 After a lot of indecision, I decided on a method of examination, which is going to be to ask you to give you some questions of which you will be able to opt for four out of more and to ask you to describe in your words in no more than ten lines what you understood of it or what you think of it. You can use your notes. Now, if you're going to prepare super notes so that everything is covered, better still, I've achieved what I wanted, and if you prefer, because this is not a test of English, if you prefer to do it in French or in Spanish or in Italian, you can do it, okay? Not other languages. My resources are limited. So those are the languages I can more or less speak, but that's all. Yeah, okay, and let me tell you, now I see the end of the tunnel. So what we're going to do is the following, and I hope that we will make it. We will see the equilibrium theorems that are important, a linear response in equilibrium today, and then we will do a little bit of large deviations, and my desire is to show you the fluctuation theorem, which is one of the most popular results in the last 30 years in auto-equilibrium, and which will also give you a clear idea of the second principle and entropy production. So it's a very nice exercise, and then the last day or so, we are going to see a bit of active matter, which is the violation of all the things that equilibrium gives you, and I'm going to be very brief, but you will see more or less how it goes, okay? So that should be it, okay? So now, linear response slash equilibrium theorems. So imagine you are in a situation where your system is in equilibrium. Somebody wants to ask? Ah, yeah. Please, before you start, I have some quick questions. Does every equilibrium system satisfy the detailed balance? Oh yes, but let me insist on this point. As I told you a few times, I am using equilibrium, and everybody uses equilibrium in several different senses, so let's be clear. First of all, sense number one, your bath is a good equilibrium bath. This is given to you by all the things we did to construct a good decent bath with response with noise correlated in a way it has to do and so on, let's say of temperature T. Then here is your system, which is in contact with a bath, and we want the system, I remember I used as a notation and I was consistent in doing it in blue. I don't know if you remarked that, where forces that derive from a potential do not derive from a potential as zero, and no time dependence. This you need because if not, your system is breaking equilibrium. That's condition number one, condition number two, and condition number three is that you prepare your system. Now when you prepare your system, sometimes it is not in equilibrium. For example, I pour a glass of water, which is at 30 degrees and mix and whatever, and then I put it in a bath at 20 degrees, which is a perfectly nice bath, and let it stay. Well, the system, for example, it's energy, for example, will go to a value, and by the time I'm here, I can say that my system is in equilibrium. So there are three things. The bath has to be good. You don't have to spoil it with your system, and then you have to give it time to happen. Now, give it time to happen depends on the system. There are systems that equilibrate very quickly, water, and there are systems like glasses that we know that can take, I don't know, a few billion years. So it depends on the system because it has to explore its face-to-face, and it's very bad at doing it. So when we say equilibrium, we mean a bit of everything. So when people say the word equilibrium, it could be a mix of all these senses. But bear in mind this. So for example, yesterday somebody asked me afterwards, I don't know, I think it was you, about entropy production, of which we will talk more. So let's say I have no forcing and no time-dependent nothing, and I prepare my system. There's going to be entropy production because my system is wanting to explore all face-to-face. Once it's in equilibrium, it's very happy. And if there is no forcing, entropy production stops. So it would be something like this, measured in some way. On the contrary, if there is forcing or there is, for example, an alternating field, your entropy production goes on forever and forever. And you are putting work into the system, okay? And what about detail balance? So detail balance is the fact that your bath is a good bath. Detail balance is a property of the bath, not of you. As we shall see, I hope, at the end of the day, there is something very nice that happens that you have a bath that has detail balance connected to a system, and then the system, if it's an okay system, will equilibrate. And then I could ask, okay, can I use this as a bath for a new system? And the answer is, does the dynamics of this in contact with my system now obey detail balance? And you will see that in half an hour, the answer is, this is what the fluctuation dissipation theorem tells you, which is one of the objects of today. But also to get detail balance, you should have no forcing, right? No forcing here, and of course the bath has to be decent. Also two should be true, right? Yes, yes, sorry. Yes, yes, yes, of course. Your system