care to which place I'm going to go after I measure. So here, I don't want to put Q because I don't care where I'm going. I'm going to put anywhere. Anywhere means something that is flat in all the space, and this I'm going to denote like this, which means a constant. So I'm summing over. So I'm going, measuring A, and then do whatever you want, I don't care. Another trajectory, I measure A, then do whatever you want, and I don't care. And then this thing here is just a constant that says that I am just summing on. This guy is the flat wave function. So in practice, this means, to be very clear, that what I'm going to do is the probability of ending in Q prime, having started in Q at time t, then I measure A of Q prime, but then I don't care which Q prime, so this is the point. I integrate over Q prime. This integrating over all Q primes is the meaning of that. And then... So here Q is the initial value. Q is an initial value, which could be also a distribution in which case I would integrate over many Qs. A can be, for example, let's invent an A. A could be an observable like, for example, I don't know, maybe I want to calculate this, something. So it's a function of which you want to measure, the magnetization, something. It doesn't really matter, but something that depends on the coordinates at that time. So you want the history, you get there, you measure, and then you let the system do what it wants. Sorry. May I ask you a question? When you did the average of the observable A, are there possible... I mean, because A is a function only of the Qs, right? Because you're asking about velocities. Yes. I mean, like in quantum mechanics, you cannot... No, but if... Remember that because I'm doing Fokker Planck and not Kramers, remember that Fokker Planck, once you throw away the inertia, the velocity becomes a disaster. So you don't have... It's forbidden to ask questions about the velocity if you're doing Langevin because the thing is being kicked, you're assuming in such a mad way that it's not even defined. Meaning that the overdamped system is a very bad model for things where you need the velocity. We will... I hope in a couple of days I can give you at least one version of the fluctuation theorem, and for there we will need a velocity because we will need a power and you will see that this is not a very good way to do it. So we will do it with Kramers. Okay, so now a silly remark if what I'm measuring is the number one, A is simply the number one, so I'm doing all the trajectories, then measuring the number one, which gives me one, and then let the system do whatever it wants. So what is the expectation of that one? So what we conclude is that if I put a one here, I should get one. But that is true whatever the initial condition. So it should be the case that the number one times e to the minus th, whatever I put here is one, which means that e to the minus th acting on the left is one. This is conservation of probability, which in fact means that h itself acting on this is equal to zero because it is. So it should be a vector, right? So one acting on e to the minus th, h f p, that this should be a vector. No, below, I'm below. This is a vector. Yes, that is an operator. Yes, and this operator acts to the left. Exactly. Which means... And then now what I'm saying is what you have on the other side of the cosine should be a vector or not. Ah, yes, sorry. It's, sorry. So this is the identity, so that's sorry, sorry. Thank you. In other words, this means that if I integrate this without dA, I integrate with respect to the target time, it should give me one, which is very normal. I have a stochastic process and I just ask, what is the probability that I go anywhere? Well, it is one, no? Because I'm summing all the possibilities. So this is a bit obvious. But it's nice to see it in... What does it mean in the Hilbert space? Okay, and then one last property that is also a bit obvious. I'm going to erase this. Am I good to erase this? So one last property is that imagine I start with a distribution that is in equilibrium. So it's proportional to e to the minus beta v. So I am in equilibrium. This vector, I am going to call it in the notation I am using. Oh, let's call it Gibbs Boltzmann. This is in the notation of brackets. So what should be the case is that if I want to make it evolve, it doesn't, which means that it's killed by... So a stationary state is something, a stationary probability is something that is annihilated to the right by the Fokker-Planck thing. And so that you're sure you see if f is not there, it's easy to see that this guy is going to be killed by this one, simply because this derivative and this cancel out. So here we are good. I'm not using the F, okay? Of course, it has to be true with the F, but yes, I didn't hear it. I said, what do you mean there by zero? Where? Here? Yes, the last line. Zero. It means that this vector here is killed by this linear. So it means that zero is... If you want this guy has a vector with zero... It's an eigenvector with zero eigenvalue. Zero probability. In this case, it means it does not evolve because, you see, when I apply the evolution to it, I get it. This and this are equivalent. One is the exponential of the other. So it means that when I make the Gibbs measure evolve, it simply stays independent of time, it stays where it is, which is what you expect. So it's a very nice situation. It means that when there is a stationary distribution for the conservative problem, it is the Gibbs-Moltzmann