 Sit we start, Benedetta has kindly set up a WhatsApp chat and we agree that this will be for you students so you can say whatever you want about organisers and lecturers. Please take a photo of this. Yes, take a photo because it will be erased in one minute. Yes, in less than one minute. But otherwise, Benedetta is the manager of this thing. So with this, we start with Jorge's first lecture. So please, Jorge. Okay, are you done with this? Everybody's done? Okay. Mm. Mm. Mm. Mm. Mm. Mm. Mm. Mm. Mm. Mm. Mm. Mm. Mm. Mm. Mm. Mm. Mm. Mm. Mm. I am a specialist of statistical mechanics and this is what I am going to try to transmit to you. I realise, and this was useful for me, that many of you want to do things that are more like applications of physics or are in the margins of physics bordering with other things like biology. I will not be able to tell you anything about biology, but this already tells me some constraints. For example, I was thinking at the end, perhaps do a bit of quantum, but I see now that maybe we use the energy in something else because I don't think many of you will be doing anything to do with quantum or are interested in that. So two things. The first is that I will try to give you a kind of introduction to let's say statistical mechanics slash, just a second, statistical mechanics and dynamics. I'm going to try to tell you things with an angle that is not exactly what you will find in books because if you, if I do exactly the books, I mean you can just as well read the books, but okay, please ask me as many questions as possible. You can stop me at any time and it's very useful if you do. So really, really stop me. Okay. So today is going to be a bit more chat and the next days, if you are worried that there are not enough actual calculations, don't worry. They are actually going to come later. But today I would like to give something that resembles a bit of an overview so that you get an idea what we are trying to face. So the story usually starts with you have a system typically following Hamilton's equations or Newton's equations if you want. Okay, what are my expectations? Well, first of all, I will modify them as we go, my expectations, but what I think that you should come out with is the following. Okay, ask me the question again in half an hour. Okay, ask me the same question in three quarters of an hour. Okay, because you will see a bit more if I, if you let me tell you a couple of things and then we will discuss the expectations. What I am trying to transmit to you. I'm dedicating this lecture basically to make an overview of what I'm trying to transmit to you. I hope that what I can do with this is to give you an idea of what physicists have done to rationalize dynamic equations. How do you treat them? But especially, how do you treat them when you have many actors that are interacting? This is what statistical mechanics is about. It's a science of many things that are interacting. And the idea, and that's complex systems, is that when you have two things interacting is easy. When you have five things interacting, it's hard. When you have two million things interacting, it might get easy again. This is the idea of complex systems. That sometimes there are simplifications that come from large numbers. So whatever we will be discussing here will have some large number in it. Okay, and this, if you're doing biology necessarily, while biology contains everything, but in particular it also contains a large number of parts that interact. So when you are taught classical mechanics, you're taught that you have Hamilton's equations which are nothing but four equals mass times acceleration if you want. And at some point, the book, the first things that they tell you, books of statistical mechanics is, well, you know, if I have a system with many particles, 10 to the 24, then I don't want to know the story of the life of each one of those particles. It would be impossible, but it would be also useless. And this is the chapter 101 of statistical mechanics books. If I had to solve the equation of motion of these guys, it would be hopelessly complicated for all the molecules in this room, but it would also, apart from being hopelessly difficult, it would be completely useless. What do you do with a list of 10 to the 24 trajectories? So this is not the approach. For example, if you have by chance in this room that all the molecules of air of nitrogen and oxygen are in one half of the room, you don't need to solve Newton's equations to see that very soon the air will fill in the room and this will lead you to a situation where they are more or less homogeneously distributed. For this, you don't really need to solve the whole of the equations of motion. And you know perfectly well that this is going to happen. So the people who founded statistical mechanics said, well, how do I do a science of this kind of phenomenon? It should include, of course, the equation of motion because they are there, but it should be something that uses the fact that the system is terribly complicated and this helps you, you know? Because in this case, if there were three molecules deciding whether the three are on one side or not is a complicated equation of motions, you have to follow the three particles that interact, it's a mess. And if there are five, it's a mess. If there are 10 to the 24, it's not such a mess. You just realize, you know intuitively that, well, to a very good position, they are spread very soon spread uniformly. Okay? So morals of this is there are things we can say that are become simple out of how complicated they are. And you have this paradox. And this is the justification of the word complex systems that this master bears. This is, if you want the simplest example of a complex system and surely the first that was really studied. And, okay, so the first aspect and this is point one that you see in books is large number of interacting parts and the simplifications that may arise thanks to that. Okay? And then if you made a course in statistical mechanics, there is a second aspect and the books don't tell you that this aspect is not completely, but it's quite independent from the first ones. This is something that you don't find in traditional books, which is a notion of equilibrium. We will go back and forth to this word equilibrium a hundred times. Equation means that if you remember when you studied this, you started with equations and then at a certain moment you were taught that thanks to the