plus the bath have to have detail balance, yes, sorry. Okay, then if you're going to do Langevan or you're going to do Monte Carlo or you're going to do what is called heat bath, these are, they all have detail balance or not, but usually they do, and that's it. Okay, so what we're going to do is the following. What does one want to know? What does Mahesh want to know when he does an experiment? One of the things that you want to know is you prepare your system, and now we will assume it is in equilibrium. So we are in equilibrium. The bath was good, the time was enough, I didn't force it, it was in equilibrium. Remember that equilibrium is tightly related to time reversal. So if I wouldn't be lazy and I wanted to write a book on stat mech, I would divide it in two sections. I would first talk about macroscopic city or not and what can we do with the fact that we have many degrees of freedom as I did with you. And then the book would be divided in two sections. Statistical mechanics with time reversal, statistical mechanics without time reversal. When there is a form of time reversal, your train takes you to equilibrium properties, partition functions, everything where the time has to be taken out. If there is no time reversal, as I told you many times, you do what you can. In the books, the second part, if it occupies five pages, I'm exaggerating. I mean, there's nothing about it. But now with active matter and things like that, we are much more into that. Okay, so now I have my system in equilibrium and I have, this is my system. Let's say it's in contact with a bath that took it to equilibrium at temperature T. And then I have an observable, I don't know the position or the density or the magnetization or whatever. And I have another observable. For example, this could be the magnetization and this the density, anything. And as you know, we have the Langevin, the bath, et cetera, et cetera. So we have an ensemble of trajectories. So let's say we start from a point and then according to the noise, different trajectories and the probability that these do is represented by the Fokker Planck or the Smoluchowski and all the Cramer's equation. So at time T prime, I decide to measure A. Fortunately, it's a classical system, so I can measure it without destroying it. It's okay. And then I continue at a time T prime, I decide to measure B. At time T, sorry, I decide to measure B, okay? Now, of course, this is one experiment, experiment number one. This is experiment number two, which I did exactly the same protocol and I started again. But of course, the noise realization is slightly different. And I do it again and I get another one. So I want to measure the correlation. So it's going to be CAB of TT prime. I always put second the smallest time. I think everybody does, but I'm not sure. Which is the average over all the realizations of the experiments of the noise of A of B of T, let's say, A of T prime. So I measure this, I measure this, I record the value. Redo the experiment, measure this, measure this, record the value, measure again, redo the experiment, and then I take the average. This is what is called the correlation function. It tells you how much B at time T depends on what A was at time T prime. Please have a question about the type of interaction. So when you have your system and you have the buff, so for all the type of interaction, are we going to have equilibrium? Always have equilibrium. Yes. They are type of interaction that doesn't allow my system to. Your system keeps on being in equilibrium. Yes. All type of interaction. Yes. For the interactions that are okay, knows that you're not. For the moment, we are doing things that are, I'm not violating equilibrium. So I start in equilibrium, measure something without disturbing it. Measure again, but the system is like a glass of water at 20 degrees. That stays all day, a glass of water at 20 degrees. I'm not stirring it or doing anything strange to it. Okay, this is a correlation. And then there is another measure that is also interesting, perhaps even more interesting, is I start my system at a time, then go on. And when I arrive here, I am going to perturb it, give it a little kick. So V, or let's make it integrating, starting from minus infinity up to time t, I apply a tiny little field, which means that instead of V, I will have V plus delta H times A. So I am perturbing a little bit my system up to time t prime. And then at time t, I measure B. Yes, the same as it was above. Yes, yes, the same. So this is an important measure. It tells you how much the system responds to a little kick. Okay, this quantity here I'm going to denote. And so it is defined as, I will tell you in a second. Just to be sure, so delta H is non-zero up to time t prime. And then it's set. Then I cut it off. And the calculation we are going to do is in the limit of the perturbation very small. So it is an expansion in powers of delta H, and we keep the first. This gives the name to the topic, and this is how the books call it, linear response. The linear means linear in delta H, okay? And because it's linear, if I do different things, I cannot be effect of different stories of the field. Another