distribution, which is something we wanted. So all the mess we made with the bath, et cetera, was well done. What about when there is a forcing? Well, this is not killed. So can we find who's the zero of all these things? No. I mean, there is no easy solution to this. You have to solve it in your computer. So once again, when there is forcing, there is a stationary distribution, the one that is killed by this object, like here. But it's not the Gibbs-1 because there is this F that is bothering me. So this part will be killed, but not the part with the F. And I have to solve it. Other methods, is there a single formula that tells me what the stationary distribution is? No. So you're on your own. You have to resort to some approximations or God help you. This is very important because it means that, apart from the many things we will see that equilibrium means, it also means that getting rid of time is very easy. You just say, okay, my distribution in a stationary case is either micro-canonical or Gibbs. In this case, it's Gibbs, and that's it. When there is an F, there is no easy formula, no generic formula that will tell you. If the system precisely is quadratic here, then you can do things. If not, you can do some perturbation expansion. Perhaps we're going to do a bit of low noise limit, but it's always approximate and it's always case by case. For example, in the next two days, we will see some examples of active matter, which is an extremely fashionable subject, and active matter is by definition something that is driven. Well, you do simulations, you do approximations, but there is no measure that tells you anything, and there is no time reversal. There is nothing like this. So as soon as you step a foot outside an equilibrium bath, a lot of what you know collapses. I'm going to go back to this two or three times. So Jorge, is that a question equivalent to detail balance? We are coming. We are doing it in detail now. Yes, just to time reversal in some form, exactly in which form depends. In Hamilton, it's just t minus t. It's easy in Newton. But here you see this is going to become time reversal in a second. In Cramers, again, it's a little bit different. But the point is that there is something that allows you to reverse the time. If you have that, then there is a chance that the system will be able to equilibrate. Who can violate that? Well, first, the bath. The bath can be a bad one. For example, if the bath is two baths with different temperatures that are talking to you at the same time, you don't have time reversal. You don't know where to equilibrate. It could be the system because it has forcing. Or it could be the system because it has time dependence. Also, time dependence in the fields. For example, a system that is subjected to an alternating field. All these things are going to break detail balance or break whatever. Detail balance is one form of time reversal. And this is something that is implicit everywhere, but older books don't have it so clear. It is thanks to the fluctuation theorems, which I hope I can give you one, that this became very, very clear. Good. So now, we are going to use this to show detail balance. This one, keep it in mind if you want as a curiosity. And if you want more details, go to Paris's book or maybe a little bit to my notes. It shows you that, technically speaking, it's easy to transform a Fokker-Plank equation that derives from a potential. Because if there was an F, this would be ugly. It's easy to transform it into something that looks like Schrodinger. In particular, you can use the same diagonalization programs that you have used in quantum mechanics. Okay. So this one we will not talk more about. This one is going to be the important one for us. Okay. This calculation that we're going to do now, not every detail, but the final result is very important. So, we started by saying, next, this is a change of basis, p, m, p minus one. So you can just as well change the basis in the same way with this guy. If you don't believe me, you expand this in series and when you get powers, you put e to the plus b minus b many times, you did it in algebra. So a change of basis downstairs is the same as doing it upstairs. Okay. Now I'm going to sandwich this with a q prime here and a q here because remember that I want to use this guy. I would like to use this guy. So I'm going to sandwich like this, this expression and of course I have to do the same on the other side. Sandwich. Okay. So when I sandwich this, q prime here simply evaluates. I get this and this one evaluates. So I'm going to get, and here I'm going to sandwich e to the minus th dagger. But you know that the dagger is transposing, so this will transpose and this will be simply h. We're almost there. Please ask me questions if you don't. Yes. Sorry, but in the second step I didn't understand why is to the power negative beta it should transpose. This one. The second, the down one, he has the right hand side. Yes. Ah, okay, okay. So it's a property of linear algebra, let me say. Let me say that a dagger is equal to b. So I want to show to you that a to the a dagger is equal to e to the b. This is the property I used there. This is the one that you don't see. If something was true downstairs, why is it true upstairs? Sorry, sorry. Here I should have p, p minus one, p, p minus one. When we conjugate a, we get b dagger. When we conjugate e to the a, we get e to the b dagger. This is the one you're asking. Sorry, it's negative th and then the other