structure of Newton's equation, you had a situation where if this is in phase space, your energy surface, energy is conserved, you were told that yes, my system, this is a, let's say, three-end dimensional system of three-end particles is going to move around the energy shell and visit it democratically with equal probabilities. When I say visited democratically, you have to think not that you visited every single point, but it's the same as when you want to in an election know who's going to win. You only ask a thousand or two thousand people, but if you choose them correctly, you are going to get a good idea of how the election will come out. So it's a few points. You never, this is very important to realize, you never go and visit all the phase space which is gigantic, but you do it so democratically that just like when you do correctly a polling system for an election. Now this democrativeness is called equilibrium, technically it means the following. If I do, if I have an observable, could be the magnetization, could be anything that depends on the coordinates, then I can do two things. Democratically means that I can do two things. One is to calculate the average over time. So I simply measure what the magnetization gives me at every time, and then I divide, I integrate. I divide by the time, I sample it and I divide by the time. This would be a time average, and to say that it is democratic means that there is a correspondence between this and the average of this thing without any time over the energy share, which is called micro-canonical. Here is how, this is how books in statistical mechanics start. Number one, they tell you, don't, we will not attempt to study individual things and we are hence going to do some form of statistics. Remember that when I said I cut the room in two and I leave the molecules to one side, at the beginning they were to one side and then I know that they're going to be mixed, but I'm not telling you if molecule number 384 is on this side or on that one. It's a statistical thing that we're saying. Here comes the second part, which is the passage from something that contains necessarily time, dynamics, and somehow by a pass of magic we end up because of the properties of the dynamical equations, that we were very lucky in this universe that Newton's law are what they are, that allow you to say that the sampling they do, we're going to come back to this a hundred times, that the sampling they do of the energy shell becomes democratic so that I don't need to solve the dynamics anymore and for certain things I just calculate the average over the energy shell. Here is where thermodynamics starts. Remember that 90% of the thermodynamics books, but this is a defect of these books, don't have time. Most of the book you're calculating entropies, energies, magnetizations, phase transitions, but there's no time at all. Usually there's a little chapter at the end where as an afterthought they say, oh well, we should also think of how things happen in time. But why has time disappeared from the whole thing? Because you get at some point this correspondence. This correspondence is referred to as equilibrium. I'm going to use the word equilibrium a lot of times and it's an ambiguous term because it means several different things so you will be warned about this sufficiently. But just to give you an example to which we will come back, imagine the particles instead of crashing against one another and following Newton's law, imagine they have a little propulsion. This is what today we call active matter, like a bacterium. But still you have an enormous number of them and they follow equations of motion. We can model this very simply but with a bit of propulsion. They have a source of energy and also a source of course of friction that will take the energy away, otherwise they would heat up. So they follow very nice equations of motion and item one is perfectly well satisfied. You still can make the argument that if I put all the bacteria on this side they will mix up and so on. But because they don't satisfy Newton's equation and the equations they satisfy are a little bit different, just this already breaks completely this fact. And everything that you learn in a stat mech book about micro canonical, you cannot apply. So you're lost. So this is the first thing to realize. First of all the construction of thermodynamics from the usual thermodynamics from the microscopic basis. There is an enormous part of it which is completely based on specific properties of Newton's equations. And what's I going to say? Yeah, of specific properties of Newton's equations. And you get equilibrium. We will see. This is another thing that older books don't tell you very much. What is it that Newton's equation have that allow you to say that you're going to visit democratically the energy thing? You will find, and we will discuss this a lot, that the property is that these equations of motion have a form of time reversal. They have a symmetry that if you pass the film backwards, it's still a valid situation. Always that you can do equilibrium. It's because there is this property of time reversal. Question? Yeah. So what is the time scale tau you are concerned about to achieve equilibrium? Sorry, I did. I should have. It has to be a long time to give it the possibility of exploring, yes. It's a long time limit. So give the system a long time, and it will explore the whole. How long is long depends on the system. For a glass, for water, it's a few minutes. For the window glass, it's a few billion years. So it depends on what you're doing, what system you have. Yeah, I'd just like to ask if time reversibility is a necessary condition. Time reversibility will turn out, and in each case we will have to define it. It will end up being necessary for what? For this thing to happen. But still, for example, the bacteria I mentioned that are propelled do not have it. But you still want to study them, and it's true that large N matters, and it's still true that you don't want to know the destiny of a single bacterium. But you lost the easy bridge that took you here and that allowed you to get rid of time for certain things. So for that, it is necessary. You will see, we will do little examples of cases where this is broken, and life, you have to reinvent it again. Yes, he had a question. So I have a following question about the time reversal. So I would like to know whether it's always possible, because if it's not always possible, that