 We have a seminar in QLS, Continuative Life Sciences. It will be in former Fermi building. It's on stochastic thermodynamics of quantum systems. So it's sort of related to the course of David Wolpert. So you are welcome to join if you want. It's a half past three. Half past three, you should go to the Leonardo building and then to the Ex-Sissa building, second floor, and it's in the open space common area. Okay. Are we all set? Yes, it's okay. Okay. So yesterday we have seen some question of principle that are important for the thermodynamics and the statistical mechanics of systems with long range interactions. We have talked about additivity, extensivity, and we have started to see the consequences of this. So my objective yesterday was cover 26 slides and I covered 12. So I was going very, very slow. So today I will try to get more into the game. And last night, when I was preparing the solution of the exercises that I gave you, I found this sentence in the book of one of the most prestigious book in Statmec, the Wang, that I recommend that you should read, although it is sometimes some of his remarks can be doubted. And in fact, sorry. What's the book? Kerson Wang, introduction to statistical mechanics. So I found the sentence in the chapter on the perfect gas that I was consulting to see what he had in the book. There seems little hope that we can straightforwardly carry out the recipe of the micro-canonical ensemble for any system but the ideal gas. So I will try to convince you today that this is not true, so that you can do it. And specifically for the Blumenhafel model, you will see that it's possible to solve the model and many other models in the micro-canonical ensemble. And why I insist on the fact that I would like to solve them in the micro-canonical ensemble, because you will understand today there is a property of the Legend- Fenchel transformer that is important for non-concave and non-convex functions. The Legend-Fenchel transport does not have the dual property. So if you Legend-transform twice and the Legend-transform as a singular point, twice the Legend-transform is not the function itself. And this is a very important property of Legend-Fenchel transforms and it's important also for stat mech. And so today the lecture will be essentially devoted to that. So since the Legend-Fenchel transform does not have this property, then you cannot get the micro-canonical entropy from the free energy if the free energy has a singular point. So if the free energy, in fact you can say that the singular point must be a point of discontinuity in the first derivative. If it is a point where the convexity is zero, you can still go on. But if it is a discontinuity in the derivative, then you are in trouble. Okay, so I learned all these from a good friend of mine, Ugo Tushet. In fact, I was the referee of his PhD thesis in McGill University long ago and which constitutes also the subject of a beautiful review paper that he wrote in Physics Reports. And I recommend if you want to go deeper into this problem that you read his Physics Reports on large deviations where you will find some of these statements. Okay, so today I will try to contradict Kersen-Wank, which is not an easy task because he is a very tough guy. Okay, so first of all I had a slide and it's important that I give a few details on this slide because Nicolò will cover this issue more deeply. So this is a plot where I put on the x-axis the dimension of the system and on the y-axis the variable sigma. I remember to you that sigma is, I never remember if it's alpha minus d or I think it's alpha minus d. Yes, alpha is the decay of the long range interaction. d is the dimension and this is the definition of sigma, which is more used in the field theory literature. Okay, so I will be very brief but just to give you a scheme that might be useful for you to remember. So of course if sigma is negative, that means that if d is larger than alpha you are in the so-called non-additive region, which is the region, the sector here. And I don't consider exponents that are positive so I limit myself to this non-additive region. And then, so there is no proof but in this non-additive region if you go to a critical point, the critical point will be of the mean field type. So you don't expect to have critical exponents that are different from the mean field. There are proofs. There's a series of paper by Ithetoshi Nishimori and also some paper by Takeshi Mori and others that are more formal, but okay. So this is literature that I will not cover. It's an interesting literature that I will not cover. So if sigma is positive and you are in this sector and you are below d over 2, there is a very easy argument that is based on fluctuations that you find in sort of Landau-Ginsburg argument that you find, for instance, in a paper that we wrote together with David Mukamell that says that the exponents are mean field. So you can extend from the non-additive region because in this region, as I showed yesterday, the energy is additive. The energy is non-additive only if you are in the sigma-negative sector. And then there is a very interesting region here. You see in this sort of sector where there are exponents that are non-trivial. So there's a critical point here which is non-trivial. And there are a lot of studies of these including using field theory, using numerics and anything that you want to try. But there are no rigorous results apart from a region where you can use a renormalization group which is near four-dimension here. So there have been studies of this line near four-dimensions and there is a point here which is quite well known in two dimensions and in one dimension very likely it is one. The region is a limit of this region and then in this region you have the usual short-range