side is beta h. This is what I don't see. Okay, thank you. Because you cannot see it. You don't understand it or you don't see it? Okay, okay, okay, okay. Wait, I, okay. That was an illuminating remark. Is it okay now? Yes. Okay, finally, finally, when you look at these things and you look at this thing, what does this property mean? You take the last step now, one last step and now you will see that it is indeed detailed balance because just rewriting this one means that the probability of going from q to q prime in time t are the beta again. And this is detailed balance which in very old generality is probability of going to from anything to anything else times e to the minus beta v of this thing is equal to probability from anything else to anything. And in this form, it is super, super, super, super general, much more than Langevin, much more than Fokker Planck. This is the form you're going to see in books but this is already telling you that it's true for Fokker Planck. Shouldn't that be a TEHFP in the exponent probability of q to q prime? No, no, no, here there is beta v because it's this guy here and this guy. I just passed that one to this side and used this thing for these. Okay, time has gone away. This is a coefficient, probability of q to q prime that coefficient then. Yes, the probability that whatever you do, whatever the noises are, take you from a to b and the probabilities that they take you from b to a, the time reversed, they, with these weights, they are okay. And this is, so as an example, if the probability of going from a to b, you put in the Gibbs weights of p of a and p of b and here too, this already tells you that equilibrium is consistent with this. I don't understand why this equation is time independent. The last equation you wrote. Okay, time went away because time was in this number. Here is time, but also here. So whatever the value which depends on time, so this side depends on time here. So to be more clear, this depends on t and this depends on t. But it's true for all t's. Whatever happened in the middle, it's true that the probability of going from a to b in an hour is the same as the probability of going from b to a in an hour. Of course, the times are different, no. Thanks, because this is a clarification that is important. So yes, here one should put t. I took it for granted, but it's better to put it, yes. Okay, so those of you who have ever who did a Monte Carlo program in or saw a Monte Carlo program in his or her life, hands up. Okay, so you've probably, if you were doing equilibrium, did this, if it was an equilibrium problem. For example, if you're doing active matter, this is not true. Which means, let me repeat this for the thousandth time, that there is no easy solution for the distribution. This allows you to say that the Gibbs, this implies the Gibbs measure, and you have the Gibbs measure, you can forget dynamics. So this is the famous step. When there is no time reversal of this kind, the first thing is that there is no easy way, no model independent way of finding the Gibbs measure. So statistical mechanics, as you know it in the books, in the first three quarters of all books, you cannot apply. Statistical mechanics that consists of calculating partition functions, which is 90% of all books, it's completely useless once you have some forcing. For example, you want to calculate a phase transition in a ferromagnet. You go to the books and they show you how to do it, either by simulation or whatever, but always using the Gibbs measure. Now, tomorrow or the day after, we're going to see a phase transition that happens in active particles that have some kind of like a little motor that allows them to go in the direction of motion. And there is a phase transition. How do you calculate, how do you study this phase transition? There's nothing. You have to do the dynamics and see what happens. There is no partition function that will help you, only as very particular approximations, et cetera, et cetera. Okay? And when you are asked why is this, well, because there is at the bottom of it the time reversal. Okay. What was I going to say now? Ah! Just a little application of this is that we can do it better just by applying this and multiplying several times. You get this one with the appropriate times. The times are the same in every, what I will show you now. Sorry, I have a tendency to read these things the other way around. So going from in the same times here, going, you multiply by the Gibbs measure here and calculate this one, or you multiply and it's true for a complete chain. You just substitute what you get here and you add and divide by these and you get it for any number of things. I did five just because five is as close to infinity as one can get. Okay? And then even nicer, and this sometimes is given as the definition of equilibrium and to my mind it's the most elegant definition because if the end is equal to the beginning then these two cancel. No! Okay, that's a tough question in the following sense. I'm using the fact that it is Markovian because I am saying P of A and I'm not telling you the previous history but you, I take it that you're asking me is there a detailed balance in the non-Markovian case if it is equilibrium? So if it is above a decent path, remember the first Langevin equation we wrote was non-Markovian. Does it have a detailed balance property? The answer is yes. We wrote it once in a blackboard and I don't know, unfortunately Edgar is not around, I don't know if anybody