means the system that we are going to study, we should take in consideration some post-it for that. Some post-it, so I don't know if you can comment about that. About the time reversal, is it always possible, or if it's not, that means we are going to study for some system, we are going to take some consideration. You mean what postulates, what do we have to postulate? I mean like for the time reversal, I want to know firstly whether it is always possible. So that means if I want to study a particular system, there are some considerations like I have to take for that to eliminate the time, so to reach the equilibrium. So I don't know if you can comment in detail about that. So you will see, we will see that the beginning of the story is with Newton's equations. You know that if you have a system of particles that are nicely interacting and you film it, and then you pass, you take two physicists and show them the film as it happened, and you show them the film in reverse, and you tell them which is the good one, they cannot distinguish. Because both the equations of motion, the dynamics, is completely time-reversible. You can take it back, turn around the velocities, and everything will redo exactly what is in the reverse. Now this is the beginning of the story. Then you will see that there are other systems that have this time reversal that are a little bit more complicated. How are they? You have a system that is coupled to a much larger system, and they all together satisfy these equations. But now you do something, this is what we're going to do in the second lecture. You take averages over the big part, and this big part is what you call a thermal bath. If your thermal bath is well constructed, with a good energy and everything, and has all the good properties, your particle now is not purely following these equations because it has all the interactions with the bath. If the bath is good, you still have a time reversal property that is a little bit more complicated. You don't have to simply turn the time around, but you will see that it is a time reversal property. For those of you who have done programs in detailed balance, which maybe if you did Monte Carlo ever, this is what it's called, is a time reversal property. This means that the bath allows you also to do something like exploring democratically things. It's not only pure Hamilton, it's your system in contact with a good equilibrium thermal bath. But if you have bacteria, even if they are in contact with a thermal bath, this is not good enough. One thing that we will do in a while is the following. The next step that the books, so the books of Statmec do not make this transition usually, the old books at least, don't make this transition very explicit. But now we make them explicit because we have granular matter, grains of sand, or we have active matter, like for example bacteria, and we know that we have this but we don't have this. It's very explicit. So now the books typically have another property that they tell you is that if I measure this in the micro canonical ensemble, it is a property of large n that this gives you the same when n goes to infinity as the canonical ensemble, which is better to use, but instead of keeping things on the surface of constant energy, you do an average which is, so the canonical measure, one way to write it is to say that you are going to integrate A over all the coordinates of, I write it as a delta function of the energy minus a certain value. This means integrate over the energy shell. This is just symbolic and you normalize it. So this means it's an instruction, the delta is an instruction to only count the points that are on the surface energy and then integrate over all the points and divide by the normalization. The next step in every book in statistical physics is to show that this, which is a nasty thing to do because keeping on a surface is not nice and so on, can be replaced by a nicer, we will do it in detail, by a nicer calculation which involves this thing which is the canonical measure, where beta is one over the temperature by definition. Who of you have seen this, okay, it's a moderate amount, okay. This is the next step and again, the fact that there is no time here, so this means that you already made this passage that you are going to calculate things in this average. Now is this enough, okay, I gave you already, oh no, before going on, okay. So I told you that we will always consider large number of particles. When you consider a small number of particles really isolated from the world, the discipline is called dynamical systems or nonlinear science or dynamical systems. There is a whole branch of physics that studies the behavior of few particles that are interacting. It's a very mathematical thing because you have to really study the properties of these differential equations that are really tough, but we are not going to do that. In our case, what we're going to do is large n, with one exception that will be what most of our energy will go into, that we will also study the following. Imagine I have a system in some potential, oops, sorry, this is x, my particle is moving in this potential, but this would be a problem of dynamical systems, but we are going to couple it to a thermal bath. A thermal bath is a very big system having an energy that is going to couple to that system exactly like a particle that is here, this chalk is interacting with all the air in the room. By definition, a bath is nothing special, just a system, usually that is itself in equilibrium, but the peculiarity that makes it a bath is the fact of being enormous. Bath means big system. What is the easiest bath you can conceive? What is the easiest system one can conceive? Well, a lot of oscillators, a lot of harmonic oscillators. These we're going to couple with my system. So if I call the coordinates of all these oscillators, and this one is x, what I want to do is couple all these coordinates with these ones, but I'm going to try to eliminate all these and remain with an effective equation that tells me how x moves in contact with a bath, but off the bath I don't want any detail. I just want to know how it affects this thing. So x is not going to satisfy Newton's equation because we have to, it does satisfy Newton's equation, but they are modified by the fact that they're interacting with all these guys. So in this case, and it's an exception, we will study systems with not many degrees of freedom sometimes, but it's only