 Okay, so nao, kar sem... Zato, da je energija kaj je tega faktorja, in nekaj sem tako vzal nemazljenje, kar sem zelo vzal drugi g, V svoju hroju, zaznam, zaznam tukaj minus, ok, ker tukaj je tukaj tukaj, je plus, ok. ampak tudi. Zazim, da se izgovamo, zelo skupaj Snbobe. Kaj mi v drugi vesi po moh zobučenju glasba, bom prosim tudi objekte, jaz drugih tudi objekte. obreženja taj obreženja. Zato... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... Zelo, da je še sklatel prej dnega vso vsezvost, je zvedna vsezvost, je tudi zvok, še je tudi vso vsezvost. Je to vsezvost? To je vsezvost, vsezvost? to je, to je, svega. Prokrojef F of m, nekako, nekaj učaj almost, je, nekaj čest, rečno, nekaj je, nekaj, nekaj, nekaj nekaj, nekaj... energi, minus, so let me remind. Fizikalne zelo je zelo fri energi, zelo zelo fri energi. Če voleće zapos없it ponužilo听C, pride Rod这里 is partial-magnmizacije. Ljte se ovo tudi. Kajte te dobro sreži, m د sweet? Če , pa sem bilo, če všeč jazim, ... ... ... ... ... ... Zelo, da je to energija, zelo je to funkcija m. Zelo je to početno 1 over z of t e to the minus e to the plus, J over 2, and then there is n, and there is t, m squared, so n, let me take it, n over t, j half m squared plus hm, so this is this object here, and then you have just the sum over the number of configurations that have magnetization equal to m, and this gives you the enthalpy, so it's what we call the n times this small s of m, this s of m is 1 minus m divided by 2 log 1 minus m divided by 2 minus 1 plus m over 2 log 1 plus m over 2. Now what is z of t, so z of t is just the normalization, so z of t is just the integral between minus 1 and 1 in the m of this object here. So this is what I call, so this I can call it, I can put t here, and what is in this parenthesis is what I call f, so sorry, is what I call minus f, I'm calling it minus f. So this object here is minus f, so this is n to the, so the normalization, which is the partition of i, is just this integral that you do with the saddle point integration. So this thing will be dominated by the m, which makes f of m maximal. So this is just e to the minus n over t f of m star. And if you look what is the m star, it's precisely given by this equation here. Now already here you see that the m star that dominates this calculation here has to be the maximum, sorry, it has to be a minimum of f, because it has to be a maximum of minus f. So it can only be this point or this point. So this point will never come up from this calculation. And then say if you put this back here, you get this formula here. Now so what we get is that the physics, which is described by this model, is one where if you are at high temperature magnetization, which is a smooth function of t, if you are low temperature, instead the magnetization is essentially this hysteretic behavior. So this is hysteresis. This will be h equal to zero. And if you go to negative h, this minima will be at zero, and this other minima will be up. So one of these two states is a metastable state, and the other one is a stable state. I wanted to make on this thing, on this set of results. So the first one is that actually in a, so mean field model are very particular, I mean this is a behavior, in particular this is a behavior, which is typical of mean field model. So when you are in, so as we said, it is a phenomenology for t less than tc, you have a phase transition between, as you change the field, you have a phase transition between two different states. So the question is how, when you decrease the magnetic field and you go from say positive h to negative h. So this state here becomes metastable. So how do you go from a stable state to a metastable state? So physically what happens in real magnets is that you create by a small fluctuation and droplet of say the negative spins aligned in a negative direction and this droplet will start growing. So this is spin over the composition. And what happens is that essentially as soon as you are on the negative, small negative magnetic field, you will just cross to negative magnetization. Because essentially what you need is just a droplet of a magnet size. And if this droplet is larger than a critical size, then it will expand and it will make the whole system. And the fluctuation and the energy that you pay, I mean the probability for this fluctuation is related to the energy difference. And so it's a finite, actually it's a probability that depends only on the surface energy of the droplet. So it's something that could be done. So instead these mean field models, since you cannot create surface because everything is connected to everything else, every spin is connected to everything, so there is no bulk and there is no surface, you cannot create any surface. So there is no way in which you can define a subregion of a fully connected model. Okay, which does not interact with all the rest. Then this droplet picture does not apply. And so what happens when this, physically when you have this mechanism of these droplets, is that essentially what you reach, I mean at this transition point, this transition point, what you have is what corresponds to the Maxwell construction, what corresponds physically to the Maxwell construction, which is essentially to substitute these functions with the function, which is a linear interpolation between these two minima. So the function, the green function is replaced by a function, which is linear between these two objects. So that it always have just one minimum. And the interpretation