published it but the important thing is detailed balance, not the number, it's ugly if it's non-Markovian but the conceptual thing is that yes there is but because it's non-Markovian you have to specify all the past so it's a nasty one. Maybe Leticia wrote it somewhere, I don't know but yeah, you can do it. Is the idea that you take a system which is Markovian satisfied with the balance and then you integrate some of the variables? Exactly, one way of getting a non-Markovian thing which is typically happens in all physics is when you have a system that has ten variables and is Markovian has detailed balance, everything but then you say okay I'm going to integrate away variables one and two because I don't care what they do so I'm going to do the marginal probability over variables three to ten then immediately the system loses the Markovianity because it has a memory term that somehow encodes that you integrate. Does that system have a form of detailed balance? Of course it has to have it but it's nasty but yes there is one. I was even planning to perhaps re-derive it but it's a lot of work and better not confuse things but yes the short answer is the difficulty the Markovianity makes your life a bit easier but what makes your life substantially easier is in fact the fact that you have a detailed balance or found some form of detailed balance. Okay this way of writing it which means the probability that you follow a closed circuit at given times and that you follow the reverse cycle at those same times reversed they are the same and it's a beautiful thing way of introducing equilibrium equilibrium now I'm using equilibrium in the bath so what the bath is doing to you is that the probability of doing this and the probability of doing this is the same for any circuit you can think of you can start there and re-derive all the equilibrium properties from there and it is the most beautiful way because you see it's completely this is completely general and doesn't depend on the system on anything so elegant people introduce equilibrium this way which means that not very often okay so I need to say two words about krammers because we will need to use it what happens with krammers do I have a property like this so remember krammers equation I don't know if I really want to copy it again yes I will but you don't need to copy it again if you have it remember that it was the same idea exactly the same only that now we have the full space space and we have very similar equation then most of it is the same and then H of krammers is something like this this is the energy this is the Hamilton part the Baff part is this one and then you could force it this was what it was and now I want to know if this one has a detailed balance property and so I'm not going to do it but using exactly the same way as before you want to know if something like this is true okay so is it true well let's start with a case f equals zero so it's a conservative case this is the equation we wrote the other day almost true you can do the exercise the same way almost true but you have to reverse the velocities both in the derivatives and in whatever you have this is all for f equals zero so this I don't know if people call this detailed balance but it's certainly a form of time reversal it's almost the same as you had with Langevin but of course when you take time when you reverse time okay the coordinates are the same but the velocities have to be reversed this is true of ordinary Hamilton or ordinary Newton if you want you can try the exercise H this H remember was B squared plus V and you just using the tricks I did take the derivatives and you get the same okay and then reasoning in exactly the same way exactly the same way you get the same thing but now here it's not only the potential but it includes also the kinetic energy but otherwise it is almost the same but now you start from Q and P and go from Q prime and P prime which is normal and then you go from Q prime P prime and you go to Q and P so it's exactly the same considering that you now have more coordinates but and this is the but the sign has to change of the velocities H is B squared over 2 plus V okay it also depends on Q and P right oh sure thank you and it's this one so morally is very very similar the only thing that you have to take into account is that you're going to see the film the other way around not only you reverse the steps which is logical but you also have to reverse the velocity but other than that the idea is the same and here the same thing happens so the same thing can be said about this equation and just to make the full story you can do the same with your circuits only that a little bar means that if A is QP A is Q minus P so when you reverse the circuit you have to reverse the positions and the velocities but the whole calculation is the same except for that little detail can we write the same detail balance condition for Brownian motion or other stochastic motion Brownian motion is an example of a poker plank normal poker plank without a potential okay so detail balance is maintained there yes only yes yes yes absolutely absolutely if you have Brownian motion in a box then it equilibrates its detail balance the measure is flat because there is no potential and that's it of course if you have Brownian motion in the whole space the system never equilibrates because it's infinitely large but apart from that it has detail balance yes so in which kind of system we lost this detail balance in which in this