exponents of the renormalization group in short-range interactions. So it's a very general type of... So it's a sort of a scheme. I'm collecting a lot of information on this. For instance, you might think what about the cost of the Stowler's type of phase transition. For instance, this point here is cost of the Stowler's so it's not really icing. And also there is randomness so I didn't put randomness on that. And if you put randomness then the lines are slightly different. So no disorder, no disorder, no order, land-out type and with some type phase transitions including some aspects of x, y, or... Equilibrium? Equilibrium? If you want, yes, it's sort of generic and I don't have an exact line. This point is for icing and they are extensive and it is related to the anomalous dimension of the icing model in two dimensions. So there are field theory arguments. I would essentially say the universality class of phi-4 theory plus maybe something of x, y. So and of course you can super-post to this quantum and all that. So this is just to give you the feeling of the... There are exactly solved models for the exponents for instance the spherical model that gives exponents in this region. So essentially you have non-additive, mean-filled exponents, non-trivial long-range exponents and short-range exponents. This is more or less the picture of these long-range... Okay, and you will see in the talk by... In the talks by Nicolò he will come back in the quantum domain and will show some calculations of these exponents. In particular, together with Nicolò who was the subject of his PhD thesis we have tried to understand this line in terms of what we call an effective dimension. In fact, this idea of the effective dimension goes back to Parisi and collaborators and we have tried to give precise estimates of this effective dimension that is sort of calculating the line that separates long-range exponents from short-range exponents. This has to do with criticality not with phase diagrams and so the lectures will be mostly on criticality. Yes? I mean the usual one of the Ising model in two dimensions. Yes? The lines separate four regions. This region, this region, this region and this region, okay? And it's just to give a picture, a global picture of what you can expect in terms of critical points in long-range. Critical points in long-range for values of the exponents. Values of the exponents. Nu, beta, the usual exponents of the critical points. So you will find different values here but if you go here you will find mean-field exponent. So this line is very likely d over 2. This line is obtained by several different methods and this transition is the additive-non-additive that is alpha equal d and here you will find the mean-field exponents. So I'm sorry, it's very imprecise. So it's just to give you a feeling of what's going on in this field in terms of critical exponents. So not saying anything about phase diagrams and all that which is much more complicated. Okay, so this announcement that the three lectures of next week will be for the quantum system will be devoted in understanding some aspects of this picture. Okay, so let's go to now to the Legendre-Fentzel transform. So I don't know, maybe I... Do you know everything about the Legendre transforms? So, yes? Okay, so I would like to help you with the graphical representation which maybe is useful. So let's take an entropy as a function of epsilon and let's suppose I am in the growing part of the entropy with respect to energy and so for a given energy I have a given entropy, okay? So and you know that the derivative of entropy with respect to energy in fact here is not a partial derivative it's just a total derivative because S depends, is better, is the inverse temperature, okay? Sometimes I will use the Boltzmann constant equal one, sometimes the Boltzmann constant will be explicitly indicated in the fence. Okay. Two days ago was the anniversary of Boltzmann so I will not put Boltzmann on the blackboard. And then I use what is called the Massier potential which is better times the free energy and this is... Phi is better, better f of beta better f of beta, okay? So the slope of the... the slope of the... I will draw it differently. The slope of the... is the temperature, okay? The slope is the temperature and if you write down... essentially what happens graphically is the following maybe you know but okay. This is the entropy, okay? And if I consider the intercept here, okay? To the y-axis and I go down this is better f, okay? This is better f and since the slope is better and epsilon is this axis so this is better epsilon, okay? This is useful to remember and I have drawn an entropy which is convex, okay? I have drawn an entropy which is convex. There are problems if the entropy is... Okay, so how do you get all these? It's on this slide. Okay, and sometimes you find two different definitions of the... the general transform. In one definition... in one definition this is positive... this is measured top down in other definition it is measured down to top and there is no difference if you stay with concave or convex functions and of course if you go from concave to convex you can just change the sign of the function you can change the sign of the legendre so there is a problem of sign in all of the legendre transform issue and you will be always in trouble to understand what is the correct sign. This is what I tell you. For instance, in large deviations the large deviation function is minus the entropy but there is no problem because you can formulate the legendre transform in the right way and you adjust this sign. All this comes geometrically from the fact that either you measure from zero to infinity like this or you take this as a negative value. So it depends... So essentially you see