apparent the fact that they don't have many degrees of freedom because they are implicitly interacting with a thermal bath. This thermal bath will do to this system two things. Just think of a particle that is sitting in this room or in water, for example. What does the water as a bath do to the particle? Two things. One thing is that if the particle wants to move, it opposes friction. The other thing is that if you're looking with a microscope at the particle, you will see that it's being crashed all the time by molecules and it will do Brownian motion. It will sort of diffuse. These two things, Brownian motion and diffusion, are the way that the bath speaks to you. We will discover these here in an exercise we will do where we will couple one particle with a lot of oscillators, integrate them away, and we will see that indeed they produce on this particle friction and noise, and we will have a particle apparently alone in the world, but now it doesn't satisfy Newton's equation only. It satisfies Newton's equation plus noise plus friction. Even when we study a few degrees of freedom, there are never a few, because the bath is there and for us it's going to be important. I just want to know what's better than the pen of K. Oh, sorry, sorry. Good. You want a Boltzmann constant here. It's one in this course. Why? Because I chose the units to make it one. So I chose how I measure energies and things, but yes, okay? In what case we consider that this case equal to one? Well, wait, I didn't change the value. What I did is because Boltzmann constant has units of one over energy, I decide to measure the energy. My unit of energy is one Boltzmann constant, one over. So it's like saying that I will measure this thing in units of the length of this. So it's one by definition, but don't think that I'm changing Boltzmann constant. Yes, sorry, in thermodynamics books, of course, temperatures and energies are measured in Celsius and Joule, et cetera. So you have the Boltzmann constant in the middle, okay? Thank you, sir. Sorry, I haven't written a Boltzmann constant since I was 20. Yeah, please, sir, I have a question about the BAF. So I would like to know whether it can do the conversely effect. Instead of making the system in equilibrium, maybe more than make the system in dynamic. We will do a lot of dynamics. We will do a lot of dynamics. Most of what we're going to talk is going to be dynamics. You said that when the BAF speaks with dissipation and Brownian motion, so I would like to know whether if I can, you can use the BAF and instead of making your system in equilibrium, it doesn't make the system in that state. Ah, you want to know if with a BAF I can put the system out of equilibrium. Yes, we will play with that and let me tell you how. Imagine you have a system that has somehow two baths that are talking to this. Well, I will do it now. Okay, so can a bath keep your system out of equilibrium? Okay, let us do the following. This is a system that has been studied a lot. You have particles that are connected by nonlinear springs. So that the interaction, these are, let's say, n particles. This is called a chain of particles. The interaction is some function of the distance between the particles, as you would have with a spring. Let us suppose that it's a nonlinear function. Okay, so that if I pull from this particle and let it go, it will vibrate and transmit the vibration on words in a nonlinear way. I can do many things. One thing I can do is just give this system energy, disconnect it from the world, and let the system do what it wants to do. Okay, I can also connect this system somehow to a bath. We will see how, but let's say a bath is a lot of particles that are here connected to that end, which has a temperature T1, and leave the other end as it is. And then the system will become eventually an equilibrium system. We will see this what it means. But I can do what you were asking. I could connect here with a bath that has the same temperature and then no problem, the system. But what happens if I connected to a bath that has another temperature? You've seen this happening in your kitchen. You have the ends of something that have two different temperatures. Well, now there's going to be a flow of heat. Let's say T1 is larger than T2. So this answers your question. You're using two baths to ruin things. And so what happens here? Well, here you have an example that is very nice because... So how do I decide? So the coordinates of these particles is QI and PI are their momenta. And if you want to know how heat is transferred, it's very easy. You take a particle and the work this side does on the particle, remember it's forced time velocity, the power. And the work that this part is doing on the particle is also forced time velocity. And if there is an imbalance, then heat is flowing. So there is a heat current, which you can write this way. I will write it first and then explain the two terms. Okay, so this one is a force to the right. This one is a force to the left. This is the derivative of the potential. And this is the velocity. So when you write this, when you think it, what you are calculating is between the particle L and the particle L plus 1 and there's the L minus 1. What is the balance of heat? Okay, so this is just a quantity that you can solve. If you want the average and you suppose that you are in equilibrium, you do what I said before, you compute it. So statistical mechanics works okay. You can get rid of time. But imagine I want to calculate the thermal conductivity. How much this chain, when there is a difference of one degree between the ends, how much heat passes? Sorry, maybe this was a little too far. So heat, you compute heat as force times displacement. Times the velocity. This is the power, no? It's the power, yes. But say shouldn't this be the difference between these and the change in energy? So this will give you a change in energy in the inside. But if you want instantaneously, ah, sorry, the particle doesn't keep any energy. It receives and it gives. And if you want to calculate on average, how much is the balance between what it receives and it gives, it's not heating up the particle. Okay, okay. So essentially heat is equal to work. Heat is equal to work. But why am I giving you this example? Because this you cannot hear now, ah, understand the details. But take my word for it. If you want to calculate the thermal conductivity, the rule is the following. Just take my word for it. You