way of putting the same is to say, instead of doing this kind of experiment, if I write here, will you be able to see it? Yes, okay. So another way that is often introduced would be I do nothing up to time t prime, and there I give it a kick with A. So this depends on time. It does blip. I turn on the field and turn it off very quickly. And at time here I measure. This one is called RAB, which is how does B change at time t when I gave A a kick at time t prime? And you see, because of linear response, if I consider this thing, I can think of it as blip blip blip blip blip blip blip. A lot of little kicks continuously in time. So this is just the integral between minus infinity and time t prime of R of tt prime d, sorry, tt second dt second RAB. Any other story you can reconstruct from knowing this one? Because it's linear response, meaning that the effect of doing kicks at different times to linear order, they're additive, okay? These two things are super important, super important because this is what an experimentalist can do. An experimentalist can hardly go and look at what every particle in the system does. But these are the things that you can measure, okay? And now we're going to show two results that are for an equilibrium system, only for an equilibrium system, meaning that it's a system that stays, is and stays in equilibrium since time minus infinity, forever and ever. And yes, so, and as you will see, these results are intimately related to detail balance. So first result is called Onsager reciprocity. I believe that Onsager got the Nobel Prize for this, basically. Although you probably, or some of you probably know him more for having solved the easing model, which is a Tour de Force or was a Tour de Force at his time, he was a great chemist. And this is a result. I think that this is why he was given the Nobel Prize. So the thing is, sorry, before that, let's zero is, this one is very easy, so I always forget it. If you think a bit of what you're doing here, remember that the system is in equilibrium since minus infinity. Now you measure at time t prime, then you measure at t. You see that the starting point doesn't really matter. So if I do at t prime plus an hour, and I measure at t plus an hour, because I am in equilibrium, you see this glass of water, if I do an experiment now and one minute later, so I measure, and I do the experiment tomorrow and one minute later, you expect exactly the same thing because it is in equilibrium. So time translational invariance means that CAB is going to be, and the same is true of the responses. If I have a system in equilibrium, give it a little kick at time t prime and measure it one minute later. It's the same as if I do the same thing tomorrow. What matters is the minute, not the starting time. So time translational invariance is not the same as saying equilibrium, but equilibrium implies time translational invariance. Why is it not? Because if I have a system that I'm constantly bothering and it has reached a stationary state, you get time translation, but you're not in equilibrium. Okay. Property number two, which we're going to prove, I'm going to do it a bit quickly, but the most important thing is that you understand the logic. This is an amazing result. If I measure A and then I measure B and calculate the correlation, it's the same as first measuring B and then measuring A. It's not obvious at all. And for example, with active matter, it doesn't happen. Even stationary active matter, for a system that is conducting heat, even if it's stationary, it doesn't happen. For economy, the market fluctuations of the prices, it doesn't happen. You could do correlations of the market, not the price of the kilo of wheat today and tomorrow. It's a fluctuating quantity. Well, and now you correlate two quantities, price of maize and price of wheat. And then you do the other way around. There is no property like this. So of all the dynamics of the universe, equilibrium is very specific. Remember that we have said many, many times that it's time reversal. So okay, I'm going to do it very quickly. You will get the logic. So CAB. Okay. Isn't this because it works only in equilibrium because the quantities A and B have to be canonically conjugate? No, it doesn't matter at all. Any A and B will do. What is important is that systems in equilibrium have detailed balance as you will see and they are in equilibrium. You will see it now. So CAB of t minus t prime, I can write it as. So this process, instead of doing it with Langevin, I'm going to do it with Fokker Planck, yes. The first here, translation. Ah, here, on Sager, that's the name of the gentleman. He was Norwegian. So I'm going to write it and then I'm going to explain it. This is the Fokker Planck thing. So this is the Gibbs measure. We are in equilibrium so we can use it. This is my notation in bracket notation. I measure B, I evolve in t minus t prime. I measure A and this is, remember, the constant. So yesterday somebody asked me, this is what? This is the Qs to which you can go, integrate over everywhere you can go. This is just telling you that after I measure, you can do whatever you like, I don't care. I'm going to sum up