 OK. OK. OK. So let's perform the exercise for the three-stage POTS model. My friend, Angelo Urpiani, since he will give talks on a couple of weeks when we were younger together, we were saying that the POTS model is the model who saves the poor theoretical physicist who cannot compute complicated things. Let's try with the POTS model, which is... OK. But apart from jokes, it's a very interesting model. And I take it in the case of all-to-all interaction. So there's a Kronecker delta over the spins and the values of the spins are three. And in the POTS model I'm free to call the three values one, two and three. I'm not obliged to call them minus one, zero and one. So I can call them one, two and three. And then there is a Katsi scaling minus j over 2n. So this is the Hamiltonian. Step one, write the Hamiltonian in terms of the global variables. That's very simple. It's minus j over n over 2 times the number of spins a plus square plus the number of spins b square plus the number of c square. And there is no rest. So this is a reply to the question that was in the chat. So in this case, which is an exact mean field model, the rest is zero. In other cases where the rest is non-zero, for instance, we have in one of our paper we studied the Dyson model. And in that case the rest is sub-leading with respect to the main part of the Hamiltonian. So it's enough that the rest is sub-leading with respect to n. And then you can apply the method. In this case the rest is zero. And what is the random variable? The random variable is a sample mean given the configuration. So given the configuration of the spins, the number of spins a, b and c fluctuates. So the random variable is a three-dimensional vector where I count the number of spins a, the number of spins b and the number of spins c. So in the local random variable is the delta on site k. On site k the spin, yes? Why is an uppercase a denominator and non-denominator in the first step? In this one? The energy is linear in... This is a fraction of... Oh, it's a fraction. Sorry. This is the fraction of spin a, the fraction of spin b, and this is linear in n. Otherwise if I don't use the fraction there should be an n. So this n here fixes the n there. If I divide by n and that's a fraction, sorry, I was not unclear about that. OK, so then what I do, I do cramere and I compute the generating function. The generating function is the average over the spin a, b, c of the exponential of lambda, scalar, the random variable. The random variable is the Kronecker delta in this case. Kronecker delta in a, b and c. And of course it is one third exponential lambda a plus exponential lambda b plus exponential lambda c. Then the rate function is the sup of these three-dimensional... ...legend, ...transform where I take the log of psi. Now this sup can be explicitly solved. The solution is lambda l equal log nl with l equal a, b, c and I get the expression for the rate function. So the rate function is explicitly known for the POTS model. So the entropy is minus the rate function plus a correction, which is a number in this case. It's log 3, the number of states. Because the entropy is the log of the volume, it's not the log of the probability, so there is always a constant that can fly away. So this is the microcanonical entropy. The microcanonical entropy is the sup over na. So the total fraction is 1. So nc is 1 minus na minus nb. So I can write the entropy as a sup over na and nb only. And I get the function of the energy because the energy has this form. So by taking this sup, I get the microcanonical entropy by taking the inf over na and b and nc, I get the canonical free energy. Now I can do the same as I did with the other models, the Brume-Cappel and the... So I can draw the so-called caloric curve. This is beta and this is energy. There is an upper energy. You cannot have energy in this model above minus 1 over 6. And if you are in the canonical free energy case, so if I fix the energy, if I fix beta and I compute the average energy, the average energy is minus 1,6 until a given inverse temperature. And then there is a jump. There is a first-order phase transition. And then there is the... This is the case for the canonical. So in the canonical ensemble, the first-order phase transition at the temperature, beta t that can be computed explicitly. And this is the Maxwell construction. The two areas are the same. While in the micro canonical ensemble I vary the energy and I compute the micro canonical temperature. And you can see that it decreases the inverse temperature. So I am here in a region of positive specific heat because then I enter a region that would be unstable in the canonical ensemble. So it's a metastable in the canonical ensemble. And then I have a region of convex entropy where the inverse temperature increases. So the temperature decreases when the energy increases. So all this branch starting from here is a negative specific heat branch. So this is a very, very simple... I don't know any simpler model than this one to show ensemble in equivalence. In fact, when I published this paper, there was a series of paper by Richard Ellis who go to Schett and others where they formulated this model. Since they had worked a lot with the Q-revis, they formulated this model in the language of probability. And for me it was a big problem because I cannot read such papers. It's very complicated. But they published a lot of papers in annals of probability because this model is extremely simple. So you can... There are several examples of the different theorems of the large aviation theory. And if you want, I can give you the references of these papers if you are interested in the more type of probability setting for this type of model. There is a question? Yes, so actually