of a point here is that any point here will be a mixture of this state, of a system in this state, so infinite dimensions, so you can create a system which has, where a certain fraction can be in the plus state and the rest is in the minus state. And by changing this relative size of these two objects, you can go from any point here to any point here. So this is essentially the physics of Maxwell construction. So in the mean field models, this, of course, is not possible because essentially there is no way to define a surface. However, there are situations where actually metastable states and actually most of the course of Federico Richarducci will deal with systems where, like glasses, where essentially the system is trapped into metastable states essentially forever. So the physics is essentially the one of metastable states essentially because these are systems which have many metastable states. So for this system essentially mean field model can provide a good approximation. So the other thing I wanted to remind you is why is it that we have this phenomenon of ergodicity breaking. So let me consider the case where you are at h equal to zero. So actually now let's consider a general case where, let me redraw this, let's consider a general case where you have two minima, so you have two minima and let's imagine that your system is in this state, in the metastable state and you ask yourself how much time will it take for the system to go into the stable state. Now, in order to think about how much time it will take you have to think about what is the reasonable dynamics. The reasonable dynamics is a local one. It's a local one where essentially is described in terms of events and the simplest event is an event where one spin flips from up to down or from up down to up. So you have generally your dynamics will be a subsequence of events where spins flip. So you have a change from a configuration C to a configuration C prime to C but say with a particular spin which is flipped. The question is how much time so if you think at two general configurations C and C tilde then any if you think so you are in this configuration here at time zero and you want to ask yourself how much time will it take you to go to a configuration that has the equilibrium magnetization. Essentially the issue is how you have to think of all possible sequence parts of configurations of spin flips that can lead you to a configuration here to a configuration here. And in order to define these so in these dynamics what you have to define is the rate at which these transition will take place. And so the general idea of how you define this transition is so what type of dynamics can be meaningful for this system is the basic consideration is that you want to look at dynamics that will lead to this to the equilibrium of one equilibrium. And so in order to do this you you want to enforce detail balance. What is detail balance? Detail balance tells you that if you have a certain probability distribution the probability that from a certain configuration C to a certain configuration C prime must be equal to the probability of the reverse transition. So the way you have to think this is that if you have let me rewrite this if you have your set of states so imagine that you have a set of states which are the configuration of your systems so you want to now you want to find a dynamics that goes from one state to the other which is made of single hop from one state to the other and you want to ensure that these dynamics defined by these rates is such that in equilibrium you will have a certain probability distribution which is this equilibrium distribution. So the first thing you have to make sure of is that the set of transitions that you allow result in a dynamics which is ergodic ok which means what does it mean ergodic you know what is ergodic so for any two states there is a sequence there is at least a sequence of transition with non-negative rates that connects one to the other you can go from any state to any other state with positive probability ok if this is so then then this is a necessary condition to be ergodic and of course these dynamics here satisfy this condition because for any two configurations we differ by say the orientation of the spins I can think of just flipping the spins which do not agree in the two configurations so one by one so the second thing so this tells you what are the transitions yes yes yes you can also define a dynamics by flipping a group of spins is actually something which is very useful but physically what you think is that essentially real dynamics is by just flipping because essentially every spin is subject to thermal fluctuations and so every spin will flip more or less independently from the other so flip in a correlated manner ok and of course I mean they can flip over longer time scales you can have this correlated move because essentially once spin flip then because of the interaction the other neighboring spin can flip and so on and and this is actually this is actually this is actually the basis of what I call the cluster algorithm methods for simulating these systems and so the point is that these dynamics are used also to do numerical simulations to do numerical simulations to sample equilibrium distributions ok so and typically what happens is that in order to reach the equilibrium state you start from a certain initial condition that may be completely random and then you let the dynamics evolve a little bit and then you will be in the