kind of system you have almost detail balance except for the reversal of the current of the velocities but in Brownian motion you normally neglect the inertia so if you have over-damped Brownian motion you have the poker plank equation if you have under-damped Brownian motion where you see some inertia in the particles then you go to this case can you say some systems where this detail balance is found there is no detail balance oh yes yes yes we will see for example if you have bacteria they swim and so they have Brownian motion but they also are propelled like a rocket this propulsion breaks detail balance and we will see so that's a typical system without detail balance and it's active matter so presence of that F term will break the detail balance exactly but we will do a concrete example and you will see it actually one way to think about it in a much simpler setting than going to active matter imagine you have a colloidal particle of one micron size therefore it is subject to thermal fluctuations so you view it under a microscope you will see the Brownian motion now you put it under an optical trap and you drag it from point A to point B now you are forcing it and then if you drag it from point B back to point A and you are calculating all the fluctuations along the way this will break detail balance so there is a slight change in the protocol where you are applying an external force and you have caused detail balance and this has been worked out both theoretically and experimentally yeah so you can force all the particles may be motorized so they force themselves so all the physics of active matter is physics of things that don't have detail balance ok one last thing before we stop what happens if I have an F well you can do the calculation it's not hard and you get this beautiful result I'm going to time reverse the left hand side just this is the same exactly as above only that I time reverse the left and now it doesn't quite work there is an extra term which is you can do the calculation with this term and you will get this so there is no I don't know if I should call it detail balance but there is no time reversal that we already said but and this is super important look at what is it that it is violating detail balance it is exactly the power you are putting in because this is force dot velocity so and this is going to be crucial and understanding this this was the basis of all the fluctuation theorems that are have received an immense activity in the last 30 years time reversal which only holds when you have a decent bath and when your system is decent both things now you have a good bath but you have also forcing when you try to do time reversal here you discover that it is broken and it is broken by precisely the term that is the forcing that you are doing am I forgetting oh sorry I forget a little thing with a temperature down here so it is even better it is not the work you are doing it is a work divided by a temperature also known as entropy production so the entropy that the system is producing is the breaking of time reversal and it is perfect because now we understand perfectly well this is at the basis of everything now that one wants to do so forces that do not derive from a potential explicitly create entropy remember the example of the C9 billiard when we applied the force we said that the volume was contracting this was entropy production but it only happened when we applied the force this is the same in this context yeah wait a moment the second equation was it not the one downwards in the right side the HK is with the dagger and with the minus p yes I am sorry so this is really very important this is perhaps the most okay we did a lot of technique at the beginning but these last three blackboards are the most important thing I am teaching you the importance of time reversal and why time reversal is what allows you to say that you are going to go to an equilibrium measure which is what allows you to forget time for certain things and only do statistical mechanics and time reversal is lost when you have systems that for some reason or rather are driven like Mahesh was doing by pulling from it or because the particles are motorized like bacteria or because you decide to apply an AC field to them or whatever in that case you don't have time reversal if you don't have time reversal you don't have a Gibbs measure if you don't have a Gibbs measure God help you you have to simply solve the whole problem with the dynamics there is nothing else that can be done except various approximation schemes but there is nothing that you can write and this is the other part the violation of time reversal is precisely and we saw it already in the example of the Cynibilia is the entropy production which is work divided by temperature in this case which is the usual definition so the work you are pumping into the system or the bacteria pumping into themselves by spending energy with their tails is precisely the violation of time reversal okay so that's the message the most important message of these two weeks and then we will go on what I'm going to do the next time is play a bit with the notion of active matter so that you get a very tiny introduction I'm not even an expert but you will get some idea and then probably I'm going to try to do a fluctuation theorem so you see one of them which is one of the big results out of equilibrium in the last 30 years and then that's going to be more or less it okay so we reconvene at 11 in the computer lab