there is... Sorry, there is a question in the chat, I think which refer to the previous subject before the break. Could you please comment on the convergence of the methods error from a computational point of view? From me? Computational point of view. I think it refers to the previous calculation. The method, I mean the step one, step two, step three. When I approximate the Hamiltonian with the rest, the rest can be estimated both analytically and in some cases and computationally in other cases. So in one of my papers I can give reference of one paper where I do an estimate explicitly of the rest. Paper in JSTAT fees where I do explicitly an estimate of the rest. OK, yes? I have a question on the POTS model. In the first place you compute the rate function by taking the sub, the sub between two things. Yes. Why? Because before the rate function... I use Kramert theorem. Kramert theorem tells me how I computed the rate function. So in the Kramert theorem you can prove it by Laplace method. It is not difficult to prove. Yes, but can I say again... So you see i of x is sub of lambda x minus psi. So it goes in steps. So first you have to compute this average and this average you compute with the local probability distribution function. Yes. So it's like the spins were independent. Yeah. Then you have the rate function, OK, which is... And then you have... Then you have a large variation principle. If with a function i, which you get from this, from the... from the sub. OK, and if I can see the expression of the rate function that we obtain in the POTS model. Yes. I hope there is a mistake. I just want to check something. Yes, this is the rate function. Yes. Lot of psi and then you have to... Legeant transform. OK. In this case it's simple because you derive... So you derive log of psi. OK, so let us now turn to a more complicated model and we try to use the same tricks. Now I will introduce, discuss three types of different models. One is the... So I include in this model just to show that it's possible to obtain results. Also the kinetic energy. Also the kinetic energy can be thought as a global variable. It's an average. It's a sum. OK, so this is order n. This is order n. I am summing vectors. And then I take a square of the square. Additional term. Without the additional term, this Hamiltonian will be like the Q-revise Hamiltonian. And there is no interesting... Instead, if you add this term, OK, then you get a phase transition and you can get non-trivial behaviors. So let's... I took this slide. So what are the global variables in this case? The global variables are the X component of the spin, the Y component of the spin, and the kinetic energy. These are the global variables. And then I can express the energy as a function of all the three global variables. This is the kinetic energy. This is the m-square. And this is m-4, OK? So then what is the local variable? The local variable, the local random variable is the Isenberg vector, cosine of theta, sine of theta, and p-square is the local random variable. And then I can get the expression for psi. And then these I leave to you. It's not difficult. Essentially, you have an exponential of lambda, lambda x cosine, so it's a better function, a reduced better function of zero order. There is a problem with the kinetic energy. I leave to you to comment on that. But OK, it's not a problem. Psi of lambda is a... So I have to do an approximation, but the terms here are subleading and when you take the log, it's OK. So then the inverse of the entropy as this expression, you see the usual form of the Legendre transform, and then as a step three, I have to take the soup over this function, over s, which is minus i, OK, at fixed energy. So I have to solve this variational problem. And if I solve this variational problem, I can show you the result. So this is the phase diagram. This is phase diagram, so this is k, k over j, and this is t over j. At k equals zero, there is a second order phase transition at a temperature 0.5, and then there is a line of second order phase transition and if I would have solved besides the canonical problem and also the micro canonical problem, also the canonical problem, I would have found a line of the first order phase transition here, but in the canonical ensemble, the line of second order phase transition continues, is a straight line in this case, it's very simple, and then there is a micro canonical trichritical point and there is a split with the two temperatures, the upper temperature and the lower temperatures, and then it continues. So you see that now the region of inequivalence that was very compressed in the Blume-Capelle model, here the difference is order one between the two points, it's not order ten to the minus three, so we can easily reach regions where the two ensembles are in equivalent results. For instance, here you can see that the micro canonical and the canonical agree in saying that this is a magnetized phase, but here, for instance, the micro canonical will tell you that it is a paramagnetic phase while the canonical will tell you that it is a ferromagnetic phase, so they disagree in the value of the order parameter in these two regions. While here they agree on the fact that they are both in the paramagnetic phase. This numerical simulation performed with molecular dynamics. Now that I have kinetic energy instead of using the Monte Carlo by Croids, I can directly integrate the equations of motion of the model and I can average the quantities temperature will be kinetic energy and you see that the agreement between the simulations, these are the points and the theory, the continuous line is quite good the system with 100 particles under spins you see the negative specific heat. So this is a molecular dynamic simulation