equilibrium state now the time to reach equilibrium it happens that it diverges at critical points ok so this is called critical slowing down which is actually related to physical I mean real physical phenomena that occur in phase transition so if you take a liquid close to a critical point it will show this critical opalescence in the sense that it is no more transparent this is related to the scattering of different wavelengths ok and and so if you think about the dynamics like this one to reach the equilibrium then closely if you want to sample the equilibrium distribution dynamics based on single spin flip will be extremely slow, very very slow it takes times to reach the equilibrium which scale with non-trivial power of the system size so if you want to make your algorithm faster what you do is is to integrate time to find say more clever definition of this transition that involves flipping whole clusters of spins ok and this is what this once in one algorithm et cetera do it's a very interesting idea but this is ok let me go back here so essentially once you ensure that every state can be reached by any other state then in equilibrium what you want to have is that the flow from any state to any other state which are connected by a transition should be equal to the flow probability in the other direction you can think of this as a fluid probability has been a fluid and you don't want that in equilibrium there is any flow in the system there is you don't want any probability flow so first of all for any transition one direction you should have the transition in the opposite direction because otherwise detail balance is not concerned transition should be reversible or should allow for the opposite transition and then if you want to impose that the flow probability in one direction equals to the flow probability in the other direction then this is the equation that you have to satisfy and this essentially tells you that the rates the ratio of the rates in the two directions should be equal to the rate of the probability of the probabilities which is essentially equal to a to the energy differences e of c prime minus e of c okay so there are different ways in which you can realize this so this is the condition of the demand so there are different ways in which you can realize this one is metropolis where you take these rates as being the minimum between one and e to the minus delta e over t where delta e is this difference okay so this so this tells you that if you are going from a state to a state with lower energy if you are proposing to go to a state with lower energy you always accept the move otherwise you don't accept the move you accept the move only with probability e to the minus delta e over t if your energy increases okay another choice is Glauber where this rate is equal to e to the minus energy of the new configuration divided by t divided by sum of the two energies et cetera et cetera you can think at different dynamics but to say once you have defined your dynamics the issue is how long should you wait before you go from this state to this state okay so one simple argument for this is the following so that if your dynamics so let's imagine that they plot the magnetization so the magnetization is a function of time and so essentially what will happen in a finite system is that under this dynamics the system is ergodic so it will visit every configuration so it will be in the negative configuration for some time then it will jump to the positive then it will jump to the negative again it will stay for a little while there jump again et cetera et cetera now what to ask is how long is this time okay so if you really look at very long times then this will be your dynamics okay now essentially the one simple argument is that say if you yeah lower temperature lower but this is here I have say non-zero magnetic field okay in this particular case so now essentially the simple argument is that essentially the time I have to wait for a transition is essentially the time it will take me to reach a state which is up here because once I am up here I have half probability to go one side on the other side so if I am on this in this state and I have to go to this state essentially the time it will take to me to go here is essentially proportional to the time it will take for me to go from here to this point because then with probability final probability I will go on the equilibrium state okay so this probability here is equal to the relative fraction of time points where I am at those states if I consider a very long time compared to the time that I am in the low energy in the negative end state okay and since this the fraction of the time I spend at this point is proportional to the probability okay and the fraction I spent this point is equal to the probability then this time here will be proportional to the ratio between the probability of in c0 and the probability of being in say configurations which are up here or let's say which have essentially which are on this unstable state so it is essentially since this probability are given by Boltzmann distribution it is essentially equal to e to the minus e to the the energy difference divided by t where the energy difference is this one okay so and now you see why when I take the limit as n goes to infinity before taking h0 to 0 I get stuck into metastable state or say because essentially these times are proportional to the volume proportional to exponential of the volume so this time diverge very very fast