because once I do the second measure, I ignore the system. So this is how you write a correlation in the bracket notation. You will find it also just by doing integrals. If you prefer that one, you can do it with that one. Now the trick we have to use for all these things is to see that there is time reversal. So now to use time reversal, I want to reverse here B and A. And how do I do it when I have a sandwich of operators? I take the transpose of what is inside and the transpose of this is itself. It's a function of the coordinates of this. This is a function of the coordinates. This is a function of the coordinates. And the transpose of this, remember, is the transpose of this. But the transpose of this, we did it and we can apply detail balance. So let me first write, so Gibbs appears now in the left. Then I have B, remember when I transpose the order changes, then I have E, T minus T prime, and then I have H dagger, poker plank, then I have A and then I have the flat thing. Okay, up to now I have done nothing almost. And now we're going to use the time reversal property, detail balance. So remember that this one, we did it yesterday, is equal to E to the beta B E to the minus T minus T prime H E to the minus beta B. And then I have A and I have B. This, I have just now used the detail balance property, which is, you see, and then here I reverse times and now I put them straight by using detail balance. And now it's very easy because this one multiplied by a constant gives me the Gibbs measure. And the Gibbs measure E to the minus B times unexponential gives me the flat thing. B is a function of the coordinate so I can commute it with these. These are functions of coordinate. So at the end of the day, this gives me, tuck, tuck, I'm sorry, did I, no, I'm okay. I'm using this one. This is a magic of having used detail balance. And so this here, you recognize exactly CBA. If you, okay, ask me questions on this, but the important thing that you have to bear in mind is that I use the time reversal that I have in my problem, which is detail balance. So sorry, just this one. So B and E to the minus beta B commute. Commute because there are some simple functions of the coordinates. And Gibbs times E to the beta B is just flat. You see how important it is that at the beginning I start with Gibbs. If I don't start with Gibbs, I don't get that, which means that this thing works to the extent that you start in equilibrium. It is important that I start in equilibrium. It is important that I have detail balance. Maybe would it help to write explicitly that Gibbs is the same integral times E to the minus beta? Okay, so yes, okay. So this is the flat guy. So if I multiply the flat guy, well, let's do it to the right, by E to the minus beta V exponential times flat, except for a normalization, I get Gibbs. Also, E to the minus beta V should be inside the integral, right? Because it depends on the coordinates. Yes, yes, because you apply it to this one, so you can put it in here. E to the minus plus beta Gibbs, this has a negative exponent, and this is proportional to the flat one. This is the property that we have used for the last step. But the important thing is that I just put the things and look what I've done and everything falls in place. Sorry, I didn't understand why the time integral in that equation is negative infinity. Sorry, here? Yeah, in X, X, V. Why did you write time to be negative infinity to D prime? I decided. You decided? I decided. To define this function, so I can tell the experimentalist, this function tells you what happens if you have a field on until time D prime and you cut it there. If, and then this one is the same thing, it's a derivative because if I put the field up to here and I cut it here, the difference is this. And because I have linear response, it's the difference corresponds to a little kick less. This is why this is the integral of that. So if Mahesh wants to do an experiment where for five minutes he turns the field on and then for two minutes he puts it with a different sign and for five more minutes he puts it twice as strong, he just has to integrate a lot of these blips and within linear response it's additively effect. That's why it's called linear response theory. Thank you. Okay, this is quite amazing, no? I mean, think of it, you have a complicated system of which you don't know anything. You have thermodynamic quantities and now by knowing the microscopy that it's at the bottom of it, you have discovered that if I measure magnetization now and density later or vice versa, I'm going to get at the same times, I'm going to get the same number. It's quite remarkable, no? Okay, now we're going to do the second property which is a bit more difficult but the idea is more or less similar. So we want to calculate this guy, no? So first thing we have to bear in mind is that chi AB of TT prime is going to be chi AB of T minus T prime because if you translate both times the experiment because you're in equilibrium is exactly the same. And now remember what chi is. It's the result of having a little field from time zero to time T prime. So, and then we measure A. So with the same logic as before, I start with