compute this at every time, which is a quantity. And then what you have to do is integrate the current at time t, because this is a function of time, times the current at time t prime. And then this quantity integrated between t prime and t. And then you divide by the total time. And this average, believe me, is the conductivity. Why am I telling you this? Because we will probably get to the point where we will prove this formula. But there is one thing that is important and the only thing I want to transmit about this formula. If you want to compute the thermal conductivity of something, this is a quantity that contains two times. It's what we call a correlation. And there is no way that I can eliminate time here. So the message is, if I want to calculate a thermal conductivity, even if, sorry, and this, sorry, and this you'd calculate it with t1 equals to t2. So you could calculate it in equilibrium. This is important. So this, you get this proof, but why am I telling you this? Because the point now is, if you want to compute a thermal conductivity or a viscosity, there are quantities that even if you are in equilibrium, you cannot get them just eliminating time. With this, you do not escape calculating the dynamics because you need to multiply a thing at two different times. So this is a time correlation. So the point is that even if you are in equilibrium in a system that is perfectly, nicely, a system that has all the properties of equilibrium, there are quantities that the usual calculation, based on eliminating time, will not give you. And so even if your system is not for some reason like the bacteria propels, propelled, or explicitly non-equilibrium, a system like this room, which we can suppose is in equilibrium, if I want to calculate the thermal conductivity of the air, it's not just a calculation of statistical mechanics in the sense of averaging over the energy shell. Okay? Yes? Can we get this heat current even when we are in equilibrium? No, there will be no current when you can check it. This average gives you zero if the T1 is equal to T2. Yeah. Because when you average this quantity, you will see that because of equilibrium, I leave it as an exercise if not we do it together, this quantity gives you zero. Because when this, you see that it's a total derivative and then a total derivative over time gives you zero. So if the temperatures are the same, this quantity here, the current is going to give you zero, on average. But we calculated the conductivity when we are in equilibrium. That's a paradox, yes. This formula miraculously allows you, because you see this is a current at two different times. And so, yes, this formula miraculously allows you to calculate using the equilibrium values at two different times. This is a current that on average gives you zero. That's a very good question. This is a quantity that even in an equilibrium system, you're just looking at the fluctuation. So you have J of T, which on average is zero, as we said. But if you take it at two times and you multiply them, it gives you something. This is the calculation you're doing here. It's a time correlation. And there is a formula which we will probably derive that allows you to use this, which is an equilibrium calculation, transport coefficient, transport of energy, the conductivity. The point is that even if this formula is in equilibrium and the system is in equilibrium, StatMec doesn't tell you how to calculate it. If you want to be convinced about this, imagine that this chain is in water and it has a thermal bath interacting with it. And imagine the same chain in honey at the same temperature. Statistical mechanics tells you that if the temperature is the same, the statistics of the problem is the same, because only temperature matters. But of course the motion of the thing is very different in honey and the conductivity of heat is going to be very different. So the morals of this is just to say, there are two ways where we step out of equilibrium. I already told you one, which is when the particles are propelled so that no longer we can apply the idea that the surface of energy is explored democratically. Then there is the case where you're applying two different temperatures and you're passing heat. This is a situation that is not equilibrium and you cannot do with the techniques of equilibrium. So even for a system like an ordinary metal bar, if you want to compute its conductivity, pure thermodynamics without time does not do the job. And there is a third thing is that, well, if the actual calculation you have to do even for a system that is in equilibrium, it doesn't work. And on top of it, if you apply two very different temperatures, then the system is very far from equilibrium and then you have to completely forget about doing a thermodynamic calculation. So I just wanted to say that I'm working on that in my master project. So we are trying to replace the chain of particles by some quantum dots between two thermal bats. So my question is, what is the difference between using, in your example, quantum mechanics and using statistical mechanics? At this level, not so much. In the sense that when you do it in quantum mechanics, the formulas here are quantum, which is not the case here. But at the end of the day, the concepts in quantum and the ideas of abying equilibrium, am I not? Can I use, can I not? The thing is very similar. But of course the underlying equations are quantum. Thanks. I have a question about the heat capacity. So because if you write it this way, it seems like it depends on time, which shouldn't be the case, I guess, or... The current, yes, yes. The capacity, the kappa. Oh, good. Excellent. Yes, but the thing is that you will repeat the experiment many times, compute this quantity. So in fact, I should have written, I didn't want to get into the details, we will, but if you repeat this experiment many times and you compute this thing at these two times, so you're doing in fact average over many runs of the experiment, then it no longer depends on time. You just redo in your head, of course, the experiment many times, and at every times TNT prime, you take the histogram of all your experiments and you do the average. Or the other possibility is you integrate over a long time and then the fluctuations go away. So the two things are possible. Yes, there are quantities that depend on time, but they tend to an average that is unique. Okay, so may... Because thermal conductivity is an equilibrium