as n goes to infinity so essentially the probability that I will go from the metastable state to the stable state in a mean field model when n goes to infinity is is going to diverge because there are these energy barriers between states which are proportional to the volume okay there was a question somewhere so the time it takes I mean this time makes to make a transition is essentially the time is proportional scales with finite power of n okay whereas this time is exponential in n so in this picture it is really sharp when n is large okay the time it takes is really the time it takes to flip and spins typically it is a sort of a diffusion you can think of it as a driven diffusion process so it takes over n to some power alpha depended but this is very very short compared to the residence time in two states now so this fact that you have this fact that you have this ergodicity breaking when n goes to infinity is related to the fact that you have energy barriers which go to infinity when the system size goes to infinity so let's take this break and then we continue so in particular this probability is 1 minus m star alpha times sigma i divided by so this this probability is 1 plus m for sigma equal to plus 1 minus m divided by 2 for equal to minus so this means that essentially for mean field model if you look at the pure state of a mean field model the probability distribution factorizes over the spins ok so and this is essentially now ok so there are more rigorous ways to see that this must be true for mean field models but essentially this I think so the fact that the correlations are essentially zero means that suggested probability distribution factorized so in general this is the even this is this assumption can be used also for five dimensional models as an approximation ok and it is the mean field approximation it may be useful ok so now the last thing I wanted to tell you how how do you compute this function so essentially the idea in the in the mean field model computing the probability of the magnetization is easy because essentially the energy is directly a function of the magnetization so it's a very easy task but in general say if you want what I think you are going to discuss with Federico when it comes is how can you deal with mean field model models with infinite range interaction but where the interactions can be anything in particular where the interactions are drawn at random from some distribution ok so these are essentially spin glass models essentially also for these models you would imagine that your probability distribution or the configuration can be divided into pure states that for any pure state you have so that essentially you can still write this thing and that on any pure state you have the clustering problem so that any of this can be written as a product of probabilities and individual spins ok now the question is how do you how do you compute these things so how do you so in what how can you access these quantities so in finite dimensional systems a pure state is selected by choosing the boundary conditions so if you choose say for example in a ferromagnet if you impose that the spins on the boundary are positive then I mean the magnetization will be positive if you are at low temperature if you impose that the spins on the boundaries are negative then magnetization will be negative but if you are in a mean field model you don't have a boundary ok there is no surface ok so what do you do so what you can do is to do what we have been doing last time it is to say you introduce a small field then take the limit as n goes to infinity and then take the the field to zero ok so you select in mean field model you can select different pure states with external fields ok then letting these fields go to zero ok now the last question that I want to address is the following so the general idea of how to compute free energies normal is that you have the partition function let me now use beta is 1 over t which is a sum of all configurations v to the minus beta the energy and essentially what you want to what you can do is what you can do is to write a generating function which depends on beta and on external fields let me call them b which is just the sum over all configurations of e to the minus beta energy of the configuration plus times sum over time b i and sigma i ok so if I do so you see one cycle if I can compute this object then I can take the log of of this thing and if I divide by beta so this this guy that I can call let me call it g of gamma of beta let me call it g of beta g of b so this thing is by taking derivative of this function as a function of bi and since it is a pure state the magnetization will give me the full probability distribution because this probability as we said this probability are just 1 plus mi alpha sigma i divided by 2 so if I can compute the magnetization on a pure state then I can compute the full probability distribution I can compute everything so and if I can compute this function here then the magnetization mi will be just equal to the derivative of g with respect to bi that's great now the problem is that here essentially I'm summing over all the configurations and I'm not summing so the the probability with which I'm waiting the configuration is given by the p is not given by the pure state so I did say for example if I want to compute if I want to compute so one thing that I could compute from here is the let's say the free energy the analogous of this thing as a function of the magnetizations