Gibbs but with Gibbs with H plus, sorry, V plus delta H times A. This is this part. Then I evolve it, then I measure B and then I don't care, it can go anywhere. And this is chi, okay? Okay, so now we're going to make an auxiliary calculation. Auxiliary but very simple. We want to calculate how is Gibbs of V plus delta HA. So it's Gibbs. Yes, in here I reverse the orders of A and B but okay, I think that I can leave with that. So it's what? It's Gibbs is what? It's E to the minus beta V plus delta HA. And I have to divide it careful with this one by the partition function, the integral which I'm going to explicitly write it. This is what it is. So this is this chi here. And now my job is to expand for small this. Okay, so two places. In one place when I expand, okay, so this is going to be E to the minus beta V divided by DQ, E to the minus beta V, okay? DQ prime, VQ prime and this is V of Q plus things. Okay, this is a bit painful but we shall do it. We first differentiate with respect to delta H plus delta H times. On one hand, when I take this one down, it's minus A times this thing, okay? This is the first term times beta A, sorry, times this thing, okay? It's just that I took this guy down and this is first term, second term. And then I have to differentiate this one. So I take this one down and when you do it carefully and then, okay, we will read it together, I get also the same E to the minus beta V of Q over the integral, this same thing. And here I get the following thing and here it's the same integral. So it's twice and here it's finished. It's just, I'm studying the small modification this does in orders of delta H. So here I took the derivative of what's below so I get a square with a negative sign. So this is what, and then I get the derivative of what's inside, I split it in two so that this and this are the same and this is what I have. And there is a beta that I have, as usual, forgotten. It's just a derivative. Sorry, here I forgot something important. It's the variation when I change H. Okay, and when you look at it carefully, sorry, this is a bit messy. This is the same Gibbs measure and this is simply all this thing here which is minus beta A and this here, you see that this is the expectation of A. So here I put minus A and what I'm multiplying by is precisely the Gibbs without delta H. So all in all what we get is that Gibbs without it. So this is the correction we were looking for. It's long but trivial, just a derivative. So just to recap, I have been applying this field up to time t prime and what I find at time t prime is this. And now I just, because this is a small correction, linear response, I am developing it and I discover that it is the ordinary Gibbs but it has this correction term. And so now I want to plug this into here. The first term is going to be the ordinary thing but we are looking at the variation with respect to delta H. So the result we are seeking for is I need to plug this thing with a sign into here and then I will be okay. Okay, because it's the thing with the delta H, without the delta H, without it's simply this piece and with is this additional term. So this additional term modifies the thing and it will come here. So now I use that formula and I replace this thing here. The first term is going to give me the thing without the delta H. The second term is proportional to delta H and it is the one I'm looking for. And what is it? I have to take a beta, so, and we're done. Just half a step more and we're done. Assign, I think, yes. Really sorry but I'm totally lost since you defined the Gibbs and up to the end. Okay, okay, so when I start my experiment here, I am starting the experiment but I have equilibrium but under a small field. The equilibrium under a small field is this guy here. So it is the Gibbs formula with a delta H A. It's almost like the old one because the field is not very strong but it has a slight modification. And this is going to be important because at time zero, t prime, sorry, I am sort of, I have been disturbing the system up to that time. So the measure of Gibbs was something like this but now it had a field. But the rule of the game is that we're doing everything with very weak small fields. So the correction of the Gibbs measure is going to be small. So small, how small? Well, I just write it down which is what I did and I expand all the exponentials I have. I use E to the minus beta V plus delta H A. I will expand them or take the derivative as E minus beta V and then I have E to the delta H E. So it's one plus minus beta delta H A and then higher orders, I don't care. What I have done here is just systematically use this one. If you use this one everywhere and you keep things properly and you have to do it on the top and in the bottom. So the top is very easy because it's just a term like this but the bottom is a bit more complicated because you are taking the derivative of this. So it's the square of this times the derivative of what's inside and then you get this term and that's it. So when you do all the, turn the crank you get the old Gibbs plus this extra variation that you have to take into account to first order in delta H. Now I want to replace