property, right? So you measure it in linear response theory, essentially. Which is the name of this. Yeah, so what you are saying is that essentially even equilibrium quantities, they need time. There are some quantities. If you ask me, can I compute the energy, the magnetization, things that depend on only one time, then statistical mechanics teaches you how to do this without using time. But there are quantities that require two times, like all the transport coefficients, and then even if you are in equilibrium, you need to solve the dynamics of the problem. Or to be clear, you could have two systems that have exactly the same thermodynamics, but they have different conductivities. Because one happens in water, the other one happens in honey. And so from the point of view of thermodynamics, let's say honey and water are the same because their thermal capacity or whatever is the same. But their electrical properties are the same. But if you're going to study, for example, the viscosity, obviously honey is very different from water. And this is a quantity that even if they are both in equilibrium at the same temperature, you need to use two times. And for that reason, they do not belong to a calculation that can be done with fewer thermodynamics without time. What Jorge said regarding your question. As an experimentalist, I know that even though viscosity is defined, the number associated with viscosity is temperature dependent. So for every temperature, there is a number. But I cannot measure that viscosity unless I disturb the fluid. So there has to be a time involved in the measurement. Yes. In practice this arises this way. Okay, let me continue a little bit and give you another example where pure statistical mechanics doesn't work. By pure statistical mechanics, I mean you get rid of time. This is a bit more technical, but it's again the same thing. Imagine I have a system that satisfies Newton's or Hamilton's equations. But this is the force which usually I could write with a potential. And this one, let me make it more easy. This is a system whose energy is... Okay, this is just a system force equals past times acceleration. In, let's say, two dimensions. And now let me say I add a force here that depends on the cues. But this one is not the gradient of anybody. Imagine this force is not a gradient like this one. So it's non-conservative. Okay, there I already stepped a foot outside the properties that led to the democratic exploration of my energy shell. And now we will see an example that is perfectly clear and you will see how this works. So again, when your system is perturbed by forces, I gave you the example of propelled cases, but any system whose force is not conservative already you step away from this thing. I have a question that comes from before. When you say democratic expansion or whatever, is the same as ergoicity? Yeah, let's say yes. But I prefer not to use ergoicity because ergoicity is something that has a mathematical definition that if a mathematician is in the room, we have a problem. Here we understand what we are talking about. Okay, great, thank you. Okay, let me go a little bit more in detail and then I will finish with an example where you can understand perfectly well what's going on when you have a system like this that is being forced. The system is forced and it could also be, and we will see it now, kept on the energy by a thermostat, you will see how it works. So the system is on the energy shell, but it loses the democracy once it is forced. You will see, it's an example that can be worked out in all detail. So what happens in practice, how does thermodynamics work when such terms do not exist and I have a democratic thing? So what happens is you can think of it this way. This is the whole energy shell and you start, imagine that you consider a set of initial conditions. Dynamics is perfectly deterministic, but each initial condition will move its own way. So think of it, if you think of all the possible initial conditions as an ink blot. Now, when time evolves, different initial conditions move to different places and then it's as if they explore different places and you get something like this. Because of the properties of Hamilton's equations, of Newton's equations, actually you can show that the area of this blot doesn't grow. So it's just as an ink, perfectly non-mixing ink blot that is sort of exploring the energy surface, each one corresponds to a trajectory that started here and each one is a different trajectory that you're exploring. So we are trying to see the ensemble of possible trajectories of the system. If we go further on, the idea that what happens is that this is going to become something really very thin and stretched and it's like an ink blot that although it preserves its volume, is democratically spread, but the democracy is not complete but as time passes, it spreads more and more over the energy shell. So if I consider a little part of this, it will be here, but if I consider another little part of the initial conditions here, they will be here and so on and so forth. And now we look at the energy shell and this is E equals constant and for example, I want to know what is the magnetization. So of all this energy, the magnetization or any other quantity. So of all this energy shell, these are the points that have a certain magnetization. Let's say these are all the other points that correspond to a different magnetization. Okay, so as my thing explores my possibilities involve more and more sampling of this energy shell, I can have, depending on where I started in this little blob, I can have this value of magnetization or this one and so on. Okay, and then what happens and what saves your life because this would mean that you have a whole distribution of possibilities and then what saves your life is a property of large dimensions which is that almost every point has the same magnetization. We will see this. So there are two elements. First, I explore democratically the energy surface. For this, as I told you, not always happens and it doesn't happen with active matter. It doesn't happen with driven systems. It happens for systems that are like that one or in equilibrium with a thermal bath. When this exploration is done, then you are with, depending on very different, very similar initial conditions, you can be in many different places in the energy shell and then what we will do a simple exercise is to show that these are much more numerous, these points