here what the legendre transform tells you is this, the better epsilon is s epsilon plus beta f of beta and of course you have to interpret this because this is better than epsilon and of course this is a meaning only if you represent epsilon as a function of beta or beta as a function of epsilon this is something that is well known but I want just to stigmatize this fact. What happens? So in this slide I simply say this and I add some more graphical information on the blackboard and if the function is concave so Hugo told me that a way to remember this is concave because it's a cave. It's a joke. I don't know if you want to laugh at that. This is concave. Convex is the opposite. In fact mathematicians sometimes say concave down convex up in order to remind... Okay usually the entropy is concave and you obtain the massier potential so beta times the free energy here I have added also the number of particles per volume because you can also think of... as Matteo was remarking yesterday of defining the Legendre transform on the volume which gives the pressure. So here it is defined on the energy and it is essentially the application of the Laplace method. You start with the exponential of minus beta n times f of beta n which is the partition sum at fixed n. Okay this is a Gibbs factor that you... I don't know if you had a course in Statmec about the Gibbs... and I don't have time to enter in this but you should look at the Gibbs contradiction. Sometimes he's told that it comes with quantum mechanics but it's not true. It's present also in classical mechanics. It has to do with the mixing entropy of identical particles. So you should really have this term. There is a beautiful discussion of this in a paper by James that I found very interesting. Okay and then you express the delta function in terms of Laplace representation and in the large n you get this. So in the large n you see that it has to be the minimum. The exponent here has to be the minimum of the... So this sometimes you find it in a couple of pages in books but it's condensated in these three steps. It's very simple. It's the Laplace method to show that in the large n the variational problem to the right gives the massier potential, the free energy and this variational problem gives the entropy. There is a problem if... So what happens at a phase transition? At a phase transition there is a line here and there is a line with constant slope. So you are in a limiting situation where the entropy is limit convex. So there is a region where the... And if you legend transform this, if you legend transform this, you get a function which is massier potential in shape which has a singular point, a point where the two derivatives are not the same and the two derivatives are indeed... You see the derivatives of the massier potential are energies. The derivative of the massier potential are energies. So there is a derivative here which is an energy and the derivative here which is an energy. And these are these two energies, epsilon 1 and epsilon 2. So these two slopes are connected with the energies of the linear region. Okay? And so you get all the information as is the scope of the legend transform. The information that is contained in the entropy is the same information that is contained in the free energy and if you repeat the legend potential on that, you get that. So you can prove that it's a reflective invariant. So if I repeat the legend potential, I get this entropy. It's okay. Have you ever seen a phase transition like this? Raise hand who saw a phase transition in this way. Okay, one, only one. And for the others, how did you see phase transitions? So never like this, never like this. Okay, so you will have to think a little bit and digest this as a description of a phase transition. And in fact, it's a description of a first order phase transition. The one that of course more often in nature. Because if you want to go to a second order phase transition and there are more complicated critical points, you have to adjust the parameters in order to reach the critical point. And you can do it. You can do it by adjusting a parameter. You can reduce this. And when this flat part reduces to a point, which is a singular limit, then you are at the second order phase transition. Okay, and again, there is no problem. Although it's a singular point in entropy, you can get it from the free energy. Okay, simply the two slopes will be the same. Okay, the two slopes in the free energy will be the same. It means that there is a discontinuity in the curvature of the free energy. Okay? So according to Ehrenfest classification of phase transition is a discontinuity in the second derivative of the free energy. So it is a second order phase transition, which in entropy is the reduction to a point of this flat part. Okay. And now I go into the domain of... This I took from... Okay, so let's first go here and then go to the mathematical description. So for long-range interactions, you have a new phenomenon that appears that you can have nonconcave entropies. And there is nothing that contradicts the principles of thermodynamics that avoids entropy from being nonconcave. So it can happen, and I will show in the example of the Blume-Capelle that entropy can be nonconcave. So I will draw the simplest situation, which in fact it is very hard to obtain in spin models, where the entropy is a continuous differentiable functions and it has a region of convexity here. Okay? And if you try to draw the better energy relation in this case, okay, you will find a relation where there is not an invertible relation between energy and temperature and inverse temperature. In particular, you see here at the given temperature, you can have several different energies. The slope, if you think of the definition of transform