this for the Gibbs without. So I use the Gibbs without but I need to do the correction. So the correction is, well the first term is what it is without the delta H, the second term is this, this is the correction. So I take this thing here and put it here, here. So the Gibbs is this Gibbs here and then I have this extra term which is the one this and this which if you rewrite it is exactly this term here. So if you want this here is the correction to Gibbs. Okay, is it okay? I can, so the difference between the two is going to be by definition this object here. Now this object here I can distribute. So I get BA, term number one and term number two is B times the expectation of A but the expectation of A is a number that I can take it out. So at the end of the day what I get is this, B and B and this is equal to and this is the remarkable result. So what have we done? We wanted to understand how a system responds to a field and after doing some calculation where we have used equilibrium, we get to the point where we discovered that the response to a system of, to this field is given by the correlation, by a correlation. This is very bizarre. Let us write it in the most usual way. You see that chi is the integral of r. So if I take the derivative here I can write it as RAB of TT prime is equal, I'm going to differentiate because I like it with respect to the second one so this will absorb the minus sign. And this is the fluctuation dissipation theorem as you read fluctuation dissipation theorem. And sorry here I could use minus because we are in equilibrium. This is technical but this is the simplest proof there is in the market. You won't, I think, find a simpler proof than this one except for the bracket notation which if you like it is easy and if you don't it's not. Okay, so now let me keep this one on saga we have seen and let me, let us discuss a bit what this means. I'm going to erase, is it okay? Just for remembering you don't need to copy what I'm going to write. Remember when we wrote, when we got read at the very beginning or some days ago of the thermal bath and we arrived at this equation you don't need to copy this one. This was a noise and we said that if I remember correctly this was okay. And we said that this, we said that this, the quality of this guy multiplied by the temperature with this guy was called fluctuation dissipation theorem or relation of the first kind. So the one we have just derived is of the second kind but they are exactly the same thing and this is what I want to transmit to you because it's very, very important. So what is the correlation of noise, noise? Well it refers to the bath which was made of a lot of oscillators, whatever but it is a correlation just like the one we found and what is the friction? This is the hard part. What is the friction? The friction is the response because when I try to move in water, the water opposes me and this is the response that the water is doing to my motion. So the point of this is that the fluctuation dissipation theorem which is completely general for an equilibrium system is what we found before when we integrated all the oscillators was nothing but the fluctuation dissipation theorem as applied to this bath of oscillators. So and this calculation, you can find it in my notes. You take a system that is in equilibrium now. It is in contact with a bath and the system has A and B or whatever. Let's say only A equals B, only one and to this A we connect a thermometer. What is a simple thermometer? For example, an oscillator. It's a thermometer because an oscillator will catch some energy and the energy it catches is a measure of what the temperature was of the system. When you compute how much energy does it get when you're coupling these two as a thermometer, you will discover that you get exactly this equation for the oscillator with the fluctuation dissipation theorem of the system telling you exactly that these two things are the same. It's the relation between the correlation which is the noise correlation. Why the noise? Because for me, if I am looking at the system, my noise is coming from the system and for me, if I look at the system, my friction is coming from the response that the system gives me. You will see this calculation in detail but the important thing to bear in mind is that fluctuation dissipation is telling you that if I am in equilibrium, I can connect, for example, an oscillator but any thermometer will do. This is an old fashioned glass thermometer. And to any observable I like and it will exchange energy exactly with the mechanism that Einstein discovered for the Brownian motion. Exactly, so the fluctuations of this system which are measured by the correlation are going to give me energy and the friction the system applies to me which is given by the response. If you want the detail of how you show that, you will find it in my notes but it is clear that they are the same thing and they satisfy this relation which is nothing but this relation because you see here, yeah, this is the temperature goes there, yeah? So, so. Oh, okay, so, sorry. Yeah, yeah, yeah. So, if I want to interpret this, so it looks like this eta is playing the same