with zero magnetization, for example. So for almost every experiment, I am going to measure this quantity. So this is how thermodynamics kicks in in the cases where I can apply it. In other words, if I make a histogram of every initial condition and I look at the magnetization I have at this time, I will get a curve like this, probability of magnetization, depending on which of the initial points landed where but then what brings, if this sphere is two-dimensional, then you are dead, you have simply a distribution. When you have a system that has many degrees of freedom, we are going to do the exercise, you see that the results are concentrated because as we will see, a sphere in many dimensions, when I am standing in the pole, almost every point, all of the points are in the equator for me. So this is the property that makes that this quantity does not fluctuate and it tends to a value. It's the same argument I made at the beginning of the room that has air on one side and nothing on the other. Almost every possibility makes that the air is roughly divided equally on both sides. I remember there was a question. Yes. You asked him to ask the question. Ah, okay. Yes. The question, if I remember correctly, what would you expect of us? So what I want to transmit to you in this course and this is what you remind me, is there is thermodynamics and if you are going to do complex systems, first of all, most of you want to do some form of complex systems. Complex systems means that there are many actors. That's the definition of complex. Many things that are interacting with one another. The science of many things interacting with one another when there are many is thermodynamics. Now, what of thermodynamics can help you and what cannot help you? If you, and now we can say it on the basis of what I said, books of thermodynamics will tell you thermodynamics can help you when you have n large and the statistical properties of n large and the situations in which you do not ask the question of what happens with everything. So if you're going to do complex systems, for example, you could want to do economics where you have many agents that are buying or selling. In your theory, you don't want to know the dynamics of one specific agent that will go bankrupt or become rich. You want, presumably, to study the dynamics of the whole ensemble. So can I apply thermodynamics? Well, more or less you can, you have certain ideas, but then equilibrium thermodynamics understood as calculating partition functions, free energies, et cetera, has the limitations that for equilibrium you need the idea that something is being explored democratically and then thanks to that you get rid of time and only calculate the average over these things. If not, what do you do? Well, that's going to be the next question. What do I do when I don't have that? Well, there are things I lose and what is it technically that I lose? You will see that technically what I lose is a form of time reversal. So the equations that are satisfied by bacteria that interact or by agents that buy and sell stocks to not satisfy these things. So you cannot do a thermodynamics of that. What can you do? Well, you can still perhaps do a thermal bath, but as you asked me a while ago, can a thermal bath be bad for you? Yes, the thermal bath of economics is bad for you. You're not going to be in any form of equilibrium interacting with the stock exchange, which is your thermal bath. So what do you do in those cases? First of all, what question should I ask? This is a very important point, because sometimes out of equilibrium, we don't know exactly what is the good question to ask. And I will try to convince you that there are bad questions to ask. And some of the bad questions are bad because they come from our prejudice that equilibrium systems are one way and not the other. So if I have a dynamic system, so we will retain complexity, but we will lose in many cases a lot of properties that are specific to equilibrium. But for this to make sense, you have to understand what is it that makes equilibrium so special? So special means, well, we will see this later, but for example, it means that I cannot produce work just by taking it away from the water in the sea because this violates the second principle. This is an equilibrium thing, and you don't have that if the sea would be made of bacteria, I would be able, because they would be propelling, I would be able to extract energy from them. Why is this? How do I see this mathematically? Okay, these are the kind of things we will say. So for me it was important, sorry not to have spontaneously asked with the delay, for me it is important to state the problem first, okay? This is why I answer you now. So in a word, what I wanted to do is to present this, and then we will see what is the role of equilibrium playing, what are the limitations, what do we do when we don't have it, what can we expect to do, and especially the most important thing is what kind of questions should I try to ask? Okay. Can I ask another question? So now you wrote the equation of motion with this force, which is non-conservative, but you crossed it out and you showed what is the picture. Now what happens when you have... This is what we're going to do now. Maybe at the end, I'm not going to do it today, but okay, if you... Yes? Go, go, go. Okay. So here usually we expect the expectation value of some observable to be equal to the point where the trajectory we see the most, is that right? But can we have a system where there's like some certain point that contributes so much like proportional to the system size, and so that... Yes. Yeah, like last duration. Yes. This we're going to do in all detail. We are going to see, and this answers Matteo's question. If you believe me here, now his question is, this is like you're sending 10 people here to all provinces of Italy and ask them how they're going to vote, and then you take the maximum vote or you send a thousand people, you have a sampling of the voting and then you publish it this party. Now what he's asking is, what happens if for some reason all these thousand guys decided to go to Reggio Emilia and ask there? Well, the thing is going to be very biased and you're not going to sample the whole thing. Can a dynamic system do this? Yes. The answer is whenever your system is not in equilibrium, what happens, and this is Matteo's question, is you do the same experiment, but these are now, I don't know, propelled bacteria