is not directly related to the... So there are regions in different energy... different energy regions with same slope. You can do it by looking at this function here, of course. For instance, this is an example. Okay, this is a point that has the same slope. In fact, in the theory of nonconcave functions, the Jean-Fanche transform for nonconcave function, this line as the name is called the supporting line. And this line that you adjust on the curve is such a way that the curve is below this line. And of course, if you replace the function by the function itself plus the supporting line, you get exactly the situation of the first order phase that I was describing before. So... And if you do this in the corresponding beta-epsilon plane, you do the Maxwell construction. You can prove it. So it's on the slide. I will not prove it. I leave it to you. So I leave it to you to prove that the equal area condition of Maxwell can be expressed in this way in energy. Okay, the integral between epsilon 1 and epsilon 2. Epsilon 1 and epsilon 2 are the two points where the supporting line meets the entropy function, which is wider than the region of convexity. Because you see that the convexity region is here. It's between epsilon... I don't read it. Epsilon A may be an epsilon B. And in this region, since they can write down... It's not written in terms of beta. I'm sorry. This is a mistake. I will correct. So T squared is 1 over beta squared. Okay? So I should have replaced this by 1 over beta squared. And you see that since this is positive, if the second derivative is positive, you get the negative specific heat. Okay? This relation can be true only if the specific heat is negative. So the region of convex entropy is a region where the specific heat is negative. It's a region where the specific heat is negative. So which is unstable in the... I will get to this later several times. It's unstable in the canonical ensemble. So it's not stable in the canonical ensemble. If you prepare states in this region in the canonical ensemble, they will be unstable. The region where the entropy is still concave here and here. Okay? You see there is an inflection point here. The region where still the entropies concave are metastable regions in the canonical ensemble. They are stable in the micro canonical ensemble. They are metastable in the canonical ensemble. And the convex region is instead unstable in the canonical ensemble. This is an example. You can construct several of these Legend-Frenchel duals. For instance, you can do it for pressure and volume and others. And you get a very, very similar... Very similar. Okay. I leave you to prove that if you use the relation ds, depsin, or equal beta, you also get the Gibbs rule. So the fact that the free energies are the same at epsilon one and epsilon two. This is a generalized free energy where you don't perform the maximum over epsilon. You keep epsilon. So it's a representation of free energy that uses the Legend-Frenchel transform. So it's a function of both the energy and the temperature. You have not yet maximized. And if you use the Gibbs rule, you get the Maxwell construction. Yes. Yes. Yes. Yes. Yes. Exactly. Yes. Sure. Sure. Or if you want to think in terms of entropy, there is always an entropy access. So if I mix again the construction of yesterday, I take a system with energy epsilon one and a system with energy epsilon two and I mix them with lambda, okay? Lambda epsilon one plus one minus lambda epsilon two. So this system will have an entropy corresponding to the bounding line, to the supporting line. And these are the higher entropy than the system where the state is mixed. So where there are, what is the alternative? The alternative is that you have a mixed system, no? And you have the two phases, epsilon one and epsilon two, and they are mixed, okay? And if they don't unmix and they go here, and this is possible if there are long range interactions, then you stay on the line of below the supporting line. This is essentially the idea. Now I have to show you that this really happens and when I got these ideas, I didn't have a model. So the point. Okay, so then came this model and this is a very essential contribution of David Moucamel. After living in my office for six months, he came out with this proposal. So why don't we try to solve this model? So he had worked on this model in the 70s. Yes, it's the convex part, yes. It's a non-concave part. Sure, perfect. So you are the only one who understood phase transition in this way, so you follow. The others are lost. I suppose, okay. Don't worry. You will understand with the model what goes. Yes? Because I can reach this region also in the canonical ensemble if I do it very cautiously. For instance, this is a super saturated region of a fluid or this is why I call it metastable. The one that I have raised. No, that's the one in the slide. Ah, the one in the slide. No, no, no. Mean field does not work for always for wiggling long range. It works only if you are below the D-halve line and to prove it, I would need half an hour, but it will be proven by Niccolò. In a different way from the proof based on the Ginsburg criterion, you will see a proof of that. There is a line D-halve below which the exponents are mean field. It will be done in the first or second lecture next week. I can give you a reference if you want to have a look at why. I can give you the reference of the Lezouche lectures of David Moucamel that proves this. So the question is, do you mean that these regions are actually accessible physically while the region between epsilon 1 and epsilon 2? No, epsilon A and epsilon B are not. In the microcanonical ensemble, you can access all the regions between epsilon 1 and epsilon 2 and it is stable in the thermodynamic sense. So you