and and the way I can do this is by choosing is essentially by the transform by essentially you know when you have a function which depends on its variable or some variables I can go from this picture to another picture where the variables the independent variables are the derivatives of the original variables by the transform what I have to do is to essentially take the function g of b minus sum over i mi bi and then just take an extremum over b which in this case I think should be a minimum okay a minimum so you see that this well put a bit here so this is the usual recipe of lejant transform if you have a function that depends on certain variables you want to pass to a description which depends on the conjugate variables which have the derivative of the state function with respect to these variables then what you do is the lejant transform okay now the problem with this is that a lejant transform transforms a convex function into another convex function okay so if you take for example if you want to take the mean field easy model as we define it and apply this strategy what you would get is not this function here but a function which is like this then like this and like this because it is the convex hull of the of the function I mean if you think the function f you do a lejant transform you do this essentially the z of beta if you want this will be given by integral from minus 1 to 1 dm of e to the minus minus if I take b to be just to b so this is what I should have I mean this should be the equivalent function generating function I should compute for the mean field is in model and you see if I do this this integral here this will be dominated by the by I can get this as e to the minus n times n times beta minus b times m star ok is the usual what we have just done so this and this would be my function g of b in this case but if I take this and I do another lejant transform I will not get back the function f I will get back a function f tilde which is the convex hull of the function f ok so this problem so this tells you that essentially you are not this calculation which is using generating functions and lejant transform is not giving you the metastable state and since the metastable states are very important to describe the physics of these other systems and glass systems then one has to do something better than this ok so the main idea is the following so that so if you look at the function f of m for the mean field is in model what you find is that so this function beta f that you are interested in is essentially equal to say minus beta f is essentially equal to the entropy minus the energy beta times the energy which is beta times yes, the energy is a function of m ok or say if you want now what you see is that this function is a very simple function of beta if you keep m fixed it's a very simple function of beta it's a linear function of beta so you can think which is sort of surprising because essentially you have a phase transition a singularity out of a function which is completely trivial in beta so you see what we discussed is say that you have this phase transition the magnetization as a function of temperature which is 1 over beta there is this critical temperature and then there is this singularity at this point now this is not a consequence of the free energy being singular at this point here it is the consequence of the fact that this m star is the solution of a minimization problem ok so this m star is the solution of is the essentially what minimizes of m of m actually minus phase transition ok so even if this function here is completely trivial as a function of beta this object can have a very non-singular behavior as a function of beta so the idea is what if we compute the free energy instead of the rejaltas form rejaltas form as as a power expansion in beta ok the idea is precisely this so essentially you want to define minus beta f of m as the log of the sum of all the configuration of e to the minus beta times the energy and then you want here a delta function that turns you that 1 over n somewhere i that the magnetization must be equal to n now let me call this say another function that's a gamma of m and beta and now I want to expand this function as a function of beta ok so now the first term will be gamma of m and 0 ok gamma of m and 0 is just I have to put beta equal to 0 here so I have just the entropy the log of the number of configuration with magnetization m ok this is precisely equal to our entropy ok and the second term then you have beta times the derivative with respect to beta of gamma of m and beta in beta equal to 0 and now when you take a derivative with respect to beta here what do you get you get minus the average energy ok so ok so you get minus the average energy and the energy is precisely equal to this object here precisely this object ok and what about higher order terms higher order terms will give you the fluctuations of the energy when m is fixed but the energy is a function of m so it does not fluctuate if m is fixed ok so everything else is 0 ok so you see that if I if instead of doing the usual trick of Legendes I take this function here and I try to compute a high temperature expansion then I can recover the full function without recovering without getting stuck with the convex problem so this is what you are going to see next week in much more detail it's called the George did the expansion and it can be done in general for all mean field models ok so with this I think we can go and have lunch