role of... Correlation. ...of the H. In the sense that I can think of that equation as being an equation where essentially you perturb the potential by minus eta times q, right? Yes, the eta works like an H for me that I am a q, no? I am q, so it's working for me. But if I couple this thermometer to this bottle, what I will see is that the eta is the noise that the system is doing to me, the noise of A, the magnetization, but the correlations of the noise of the magnetization, so it's a correlation of the magnetization minus the expectation value, is precisely the amplitude of the noise that the system is giving me. But there is a second term that comes from the fact that when my system moves, it acts on the bath, let's say, or when my thermometer changes it. And there is a retro feedback, and this is given by the response function. And when you do it with a bath with oscillators, you do it and we did it and it was okay. But if you want to couple to any system, you can use the fact that it satisfies fluctuation dissipation to show that any system in equilibrium will be a good bath for you. This is very, very important because now you see that what Einstein was doing is that instead of an oscillator, he had a particle of pollen or Brownian particle, and instead of whatever, he had the water where this was dancing. And the correlation and the response of the water, meaning the noise you get from the water, plus the friction, they are precisely given by these two terms. So fluctuation dissipation that sometimes is given to you as, wow, look at this nice property that allows you to measure correlations just by measuring. It's much more than that. It's at the heart of what equilibrium and measuring a temperature means. And what have we used to prove it? This is the important part, the calculation, you can check it tonight, for example. What we have used is that the system starts with a Gibbs measure or if it was under a field, it starts with the appropriate Gibbs measure and that it evolves in equilibrium fashion. So please, you didn't make a comment about the fact that it may happen if in the case where we have a strong field like for the perturbation, because in our case, we just consider the fact that the field is weak. So what's happened if in the case of a strong field? If you apply a strong field, so you could, you see that I did everything to order delta H. And I said that if you computer pulse like this of field and another one like this and you, this one where you do the two are just the sum. This is linear response to order linear. If you go to delta H squared, which a little bit stronger field, you're lost. There's nothing easy, you can say. It all becomes nonlinear, horrible. So beyond linear response, when the fields are a bit larger, life becomes more, you cannot do a calculation like this. So yeah, so it's called nonlinear response. Of course, in life, these things happen, but there are no, it gets really quickly very, very hard. I mean, you can, of course, in the computer compute it, but there are no, the relations are much more complicated and not very useful. You mentioned that this is useful also because it's like more simple for experimentalists to measure it. And is it even like simpler than measuring over the linear, like it's simple to do the linear response? Is it simpler than doing like a higher order response for experimentalists? I think that the experimentalists usually prefer responses because applying a field and seeing how the magnetization grows or applying a pressure and seeing how the volume changes is much better than looking at how the positions of the molecules correlate over time. No? Absolutely, because we don't have access to the microstructure of the system. So we are always trying to infer what are the microscopic responses arising due to some microscopic changes. That is what all condensed matter and statistical physics is about. Yeah, and then these things happen also in quantum mechanics and so on. But it's interesting that my feeling is that sometimes the books don't emphasize how important fluctuation dissipation is and we have to go back to the intuition of Einstein with respect to Brownian motion. He said, put it in a modern way, you would say, I put a particle of pollen inside the water. The particle of pollen measures the correlation of the crashing velocities of the water for sure. But it also measures the response because when it tries to move because of all the crashes it got, it finds that they respond and you can show that the response is precisely the response we defined here. So these two things must be equilibrated because I need that my pollen particle is in equipartition or some form of equilibrium. It cannot heat up because then I would be able to throw in pollen and get energy from water. It has to be at the same temperature and this intuition, which is for us maybe easy but it was quite clever, is the idea behind fluctuation dissipation too. Only that this way we can formalize it. Or the way my advisor taught me the first time he talked about fluctuation dissipation and frequency dependence susceptibility and all those concepts. He said, Mahesh, when you go to buy an old