will be able to reach the region of negative specific heat in physical systems with long range interactions. So this happens in self-gravitating Newtonian systems. It happens in charged systems in the Euler equation in the Onsager-Vortex model. So there are several models in which this can happen. Okay, thanks. Models of realistic natural systems. Okay, but of course I have not replied to the questions. It's a very complicated question and it needs a deep analysis. Okay, so the Blume-Capelle model. So the Blume-Capelle model is a spin-1 model. So SI can take three values and this is the way out from the Curivice. Instead of working with spin-1 half, we work with spin-1 because we have seen that for Curivice there is no problem. And so, okay, so I don't want to enter in the description of this model. It's very important for the theory of Ilium-3, Ilium-4 mixtures. It was introduced for that purpose. And in fact, the reason why there are three states is just to distinguish the two fluids. And in this, let us look at it simply as a spin-1 model. So here is the usual Curivice term. Okay, you see minus j over 2n sum over ij, all pairs of spin are coupled. And then there is a local term. It's very important that you don't keep only the Curivice term. You have to add some coupling, some coupling. And this, otherwise, you get an energy which is proportional to m squared. So, and the model is not rich enough and you don't get what you want. So this is a parameter delta that measures how much of the fluid one you have. Of course, if Si is either plus or minus, it gives you a contribution delta. And that's the one fluid. And then there is a zero fluid which is the one which is the paramagnetic fluid is as a spin zero. Okay. So, the model has been solved in the canonical ensemble by Blum and Kapell in the 70s. And there is a more general model. It's called the Blum-Emery Griffiths model which has additional couplings and they also can be solved in the canonical ensemble. The solution is very simple. So, I will not do it in the lecture, but I propose to you to solve this model as an exercise in the canonical ensemble. And what you have to do, I will, if you want, I will give you the definitions, but okay, let's, this is rather complete. It's a sort of landau type exercise, so it's not simple. Okay. You write this term as a square. Okay. You can write it as a sum of Si square. Okay. Then you apply the inverse Gaussian transformation. Sometimes it's called the Haber-Stratonovich transformation. Okay. So, and for that, you need an additional field, one additional field, which is the term in front of X sum of Si. And then you get an Haber-Stratonovich Hamiltonian which has the form X sum of Si. And then you have a sum of Si square, delta. Okay. I don't put the sign here. It could be plus or minus. I don't remember. So you have something like this. Okay. And then you have to integrate over X. And then you have to integrate, you have to trace over sigma. Okay. You have to do this. But since this is local. Okay. It's a sum. So this is a product over I of e to the minus beta delta Si square. Plus or minus. It depends on the sign of the Si. Okay. Beta Si. X Si. Okay. And this, of course, it's local. So you can trace. You can trace here simply on Si. And you get an expression in terms of beta, delta, and X. Okay. And then you do subtle point on X. You are lost. You do subtle point on X. And you get the free energy. Okay. So if you want, okay, and we can maybe I can devote the tutorial. I don't know 10 minutes to go deeper into this solution. The model is not solvable in the infinite dimension. So there is no solution of the model in dimension two, three, and so on. So it was solved in this way or slightly different by by Griffiths. Okay. What happens for the free energy of this model? So for those who are familiar with this method, it's a very straightforward application streaming straightforward application of the method. So, so what, what do you have? You have a, let's go to the face to the face diagram and you will understand. Okay. So first of all, the model, there are two parameters. Okay. Delta and J. And then there is temperature. Okay. Then there is temperature. So I fix. I fix J to one. There are three parameters. Whether you divide by J here, alternative, you can fix J to one. So the unit of energies is J equal one. So and then you are in the plane. Delta T. Delta T here is not clear, but it's just T. Okay. And there is a line of face transitions in this plane. There are extensions in other parts. So I will consider only delta positive and of course, T positive, but you can extend to the, to delta negative. And there is a line of second order face transition here. A line of first order face transition here which ends up with a steep slope. And there is a tri-critical point. You will understand more about the tri-critical point. I will stop now in two, three minutes. And in this region, both the magnetization and the parameter Q, I go back to, there are two order parameters. The reason it comes from the two fluids. Okay. One is the magnetization, which is the sum of the plus spins, minus the minus spins divided by the total number of spins. And there is the quadrupole moment, which is the sum of the two. There are two order parameters because there are three species. And both in the region of, in the ferromagnetic region, they are both positive. So here, both the magnetization and the quadrupole moment are positive. And they both have a jump at the first order face transition line. And they are continuous at the second order face transition line. And here, there is this mysterious tri-critical point. And I will tell you more about this tri-critical point in the second part of the lecture. And now we pause for five minutes. Okay. Thank you very much.