so I am integrating over orbits and I find a negative specific heat so if I were able to see this system in real physics it would have a negative specific heat so here I am increasing the temperature the energy and the temperature goes down this is exactly, not exactly but very similar to the model that brought me to study this field so I was performing numerical simulations that I found in molecular dynamics and I was finding this very curious negative specific heat region and there is a temperature jump here so for instance if I would place here this energy and I run the molecular simulation the molecular dynamic simulation I would see a bimodal distribution of temperatures so the higher temperature and the lower temperature the higher energy and I would observe a sort of bimodal distribution of the temperature this is one example then I wanted to show you some problems on which I have been working so applications one after the other so this is a paper I was trying to publish it in PRL I think it was nice to have a PRL on such a problem because it was very interdisciplinary but it was rejected and then it appeared as a PRE rapid communication so I asked a friend what is a free electron laser and then he showed me equations that are very, very long and so I decided that I could not find anything for such Hamiltonian so I decided to reduce the complexity of this model and I found this model by Bonifacio he is an Italian physicist from Milano and which is a model of so the free electron laser is there is a also here there is a new one when we were working on this problem one of the co-authors is Giovanni di Nino who is working on the free electron laser here and they were still planning the length of this region the region of the magnets and I thought that my calculation could have been useful to tell the engineers how long should be this magnetic region so we even tried to go to the experimental site when we wrote this paper to talk to the engineers but they were interested in totally different problems so it was clear that they were not following our advices for the planning of the machine so we went on our own so what is a free electron laser first of all because let's talk a little bit about physics so there is a gun of electrons and the electrons go through an electron accelerator an electric field and they get out when they are accelerated from this field and then there is a region of opposite of opposite swapping magnets so the electron is like this and the magnet is like this so if it is south north it turns like this if it is south goes like this so it's called the wiggler because the trajectory of the electron wiggles when you pass through this region and since the electron is accelerated emits radiation so in this region the electron and the field are interacting each other and this model captures this interaction so this is the self-generated field this is the vector potential of the self-generated field and theta is the position of the electron inside the wave of the self-generated field and these are indeed one starts from relativistic invariant equations and then write the equations in the center of mass and in the end it's an Hamiltonian form so these equations of motion although they are free so it's very curious but you think Hamiltonian system should have an even number of equations the reason why they are free is that the field is a dual so there is an equation also for a star which is dual to a so a star is the conjugate variable to a in fact in the Hamiltonian you have a term a a star which is gives this so this is a Hamiltonian sorry I don't remember if I written the Hamiltonian and I can use the same trick because you see that this is called the bunching bunching of electrons this is a parameter that is measured experimentally the bunching of electrons and this is called the tuning and this is the amplitude of the field and this is the relative kinetic energy in a frame that moves with the electron so you see it's a very different problem but it can be treated exactly in the same way so this is the Hamiltonian a is the square root of a square and this is the entropy computed exactly where the configurational entropy is very similar to the entropy of the X-Y model and then you have to solve this soup for the configurational entropy and you get the second of the phase transition in the micro canonical ensemble the system is isolated so the canonical ensemble is very inappropriate for this system because the system is isolated and the energy is really conserved there is the electrons travel in the empty region so it's isolated in fact this type of transition is very interesting because it's present also in cold atoms in cavities so we have a series of papers also on that but it's interesting physically that you can compute a second of the phase transition in an object of this kind but this is something that I will discuss tomorrow but this is why our paper was rejected I tried to explain to you why our paper was rejected so this is a simulation of the Hamiltonian this is the intensity of the laser the laser has a certain intensity and the intensity grows then there are oscillations that dump around this value and then if you wait long enough you see the dumping at this level if you wait long enough there are other trajectories here there is a slow drift and finally you reach this other level there is a gap what our theory was predicting was this level here not this level here so we were arguing a different theory which was inspired to the theory that I will present tomorrow why I stay around this level for such a long time it has to do with another entropy the Lindenbell entropy and I will introduce the Lindenbell entropy tomorrow if I will have time and then you see that there is a drift