or a system with two thermal baths that do not have the same temperature. This starts spreading, but the spreading is no longer democratic. So it can be very chaotic, but it will be... We're going to do an example, but it may be, I don't know, that it ends up exploring this kind of sub-ensemble. Energy is fixed, and we will do an example. Energy is fixed, so you are on the energy shell, but you do not have the beautiful, marvelous and unusual properties of these equations, and you end up by visiting a region that is something. And what is it? Well, the short answer is, in a problem that is non-equilibrium, we don't know. There is no method that tells you where it goes, and then when you measure the magnetization, doing this democratically doesn't mean anything, because you are not visiting it democratically. So who wins this championship in principle is not a simple statistics on the sphere because you are not studying the sphere. It's like the example I was giving of everybody going to Reggio Emilia. So this is the Reggio Emilia for you, and this is what a system does. So the point is that it's a miracle that the laws of that Mr. Newton discovered had this property in the first place, and I don't know why it is, but just to give you an example, if you have these equations, and here you multiply by 0.9, you broke the beautiful properties of Hamilton's equations, and you already do not have the democratic exploration, and all your thermodynamics crumbles. It falls to pieces. You cannot do anything. You cannot have a second principle. It's a mess. But you can still do dynamics, and you can still do complex dynamics because you still have the aspect, perhaps, of having many actors. So this is the situation that I insist you will face all the time when you're trying to do statistical mechanics of complex systems. You're doing statistical mechanics, but you're losing the aspect of getting rid of time. But instead there are some legion that is really visited rarely but contributes so much. Okay, okay, okay. At short times, you can have this. For example, if you... the glass in your window, there is this energy surface. Sorry, we are in equilibrium. We are trying to get to equilibrium, so we are on our way, okay? Blue, blue, blue, blue. But there is a region here in the phase space which is crystal. A crystal of... what is it? Dioxide of silicon. Silicon dioxide, no? But our system has been prepared there when this was a molten thing, and then it was cooled, and it started moving, and the one that occupies a lot is the crystal, but you don't see it, so you move, and you move, and you move, but because of how the dynamics is made, it takes you very long to get to visit this place. So it will eventually, in, I don't know, a few billion years, crystallize, meaning that finally this trajectory will find a beautiful promised land, but at the times we have hasn't yet got there. So these things happen, but of course it doesn't happen with a glass of water. It happens with things like glasses that are very nasty. Take very long to equilibrate, okay? So next time we are going to do the following exercise, which I think is very nice. Usually to prove these things is very hard, and it's the subject of ergodic theory, to prove that a trajectory is going to really go democratically and so on. One system for which this has been done by, I think, Sinai is a billiard that is constructed like this. So you do not enter here, or here, or here, or here. We will consider it with periodic boundary conditions. There is a part here which you do not enter either. So your trajectory does like this, and then it, well, here I've been a bit more symmetrical, it re-enters through the other side, et cetera, et cetera. Okay? And for this system you can prove, you let it go, let it go, and mathematicians have been able to prove that you will see this space, if you let it happen for a long while, you will visit them completely democratically. Imagine how hard it is to prove something like that. You really have to say that this trajectory, that is a complete mess, will not only fill in this completely, but if you look at the velocity vector, which is constant, but its direction in the PX, PY plane, also is democratic. So for this system you can prove perfectly well this democratic mess. Okay? And then we will do the following exercise. We will apply an electric field in this direction that pushes the particle, and we will apply a thermostat that keeps the velocity at the same value. So we will do it so that the momentum is constant, so that the energy is constant. So what happens is that trajectories, once we do this, a trajectory that started like this, will try to go in this direction, but keeping its modulus of velocity constant. Okay? And then if it crashes against this, it will rebound, and then again try to do the same and so on. This is a model of conduction. It's not so bad. You're modeling conduction of electricity, for example. You're pulling from particles that... Once you have this, you lost the beautiful property of ergodicity. And we will see very clearly, without solving any equation, how the democraticness of exploring everything is immediately lost. And no longer, as soon as I apply this, I'm kicking the system out of equilibrium because I'm applying something that is a force that does not derive from a potential because it's pulling you all the time like this, and you will see that immediately this beautiful property is lost and you will see how it is lost. It will be clear. And this is what happens as soon as you break the nice out-of-equilibrium properties. By the way, the time reversal here is broken because if you pass the film the other way around, you will see the particles moving in the other direction. So in this example, I think it's a very nice example because it's one of the very few where you understand perfectly well what does it mean to be able to equilibrate. Okay, but we will do this next time. Let's... So, questions? So, actually, I forgot to say one thing. I mean, each lecture is one hour and a half. You decide what you want to... When to finish. I mean, you decide what you want to... Break in between if you want to finish. The session is over when the therapist decides. Okay. Go on. When you see that your audience is inert. Yes, I think the audience is inert. So, thank you very much and we'll go on tomorrow. Okay. So essentially, there is nothing else this today apart from half a six. There will be the get-together. Okay.