car what do you do? You kick the tire and you lose parts that are about to fall off. That was the way he explained it. I have a conceptual question, not on the technique. There is something more constrained, there seems something more constrained about on saga reciprocity compared to FDT and I could be wrong. Which is that there are always the couple transport coefficients. You have the seabag and the peltier. But it's not as if they are response, one is response to the other. But you can do the fluctuation dissipation cross and it will be as, there is a nice way but I didn't do it, I wanted to do it but it's a little bit more complicated but a little bit to derive the same thing we did for the response of fluctuation dissipation with explicitly time reversal. So which was why I asked if A and B have any constraints on what those two observables can do? Or could they, there can't be any two observables? The only thing I didn't say and it's important that I say it now is that I did it about on saga. Let me write on saga. On saga is CAB of T minus, we are in equilibrium. In fact, when you have velocities, you have to reverse them. So if this is Q, something of QP and this is something of QP here, it's going to be of Q minus P, Q minus P. And so here I put a little bar to tell you if you're doing observables that are coordinates like the magnetization, it's okay what I said but if you're going to use velocities, you need to reverse their sign. If you do the same exercise that we did before for reciprocity, you will see that naturally this appears because when you do time reversal, remember that in kramers, there is a flip of sign. I thought it wasn't worthwhile to make it the whole solution more complicated. Okay, so just to tell you there are lots of interesting things. For example, when you think of the financial markets, for example, there you have a system with observables, as I said, the price of wheat and the price of, I don't know, PCs. And then suppose I buy wheat, a lot of wheat, so the price goes up and I want to see, let's say it's a small perturbation and then this may, for example, affect, for some mysterious reason, the price of PCs or something else or the price of wheat, of course. And can I, or I look at simply the fluctuations of the price of wheat and the price of corn and then I measure like this and I measure like that and I ask myself, do I get something like that? Do I get something like that? And I respond in response to buying. And the answer is no, which is amazing because the markets are made of millions of operators and they are fluctuations and everything, but they are not equilibrium. They don't have the specific thing that equilibrium has which all comes from some form of time reversal, detailed balance or something like that. And we will see that you don't need to go to the markets the day after tomorrow. We will see for active matter which, for example, bacteria or motorized particles, the same happens. You don't have all these magical properties. So once again, just to convince you, the moment you put one foot outside equilibrium, you lost the measure, the Gibbs measure, and you lost a formula that gives you the measure. There is no simple formula that will give you the stationary distribution. And you lose the equilibrium theorems and you lose reciprocity. So the restriction now to repeat an old joke that I think was made for linear and nonlinear systems to talk of non-equilibrium, the joke is from Ulembek maybe, to talk of non-equilibrium systems is like talking of non-elephant zoology. It's almost everything. I think that the original phrase is... Markets, okay. To talk about non-linear systems is to talk about non-elephant animals. Of course, most animals are not elephants. And here the same. I mean, thanks God we can... So all the things that are related to equilibrium occupy 99% of your books. And there is very little left for out of equilibrium, but this is a completely unfair and unbalanced situation with respect to how the universe works. With respect to what you know how to solve, then it's perfectly fair because out of equilibrium as you will see, you will see perhaps the most flashy result in three decades, the fluctuation theorem tomorrow, and you will see that it's nice, but it's not great. It's very modest thing. And it's also everything we know out of equilibrium is very modest. This one has to take into account. So I think I will stop here. Yes, just one observation that say the... I mean, this relation... If you consider this relation for the integrated response, the integrated response is minus beta times the connected correlation. When t is equal to t prime, then it's just very easy. Yes, that one you can get without time. Yes, very easy. So the non-trivial thing is that it extends two different times. Yes, yes, yes, yes. Okay, so then the exercise is... Go through the lecture notes now. There is a lot of depth here. So we are going to meet again at 2. To discuss about these things and other things. Yes, and once again, I'm sorry I cannot participate. I have to give a seminar that has been long programmed. And now we are going to take a photo if I'm going to look for the photographer.