to a new value so there is a very fast relaxation to a level and then there is a drift to a different level I can predict with statistical mechanics the higher level not the lower level and the point is that one of the referees was a person in free electron lasers and he computed how long would be the free electron laser in order to reach the higher level and the length was one parsec so clearly you cannot construct on earth a free electron laser of a length of one parsec so totally unrealistic length so the time of convergence to statistical mechanics or the length of convergence to statistical mechanics makes it totally unphysical in this problem the problem while the laser seems to relax to a laser intensity so what is this laser intensity to which the system relaxes and will be the subject of the next lecture we have constructed a statistical theory for such a level of of relaxation it introduces another problem in long range interactions which is the presence of what we have called the quasi stationary states states in which the relaxation is fast the phase space is occupied but only in parts the system is not ergodic and it's visiting only one part of the phase space and only by diffusing slower and slower it can reach the ergodicity and then give the value of the statistical mechanics the maximum entropy value so this is a story that it's a long story it has to do with the Lindembell distributions and with the fact that there is a phenomenon called vion relaxation and I will try to describe this phenomenon in the next in the next talk where I speak about dynamics not about statistical mechanics so and just to finish this is also another model theoretical physics is an important model it's a 5-4 model so I have a local potential with a Hamiltonian with a local potential which is a double well ok, locally so the variable is continuous but locally I have a double well so it means that the particle will spend most of his time either in one well or in the other so like anising spin so it will be around the minimum of a well and some point around the another minimum now if you couple with a Q-revised term this continuous variable you will have a phase transition of the type of the icing phase transition of the type of the Q-revised phase transition but with continuous variables the model was introduced by the Zion's fancy and they studied the model in presence of noise and they looked at the stationary solution of the Fokker Planck equation was totally different analysis but we did arrive the results using our method which implies computing entropies so also here you have global variables the kinetic energy the local potential and the mean field you can compute log of psi using the Kramer problem and then by taking the soup you can get the you can get the entropy as a function of the energy and of the local magnetization now I have access to another quantity which is the entropy as a function of energy and of the local magnetization so I would like to check another property for long range system which is are there other response functions that that are the wrong signature in a long range system so this is the entropy as a function of M and as you can clearly see there is a phase transition and there is only a maximum in zero and some point the maximum in zero becomes a minimum and there are two side minima and this so you see that this model the micro canonical as a phase transition exactly as but I can define the field in a different way because you know in the canonical ensemble I can define the magnetization by the free energy with respect to the field in the micro canonical ensemble the definition goes opposite so you have to the field is not present in the Hamiltonian H is not present but if you derive this is a thermodynamic formula tds equals d minus hdm if you derive the entropy with respect to M and then you multiply by the temperature like here you get the field as a function of M the opposite you don't get the M as a function of the field but you get the field as a function of M and now I can of course in the canonical ensemble would be totally different I can compute the susceptibility susceptibility is dm dh but in in the micro canonical will be the inverse of dh dm and you get this function in the canonical ensemble the curvature is always positive but in the micro canonical ensemble it can happen that this quantity is negative so the susceptibility can be negative and this is the simulation and you can see that the field sometimes if I increase M the field increases but then there is a region where if I increase M the field decreases and you can see chi the susceptibility here that becomes negative and this is a numerical simulation again molecular dynamic simulation to check that the susceptibility is negative so this specific heat is only one among the several response function that you can introduce in the context of thermodynamics which can become negative due to non-additivity of course if I change the story and make the model additive all these features will disappear so this was very fast I recognized what I had no time enough you can read the slides I put the slides on the and you please check there might be errors so this lecture showed that large deviations are a powerful tool to derive micro canonical entropies they have several applications but this is one and the first simple example was the POTS model and then we have looked at the generalized XY models called somebody fashion model for the electron laser for theory and this is the lecture today finishes now any question? I have a question on zoom so this was not just stretching ok so thank you very much Stefano I think we can close here I am Trieste in this moment in the next lecture of David Wolper Casa di cura si, era dimentika